Research ArticleOn the Uniform Exponential Stability ofTime-Varying Systems Subject to Discrete Time-VaryingDelays and Nonlinear Delayed Perturbations

Maher Hammami1 Mohamed Ali Hammami1 and Manuel De la Sen2

1Faculty of Sciences of Sfax University of Sfax Route Soukra BP 1171 3000 Sfax Tunisia2Departamento de Electricidad y Electronica Facultad de Ciencias Universidad del Pais Vasco Leioa (Bizkaia)Apartado 644 480809 Bilbao Spain

Correspondence should be addressed to Maher Hammami hammami maheryahoofr

Received 9 November 2014 Accepted 19 February 2015

Academic Editor Qingling Zhang

Copyright copy 2015 Maher Hammami et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper addresses the problem of stability analysis of systems with delayed time-varying perturbations Some sufficientconditions for a class of linear time-varying systems with nonlinear delayed perturbations are derived by using an improvedLyapunov-Krasovskii functionalTheuniformglobal asymptotic stability of the solutions is obtained in terms of convergence towarda neighborhood of the origin

1 Introduction

Theproblemof robust stability analysis of linear time-varyingsystems subject to time-varying perturbations has attractedthe attention of many researchers Explicit bounds for thestructured time-varying perturbations have been derived [1ndash6] where the stability problem of linear systems subject todelayed time-varying perturbations has been studied whileonly few papers [7ndash11] give stability conditions for lineartime-varying delay systems among those [10] dealing withthe exponential stability of perturbed systems In [5] a newsufficient delay dependent exponential stability for a class oflinear time-varying systemswith nonlinear delayed perturba-tions is obtained based on a Lyapunov-Krasovskii functionalTime delay systems can include mixed neutral discrete (orpoint) delays and distributed delays including Volterra-typedistributed dynamics [12 13] Also delayed dynamics oftenappears in real-life problems like for instance epidemicpropagation models [14 15] since they affect the illnesspropagation via the incubation process in the studied pop-ulation and the vaccination period Delays are also useful todescribe single-species population evolution models [16] andare related to certain diffusion and competition predator-prey

models [17] Conditions to preserve the asymptotic stabilitycompared to a delay-free nominal model description havebeen widely studied in the literature including the case ofpresence of possibly delayed perturbation dynamics See forinstance [1ndash5 7 8 11 12 18ndash22] and references therein Themain novelty of this paper relies on the fact that the proposedapproach for stability analysis allows for the computationof the bounds which characterize the exponential rate ofconvergence of the solution towards a closed ball centeredat the origin by extending the complexity of the system byconsidering at the same time time-varying dynamics withtime-varying time differentiable in the delays in the nominalpart by considering nonnecessarily zero lower-bounds forthe delays and by considering more general conditions thanjust to be Lipschitz for the delayed in general nonlineardynamics Note for instance that the nominal part of thesystem has no delays in [5] the lower-bound of the delaysof the perturbations is assumed to be zero while those per-turbations are assumed to be Lipschitz in the state-variablesIn this paper the nominal part is time-varying with time-varying delays the lower-bounds of the delays can exceedzero and the perturbations norms incorporate a time-varyingupper-bound apart from the Lipschiptz type one In [9]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 641268 12 pageshttpdxdoiorg1011552015641268

2 Mathematical Problems in Engineering

a global-null controllability is required while the asymptoticstability is not guaranteed to be of exponential type In thesame way the asymptotic stability is not guaranteed to beexponential in [11] Another novelty is that the delays aretime-varying time-differentiable and they are not requiredto be known Only lower and upper-bound of the delayfunctions and their time-derivatives are required for stabilityanalysis We will study a class of nonlinear system such thatthe nonlinearity is bounded by some integrable functionswhich are bounded where the origin is not necessarily anequilibrium point We deal with the practical stability of theorigin (see [23]) The asymptotic stability is more importantthan stability also the desired system may be unstable andyet the system may oscillate sufficiently near this state thatits performance is acceptable thus the notion of practicalstability is more suitable in several situations than Lyapunovstability In this case all state trajectories are bounded andapproach a sufficiently small neighborhood of the origin Onealso desires that the state approaches the origin (or some suffi-ciently small neighborhood of it) in a sufficiently fast mannerThis notion of practical stability was introduced by [24] fornonlinear time-varying systems and studied for differentialequations with delays by [25] (see also the references therein)Moreover the authors in [26 27] constructed stabilizingcontrollers to obtain global convergence of solutions towarda small ball for some classes of uncertain control systemsIn this paper some sufficient conditions are given to obtainthe exponential uniform stability of the solutions toward aneighborhood of the origin based on a suitable Lyapunov-Krasovskii functional Two illustrative examples are givento demonstrate the validity of the main result where weestablish a table of comparison with other results

2 Preliminaries

We start by introducing some notations and definitions thatwill be employed throughout the paper

R+ denotes the set of all nonnegative real numbersR119899 denotes the 119899-dimensional Euclidean space 119909denotes the Euclidean vector norm of 119909 isin R119899 119909119879119910denotes the scalar product of two vectors 119909 119910

R119899times119903 denotes the space of all (119899 times 119903)-matrices

119860119879 denotes the transpose of the matrix 119860 119860 is

symmetric if 119860 = 119860119879

119868 denotes the identity matrix

120582(119860) denotes the set of eigenvalues of 119860 120582max(119860) =maxR119890(120582) 120582 isin 120582(119860)

120583(119860(119905)) denotes the matrix measure of the matrix 119860defined by

120583 (119860 (119905)) =

1

2

120582max (119860 (119905) + 119860119879

(119905)) (1)

1198712([minus120591119889 0]R119899) denotes the Hilbert space of all 119871

2-

integrable andR119899-valued functions on [0 119905]

119862([minus120591119889 0]R119899) denotes the Banach space of all R119899-

valued continuous functions mapping [minus120591119889 0] into

R119899 with 120591119889gt 0

119909119905= 119909 (119905 + 119904) 119904 isin [minus120591

119889 0]

1003817100381710038171003817119909119905

1003817100381710038171003817= sup119904isin[minus120591119889 0]

119909 (119905 + 119904)

(2)

119861(0 119903) = 119909 isin R119899119909 le 119903 with 119903 gt 0 denotes theclosed ball of center 0 and radius 119903

Consider a linear time-varying system with nonlineardelayed perturbations of the following form

(119905) = 119860 (119905) 119909 (119905) + 1198910(119905 119909 (119905)) +

119898

sum

119894=1

119860119894(119905) 119909 (119905 minus 120591

119894(119905))

+

119898

sum

119894=1

119891119894(119905 119909 (119905 minus 120591

119894(119905))) 119905 ge 0

119909 (119905) = 120601 (119905) 119905 isin [minus120591119889 0]

(3)

with 120591119889= max

1le119894le119898(120591119894119867) where 119909(119905) isin R119899 is the vector

where119860(119905) 119860119894(119905) isin R(119899119899) 119894 = 1 119898 arematrices functions

continuous and bounded in 119905 ge 0 120601(119905) isin 119862([minus120591119889 0]R119899)

is the function of initial conditions with the norm 120601 =

sup119904isin[minus120591119889 0]

120601(119904) 120591119894(119905) (119894 = 1 119898) is a given time-

differentiable time-varying delay function satisfying

120591119894119871lt 120591119894(119905) le 120591

119894119867

120591119894(119905) le 120583 lt 1 (119894 = 1 119898) forall119905 ge 0

(4)

where 120591119894119871 120591119894119867(0 le 120591

119894119871lt 120591119894119867) 120591119894119867119871

= 120591119894119867minus 120591119894119871

As a matter of fact the case that the delay derivative islarger than or equal to 1 is universal For example in networkcontrol systems the delay 120591

119894(119905) denotes 119905 minus 119895

119896 where 119895

119896(119896 =

1 2 ) are the sampling instants Thus this kind of delaysatisfies 120591

119894(119905) = 1 almost for all 119905 ge 0 (see [12]) For the case

of 120583 ge 1 if choosing a positive scalar 0 lt ] lt 120583minus1 then it

follows that

(]120591119894(119905))1015840

= ] 120591119894(119905) le ]120583 lt 1 (5)

and the nonlinear perturbation 119891119894(sdot sdot) (119894 = 0 119898) satisfies

1003817100381710038171003817119891119894(119905 119910)

1003817100381710038171003817le 1205751198941

10038171003817100381710038171199101003817100381710038171003817+ 1205751198942(119905) forall119905 ge 0 forall119910 isin R

119899

(6)

where 1205751198941gt 0 and 120575

1198942(sdot) are nonnegative continuous bounded

functions for 119894 = 0 1 119898

Definition 1 The system (3) is said to be globally uniformlyexponentially practically stable toward a ball 119861(0 119903) of radius119903which is a neighborhood of the origin if there exist positivenumbers 120572 120574 and 119903 such that every solution 119909(119905 120601) of thesystem satisfies

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le 120574119890minus120572(119905minus1199050)

10038171003817100381710038171206011003817100381710038171003817+ 119903 forall119905 ge 119905

0ge 0 (7)

The following technical proposition is needed for theproof of the main result

Mathematical Problems in Engineering 3

Proposition 2 Let 119876 119878 be symmetric matrices of appropriatedimensions and 119878 gt 0 Then

2119909119879

119876119910 minus 119910119879

119878119910 le 119909119879

119876119878minus1

119876119879

119909 forall (119909 119910) isin R119899

timesR119899

(8)

3 Main Result

Theorem3 Suppose that there exist some positive constants 120572120573 ] 120581

1 12058121119894 12058122119894 12058123119894 12058124119894 12058131119894 12058132119894 and a symmetric bounded

positive semidefinite differentiable matrix function 119875(119905) for all119905 ge 0 satisfying the following Lyapunov differential matrixequation (see [18])

(119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) + 120598119868 = 0

(9)

with

120598 = 2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811

120598 gt 0

1205812119894= 12058121119894+ 12058122119894+ 12058123119894+ 12058124119894

1205813119894= 120581311198941205912

119894119867+ 120581321198941205912

119894119867119871

119875120573(119905) = 119875 (119905) + 120573119868

119901 = sup119905isinR+

119875 (119905)

(10)

where

120578 = 1205811minus 2120573120583 (119860)

minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

gt 0

0 lt 119902119894lt 12058123119894119890minus2120572120591119894119867

(1 minus 120583) 119894 = 1 119898

0 lt 120583 lt ]minus1

(11)

with 120583(119860) = sup119905isinR+120583(119860(119905)) and 1205751 = max

1le119894le1198981205751198941

If

1205751lt

1

(119901 + 120573)

sdot (

119898

sum

119894=1

(12058123119894119890minus2120572120591119894119867

(1 minus 120583) minus 119902119894) (1205811minus 2120573120583 (119860)))

12

(12)

with 1205811minus 2120573120583(119860) gt 0 and 120575

2(119905) = sum

119898

119894=01205751198942(119905) then the system

(3) is globally uniformly exponentially practically stable towarda certain ball 119861(0 119903)

Moreover the solution 119909(119905 120601) satisfies an estimation as in(7) with size

120574 = ((119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))

sdot (120573)minus1

)

12

(13)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) 119888119894119867

= (21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus

1)(2120572120591119894119867)2 119888119894119871= (2120572120591

1198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 and

(i)

119903 = 1199031= radic

1198721

2120572120573

(14)

if 1205752is bounded by a scalar positive 119872 for all 119905 ge 0

with

1198721=

(119901 + 120573)2

1198722

120578

(15)

(ii)

119903 = 1199032= radic

11986812

2120573radic120572

(16)

if int+infin0

1205754

2(119904)119889119904 lt +infin with

119868 = int

+infin

0

1199032

(119904) 119889119904 119903 (119905) =

(119901 + 120573)2

120578

1205752

2(119905) (17)

Proof Consider the following Lyapunov-Karovskii func-tional

119881 (119905 119909119905) = 1198811(sdot) + 119881

2(sdot) + 119881

3(sdot) (18)

4 Mathematical Problems in Engineering

where

1198811(119905 119909119905) = 119909119879

119875 (119905) 119909 (119905) + 120573 119909 (119905)2

1198812(119905 119909119905) =

119898

sum

119894=1

12058121119894int

119905

119905minus120591119894119871

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

119905

119905minus120591119894119867

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

119905

119905minus120591119894(119905)

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

119905

119905minus]120591119894(119905)1198902120572(119904minus119905)

119909 (119904)2

119889119904

1198813(119905 119909119905)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

119905

119905+1199051minus120591119894(119905+1199051)

1198902120572(119904+120591119894119867minus119905)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

119905

119905+1199051minus120591119894(119905+1199051)

1198902120572(119904+120591119894119867minus119905)

119909 (119904)2

119889119904 1198891199051

(19)

with 120572 gt 0Let us consider the time derivative of 119881

1(119905 119909119905)

1(119905 119909119905) = 119909119879

(119905) (119905) 119909 (119905) + 119879

(119905) 119875120573(119905) 119909 (119905)

+ 119909119879

(119905) 119875120573(119905) (119905)

(20)

with 119875120573(119905) = 119875(119905) + 120573119868

Thus

1(119905 119909119905) = 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)) 119909 (119905)

+ 2119875120573(119905) (119891

0(119905 119909 (119905)) +

119898

sum

119894=1

119891119894(119905 119909 (119905 minus 120591

119894))) 119909 (119905)

+ 2

119898

sum

119894=1

119909119879

(119905) 119875120573(119905) 119860119894(119905) 119909 (119905 minus 120591

119894(119905))

(21)

From Proposition 2 we have

2

119898

sum

119894=1

119909119879

(119905) 119875120573(119905) 119860119894(119905) 119909 (119905 minus 120591

119894(119905))

le

119898

sum

119894=1

(

1

119902119894

119909119879

(119905) 119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) 119909 (119905)

+ 119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

)

(22)

This implies that

1(119905 119909119905)

le 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)) 119909 (119905)

+ 2( sup119905isinR+

119875 (119905) + 120573)

sdot (1198910(119905 119909 (119905)) +

119898

sum

119894=1

1003817100381710038171003817119891119894(119905 119909 (119905 minus 120591

119894))1003817100381710038171003817) sdot 119909 (119905)

+

119898

sum

119894=1

1

119902119894

119909119879

(119905) 119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

(23)

It follows that

1(119905 119909119905) le minus2120572119881

1(119905 119909119905) + 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+2120572119875 (119905) + 2 ((119901 + 120573) 12057501+ 120572120573) 119868)119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573)

sdot

119898

sum

119894=1

1205751(119905)

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

(24)

with

119901 = sup119905isinR+

119875 (119905)

1205751= max1le119894le119898

1205751198941 120575

2(119905) =

119898

sum

119894=0

1205751198942(119905)

(25)

Mathematical Problems in Engineering 5

Next the time derivative of 1198812(119905 119909119905) is given by

2(119905 119909119905) le minus2120572119881

2(119905 119909119905) +

119898

sum

119894=1

1205812119894119909 (119905)

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(26)

with 1205812119894= 12058121119894+ 12058122119894+ 12058123119894+ 12058124119894

The time derivative of 1198813(119905 119909119905) is given by

3(119905 119909119905)

= minus21205721198813(119905 119909119905)

+

119898

sum

119894=1

12058131119894120591119894119867(1205911198941198671198902120572120591119894119867

119909(119905)2

+ int

0

minus120591119894119867

(minus (119905 + 119904) 1198902120572(119904minus120591119894(119905+119904)+120591119894119867)

sdot1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

) 119889119904)

+

119898

sum

119894=1

12058132119894120591119894119867119871

(120591119894119867119871

1198902120572120591119894119867

119909 (119905)2

+ int

minus120591119894119871

minus120591119894119867

(minus (119905 + 119904) 1198902120572(119904minus120591119894(119905+119904)+120591119894119867)

sdot1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

) 119889119904)

(27)

By using the following differentiation rule (see [5]) oneobtains

119889

119889119905

(int

0

minus120591119894

int

119905

119906(119905+1199051)

120595 (119904) 119889119904 1198891199051)

= 120591119894120595 (119905) minus int

0

minus120591119894

(119905 + 119904) 120595 (119906 (119905 + 119904)) 119889119904

(28)

with

119906 (119905 + 119904) = 119905 + 119904 minus 120591119894(119905 + 119904)

(119905 + 119904) = 1 minus 120591119894(119905 + 119904)

(29)

It follows that

minus (119905 + 119904) le 120583 minus 1 (30)

So

3(119905 119909119905) le minus2120572119881

3(119905 119909119905)

+ (

119898

sum

119894=1

120581311198941205912

119894119867+

119898

sum

119894=1

120581321198941205912

119894119867119871)1198902120572120591119894119867

119909(119905)2

minus

119898

sum

119894=1

12058131119894120591119894119867119890minus2120572120591119894119867

(1 minus 120583)

sdot int

0

minus120591119894119867

1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

119889119904

minus

119898

sum

119894=1

12058132119894120591119894119867119871

119890minus2120572120591119894119867

(1 minus 120583)

sdot int

minus120591119894119871

minus120591119894119867

1003817100381710038171003817119909(119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

119889119904

(31)

Since the last integral term is nonnegative we obtain thefollowing estimation on

3(119905 119909119905)

3(119905 119909119905) le minus2120572119881

3(119905 119909119905) +

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(32)

with 1205813119894= 120581311198941205912

119894119867+ 120581321198941205912

119894119867119871

Therefore from (24)ndash(26)ndash(32) it follows that

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 1198810(119905 119909119905) (33)

where

1198810(119905 119909119905)

= 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ 2120572119875 (119905) + 2 ((119901 + 120573) 12057501+ 120572120573) 119868) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

1205812119894119909 (119905)

2

6 Mathematical Problems in Engineering

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(34)

It follows that

1198810(119905 119909119905)

le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

)119868)119909 (119905)

+ 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(35)

Now using Proposition 2 we have119898

sum

119894=1

(2 (119901 + 120573) 1205751

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817sdot 119909 (119905)

minus (12058123119894119890minus2120572120591119894119867

(1 minus 120583) minus 119902119894)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

)

le

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

(36)

with

0 lt 119902119894lt 12058123119894119890minus2120572120591119894119867

(1 minus 120583) 119894 = 1 119898 (37)

Hence the above expression in conjunction with (35) yields

1198810(119905 119909119905) le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573

+

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811)119868)119909 (119905)

minus 1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(38)

Since 119875(119905) is a solution of (9) with

120598 = 2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811 (39)

Mathematical Problems in Engineering 7

then

1198810(119905 119909119905)

le minus1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(40)

Note that we have

119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

le 120582max (119860 (119905) + 119860119879

(119905)) 119909 (119905)2

le 2120583 (119860 (119905)) 119909 (119905)2

le 2120583 (119860) 119909 (119905)2

(41)

with 120583(119860) = sup119905isinR+120583(119860(119905))

Then it follows that

1198810(119905 119909119905)

le minus(1205811minus 2120573120583 (119860)

minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

)119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(42)

Let

120578 = 1205811minus 2120573120583 (119860) minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

(43)

One has

120578 gt 0

997904rArr 1205751lt

radicsum119898

119894=1(12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894) (1205811minus 2120573120583 (119860))

(119901 + 120573)

1205811minus 2120573120583 (119860) gt 0

(44)

Then

1198810(119905 119909119905) le minus120578 (119909 (119905) minus 120585 (119905))

2

+ 119903 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(45)

with 120585(119905) = ((119901 + 120573)120578)1205752(119905) and 119903(119905) = ((119901 + 120573)2120578)1205752

2(119905)

Taking condition (44) into account we have

1198810(119905 119909119905) le 119903 (119905) (46)

Therefore from (33) and (46)

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 119903 (119905) forall119905 ge 0 (47)

which gives

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+ int

119905

0

1198902120572(119904minus119905)

119903 (119904) 119889119904 forall119905 ge 0

(48)

Now if 1205752(119905) is bounded for all 119905 ge 0 then there exists119872 gt 0

such that 1205752(119905) le 119872forall119905 ge 0Therefore 119903(119905)will be bounded

so there exists 1198721gt 0 such that 119903(119905) le 119872

1 forall119905 ge 0 with

1198721= (119901 + 120573)

2

1198722120578 So

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

1198721

2120572

(49)

and hence using the fact that

120573 119909 (119905)2

le 119881 (119905 119909119905) 119905 isin R

+

(50)

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199031 forall119905 ge 0 (51)

with 1199031= radic119872

12120572120573 This implies that the solution converges

to the ball 119861(0 1199031)

8 Mathematical Problems in Engineering

Next if 1205752satisfies

int

+infin

0

1205754

2(119904) 119889119904 lt +infin (52)

then 119868 = int+infin0

1199032(119904)119889119904 lt +infin

It follows that

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

11986812

2radic120572

forall119905 ge 0 (53)

Hence using (50) one gets

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199032 forall119905 ge 0 (54)

with 1199032= radic119868122120573radic120572

In this case the solution converges to the ball 119861(0 1199032)

Remark that from (45) if we suppose that 1205752(119905) tends

to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr

+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909

0) as follows

(1) Estimate 1198811(0 1199090) as

1198811(0 1199090) le (119901 + 120573)

10038171003817100381710038171206011003817100381710038171003817

2 (55)

(2) Estimate 1198812(0 1199090) as

1198812(0 1199090) =

119898

sum

119894=1

12058121119894int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

(56)

(i) Considering int119905119905minus120591119894119871

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119871

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119871

10038171003817100381710038171206011003817100381710038171003817

2

(57)

(ii) Considering int119905119905minus120591119894119867

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119867

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119867

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(58)

(iii) Considering int119905119905minus120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894(119905)

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(59)

(iv) Considering int119905119905minus]120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus]120591119894(119905)1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572]120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le ]120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(60)

1198812(0 1199090) le

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867)10038171003817100381710038171206011003817100381710038171003817

2

(61)

(3) Estimate 1198813(0 1199090) as

1198813(0 1199090)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

(62)

The first member of 1198813(0 1199090) is

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867

2120572

(1205911198941198671198902120572120591119894119867

minus

1

2120572

(1 minus 119890minus2120572120591119894119867

))10038171003817100381710038171206011003817100381710038171003817

2

le 1198881198941198671205913

119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(63)

with ((119909119890119909 + 119890minus119909 minus 1)1199092 ge 15) 119888119894119867= ((2120572120591

1198941198671198902120572120591119894119867

+

119890minus2120572120591119894119867

minus 1)(2120572120591119894119867)2

)

Mathematical Problems in Engineering 9

The second member of 1198813(0 1199090) is

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867119871

2120572

(120591119894119867119871

1198902120572120591119894119867

minus

1

2120572

[119890minus2120572120591119894119871

minus 119890minus2120572120591119894119867

])10038171003817100381710038171206011003817100381710038171003817

2

le

120591119894119867119871

(2120572)2(2120572120591119894119867119871

1198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

(

21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 1

(2120572)2

minus

21205721205911198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1

(2120572)2

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

119889119894119867119871

10038171003817100381710038171206011003817100381710038171003817

2

(64)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) and 119888

119894119871= (2120572120591

1198941198711198902120572120591119894119867

+

119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 Consider

1198813(0 1199090) le

119898

sum

119894=1

(120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

)10038171003817100381710038171206011003817100381710038171003817

2

(65)

Therefore from (55) (61) and (65) it follows that size

119881 (0 1199090)

le (119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+ 120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))10038171003817100381710038171206011003817100381710038171003817

2

(66)

Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575

(sdot)2char-

acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575

(sdot)2

increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin

Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575

(sdot)1are large enough then there is trade-off

between the values of 1205811and the maximum matrix measure

120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581

1is large then the constant 120598 is

requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)

Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes

4 Examples

Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591

1cos2(045119905)

1198911(119905 119909 (119905 minus 120591

1(119905)))

=[

[

minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575

2lowast cos( 119905

1 + 1199092

1

)

120575 cos (119905) 1199091(119905 minus 1205911(119905))

]

]

(67)where 120591

1 120575 gt 0 120575

2ge 0 will be chosen later and

119860 (119905) = [

119886 (119905) 1

minus1 119886 (119905)

] (68)

where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0

and 119902119894= 0 for (119894 = 1 119898) By computation we obtain

120583(119860) asymp minus81033741

Let120573 = 001 120572 = 1 120583 = 09 ] = 11

1205911119867

= 1205911 120591

1119871= 0 997904rArr 120591

1119867119871= 1205911119867

1205751= 12057511= 120575 120575

12= 1205752

1205811= 11989010

120581211

= 0 120581221

= 0 120581231

= 11989010

120581241

= 0 997904rArr 12058121= 11989010

120581311

=

1

1205912

1119867exp (2120572120591

1119867)

120581321

= 0

997904rArr 12058131=

1

exp (21205721205911119867)

(69)

Then120598 = 2 (119901 + 120573) 120575

01+ 2120572120573 + 120581

21

+ 1205813111989021205721205911119867

+ 1205811= 102 + 2119890

10

gt 0

(70)

10 Mathematical Problems in Engineering

Table 1

Delay bound (1205911119867) 2 4 6 8 10

Perturbation bound 1205751

3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245

We can verify that a solution 119875(119905) is given by

119875 (119905) =

[

[

[

[

119890sin(119905)

10

0

0

119890sin(119905)

10

]

]

]

]

(71)

We have 119901 = sup119905isinR+119875(119905) = 11989010

1205751lt

radic120581231119890minus21205721205911119867 (1 minus 120583) (120581

1minus 2120573120583 (119860))

(119901 + 120573)

1205811= 11989010

gt 2120573120583 (119860) = minus162067

120578 = 1205811minus 2120573120583 (119860) minus

(119901 + 120573)2

1205752

120581231119890minus21205721205911119867 (1 minus 120583)

gt 0

120578120577= (1 minus 120577

2

) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575

1 120577 lt 1

120574 =radic

119901 + 120573 + 1205812311205911119867

+ (1198881119867120591111986711989021205721205911119867

)

120573

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 = radic(119901 + 120573)

2

2120572120573120578120577

1205752=

1295910minus3

radic1 minus 1205772

1205752

(72)

If we take 120577 = 09 then 119903 = 297310minus31205752

We see that the perturbation bound 1205751in this example is

the same as in [5] if 1205752= 0 and is better than [4] as shown in

Table 1 The simulation of this example is shown in Figures 1and 2

Example 2 Consider the following second-order differentialsystem

1= minus5119909

1+ 1199092(119905 minus 01) + cos( 119909

1

1 + 1199052)

+ 002 sin (1199091(119905 minus 01) + 05)

2= minus1199091(119905 minus 01) minus 5119909

2

(73)

0 001 002 003 004 0050

001

002

003

004

005

006

007

008

009

Solu

tionx1

x1

Time t

Figure 1 The trajectories of 1199091

with

119860 = [

minus5 0

0 minus5

] 120582max (119860) = minus5 120583 (119860) = minus5

1198601= [

0 1

minus1 0

] 1205911119871= 0 120591

1= 1205911119867

= 01 120583 = 0

1198910(119905 119909) =

[

[

cos( 1199091

1 + 1199052)

0

]

]

12057501= 0 120575

02= 1

1198911(119905 119909 (119905 minus 1)) = [

002 sin (1199091(119905 minus 01) + 05)

0

]

12057511= 002 120575

12= 01

(74)

119909 = (1199091 1199092)119879

isin R2 Hence using (25) we obtain 1205751=

002 1205752= 11

The matrix

119875 (119905) = [

11990111

11990112

11990112

11990122

] (75)

can be computed as follows

11990111= 11990122= minus120573 + 119902

1(5 minus 120572)

11990112= radic1199021sdot radicminus10120573 + 2120572120573 + 119902

1(120572 minus 5)

2

minus 120598

(76)

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

a global-null controllability is required while the asymptoticstability is not guaranteed to be of exponential type In thesame way the asymptotic stability is not guaranteed to beexponential in [11] Another novelty is that the delays aretime-varying time-differentiable and they are not requiredto be known Only lower and upper-bound of the delayfunctions and their time-derivatives are required for stabilityanalysis We will study a class of nonlinear system such thatthe nonlinearity is bounded by some integrable functionswhich are bounded where the origin is not necessarily anequilibrium point We deal with the practical stability of theorigin (see [23]) The asymptotic stability is more importantthan stability also the desired system may be unstable andyet the system may oscillate sufficiently near this state thatits performance is acceptable thus the notion of practicalstability is more suitable in several situations than Lyapunovstability In this case all state trajectories are bounded andapproach a sufficiently small neighborhood of the origin Onealso desires that the state approaches the origin (or some suffi-ciently small neighborhood of it) in a sufficiently fast mannerThis notion of practical stability was introduced by [24] fornonlinear time-varying systems and studied for differentialequations with delays by [25] (see also the references therein)Moreover the authors in [26 27] constructed stabilizingcontrollers to obtain global convergence of solutions towarda small ball for some classes of uncertain control systemsIn this paper some sufficient conditions are given to obtainthe exponential uniform stability of the solutions toward aneighborhood of the origin based on a suitable Lyapunov-Krasovskii functional Two illustrative examples are givento demonstrate the validity of the main result where weestablish a table of comparison with other results

2 Preliminaries

We start by introducing some notations and definitions thatwill be employed throughout the paper

R+ denotes the set of all nonnegative real numbersR119899 denotes the 119899-dimensional Euclidean space 119909denotes the Euclidean vector norm of 119909 isin R119899 119909119879119910denotes the scalar product of two vectors 119909 119910

R119899times119903 denotes the space of all (119899 times 119903)-matrices

119860119879 denotes the transpose of the matrix 119860 119860 is

symmetric if 119860 = 119860119879

119868 denotes the identity matrix

120582(119860) denotes the set of eigenvalues of 119860 120582max(119860) =maxR119890(120582) 120582 isin 120582(119860)

120583(119860(119905)) denotes the matrix measure of the matrix 119860defined by

120583 (119860 (119905)) =

1

2

120582max (119860 (119905) + 119860119879

(119905)) (1)

1198712([minus120591119889 0]R119899) denotes the Hilbert space of all 119871

2-

integrable andR119899-valued functions on [0 119905]

119862([minus120591119889 0]R119899) denotes the Banach space of all R119899-

valued continuous functions mapping [minus120591119889 0] into

R119899 with 120591119889gt 0

119909119905= 119909 (119905 + 119904) 119904 isin [minus120591

119889 0]

1003817100381710038171003817119909119905

1003817100381710038171003817= sup119904isin[minus120591119889 0]

119909 (119905 + 119904)

(2)

119861(0 119903) = 119909 isin R119899119909 le 119903 with 119903 gt 0 denotes theclosed ball of center 0 and radius 119903

Consider a linear time-varying system with nonlineardelayed perturbations of the following form

(119905) = 119860 (119905) 119909 (119905) + 1198910(119905 119909 (119905)) +

119898

sum

119894=1

119860119894(119905) 119909 (119905 minus 120591

119894(119905))

+

119898

sum

119894=1

119891119894(119905 119909 (119905 minus 120591

119894(119905))) 119905 ge 0

119909 (119905) = 120601 (119905) 119905 isin [minus120591119889 0]

(3)

with 120591119889= max

1le119894le119898(120591119894119867) where 119909(119905) isin R119899 is the vector

where119860(119905) 119860119894(119905) isin R(119899119899) 119894 = 1 119898 arematrices functions

continuous and bounded in 119905 ge 0 120601(119905) isin 119862([minus120591119889 0]R119899)

is the function of initial conditions with the norm 120601 =

sup119904isin[minus120591119889 0]

120601(119904) 120591119894(119905) (119894 = 1 119898) is a given time-

differentiable time-varying delay function satisfying

120591119894119871lt 120591119894(119905) le 120591

119894119867

120591119894(119905) le 120583 lt 1 (119894 = 1 119898) forall119905 ge 0

(4)

where 120591119894119871 120591119894119867(0 le 120591

119894119871lt 120591119894119867) 120591119894119867119871

= 120591119894119867minus 120591119894119871

As a matter of fact the case that the delay derivative islarger than or equal to 1 is universal For example in networkcontrol systems the delay 120591

119894(119905) denotes 119905 minus 119895

119896 where 119895

119896(119896 =

1 2 ) are the sampling instants Thus this kind of delaysatisfies 120591

119894(119905) = 1 almost for all 119905 ge 0 (see [12]) For the case

of 120583 ge 1 if choosing a positive scalar 0 lt ] lt 120583minus1 then it

follows that

(]120591119894(119905))1015840

= ] 120591119894(119905) le ]120583 lt 1 (5)

and the nonlinear perturbation 119891119894(sdot sdot) (119894 = 0 119898) satisfies

1003817100381710038171003817119891119894(119905 119910)

1003817100381710038171003817le 1205751198941

10038171003817100381710038171199101003817100381710038171003817+ 1205751198942(119905) forall119905 ge 0 forall119910 isin R

119899

(6)

where 1205751198941gt 0 and 120575

1198942(sdot) are nonnegative continuous bounded

functions for 119894 = 0 1 119898

Definition 1 The system (3) is said to be globally uniformlyexponentially practically stable toward a ball 119861(0 119903) of radius119903which is a neighborhood of the origin if there exist positivenumbers 120572 120574 and 119903 such that every solution 119909(119905 120601) of thesystem satisfies

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le 120574119890minus120572(119905minus1199050)

10038171003817100381710038171206011003817100381710038171003817+ 119903 forall119905 ge 119905

0ge 0 (7)

The following technical proposition is needed for theproof of the main result

Mathematical Problems in Engineering 3

Proposition 2 Let 119876 119878 be symmetric matrices of appropriatedimensions and 119878 gt 0 Then

2119909119879

119876119910 minus 119910119879

119878119910 le 119909119879

119876119878minus1

119876119879

119909 forall (119909 119910) isin R119899

timesR119899

(8)

3 Main Result

Theorem3 Suppose that there exist some positive constants 120572120573 ] 120581

1 12058121119894 12058122119894 12058123119894 12058124119894 12058131119894 12058132119894 and a symmetric bounded

positive semidefinite differentiable matrix function 119875(119905) for all119905 ge 0 satisfying the following Lyapunov differential matrixequation (see [18])

(119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) + 120598119868 = 0

(9)

with

120598 = 2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811

120598 gt 0

1205812119894= 12058121119894+ 12058122119894+ 12058123119894+ 12058124119894

1205813119894= 120581311198941205912

119894119867+ 120581321198941205912

119894119867119871

119875120573(119905) = 119875 (119905) + 120573119868

119901 = sup119905isinR+

119875 (119905)

(10)

where

120578 = 1205811minus 2120573120583 (119860)

minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

gt 0

0 lt 119902119894lt 12058123119894119890minus2120572120591119894119867

(1 minus 120583) 119894 = 1 119898

0 lt 120583 lt ]minus1

(11)

with 120583(119860) = sup119905isinR+120583(119860(119905)) and 1205751 = max

1le119894le1198981205751198941

If

1205751lt

1

(119901 + 120573)

sdot (

119898

sum

119894=1

(12058123119894119890minus2120572120591119894119867

(1 minus 120583) minus 119902119894) (1205811minus 2120573120583 (119860)))

12

(12)

with 1205811minus 2120573120583(119860) gt 0 and 120575

2(119905) = sum

119898

119894=01205751198942(119905) then the system

(3) is globally uniformly exponentially practically stable towarda certain ball 119861(0 119903)

Moreover the solution 119909(119905 120601) satisfies an estimation as in(7) with size

120574 = ((119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))

sdot (120573)minus1

)

12

(13)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) 119888119894119867

= (21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus

1)(2120572120591119894119867)2 119888119894119871= (2120572120591

1198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 and

(i)

119903 = 1199031= radic

1198721

2120572120573

(14)

if 1205752is bounded by a scalar positive 119872 for all 119905 ge 0

with

1198721=

(119901 + 120573)2

1198722

120578

(15)

(ii)

119903 = 1199032= radic

11986812

2120573radic120572

(16)

if int+infin0

1205754

2(119904)119889119904 lt +infin with

119868 = int

+infin

0

1199032

(119904) 119889119904 119903 (119905) =

(119901 + 120573)2

120578

1205752

2(119905) (17)

Proof Consider the following Lyapunov-Karovskii func-tional

119881 (119905 119909119905) = 1198811(sdot) + 119881

2(sdot) + 119881

3(sdot) (18)

4 Mathematical Problems in Engineering

where

1198811(119905 119909119905) = 119909119879

119875 (119905) 119909 (119905) + 120573 119909 (119905)2

1198812(119905 119909119905) =

119898

sum

119894=1

12058121119894int

119905

119905minus120591119894119871

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

119905

119905minus120591119894119867

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

119905

119905minus120591119894(119905)

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

119905

119905minus]120591119894(119905)1198902120572(119904minus119905)

119909 (119904)2

119889119904

1198813(119905 119909119905)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

119905

119905+1199051minus120591119894(119905+1199051)

1198902120572(119904+120591119894119867minus119905)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

119905

119905+1199051minus120591119894(119905+1199051)

1198902120572(119904+120591119894119867minus119905)

119909 (119904)2

119889119904 1198891199051

(19)

with 120572 gt 0Let us consider the time derivative of 119881

1(119905 119909119905)

1(119905 119909119905) = 119909119879

(119905) (119905) 119909 (119905) + 119879

(119905) 119875120573(119905) 119909 (119905)

+ 119909119879

(119905) 119875120573(119905) (119905)

(20)

with 119875120573(119905) = 119875(119905) + 120573119868

Thus

1(119905 119909119905) = 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)) 119909 (119905)

+ 2119875120573(119905) (119891

0(119905 119909 (119905)) +

119898

sum

119894=1

119891119894(119905 119909 (119905 minus 120591

119894))) 119909 (119905)

+ 2

119898

sum

119894=1

119909119879

(119905) 119875120573(119905) 119860119894(119905) 119909 (119905 minus 120591

119894(119905))

(21)

From Proposition 2 we have

2

119898

sum

119894=1

119909119879

(119905) 119875120573(119905) 119860119894(119905) 119909 (119905 minus 120591

119894(119905))

le

119898

sum

119894=1

(

1

119902119894

119909119879

(119905) 119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) 119909 (119905)

+ 119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

)

(22)

This implies that

1(119905 119909119905)

le 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)) 119909 (119905)

+ 2( sup119905isinR+

119875 (119905) + 120573)

sdot (1198910(119905 119909 (119905)) +

119898

sum

119894=1

1003817100381710038171003817119891119894(119905 119909 (119905 minus 120591

119894))1003817100381710038171003817) sdot 119909 (119905)

+

119898

sum

119894=1

1

119902119894

119909119879

(119905) 119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

(23)

It follows that

1(119905 119909119905) le minus2120572119881

1(119905 119909119905) + 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+2120572119875 (119905) + 2 ((119901 + 120573) 12057501+ 120572120573) 119868)119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573)

sdot

119898

sum

119894=1

1205751(119905)

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

(24)

with

119901 = sup119905isinR+

119875 (119905)

1205751= max1le119894le119898

1205751198941 120575

2(119905) =

119898

sum

119894=0

1205751198942(119905)

(25)

Mathematical Problems in Engineering 5

Next the time derivative of 1198812(119905 119909119905) is given by

2(119905 119909119905) le minus2120572119881

2(119905 119909119905) +

119898

sum

119894=1

1205812119894119909 (119905)

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(26)

with 1205812119894= 12058121119894+ 12058122119894+ 12058123119894+ 12058124119894

The time derivative of 1198813(119905 119909119905) is given by

3(119905 119909119905)

= minus21205721198813(119905 119909119905)

+

119898

sum

119894=1

12058131119894120591119894119867(1205911198941198671198902120572120591119894119867

119909(119905)2

+ int

0

minus120591119894119867

(minus (119905 + 119904) 1198902120572(119904minus120591119894(119905+119904)+120591119894119867)

sdot1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

) 119889119904)

+

119898

sum

119894=1

12058132119894120591119894119867119871

(120591119894119867119871

1198902120572120591119894119867

119909 (119905)2

+ int

minus120591119894119871

minus120591119894119867

(minus (119905 + 119904) 1198902120572(119904minus120591119894(119905+119904)+120591119894119867)

sdot1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

) 119889119904)

(27)

By using the following differentiation rule (see [5]) oneobtains

119889

119889119905

(int

0

minus120591119894

int

119905

119906(119905+1199051)

120595 (119904) 119889119904 1198891199051)

= 120591119894120595 (119905) minus int

0

minus120591119894

(119905 + 119904) 120595 (119906 (119905 + 119904)) 119889119904

(28)

with

119906 (119905 + 119904) = 119905 + 119904 minus 120591119894(119905 + 119904)

(119905 + 119904) = 1 minus 120591119894(119905 + 119904)

(29)

It follows that

minus (119905 + 119904) le 120583 minus 1 (30)

So

3(119905 119909119905) le minus2120572119881

3(119905 119909119905)

+ (

119898

sum

119894=1

120581311198941205912

119894119867+

119898

sum

119894=1

120581321198941205912

119894119867119871)1198902120572120591119894119867

119909(119905)2

minus

119898

sum

119894=1

12058131119894120591119894119867119890minus2120572120591119894119867

(1 minus 120583)

sdot int

0

minus120591119894119867

1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

119889119904

minus

119898

sum

119894=1

12058132119894120591119894119867119871

119890minus2120572120591119894119867

(1 minus 120583)

sdot int

minus120591119894119871

minus120591119894119867

1003817100381710038171003817119909(119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

119889119904

(31)

Since the last integral term is nonnegative we obtain thefollowing estimation on

3(119905 119909119905)

3(119905 119909119905) le minus2120572119881

3(119905 119909119905) +

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(32)

with 1205813119894= 120581311198941205912

119894119867+ 120581321198941205912

119894119867119871

Therefore from (24)ndash(26)ndash(32) it follows that

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 1198810(119905 119909119905) (33)

where

1198810(119905 119909119905)

= 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ 2120572119875 (119905) + 2 ((119901 + 120573) 12057501+ 120572120573) 119868) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

1205812119894119909 (119905)

2

6 Mathematical Problems in Engineering

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(34)

It follows that

1198810(119905 119909119905)

le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

)119868)119909 (119905)

+ 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(35)

Now using Proposition 2 we have119898

sum

119894=1

(2 (119901 + 120573) 1205751

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817sdot 119909 (119905)

minus (12058123119894119890minus2120572120591119894119867

(1 minus 120583) minus 119902119894)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

)

le

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

(36)

with

0 lt 119902119894lt 12058123119894119890minus2120572120591119894119867

(1 minus 120583) 119894 = 1 119898 (37)

Hence the above expression in conjunction with (35) yields

1198810(119905 119909119905) le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573

+

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811)119868)119909 (119905)

minus 1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(38)

Since 119875(119905) is a solution of (9) with

120598 = 2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811 (39)

Mathematical Problems in Engineering 7

then

1198810(119905 119909119905)

le minus1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(40)

Note that we have

119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

le 120582max (119860 (119905) + 119860119879

(119905)) 119909 (119905)2

le 2120583 (119860 (119905)) 119909 (119905)2

le 2120583 (119860) 119909 (119905)2

(41)

with 120583(119860) = sup119905isinR+120583(119860(119905))

Then it follows that

1198810(119905 119909119905)

le minus(1205811minus 2120573120583 (119860)

minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

)119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(42)

Let

120578 = 1205811minus 2120573120583 (119860) minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

(43)

One has

120578 gt 0

997904rArr 1205751lt

radicsum119898

119894=1(12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894) (1205811minus 2120573120583 (119860))

(119901 + 120573)

1205811minus 2120573120583 (119860) gt 0

(44)

Then

1198810(119905 119909119905) le minus120578 (119909 (119905) minus 120585 (119905))

2

+ 119903 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(45)

with 120585(119905) = ((119901 + 120573)120578)1205752(119905) and 119903(119905) = ((119901 + 120573)2120578)1205752

2(119905)

Taking condition (44) into account we have

1198810(119905 119909119905) le 119903 (119905) (46)

Therefore from (33) and (46)

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 119903 (119905) forall119905 ge 0 (47)

which gives

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+ int

119905

0

1198902120572(119904minus119905)

119903 (119904) 119889119904 forall119905 ge 0

(48)

Now if 1205752(119905) is bounded for all 119905 ge 0 then there exists119872 gt 0

such that 1205752(119905) le 119872forall119905 ge 0Therefore 119903(119905)will be bounded

so there exists 1198721gt 0 such that 119903(119905) le 119872

1 forall119905 ge 0 with

1198721= (119901 + 120573)

2

1198722120578 So

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

1198721

2120572

(49)

and hence using the fact that

120573 119909 (119905)2

le 119881 (119905 119909119905) 119905 isin R

+

(50)

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199031 forall119905 ge 0 (51)

with 1199031= radic119872

12120572120573 This implies that the solution converges

to the ball 119861(0 1199031)

8 Mathematical Problems in Engineering

Next if 1205752satisfies

int

+infin

0

1205754

2(119904) 119889119904 lt +infin (52)

then 119868 = int+infin0

1199032(119904)119889119904 lt +infin

It follows that

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

11986812

2radic120572

forall119905 ge 0 (53)

Hence using (50) one gets

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199032 forall119905 ge 0 (54)

with 1199032= radic119868122120573radic120572

In this case the solution converges to the ball 119861(0 1199032)

Remark that from (45) if we suppose that 1205752(119905) tends

to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr

+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909

0) as follows

(1) Estimate 1198811(0 1199090) as

1198811(0 1199090) le (119901 + 120573)

10038171003817100381710038171206011003817100381710038171003817

2 (55)

(2) Estimate 1198812(0 1199090) as

1198812(0 1199090) =

119898

sum

119894=1

12058121119894int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

(56)

(i) Considering int119905119905minus120591119894119871

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119871

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119871

10038171003817100381710038171206011003817100381710038171003817

2

(57)

(ii) Considering int119905119905minus120591119894119867

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119867

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119867

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(58)

(iii) Considering int119905119905minus120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894(119905)

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(59)

(iv) Considering int119905119905minus]120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus]120591119894(119905)1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572]120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le ]120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(60)

1198812(0 1199090) le

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867)10038171003817100381710038171206011003817100381710038171003817

2

(61)

(3) Estimate 1198813(0 1199090) as

1198813(0 1199090)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

(62)

The first member of 1198813(0 1199090) is

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867

2120572

(1205911198941198671198902120572120591119894119867

minus

1

2120572

(1 minus 119890minus2120572120591119894119867

))10038171003817100381710038171206011003817100381710038171003817

2

le 1198881198941198671205913

119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(63)

with ((119909119890119909 + 119890minus119909 minus 1)1199092 ge 15) 119888119894119867= ((2120572120591

1198941198671198902120572120591119894119867

+

119890minus2120572120591119894119867

minus 1)(2120572120591119894119867)2

)

Mathematical Problems in Engineering 9

The second member of 1198813(0 1199090) is

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867119871

2120572

(120591119894119867119871

1198902120572120591119894119867

minus

1

2120572

[119890minus2120572120591119894119871

minus 119890minus2120572120591119894119867

])10038171003817100381710038171206011003817100381710038171003817

2

le

120591119894119867119871

(2120572)2(2120572120591119894119867119871

1198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

(

21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 1

(2120572)2

minus

21205721205911198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1

(2120572)2

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

119889119894119867119871

10038171003817100381710038171206011003817100381710038171003817

2

(64)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) and 119888

119894119871= (2120572120591

1198941198711198902120572120591119894119867

+

119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 Consider

1198813(0 1199090) le

119898

sum

119894=1

(120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

)10038171003817100381710038171206011003817100381710038171003817

2

(65)

Therefore from (55) (61) and (65) it follows that size

119881 (0 1199090)

le (119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+ 120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))10038171003817100381710038171206011003817100381710038171003817

2

(66)

Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575

(sdot)2char-

acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575

(sdot)2

increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin

Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575

(sdot)1are large enough then there is trade-off

between the values of 1205811and the maximum matrix measure

120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581

1is large then the constant 120598 is

requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)

Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes

4 Examples

Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591

1cos2(045119905)

1198911(119905 119909 (119905 minus 120591

1(119905)))

=[

[

minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575

2lowast cos( 119905

1 + 1199092

1

)

120575 cos (119905) 1199091(119905 minus 1205911(119905))

]

]

(67)where 120591

1 120575 gt 0 120575

2ge 0 will be chosen later and

119860 (119905) = [

119886 (119905) 1

minus1 119886 (119905)

] (68)

where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0

and 119902119894= 0 for (119894 = 1 119898) By computation we obtain

120583(119860) asymp minus81033741

Let120573 = 001 120572 = 1 120583 = 09 ] = 11

1205911119867

= 1205911 120591

1119871= 0 997904rArr 120591

1119867119871= 1205911119867

1205751= 12057511= 120575 120575

12= 1205752

1205811= 11989010

120581211

= 0 120581221

= 0 120581231

= 11989010

120581241

= 0 997904rArr 12058121= 11989010

120581311

=

1

1205912

1119867exp (2120572120591

1119867)

120581321

= 0

997904rArr 12058131=

1

exp (21205721205911119867)

(69)

Then120598 = 2 (119901 + 120573) 120575

01+ 2120572120573 + 120581

21

+ 1205813111989021205721205911119867

+ 1205811= 102 + 2119890

10

gt 0

(70)

10 Mathematical Problems in Engineering

Table 1

Delay bound (1205911119867) 2 4 6 8 10

Perturbation bound 1205751

3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245

We can verify that a solution 119875(119905) is given by

119875 (119905) =

[

[

[

[

119890sin(119905)

10

0

0

119890sin(119905)

10

]

]

]

]

(71)

We have 119901 = sup119905isinR+119875(119905) = 11989010

1205751lt

radic120581231119890minus21205721205911119867 (1 minus 120583) (120581

1minus 2120573120583 (119860))

(119901 + 120573)

1205811= 11989010

gt 2120573120583 (119860) = minus162067

120578 = 1205811minus 2120573120583 (119860) minus

(119901 + 120573)2

1205752

120581231119890minus21205721205911119867 (1 minus 120583)

gt 0

120578120577= (1 minus 120577

2

) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575

1 120577 lt 1

120574 =radic

119901 + 120573 + 1205812311205911119867

+ (1198881119867120591111986711989021205721205911119867

)

120573

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 = radic(119901 + 120573)

2

2120572120573120578120577

1205752=

1295910minus3

radic1 minus 1205772

1205752

(72)

If we take 120577 = 09 then 119903 = 297310minus31205752

We see that the perturbation bound 1205751in this example is

the same as in [5] if 1205752= 0 and is better than [4] as shown in

Table 1 The simulation of this example is shown in Figures 1and 2

Example 2 Consider the following second-order differentialsystem

1= minus5119909

1+ 1199092(119905 minus 01) + cos( 119909

1

1 + 1199052)

+ 002 sin (1199091(119905 minus 01) + 05)

2= minus1199091(119905 minus 01) minus 5119909

2

(73)

0 001 002 003 004 0050

001

002

003

004

005

006

007

008

009

Solu

tionx1

x1

Time t

Figure 1 The trajectories of 1199091

with

119860 = [

minus5 0

0 minus5

] 120582max (119860) = minus5 120583 (119860) = minus5

1198601= [

0 1

minus1 0

] 1205911119871= 0 120591

1= 1205911119867

= 01 120583 = 0

1198910(119905 119909) =

[

[

cos( 1199091

1 + 1199052)

0

]

]

12057501= 0 120575

02= 1

1198911(119905 119909 (119905 minus 1)) = [

002 sin (1199091(119905 minus 01) + 05)

0

]

12057511= 002 120575

12= 01

(74)

119909 = (1199091 1199092)119879

isin R2 Hence using (25) we obtain 1205751=

002 1205752= 11

The matrix

119875 (119905) = [

11990111

11990112

11990112

11990122

] (75)

can be computed as follows

11990111= 11990122= minus120573 + 119902

1(5 minus 120572)

11990112= radic1199021sdot radicminus10120573 + 2120572120573 + 119902

1(120572 minus 5)

2

minus 120598

(76)

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Proposition 2 Let 119876 119878 be symmetric matrices of appropriatedimensions and 119878 gt 0 Then

2119909119879

119876119910 minus 119910119879

119878119910 le 119909119879

119876119878minus1

119876119879

119909 forall (119909 119910) isin R119899

timesR119899

(8)

3 Main Result

Theorem3 Suppose that there exist some positive constants 120572120573 ] 120581

1 12058121119894 12058122119894 12058123119894 12058124119894 12058131119894 12058132119894 and a symmetric bounded

positive semidefinite differentiable matrix function 119875(119905) for all119905 ge 0 satisfying the following Lyapunov differential matrixequation (see [18])

(119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) + 120598119868 = 0

(9)

with

120598 = 2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811

120598 gt 0

1205812119894= 12058121119894+ 12058122119894+ 12058123119894+ 12058124119894

1205813119894= 120581311198941205912

119894119867+ 120581321198941205912

119894119867119871

119875120573(119905) = 119875 (119905) + 120573119868

119901 = sup119905isinR+

119875 (119905)

(10)

where

120578 = 1205811minus 2120573120583 (119860)

minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

gt 0

0 lt 119902119894lt 12058123119894119890minus2120572120591119894119867

(1 minus 120583) 119894 = 1 119898

0 lt 120583 lt ]minus1

(11)

with 120583(119860) = sup119905isinR+120583(119860(119905)) and 1205751 = max

1le119894le1198981205751198941

If

1205751lt

1

(119901 + 120573)

sdot (

119898

sum

119894=1

(12058123119894119890minus2120572120591119894119867

(1 minus 120583) minus 119902119894) (1205811minus 2120573120583 (119860)))

12

(12)

with 1205811minus 2120573120583(119860) gt 0 and 120575

2(119905) = sum

119898

119894=01205751198942(119905) then the system

(3) is globally uniformly exponentially practically stable towarda certain ball 119861(0 119903)

Moreover the solution 119909(119905 120601) satisfies an estimation as in(7) with size

120574 = ((119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))

sdot (120573)minus1

)

12

(13)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) 119888119894119867

= (21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus

1)(2120572120591119894119867)2 119888119894119871= (2120572120591

1198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 and

(i)

119903 = 1199031= radic

1198721

2120572120573

(14)

if 1205752is bounded by a scalar positive 119872 for all 119905 ge 0

with

1198721=

(119901 + 120573)2

1198722

120578

(15)

(ii)

119903 = 1199032= radic

11986812

2120573radic120572

(16)

if int+infin0

1205754

2(119904)119889119904 lt +infin with

119868 = int

+infin

0

1199032

(119904) 119889119904 119903 (119905) =

(119901 + 120573)2

120578

1205752

2(119905) (17)

Proof Consider the following Lyapunov-Karovskii func-tional

119881 (119905 119909119905) = 1198811(sdot) + 119881

2(sdot) + 119881

3(sdot) (18)

4 Mathematical Problems in Engineering

where

1198811(119905 119909119905) = 119909119879

119875 (119905) 119909 (119905) + 120573 119909 (119905)2

1198812(119905 119909119905) =

119898

sum

119894=1

12058121119894int

119905

119905minus120591119894119871

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

119905

119905minus120591119894119867

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

119905

119905minus120591119894(119905)

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

119905

119905minus]120591119894(119905)1198902120572(119904minus119905)

119909 (119904)2

119889119904

1198813(119905 119909119905)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

119905

119905+1199051minus120591119894(119905+1199051)

1198902120572(119904+120591119894119867minus119905)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

119905

119905+1199051minus120591119894(119905+1199051)

1198902120572(119904+120591119894119867minus119905)

119909 (119904)2

119889119904 1198891199051

(19)

with 120572 gt 0Let us consider the time derivative of 119881

1(119905 119909119905)

1(119905 119909119905) = 119909119879

(119905) (119905) 119909 (119905) + 119879

(119905) 119875120573(119905) 119909 (119905)

+ 119909119879

(119905) 119875120573(119905) (119905)

(20)

with 119875120573(119905) = 119875(119905) + 120573119868

Thus

1(119905 119909119905) = 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)) 119909 (119905)

+ 2119875120573(119905) (119891

0(119905 119909 (119905)) +

119898

sum

119894=1

119891119894(119905 119909 (119905 minus 120591

119894))) 119909 (119905)

+ 2

119898

sum

119894=1

119909119879

(119905) 119875120573(119905) 119860119894(119905) 119909 (119905 minus 120591

119894(119905))

(21)

From Proposition 2 we have

2

119898

sum

119894=1

119909119879

(119905) 119875120573(119905) 119860119894(119905) 119909 (119905 minus 120591

119894(119905))

le

119898

sum

119894=1

(

1

119902119894

119909119879

(119905) 119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) 119909 (119905)

+ 119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

)

(22)

This implies that

1(119905 119909119905)

le 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)) 119909 (119905)

+ 2( sup119905isinR+

119875 (119905) + 120573)

sdot (1198910(119905 119909 (119905)) +

119898

sum

119894=1

1003817100381710038171003817119891119894(119905 119909 (119905 minus 120591

119894))1003817100381710038171003817) sdot 119909 (119905)

+

119898

sum

119894=1

1

119902119894

119909119879

(119905) 119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

(23)

It follows that

1(119905 119909119905) le minus2120572119881

1(119905 119909119905) + 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+2120572119875 (119905) + 2 ((119901 + 120573) 12057501+ 120572120573) 119868)119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573)

sdot

119898

sum

119894=1

1205751(119905)

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

(24)

with

119901 = sup119905isinR+

119875 (119905)

1205751= max1le119894le119898

1205751198941 120575

2(119905) =

119898

sum

119894=0

1205751198942(119905)

(25)

Mathematical Problems in Engineering 5

Next the time derivative of 1198812(119905 119909119905) is given by

2(119905 119909119905) le minus2120572119881

2(119905 119909119905) +

119898

sum

119894=1

1205812119894119909 (119905)

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(26)

with 1205812119894= 12058121119894+ 12058122119894+ 12058123119894+ 12058124119894

The time derivative of 1198813(119905 119909119905) is given by

3(119905 119909119905)

= minus21205721198813(119905 119909119905)

+

119898

sum

119894=1

12058131119894120591119894119867(1205911198941198671198902120572120591119894119867

119909(119905)2

+ int

0

minus120591119894119867

(minus (119905 + 119904) 1198902120572(119904minus120591119894(119905+119904)+120591119894119867)

sdot1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

) 119889119904)

+

119898

sum

119894=1

12058132119894120591119894119867119871

(120591119894119867119871

1198902120572120591119894119867

119909 (119905)2

+ int

minus120591119894119871

minus120591119894119867

(minus (119905 + 119904) 1198902120572(119904minus120591119894(119905+119904)+120591119894119867)

sdot1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

) 119889119904)

(27)

By using the following differentiation rule (see [5]) oneobtains

119889

119889119905

(int

0

minus120591119894

int

119905

119906(119905+1199051)

120595 (119904) 119889119904 1198891199051)

= 120591119894120595 (119905) minus int

0

minus120591119894

(119905 + 119904) 120595 (119906 (119905 + 119904)) 119889119904

(28)

with

119906 (119905 + 119904) = 119905 + 119904 minus 120591119894(119905 + 119904)

(119905 + 119904) = 1 minus 120591119894(119905 + 119904)

(29)

It follows that

minus (119905 + 119904) le 120583 minus 1 (30)

So

3(119905 119909119905) le minus2120572119881

3(119905 119909119905)

+ (

119898

sum

119894=1

120581311198941205912

119894119867+

119898

sum

119894=1

120581321198941205912

119894119867119871)1198902120572120591119894119867

119909(119905)2

minus

119898

sum

119894=1

12058131119894120591119894119867119890minus2120572120591119894119867

(1 minus 120583)

sdot int

0

minus120591119894119867

1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

119889119904

minus

119898

sum

119894=1

12058132119894120591119894119867119871

119890minus2120572120591119894119867

(1 minus 120583)

sdot int

minus120591119894119871

minus120591119894119867

1003817100381710038171003817119909(119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

119889119904

(31)

Since the last integral term is nonnegative we obtain thefollowing estimation on

3(119905 119909119905)

3(119905 119909119905) le minus2120572119881

3(119905 119909119905) +

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(32)

with 1205813119894= 120581311198941205912

119894119867+ 120581321198941205912

119894119867119871

Therefore from (24)ndash(26)ndash(32) it follows that

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 1198810(119905 119909119905) (33)

where

1198810(119905 119909119905)

= 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ 2120572119875 (119905) + 2 ((119901 + 120573) 12057501+ 120572120573) 119868) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

1205812119894119909 (119905)

2

6 Mathematical Problems in Engineering

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(34)

It follows that

1198810(119905 119909119905)

le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

)119868)119909 (119905)

+ 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(35)

Now using Proposition 2 we have119898

sum

119894=1

(2 (119901 + 120573) 1205751

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817sdot 119909 (119905)

minus (12058123119894119890minus2120572120591119894119867

(1 minus 120583) minus 119902119894)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

)

le

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

(36)

with

0 lt 119902119894lt 12058123119894119890minus2120572120591119894119867

(1 minus 120583) 119894 = 1 119898 (37)

Hence the above expression in conjunction with (35) yields

1198810(119905 119909119905) le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573

+

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811)119868)119909 (119905)

minus 1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(38)

Since 119875(119905) is a solution of (9) with

120598 = 2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811 (39)

Mathematical Problems in Engineering 7

then

1198810(119905 119909119905)

le minus1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(40)

Note that we have

119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

le 120582max (119860 (119905) + 119860119879

(119905)) 119909 (119905)2

le 2120583 (119860 (119905)) 119909 (119905)2

le 2120583 (119860) 119909 (119905)2

(41)

with 120583(119860) = sup119905isinR+120583(119860(119905))

Then it follows that

1198810(119905 119909119905)

le minus(1205811minus 2120573120583 (119860)

minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

)119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(42)

Let

120578 = 1205811minus 2120573120583 (119860) minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

(43)

One has

120578 gt 0

997904rArr 1205751lt

radicsum119898

119894=1(12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894) (1205811minus 2120573120583 (119860))

(119901 + 120573)

1205811minus 2120573120583 (119860) gt 0

(44)

Then

1198810(119905 119909119905) le minus120578 (119909 (119905) minus 120585 (119905))

2

+ 119903 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(45)

with 120585(119905) = ((119901 + 120573)120578)1205752(119905) and 119903(119905) = ((119901 + 120573)2120578)1205752

2(119905)

Taking condition (44) into account we have

1198810(119905 119909119905) le 119903 (119905) (46)

Therefore from (33) and (46)

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 119903 (119905) forall119905 ge 0 (47)

which gives

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+ int

119905

0

1198902120572(119904minus119905)

119903 (119904) 119889119904 forall119905 ge 0

(48)

Now if 1205752(119905) is bounded for all 119905 ge 0 then there exists119872 gt 0

such that 1205752(119905) le 119872forall119905 ge 0Therefore 119903(119905)will be bounded

so there exists 1198721gt 0 such that 119903(119905) le 119872

1 forall119905 ge 0 with

1198721= (119901 + 120573)

2

1198722120578 So

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

1198721

2120572

(49)

and hence using the fact that

120573 119909 (119905)2

le 119881 (119905 119909119905) 119905 isin R

+

(50)

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199031 forall119905 ge 0 (51)

with 1199031= radic119872

12120572120573 This implies that the solution converges

to the ball 119861(0 1199031)

8 Mathematical Problems in Engineering

Next if 1205752satisfies

int

+infin

0

1205754

2(119904) 119889119904 lt +infin (52)

then 119868 = int+infin0

1199032(119904)119889119904 lt +infin

It follows that

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

11986812

2radic120572

forall119905 ge 0 (53)

Hence using (50) one gets

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199032 forall119905 ge 0 (54)

with 1199032= radic119868122120573radic120572

In this case the solution converges to the ball 119861(0 1199032)

Remark that from (45) if we suppose that 1205752(119905) tends

to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr

+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909

0) as follows

(1) Estimate 1198811(0 1199090) as

1198811(0 1199090) le (119901 + 120573)

10038171003817100381710038171206011003817100381710038171003817

2 (55)

(2) Estimate 1198812(0 1199090) as

1198812(0 1199090) =

119898

sum

119894=1

12058121119894int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

(56)

(i) Considering int119905119905minus120591119894119871

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119871

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119871

10038171003817100381710038171206011003817100381710038171003817

2

(57)

(ii) Considering int119905119905minus120591119894119867

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119867

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119867

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(58)

(iii) Considering int119905119905minus120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894(119905)

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(59)

(iv) Considering int119905119905minus]120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus]120591119894(119905)1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572]120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le ]120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(60)

1198812(0 1199090) le

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867)10038171003817100381710038171206011003817100381710038171003817

2

(61)

(3) Estimate 1198813(0 1199090) as

1198813(0 1199090)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

(62)

The first member of 1198813(0 1199090) is

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867

2120572

(1205911198941198671198902120572120591119894119867

minus

1

2120572

(1 minus 119890minus2120572120591119894119867

))10038171003817100381710038171206011003817100381710038171003817

2

le 1198881198941198671205913

119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(63)

with ((119909119890119909 + 119890minus119909 minus 1)1199092 ge 15) 119888119894119867= ((2120572120591

1198941198671198902120572120591119894119867

+

119890minus2120572120591119894119867

minus 1)(2120572120591119894119867)2

)

Mathematical Problems in Engineering 9

The second member of 1198813(0 1199090) is

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867119871

2120572

(120591119894119867119871

1198902120572120591119894119867

minus

1

2120572

[119890minus2120572120591119894119871

minus 119890minus2120572120591119894119867

])10038171003817100381710038171206011003817100381710038171003817

2

le

120591119894119867119871

(2120572)2(2120572120591119894119867119871

1198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

(

21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 1

(2120572)2

minus

21205721205911198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1

(2120572)2

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

119889119894119867119871

10038171003817100381710038171206011003817100381710038171003817

2

(64)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) and 119888

119894119871= (2120572120591

1198941198711198902120572120591119894119867

+

119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 Consider

1198813(0 1199090) le

119898

sum

119894=1

(120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

)10038171003817100381710038171206011003817100381710038171003817

2

(65)

Therefore from (55) (61) and (65) it follows that size

119881 (0 1199090)

le (119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+ 120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))10038171003817100381710038171206011003817100381710038171003817

2

(66)

Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575

(sdot)2char-

acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575

(sdot)2

increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin

Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575

(sdot)1are large enough then there is trade-off

between the values of 1205811and the maximum matrix measure

120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581

1is large then the constant 120598 is

requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)

Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes

4 Examples

Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591

1cos2(045119905)

1198911(119905 119909 (119905 minus 120591

1(119905)))

=[

[

minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575

2lowast cos( 119905

1 + 1199092

1

)

120575 cos (119905) 1199091(119905 minus 1205911(119905))

]

]

(67)where 120591

1 120575 gt 0 120575

2ge 0 will be chosen later and

119860 (119905) = [

119886 (119905) 1

minus1 119886 (119905)

] (68)

where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0

and 119902119894= 0 for (119894 = 1 119898) By computation we obtain

120583(119860) asymp minus81033741

Let120573 = 001 120572 = 1 120583 = 09 ] = 11

1205911119867

= 1205911 120591

1119871= 0 997904rArr 120591

1119867119871= 1205911119867

1205751= 12057511= 120575 120575

12= 1205752

1205811= 11989010

120581211

= 0 120581221

= 0 120581231

= 11989010

120581241

= 0 997904rArr 12058121= 11989010

120581311

=

1

1205912

1119867exp (2120572120591

1119867)

120581321

= 0

997904rArr 12058131=

1

exp (21205721205911119867)

(69)

Then120598 = 2 (119901 + 120573) 120575

01+ 2120572120573 + 120581

21

+ 1205813111989021205721205911119867

+ 1205811= 102 + 2119890

10

gt 0

(70)

10 Mathematical Problems in Engineering

Table 1

Delay bound (1205911119867) 2 4 6 8 10

Perturbation bound 1205751

3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245

We can verify that a solution 119875(119905) is given by

119875 (119905) =

[

[

[

[

119890sin(119905)

10

0

0

119890sin(119905)

10

]

]

]

]

(71)

We have 119901 = sup119905isinR+119875(119905) = 11989010

1205751lt

radic120581231119890minus21205721205911119867 (1 minus 120583) (120581

1minus 2120573120583 (119860))

(119901 + 120573)

1205811= 11989010

gt 2120573120583 (119860) = minus162067

120578 = 1205811minus 2120573120583 (119860) minus

(119901 + 120573)2

1205752

120581231119890minus21205721205911119867 (1 minus 120583)

gt 0

120578120577= (1 minus 120577

2

) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575

1 120577 lt 1

120574 =radic

119901 + 120573 + 1205812311205911119867

+ (1198881119867120591111986711989021205721205911119867

)

120573

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 = radic(119901 + 120573)

2

2120572120573120578120577

1205752=

1295910minus3

radic1 minus 1205772

1205752

(72)

If we take 120577 = 09 then 119903 = 297310minus31205752

We see that the perturbation bound 1205751in this example is

the same as in [5] if 1205752= 0 and is better than [4] as shown in

Table 1 The simulation of this example is shown in Figures 1and 2

Example 2 Consider the following second-order differentialsystem

1= minus5119909

1+ 1199092(119905 minus 01) + cos( 119909

1

1 + 1199052)

+ 002 sin (1199091(119905 minus 01) + 05)

2= minus1199091(119905 minus 01) minus 5119909

2

(73)

0 001 002 003 004 0050

001

002

003

004

005

006

007

008

009

Solu

tionx1

x1

Time t

Figure 1 The trajectories of 1199091

with

119860 = [

minus5 0

0 minus5

] 120582max (119860) = minus5 120583 (119860) = minus5

1198601= [

0 1

minus1 0

] 1205911119871= 0 120591

1= 1205911119867

= 01 120583 = 0

1198910(119905 119909) =

[

[

cos( 1199091

1 + 1199052)

0

]

]

12057501= 0 120575

02= 1

1198911(119905 119909 (119905 minus 1)) = [

002 sin (1199091(119905 minus 01) + 05)

0

]

12057511= 002 120575

12= 01

(74)

119909 = (1199091 1199092)119879

isin R2 Hence using (25) we obtain 1205751=

002 1205752= 11

The matrix

119875 (119905) = [

11990111

11990112

11990112

11990122

] (75)

can be computed as follows

11990111= 11990122= minus120573 + 119902

1(5 minus 120572)

11990112= radic1199021sdot radicminus10120573 + 2120572120573 + 119902

1(120572 minus 5)

2

minus 120598

(76)

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where

1198811(119905 119909119905) = 119909119879

119875 (119905) 119909 (119905) + 120573 119909 (119905)2

1198812(119905 119909119905) =

119898

sum

119894=1

12058121119894int

119905

119905minus120591119894119871

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

119905

119905minus120591119894119867

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

119905

119905minus120591119894(119905)

1198902120572(119904minus119905)

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

119905

119905minus]120591119894(119905)1198902120572(119904minus119905)

119909 (119904)2

119889119904

1198813(119905 119909119905)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

119905

119905+1199051minus120591119894(119905+1199051)

1198902120572(119904+120591119894119867minus119905)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

119905

119905+1199051minus120591119894(119905+1199051)

1198902120572(119904+120591119894119867minus119905)

119909 (119904)2

119889119904 1198891199051

(19)

with 120572 gt 0Let us consider the time derivative of 119881

1(119905 119909119905)

1(119905 119909119905) = 119909119879

(119905) (119905) 119909 (119905) + 119879

(119905) 119875120573(119905) 119909 (119905)

+ 119909119879

(119905) 119875120573(119905) (119905)

(20)

with 119875120573(119905) = 119875(119905) + 120573119868

Thus

1(119905 119909119905) = 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)) 119909 (119905)

+ 2119875120573(119905) (119891

0(119905 119909 (119905)) +

119898

sum

119894=1

119891119894(119905 119909 (119905 minus 120591

119894))) 119909 (119905)

+ 2

119898

sum

119894=1

119909119879

(119905) 119875120573(119905) 119860119894(119905) 119909 (119905 minus 120591

119894(119905))

(21)

From Proposition 2 we have

2

119898

sum

119894=1

119909119879

(119905) 119875120573(119905) 119860119894(119905) 119909 (119905 minus 120591

119894(119905))

le

119898

sum

119894=1

(

1

119902119894

119909119879

(119905) 119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) 119909 (119905)

+ 119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

)

(22)

This implies that

1(119905 119909119905)

le 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)) 119909 (119905)

+ 2( sup119905isinR+

119875 (119905) + 120573)

sdot (1198910(119905 119909 (119905)) +

119898

sum

119894=1

1003817100381710038171003817119891119894(119905 119909 (119905 minus 120591

119894))1003817100381710038171003817) sdot 119909 (119905)

+

119898

sum

119894=1

1

119902119894

119909119879

(119905) 119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905) 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

(23)

It follows that

1(119905 119909119905) le minus2120572119881

1(119905 119909119905) + 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+2120572119875 (119905) + 2 ((119901 + 120573) 12057501+ 120572120573) 119868)119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573)

sdot

119898

sum

119894=1

1205751(119905)

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

(24)

with

119901 = sup119905isinR+

119875 (119905)

1205751= max1le119894le119898

1205751198941 120575

2(119905) =

119898

sum

119894=0

1205751198942(119905)

(25)

Mathematical Problems in Engineering 5

Next the time derivative of 1198812(119905 119909119905) is given by

2(119905 119909119905) le minus2120572119881

2(119905 119909119905) +

119898

sum

119894=1

1205812119894119909 (119905)

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(26)

with 1205812119894= 12058121119894+ 12058122119894+ 12058123119894+ 12058124119894

The time derivative of 1198813(119905 119909119905) is given by

3(119905 119909119905)

= minus21205721198813(119905 119909119905)

+

119898

sum

119894=1

12058131119894120591119894119867(1205911198941198671198902120572120591119894119867

119909(119905)2

+ int

0

minus120591119894119867

(minus (119905 + 119904) 1198902120572(119904minus120591119894(119905+119904)+120591119894119867)

sdot1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

) 119889119904)

+

119898

sum

119894=1

12058132119894120591119894119867119871

(120591119894119867119871

1198902120572120591119894119867

119909 (119905)2

+ int

minus120591119894119871

minus120591119894119867

(minus (119905 + 119904) 1198902120572(119904minus120591119894(119905+119904)+120591119894119867)

sdot1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

) 119889119904)

(27)

By using the following differentiation rule (see [5]) oneobtains

119889

119889119905

(int

0

minus120591119894

int

119905

119906(119905+1199051)

120595 (119904) 119889119904 1198891199051)

= 120591119894120595 (119905) minus int

0

minus120591119894

(119905 + 119904) 120595 (119906 (119905 + 119904)) 119889119904

(28)

with

119906 (119905 + 119904) = 119905 + 119904 minus 120591119894(119905 + 119904)

(119905 + 119904) = 1 minus 120591119894(119905 + 119904)

(29)

It follows that

minus (119905 + 119904) le 120583 minus 1 (30)

So

3(119905 119909119905) le minus2120572119881

3(119905 119909119905)

+ (

119898

sum

119894=1

120581311198941205912

119894119867+

119898

sum

119894=1

120581321198941205912

119894119867119871)1198902120572120591119894119867

119909(119905)2

minus

119898

sum

119894=1

12058131119894120591119894119867119890minus2120572120591119894119867

(1 minus 120583)

sdot int

0

minus120591119894119867

1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

119889119904

minus

119898

sum

119894=1

12058132119894120591119894119867119871

119890minus2120572120591119894119867

(1 minus 120583)

sdot int

minus120591119894119871

minus120591119894119867

1003817100381710038171003817119909(119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

119889119904

(31)

Since the last integral term is nonnegative we obtain thefollowing estimation on

3(119905 119909119905)

3(119905 119909119905) le minus2120572119881

3(119905 119909119905) +

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(32)

with 1205813119894= 120581311198941205912

119894119867+ 120581321198941205912

119894119867119871

Therefore from (24)ndash(26)ndash(32) it follows that

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 1198810(119905 119909119905) (33)

where

1198810(119905 119909119905)

= 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ 2120572119875 (119905) + 2 ((119901 + 120573) 12057501+ 120572120573) 119868) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

1205812119894119909 (119905)

2

6 Mathematical Problems in Engineering

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(34)

It follows that

1198810(119905 119909119905)

le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

)119868)119909 (119905)

+ 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(35)

Now using Proposition 2 we have119898

sum

119894=1

(2 (119901 + 120573) 1205751

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817sdot 119909 (119905)

minus (12058123119894119890minus2120572120591119894119867

(1 minus 120583) minus 119902119894)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

)

le

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

(36)

with

0 lt 119902119894lt 12058123119894119890minus2120572120591119894119867

(1 minus 120583) 119894 = 1 119898 (37)

Hence the above expression in conjunction with (35) yields

1198810(119905 119909119905) le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573

+

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811)119868)119909 (119905)

minus 1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(38)

Since 119875(119905) is a solution of (9) with

120598 = 2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811 (39)

Mathematical Problems in Engineering 7

then

1198810(119905 119909119905)

le minus1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(40)

Note that we have

119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

le 120582max (119860 (119905) + 119860119879

(119905)) 119909 (119905)2

le 2120583 (119860 (119905)) 119909 (119905)2

le 2120583 (119860) 119909 (119905)2

(41)

with 120583(119860) = sup119905isinR+120583(119860(119905))

Then it follows that

1198810(119905 119909119905)

le minus(1205811minus 2120573120583 (119860)

minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

)119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(42)

Let

120578 = 1205811minus 2120573120583 (119860) minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

(43)

One has

120578 gt 0

997904rArr 1205751lt

radicsum119898

119894=1(12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894) (1205811minus 2120573120583 (119860))

(119901 + 120573)

1205811minus 2120573120583 (119860) gt 0

(44)

Then

1198810(119905 119909119905) le minus120578 (119909 (119905) minus 120585 (119905))

2

+ 119903 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(45)

with 120585(119905) = ((119901 + 120573)120578)1205752(119905) and 119903(119905) = ((119901 + 120573)2120578)1205752

2(119905)

Taking condition (44) into account we have

1198810(119905 119909119905) le 119903 (119905) (46)

Therefore from (33) and (46)

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 119903 (119905) forall119905 ge 0 (47)

which gives

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+ int

119905

0

1198902120572(119904minus119905)

119903 (119904) 119889119904 forall119905 ge 0

(48)

Now if 1205752(119905) is bounded for all 119905 ge 0 then there exists119872 gt 0

such that 1205752(119905) le 119872forall119905 ge 0Therefore 119903(119905)will be bounded

so there exists 1198721gt 0 such that 119903(119905) le 119872

1 forall119905 ge 0 with

1198721= (119901 + 120573)

2

1198722120578 So

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

1198721

2120572

(49)

and hence using the fact that

120573 119909 (119905)2

le 119881 (119905 119909119905) 119905 isin R

+

(50)

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199031 forall119905 ge 0 (51)

with 1199031= radic119872

12120572120573 This implies that the solution converges

to the ball 119861(0 1199031)

8 Mathematical Problems in Engineering

Next if 1205752satisfies

int

+infin

0

1205754

2(119904) 119889119904 lt +infin (52)

then 119868 = int+infin0

1199032(119904)119889119904 lt +infin

It follows that

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

11986812

2radic120572

forall119905 ge 0 (53)

Hence using (50) one gets

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199032 forall119905 ge 0 (54)

with 1199032= radic119868122120573radic120572

In this case the solution converges to the ball 119861(0 1199032)

Remark that from (45) if we suppose that 1205752(119905) tends

to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr

+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909

0) as follows

(1) Estimate 1198811(0 1199090) as

1198811(0 1199090) le (119901 + 120573)

10038171003817100381710038171206011003817100381710038171003817

2 (55)

(2) Estimate 1198812(0 1199090) as

1198812(0 1199090) =

119898

sum

119894=1

12058121119894int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

(56)

(i) Considering int119905119905minus120591119894119871

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119871

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119871

10038171003817100381710038171206011003817100381710038171003817

2

(57)

(ii) Considering int119905119905minus120591119894119867

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119867

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119867

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(58)

(iii) Considering int119905119905minus120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894(119905)

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(59)

(iv) Considering int119905119905minus]120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus]120591119894(119905)1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572]120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le ]120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(60)

1198812(0 1199090) le

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867)10038171003817100381710038171206011003817100381710038171003817

2

(61)

(3) Estimate 1198813(0 1199090) as

1198813(0 1199090)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

(62)

The first member of 1198813(0 1199090) is

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867

2120572

(1205911198941198671198902120572120591119894119867

minus

1

2120572

(1 minus 119890minus2120572120591119894119867

))10038171003817100381710038171206011003817100381710038171003817

2

le 1198881198941198671205913

119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(63)

with ((119909119890119909 + 119890minus119909 minus 1)1199092 ge 15) 119888119894119867= ((2120572120591

1198941198671198902120572120591119894119867

+

119890minus2120572120591119894119867

minus 1)(2120572120591119894119867)2

)

Mathematical Problems in Engineering 9

The second member of 1198813(0 1199090) is

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867119871

2120572

(120591119894119867119871

1198902120572120591119894119867

minus

1

2120572

[119890minus2120572120591119894119871

minus 119890minus2120572120591119894119867

])10038171003817100381710038171206011003817100381710038171003817

2

le

120591119894119867119871

(2120572)2(2120572120591119894119867119871

1198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

(

21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 1

(2120572)2

minus

21205721205911198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1

(2120572)2

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

119889119894119867119871

10038171003817100381710038171206011003817100381710038171003817

2

(64)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) and 119888

119894119871= (2120572120591

1198941198711198902120572120591119894119867

+

119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 Consider

1198813(0 1199090) le

119898

sum

119894=1

(120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

)10038171003817100381710038171206011003817100381710038171003817

2

(65)

Therefore from (55) (61) and (65) it follows that size

119881 (0 1199090)

le (119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+ 120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))10038171003817100381710038171206011003817100381710038171003817

2

(66)

Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575

(sdot)2char-

acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575

(sdot)2

increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin

Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575

(sdot)1are large enough then there is trade-off

between the values of 1205811and the maximum matrix measure

120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581

1is large then the constant 120598 is

requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)

Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes

4 Examples

Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591

1cos2(045119905)

1198911(119905 119909 (119905 minus 120591

1(119905)))

=[

[

minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575

2lowast cos( 119905

1 + 1199092

1

)

120575 cos (119905) 1199091(119905 minus 1205911(119905))

]

]

(67)where 120591

1 120575 gt 0 120575

2ge 0 will be chosen later and

119860 (119905) = [

119886 (119905) 1

minus1 119886 (119905)

] (68)

where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0

and 119902119894= 0 for (119894 = 1 119898) By computation we obtain

120583(119860) asymp minus81033741

Let120573 = 001 120572 = 1 120583 = 09 ] = 11

1205911119867

= 1205911 120591

1119871= 0 997904rArr 120591

1119867119871= 1205911119867

1205751= 12057511= 120575 120575

12= 1205752

1205811= 11989010

120581211

= 0 120581221

= 0 120581231

= 11989010

120581241

= 0 997904rArr 12058121= 11989010

120581311

=

1

1205912

1119867exp (2120572120591

1119867)

120581321

= 0

997904rArr 12058131=

1

exp (21205721205911119867)

(69)

Then120598 = 2 (119901 + 120573) 120575

01+ 2120572120573 + 120581

21

+ 1205813111989021205721205911119867

+ 1205811= 102 + 2119890

10

gt 0

(70)

10 Mathematical Problems in Engineering

Table 1

Delay bound (1205911119867) 2 4 6 8 10

Perturbation bound 1205751

3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245

We can verify that a solution 119875(119905) is given by

119875 (119905) =

[

[

[

[

119890sin(119905)

10

0

0

119890sin(119905)

10

]

]

]

]

(71)

We have 119901 = sup119905isinR+119875(119905) = 11989010

1205751lt

radic120581231119890minus21205721205911119867 (1 minus 120583) (120581

1minus 2120573120583 (119860))

(119901 + 120573)

1205811= 11989010

gt 2120573120583 (119860) = minus162067

120578 = 1205811minus 2120573120583 (119860) minus

(119901 + 120573)2

1205752

120581231119890minus21205721205911119867 (1 minus 120583)

gt 0

120578120577= (1 minus 120577

2

) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575

1 120577 lt 1

120574 =radic

119901 + 120573 + 1205812311205911119867

+ (1198881119867120591111986711989021205721205911119867

)

120573

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 = radic(119901 + 120573)

2

2120572120573120578120577

1205752=

1295910minus3

radic1 minus 1205772

1205752

(72)

If we take 120577 = 09 then 119903 = 297310minus31205752

We see that the perturbation bound 1205751in this example is

the same as in [5] if 1205752= 0 and is better than [4] as shown in

Table 1 The simulation of this example is shown in Figures 1and 2

Example 2 Consider the following second-order differentialsystem

1= minus5119909

1+ 1199092(119905 minus 01) + cos( 119909

1

1 + 1199052)

+ 002 sin (1199091(119905 minus 01) + 05)

2= minus1199091(119905 minus 01) minus 5119909

2

(73)

0 001 002 003 004 0050

001

002

003

004

005

006

007

008

009

Solu

tionx1

x1

Time t

Figure 1 The trajectories of 1199091

with

119860 = [

minus5 0

0 minus5

] 120582max (119860) = minus5 120583 (119860) = minus5

1198601= [

0 1

minus1 0

] 1205911119871= 0 120591

1= 1205911119867

= 01 120583 = 0

1198910(119905 119909) =

[

[

cos( 1199091

1 + 1199052)

0

]

]

12057501= 0 120575

02= 1

1198911(119905 119909 (119905 minus 1)) = [

002 sin (1199091(119905 minus 01) + 05)

0

]

12057511= 002 120575

12= 01

(74)

119909 = (1199091 1199092)119879

isin R2 Hence using (25) we obtain 1205751=

002 1205752= 11

The matrix

119875 (119905) = [

11990111

11990112

11990112

11990122

] (75)

can be computed as follows

11990111= 11990122= minus120573 + 119902

1(5 minus 120572)

11990112= radic1199021sdot radicminus10120573 + 2120572120573 + 119902

1(120572 minus 5)

2

minus 120598

(76)

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Next the time derivative of 1198812(119905 119909119905) is given by

2(119905 119909119905) le minus2120572119881

2(119905 119909119905) +

119898

sum

119894=1

1205812119894119909 (119905)

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(26)

with 1205812119894= 12058121119894+ 12058122119894+ 12058123119894+ 12058124119894

The time derivative of 1198813(119905 119909119905) is given by

3(119905 119909119905)

= minus21205721198813(119905 119909119905)

+

119898

sum

119894=1

12058131119894120591119894119867(1205911198941198671198902120572120591119894119867

119909(119905)2

+ int

0

minus120591119894119867

(minus (119905 + 119904) 1198902120572(119904minus120591119894(119905+119904)+120591119894119867)

sdot1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

) 119889119904)

+

119898

sum

119894=1

12058132119894120591119894119867119871

(120591119894119867119871

1198902120572120591119894119867

119909 (119905)2

+ int

minus120591119894119871

minus120591119894119867

(minus (119905 + 119904) 1198902120572(119904minus120591119894(119905+119904)+120591119894119867)

sdot1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

) 119889119904)

(27)

By using the following differentiation rule (see [5]) oneobtains

119889

119889119905

(int

0

minus120591119894

int

119905

119906(119905+1199051)

120595 (119904) 119889119904 1198891199051)

= 120591119894120595 (119905) minus int

0

minus120591119894

(119905 + 119904) 120595 (119906 (119905 + 119904)) 119889119904

(28)

with

119906 (119905 + 119904) = 119905 + 119904 minus 120591119894(119905 + 119904)

(119905 + 119904) = 1 minus 120591119894(119905 + 119904)

(29)

It follows that

minus (119905 + 119904) le 120583 minus 1 (30)

So

3(119905 119909119905) le minus2120572119881

3(119905 119909119905)

+ (

119898

sum

119894=1

120581311198941205912

119894119867+

119898

sum

119894=1

120581321198941205912

119894119867119871)1198902120572120591119894119867

119909(119905)2

minus

119898

sum

119894=1

12058131119894120591119894119867119890minus2120572120591119894119867

(1 minus 120583)

sdot int

0

minus120591119894119867

1003817100381710038171003817119909 (119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

119889119904

minus

119898

sum

119894=1

12058132119894120591119894119867119871

119890minus2120572120591119894119867

(1 minus 120583)

sdot int

minus120591119894119871

minus120591119894119867

1003817100381710038171003817119909(119905 + 119904 minus 120591

119894(119905 + 119904))

1003817100381710038171003817

2

119889119904

(31)

Since the last integral term is nonnegative we obtain thefollowing estimation on

3(119905 119909119905)

3(119905 119909119905) le minus2120572119881

3(119905 119909119905) +

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(32)

with 1205813119894= 120581311198941205912

119894119867+ 120581321198941205912

119894119867119871

Therefore from (24)ndash(26)ndash(32) it follows that

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 1198810(119905 119909119905) (33)

where

1198810(119905 119909119905)

= 119909119879

(119905) ( (119905) + 119860119879

(119905) 119875120573(119905) + 119875

120573(119905) 119860 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ 2120572119875 (119905) + 2 ((119901 + 120573) 12057501+ 120572120573) 119868) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

1205812119894119909 (119905)

2

6 Mathematical Problems in Engineering

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(34)

It follows that

1198810(119905 119909119905)

le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

)119868)119909 (119905)

+ 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(35)

Now using Proposition 2 we have119898

sum

119894=1

(2 (119901 + 120573) 1205751

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817sdot 119909 (119905)

minus (12058123119894119890minus2120572120591119894119867

(1 minus 120583) minus 119902119894)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

)

le

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

(36)

with

0 lt 119902119894lt 12058123119894119890minus2120572120591119894119867

(1 minus 120583) 119894 = 1 119898 (37)

Hence the above expression in conjunction with (35) yields

1198810(119905 119909119905) le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573

+

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811)119868)119909 (119905)

minus 1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(38)

Since 119875(119905) is a solution of (9) with

120598 = 2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811 (39)

Mathematical Problems in Engineering 7

then

1198810(119905 119909119905)

le minus1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(40)

Note that we have

119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

le 120582max (119860 (119905) + 119860119879

(119905)) 119909 (119905)2

le 2120583 (119860 (119905)) 119909 (119905)2

le 2120583 (119860) 119909 (119905)2

(41)

with 120583(119860) = sup119905isinR+120583(119860(119905))

Then it follows that

1198810(119905 119909119905)

le minus(1205811minus 2120573120583 (119860)

minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

)119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(42)

Let

120578 = 1205811minus 2120573120583 (119860) minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

(43)

One has

120578 gt 0

997904rArr 1205751lt

radicsum119898

119894=1(12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894) (1205811minus 2120573120583 (119860))

(119901 + 120573)

1205811minus 2120573120583 (119860) gt 0

(44)

Then

1198810(119905 119909119905) le minus120578 (119909 (119905) minus 120585 (119905))

2

+ 119903 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(45)

with 120585(119905) = ((119901 + 120573)120578)1205752(119905) and 119903(119905) = ((119901 + 120573)2120578)1205752

2(119905)

Taking condition (44) into account we have

1198810(119905 119909119905) le 119903 (119905) (46)

Therefore from (33) and (46)

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 119903 (119905) forall119905 ge 0 (47)

which gives

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+ int

119905

0

1198902120572(119904minus119905)

119903 (119904) 119889119904 forall119905 ge 0

(48)

Now if 1205752(119905) is bounded for all 119905 ge 0 then there exists119872 gt 0

such that 1205752(119905) le 119872forall119905 ge 0Therefore 119903(119905)will be bounded

so there exists 1198721gt 0 such that 119903(119905) le 119872

1 forall119905 ge 0 with

1198721= (119901 + 120573)

2

1198722120578 So

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

1198721

2120572

(49)

and hence using the fact that

120573 119909 (119905)2

le 119881 (119905 119909119905) 119905 isin R

+

(50)

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199031 forall119905 ge 0 (51)

with 1199031= radic119872

12120572120573 This implies that the solution converges

to the ball 119861(0 1199031)

8 Mathematical Problems in Engineering

Next if 1205752satisfies

int

+infin

0

1205754

2(119904) 119889119904 lt +infin (52)

then 119868 = int+infin0

1199032(119904)119889119904 lt +infin

It follows that

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

11986812

2radic120572

forall119905 ge 0 (53)

Hence using (50) one gets

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199032 forall119905 ge 0 (54)

with 1199032= radic119868122120573radic120572

In this case the solution converges to the ball 119861(0 1199032)

Remark that from (45) if we suppose that 1205752(119905) tends

to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr

+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909

0) as follows

(1) Estimate 1198811(0 1199090) as

1198811(0 1199090) le (119901 + 120573)

10038171003817100381710038171206011003817100381710038171003817

2 (55)

(2) Estimate 1198812(0 1199090) as

1198812(0 1199090) =

119898

sum

119894=1

12058121119894int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

(56)

(i) Considering int119905119905minus120591119894119871

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119871

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119871

10038171003817100381710038171206011003817100381710038171003817

2

(57)

(ii) Considering int119905119905minus120591119894119867

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119867

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119867

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(58)

(iii) Considering int119905119905minus120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894(119905)

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(59)

(iv) Considering int119905119905minus]120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus]120591119894(119905)1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572]120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le ]120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(60)

1198812(0 1199090) le

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867)10038171003817100381710038171206011003817100381710038171003817

2

(61)

(3) Estimate 1198813(0 1199090) as

1198813(0 1199090)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

(62)

The first member of 1198813(0 1199090) is

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867

2120572

(1205911198941198671198902120572120591119894119867

minus

1

2120572

(1 minus 119890minus2120572120591119894119867

))10038171003817100381710038171206011003817100381710038171003817

2

le 1198881198941198671205913

119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(63)

with ((119909119890119909 + 119890minus119909 minus 1)1199092 ge 15) 119888119894119867= ((2120572120591

1198941198671198902120572120591119894119867

+

119890minus2120572120591119894119867

minus 1)(2120572120591119894119867)2

)

Mathematical Problems in Engineering 9

The second member of 1198813(0 1199090) is

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867119871

2120572

(120591119894119867119871

1198902120572120591119894119867

minus

1

2120572

[119890minus2120572120591119894119871

minus 119890minus2120572120591119894119867

])10038171003817100381710038171206011003817100381710038171003817

2

le

120591119894119867119871

(2120572)2(2120572120591119894119867119871

1198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

(

21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 1

(2120572)2

minus

21205721205911198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1

(2120572)2

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

119889119894119867119871

10038171003817100381710038171206011003817100381710038171003817

2

(64)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) and 119888

119894119871= (2120572120591

1198941198711198902120572120591119894119867

+

119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 Consider

1198813(0 1199090) le

119898

sum

119894=1

(120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

)10038171003817100381710038171206011003817100381710038171003817

2

(65)

Therefore from (55) (61) and (65) it follows that size

119881 (0 1199090)

le (119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+ 120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))10038171003817100381710038171206011003817100381710038171003817

2

(66)

Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575

(sdot)2char-

acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575

(sdot)2

increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin

Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575

(sdot)1are large enough then there is trade-off

between the values of 1205811and the maximum matrix measure

120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581

1is large then the constant 120598 is

requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)

Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes

4 Examples

Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591

1cos2(045119905)

1198911(119905 119909 (119905 minus 120591

1(119905)))

=[

[

minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575

2lowast cos( 119905

1 + 1199092

1

)

120575 cos (119905) 1199091(119905 minus 1205911(119905))

]

]

(67)where 120591

1 120575 gt 0 120575

2ge 0 will be chosen later and

119860 (119905) = [

119886 (119905) 1

minus1 119886 (119905)

] (68)

where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0

and 119902119894= 0 for (119894 = 1 119898) By computation we obtain

120583(119860) asymp minus81033741

Let120573 = 001 120572 = 1 120583 = 09 ] = 11

1205911119867

= 1205911 120591

1119871= 0 997904rArr 120591

1119867119871= 1205911119867

1205751= 12057511= 120575 120575

12= 1205752

1205811= 11989010

120581211

= 0 120581221

= 0 120581231

= 11989010

120581241

= 0 997904rArr 12058121= 11989010

120581311

=

1

1205912

1119867exp (2120572120591

1119867)

120581321

= 0

997904rArr 12058131=

1

exp (21205721205911119867)

(69)

Then120598 = 2 (119901 + 120573) 120575

01+ 2120572120573 + 120581

21

+ 1205813111989021205721205911119867

+ 1205811= 102 + 2119890

10

gt 0

(70)

10 Mathematical Problems in Engineering

Table 1

Delay bound (1205911119867) 2 4 6 8 10

Perturbation bound 1205751

3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245

We can verify that a solution 119875(119905) is given by

119875 (119905) =

[

[

[

[

119890sin(119905)

10

0

0

119890sin(119905)

10

]

]

]

]

(71)

We have 119901 = sup119905isinR+119875(119905) = 11989010

1205751lt

radic120581231119890minus21205721205911119867 (1 minus 120583) (120581

1minus 2120573120583 (119860))

(119901 + 120573)

1205811= 11989010

gt 2120573120583 (119860) = minus162067

120578 = 1205811minus 2120573120583 (119860) minus

(119901 + 120573)2

1205752

120581231119890minus21205721205911119867 (1 minus 120583)

gt 0

120578120577= (1 minus 120577

2

) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575

1 120577 lt 1

120574 =radic

119901 + 120573 + 1205812311205911119867

+ (1198881119867120591111986711989021205721205911119867

)

120573

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 = radic(119901 + 120573)

2

2120572120573120578120577

1205752=

1295910minus3

radic1 minus 1205772

1205752

(72)

If we take 120577 = 09 then 119903 = 297310minus31205752

We see that the perturbation bound 1205751in this example is

the same as in [5] if 1205752= 0 and is better than [4] as shown in

Table 1 The simulation of this example is shown in Figures 1and 2

Example 2 Consider the following second-order differentialsystem

1= minus5119909

1+ 1199092(119905 minus 01) + cos( 119909

1

1 + 1199052)

+ 002 sin (1199091(119905 minus 01) + 05)

2= minus1199091(119905 minus 01) minus 5119909

2

(73)

0 001 002 003 004 0050

001

002

003

004

005

006

007

008

009

Solu

tionx1

x1

Time t

Figure 1 The trajectories of 1199091

with

119860 = [

minus5 0

0 minus5

] 120582max (119860) = minus5 120583 (119860) = minus5

1198601= [

0 1

minus1 0

] 1205911119871= 0 120591

1= 1205911119867

= 01 120583 = 0

1198910(119905 119909) =

[

[

cos( 1199091

1 + 1199052)

0

]

]

12057501= 0 120575

02= 1

1198911(119905 119909 (119905 minus 1)) = [

002 sin (1199091(119905 minus 01) + 05)

0

]

12057511= 002 120575

12= 01

(74)

119909 = (1199091 1199092)119879

isin R2 Hence using (25) we obtain 1205751=

002 1205752= 11

The matrix

119875 (119905) = [

11990111

11990112

11990112

11990122

] (75)

can be computed as follows

11990111= 11990122= minus120573 + 119902

1(5 minus 120572)

11990112= radic1199021sdot radicminus10120573 + 2120572120573 + 119902

1(120572 minus 5)

2

minus 120598

(76)

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

119909(119905)2

(34)

It follows that

1198810(119905 119909119905)

le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894

+

119898

sum

119894=1

12058131198941198902120572120591119894119867

)119868)119909 (119905)

+ 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+ 2 (119901 + 120573) 1205752(119905) 119909 (119905) + 2 (119901 + 120573) 120575

1

sdot

119898

sum

119894=1

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817times 119909 (119905)

+

119898

sum

119894=1

119902119894

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058123119894119890minus2120572120591119894119867

(1 minus 120583)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(35)

Now using Proposition 2 we have119898

sum

119894=1

(2 (119901 + 120573) 1205751

1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817sdot 119909 (119905)

minus (12058123119894119890minus2120572120591119894119867

(1 minus 120583) minus 119902119894)1003817100381710038171003817119909 (119905 minus 120591

119894(119905))

1003817100381710038171003817

2

)

le

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

(36)

with

0 lt 119902119894lt 12058123119894119890minus2120572120591119894119867

(1 minus 120583) 119894 = 1 119898 (37)

Hence the above expression in conjunction with (35) yields

1198810(119905 119909119905) le 119909119879

(119905)

sdot ( (119905) + 119860119879

(119905) 119875 (119905) + 119875 (119905) 119860 (119905) + 2120572119875 (119905)

+

119898

sum

119894=1

1

119902119894

119875120573(119905) 119860119894(119905) sdot 119860

119879

119894(119905) 119875120573(119905)

+ (2 (119901 + 120573) 12057501+ 2120572120573

+

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811)119868)119909 (119905)

minus 1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(38)

Since 119875(119905) is a solution of (9) with

120598 = 2 (119901 + 120573) 12057501+ 2120572120573 +

119898

sum

119894=1

1205812119894+

119898

sum

119894=1

12058131198941198902120572120591119894119867

+ 1205811 (39)

Mathematical Problems in Engineering 7

then

1198810(119905 119909119905)

le minus1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(40)

Note that we have

119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

le 120582max (119860 (119905) + 119860119879

(119905)) 119909 (119905)2

le 2120583 (119860 (119905)) 119909 (119905)2

le 2120583 (119860) 119909 (119905)2

(41)

with 120583(119860) = sup119905isinR+120583(119860(119905))

Then it follows that

1198810(119905 119909119905)

le minus(1205811minus 2120573120583 (119860)

minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

)119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(42)

Let

120578 = 1205811minus 2120573120583 (119860) minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

(43)

One has

120578 gt 0

997904rArr 1205751lt

radicsum119898

119894=1(12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894) (1205811minus 2120573120583 (119860))

(119901 + 120573)

1205811minus 2120573120583 (119860) gt 0

(44)

Then

1198810(119905 119909119905) le minus120578 (119909 (119905) minus 120585 (119905))

2

+ 119903 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(45)

with 120585(119905) = ((119901 + 120573)120578)1205752(119905) and 119903(119905) = ((119901 + 120573)2120578)1205752

2(119905)

Taking condition (44) into account we have

1198810(119905 119909119905) le 119903 (119905) (46)

Therefore from (33) and (46)

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 119903 (119905) forall119905 ge 0 (47)

which gives

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+ int

119905

0

1198902120572(119904minus119905)

119903 (119904) 119889119904 forall119905 ge 0

(48)

Now if 1205752(119905) is bounded for all 119905 ge 0 then there exists119872 gt 0

such that 1205752(119905) le 119872forall119905 ge 0Therefore 119903(119905)will be bounded

so there exists 1198721gt 0 such that 119903(119905) le 119872

1 forall119905 ge 0 with

1198721= (119901 + 120573)

2

1198722120578 So

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

1198721

2120572

(49)

and hence using the fact that

120573 119909 (119905)2

le 119881 (119905 119909119905) 119905 isin R

+

(50)

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199031 forall119905 ge 0 (51)

with 1199031= radic119872

12120572120573 This implies that the solution converges

to the ball 119861(0 1199031)

8 Mathematical Problems in Engineering

Next if 1205752satisfies

int

+infin

0

1205754

2(119904) 119889119904 lt +infin (52)

then 119868 = int+infin0

1199032(119904)119889119904 lt +infin

It follows that

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

11986812

2radic120572

forall119905 ge 0 (53)

Hence using (50) one gets

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199032 forall119905 ge 0 (54)

with 1199032= radic119868122120573radic120572

In this case the solution converges to the ball 119861(0 1199032)

Remark that from (45) if we suppose that 1205752(119905) tends

to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr

+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909

0) as follows

(1) Estimate 1198811(0 1199090) as

1198811(0 1199090) le (119901 + 120573)

10038171003817100381710038171206011003817100381710038171003817

2 (55)

(2) Estimate 1198812(0 1199090) as

1198812(0 1199090) =

119898

sum

119894=1

12058121119894int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

(56)

(i) Considering int119905119905minus120591119894119871

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119871

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119871

10038171003817100381710038171206011003817100381710038171003817

2

(57)

(ii) Considering int119905119905minus120591119894119867

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119867

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119867

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(58)

(iii) Considering int119905119905minus120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894(119905)

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(59)

(iv) Considering int119905119905minus]120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus]120591119894(119905)1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572]120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le ]120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(60)

1198812(0 1199090) le

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867)10038171003817100381710038171206011003817100381710038171003817

2

(61)

(3) Estimate 1198813(0 1199090) as

1198813(0 1199090)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

(62)

The first member of 1198813(0 1199090) is

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867

2120572

(1205911198941198671198902120572120591119894119867

minus

1

2120572

(1 minus 119890minus2120572120591119894119867

))10038171003817100381710038171206011003817100381710038171003817

2

le 1198881198941198671205913

119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(63)

with ((119909119890119909 + 119890minus119909 minus 1)1199092 ge 15) 119888119894119867= ((2120572120591

1198941198671198902120572120591119894119867

+

119890minus2120572120591119894119867

minus 1)(2120572120591119894119867)2

)

Mathematical Problems in Engineering 9

The second member of 1198813(0 1199090) is

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867119871

2120572

(120591119894119867119871

1198902120572120591119894119867

minus

1

2120572

[119890minus2120572120591119894119871

minus 119890minus2120572120591119894119867

])10038171003817100381710038171206011003817100381710038171003817

2

le

120591119894119867119871

(2120572)2(2120572120591119894119867119871

1198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

(

21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 1

(2120572)2

minus

21205721205911198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1

(2120572)2

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

119889119894119867119871

10038171003817100381710038171206011003817100381710038171003817

2

(64)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) and 119888

119894119871= (2120572120591

1198941198711198902120572120591119894119867

+

119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 Consider

1198813(0 1199090) le

119898

sum

119894=1

(120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

)10038171003817100381710038171206011003817100381710038171003817

2

(65)

Therefore from (55) (61) and (65) it follows that size

119881 (0 1199090)

le (119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+ 120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))10038171003817100381710038171206011003817100381710038171003817

2

(66)

Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575

(sdot)2char-

acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575

(sdot)2

increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin

Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575

(sdot)1are large enough then there is trade-off

between the values of 1205811and the maximum matrix measure

120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581

1is large then the constant 120598 is

requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)

Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes

4 Examples

Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591

1cos2(045119905)

1198911(119905 119909 (119905 minus 120591

1(119905)))

=[

[

minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575

2lowast cos( 119905

1 + 1199092

1

)

120575 cos (119905) 1199091(119905 minus 1205911(119905))

]

]

(67)where 120591

1 120575 gt 0 120575

2ge 0 will be chosen later and

119860 (119905) = [

119886 (119905) 1

minus1 119886 (119905)

] (68)

where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0

and 119902119894= 0 for (119894 = 1 119898) By computation we obtain

120583(119860) asymp minus81033741

Let120573 = 001 120572 = 1 120583 = 09 ] = 11

1205911119867

= 1205911 120591

1119871= 0 997904rArr 120591

1119867119871= 1205911119867

1205751= 12057511= 120575 120575

12= 1205752

1205811= 11989010

120581211

= 0 120581221

= 0 120581231

= 11989010

120581241

= 0 997904rArr 12058121= 11989010

120581311

=

1

1205912

1119867exp (2120572120591

1119867)

120581321

= 0

997904rArr 12058131=

1

exp (21205721205911119867)

(69)

Then120598 = 2 (119901 + 120573) 120575

01+ 2120572120573 + 120581

21

+ 1205813111989021205721205911119867

+ 1205811= 102 + 2119890

10

gt 0

(70)

10 Mathematical Problems in Engineering

Table 1

Delay bound (1205911119867) 2 4 6 8 10

Perturbation bound 1205751

3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245

We can verify that a solution 119875(119905) is given by

119875 (119905) =

[

[

[

[

119890sin(119905)

10

0

0

119890sin(119905)

10

]

]

]

]

(71)

We have 119901 = sup119905isinR+119875(119905) = 11989010

1205751lt

radic120581231119890minus21205721205911119867 (1 minus 120583) (120581

1minus 2120573120583 (119860))

(119901 + 120573)

1205811= 11989010

gt 2120573120583 (119860) = minus162067

120578 = 1205811minus 2120573120583 (119860) minus

(119901 + 120573)2

1205752

120581231119890minus21205721205911119867 (1 minus 120583)

gt 0

120578120577= (1 minus 120577

2

) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575

1 120577 lt 1

120574 =radic

119901 + 120573 + 1205812311205911119867

+ (1198881119867120591111986711989021205721205911119867

)

120573

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 = radic(119901 + 120573)

2

2120572120573120578120577

1205752=

1295910minus3

radic1 minus 1205772

1205752

(72)

If we take 120577 = 09 then 119903 = 297310minus31205752

We see that the perturbation bound 1205751in this example is

the same as in [5] if 1205752= 0 and is better than [4] as shown in

Table 1 The simulation of this example is shown in Figures 1and 2

Example 2 Consider the following second-order differentialsystem

1= minus5119909

1+ 1199092(119905 minus 01) + cos( 119909

1

1 + 1199052)

+ 002 sin (1199091(119905 minus 01) + 05)

2= minus1199091(119905 minus 01) minus 5119909

2

(73)

0 001 002 003 004 0050

001

002

003

004

005

006

007

008

009

Solu

tionx1

x1

Time t

Figure 1 The trajectories of 1199091

with

119860 = [

minus5 0

0 minus5

] 120582max (119860) = minus5 120583 (119860) = minus5

1198601= [

0 1

minus1 0

] 1205911119871= 0 120591

1= 1205911119867

= 01 120583 = 0

1198910(119905 119909) =

[

[

cos( 1199091

1 + 1199052)

0

]

]

12057501= 0 120575

02= 1

1198911(119905 119909 (119905 minus 1)) = [

002 sin (1199091(119905 minus 01) + 05)

0

]

12057511= 002 120575

12= 01

(74)

119909 = (1199091 1199092)119879

isin R2 Hence using (25) we obtain 1205751=

002 1205752= 11

The matrix

119875 (119905) = [

11990111

11990112

11990112

11990122

] (75)

can be computed as follows

11990111= 11990122= minus120573 + 119902

1(5 minus 120572)

11990112= radic1199021sdot radicminus10120573 + 2120572120573 + 119902

1(120572 minus 5)

2

minus 120598

(76)

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

then

1198810(119905 119909119905)

le minus1205811119909119879

(119905) 119909 (119905) + 120573119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

+

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(40)

Note that we have

119909119879

(119905) (119860119879

(119905) + 119860 (119905)) 119909 (119905)

le 120582max (119860 (119905) + 119860119879

(119905)) 119909 (119905)2

le 2120583 (119860 (119905)) 119909 (119905)2

le 2120583 (119860) 119909 (119905)2

(41)

with 120583(119860) = sup119905isinR+120583(119860(119905))

Then it follows that

1198810(119905 119909119905)

le minus(1205811minus 2120573120583 (119860)

minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

)119909 (119905)2

+ 2 (119901 + 120573) 1205752(119905) sdot 119909 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(42)

Let

120578 = 1205811minus 2120573120583 (119860) minus

119898

sum

119894=1

(119901 + 120573)2

1205752

1

12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894

(43)

One has

120578 gt 0

997904rArr 1205751lt

radicsum119898

119894=1(12058123119894119890minus2120572120591119894119867 (1 minus 120583) minus 119902

119894) (1205811minus 2120573120583 (119860))

(119901 + 120573)

1205811minus 2120573120583 (119860) gt 0

(44)

Then

1198810(119905 119909119905) le minus120578 (119909 (119905) minus 120585 (119905))

2

+ 119903 (119905)

minus

119898

sum

119894=1

12058121119894119890minus2120572120591119894119871

1003817100381710038171003817119909 (119905 minus 120591

119894119871)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058122119894119890minus2120572120591119894119867

1003817100381710038171003817119909 (119905 minus 120591

119894119867)1003817100381710038171003817

2

minus

119898

sum

119894=1

12058124119894119890minus2120572]120591119894119867

(1 minus ]120583) 1003817100381710038171003817119909 (119905 minus ]120591

119894(119905))

1003817100381710038171003817

2

(45)

with 120585(119905) = ((119901 + 120573)120578)1205752(119905) and 119903(119905) = ((119901 + 120573)2120578)1205752

2(119905)

Taking condition (44) into account we have

1198810(119905 119909119905) le 119903 (119905) (46)

Therefore from (33) and (46)

(119905 119909119905) le minus2120572119881 (119905 119909

119905) + 119903 (119905) forall119905 ge 0 (47)

which gives

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+ int

119905

0

1198902120572(119904minus119905)

119903 (119904) 119889119904 forall119905 ge 0

(48)

Now if 1205752(119905) is bounded for all 119905 ge 0 then there exists119872 gt 0

such that 1205752(119905) le 119872forall119905 ge 0Therefore 119903(119905)will be bounded

so there exists 1198721gt 0 such that 119903(119905) le 119872

1 forall119905 ge 0 with

1198721= (119901 + 120573)

2

1198722120578 So

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

1198721

2120572

(49)

and hence using the fact that

120573 119909 (119905)2

le 119881 (119905 119909119905) 119905 isin R

+

(50)

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199031 forall119905 ge 0 (51)

with 1199031= radic119872

12120572120573 This implies that the solution converges

to the ball 119861(0 1199031)

8 Mathematical Problems in Engineering

Next if 1205752satisfies

int

+infin

0

1205754

2(119904) 119889119904 lt +infin (52)

then 119868 = int+infin0

1199032(119904)119889119904 lt +infin

It follows that

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

11986812

2radic120572

forall119905 ge 0 (53)

Hence using (50) one gets

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199032 forall119905 ge 0 (54)

with 1199032= radic119868122120573radic120572

In this case the solution converges to the ball 119861(0 1199032)

Remark that from (45) if we suppose that 1205752(119905) tends

to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr

+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909

0) as follows

(1) Estimate 1198811(0 1199090) as

1198811(0 1199090) le (119901 + 120573)

10038171003817100381710038171206011003817100381710038171003817

2 (55)

(2) Estimate 1198812(0 1199090) as

1198812(0 1199090) =

119898

sum

119894=1

12058121119894int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

(56)

(i) Considering int119905119905minus120591119894119871

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119871

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119871

10038171003817100381710038171206011003817100381710038171003817

2

(57)

(ii) Considering int119905119905minus120591119894119867

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119867

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119867

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(58)

(iii) Considering int119905119905minus120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894(119905)

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(59)

(iv) Considering int119905119905minus]120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus]120591119894(119905)1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572]120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le ]120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(60)

1198812(0 1199090) le

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867)10038171003817100381710038171206011003817100381710038171003817

2

(61)

(3) Estimate 1198813(0 1199090) as

1198813(0 1199090)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

(62)

The first member of 1198813(0 1199090) is

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867

2120572

(1205911198941198671198902120572120591119894119867

minus

1

2120572

(1 minus 119890minus2120572120591119894119867

))10038171003817100381710038171206011003817100381710038171003817

2

le 1198881198941198671205913

119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(63)

with ((119909119890119909 + 119890minus119909 minus 1)1199092 ge 15) 119888119894119867= ((2120572120591

1198941198671198902120572120591119894119867

+

119890minus2120572120591119894119867

minus 1)(2120572120591119894119867)2

)

Mathematical Problems in Engineering 9

The second member of 1198813(0 1199090) is

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867119871

2120572

(120591119894119867119871

1198902120572120591119894119867

minus

1

2120572

[119890minus2120572120591119894119871

minus 119890minus2120572120591119894119867

])10038171003817100381710038171206011003817100381710038171003817

2

le

120591119894119867119871

(2120572)2(2120572120591119894119867119871

1198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

(

21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 1

(2120572)2

minus

21205721205911198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1

(2120572)2

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

119889119894119867119871

10038171003817100381710038171206011003817100381710038171003817

2

(64)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) and 119888

119894119871= (2120572120591

1198941198711198902120572120591119894119867

+

119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 Consider

1198813(0 1199090) le

119898

sum

119894=1

(120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

)10038171003817100381710038171206011003817100381710038171003817

2

(65)

Therefore from (55) (61) and (65) it follows that size

119881 (0 1199090)

le (119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+ 120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))10038171003817100381710038171206011003817100381710038171003817

2

(66)

Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575

(sdot)2char-

acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575

(sdot)2

increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin

Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575

(sdot)1are large enough then there is trade-off

between the values of 1205811and the maximum matrix measure

120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581

1is large then the constant 120598 is

requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)

Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes

4 Examples

Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591

1cos2(045119905)

1198911(119905 119909 (119905 minus 120591

1(119905)))

=[

[

minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575

2lowast cos( 119905

1 + 1199092

1

)

120575 cos (119905) 1199091(119905 minus 1205911(119905))

]

]

(67)where 120591

1 120575 gt 0 120575

2ge 0 will be chosen later and

119860 (119905) = [

119886 (119905) 1

minus1 119886 (119905)

] (68)

where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0

and 119902119894= 0 for (119894 = 1 119898) By computation we obtain

120583(119860) asymp minus81033741

Let120573 = 001 120572 = 1 120583 = 09 ] = 11

1205911119867

= 1205911 120591

1119871= 0 997904rArr 120591

1119867119871= 1205911119867

1205751= 12057511= 120575 120575

12= 1205752

1205811= 11989010

120581211

= 0 120581221

= 0 120581231

= 11989010

120581241

= 0 997904rArr 12058121= 11989010

120581311

=

1

1205912

1119867exp (2120572120591

1119867)

120581321

= 0

997904rArr 12058131=

1

exp (21205721205911119867)

(69)

Then120598 = 2 (119901 + 120573) 120575

01+ 2120572120573 + 120581

21

+ 1205813111989021205721205911119867

+ 1205811= 102 + 2119890

10

gt 0

(70)

10 Mathematical Problems in Engineering

Table 1

Delay bound (1205911119867) 2 4 6 8 10

Perturbation bound 1205751

3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245

We can verify that a solution 119875(119905) is given by

119875 (119905) =

[

[

[

[

119890sin(119905)

10

0

0

119890sin(119905)

10

]

]

]

]

(71)

We have 119901 = sup119905isinR+119875(119905) = 11989010

1205751lt

radic120581231119890minus21205721205911119867 (1 minus 120583) (120581

1minus 2120573120583 (119860))

(119901 + 120573)

1205811= 11989010

gt 2120573120583 (119860) = minus162067

120578 = 1205811minus 2120573120583 (119860) minus

(119901 + 120573)2

1205752

120581231119890minus21205721205911119867 (1 minus 120583)

gt 0

120578120577= (1 minus 120577

2

) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575

1 120577 lt 1

120574 =radic

119901 + 120573 + 1205812311205911119867

+ (1198881119867120591111986711989021205721205911119867

)

120573

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 = radic(119901 + 120573)

2

2120572120573120578120577

1205752=

1295910minus3

radic1 minus 1205772

1205752

(72)

If we take 120577 = 09 then 119903 = 297310minus31205752

We see that the perturbation bound 1205751in this example is

the same as in [5] if 1205752= 0 and is better than [4] as shown in

Table 1 The simulation of this example is shown in Figures 1and 2

Example 2 Consider the following second-order differentialsystem

1= minus5119909

1+ 1199092(119905 minus 01) + cos( 119909

1

1 + 1199052)

+ 002 sin (1199091(119905 minus 01) + 05)

2= minus1199091(119905 minus 01) minus 5119909

2

(73)

0 001 002 003 004 0050

001

002

003

004

005

006

007

008

009

Solu

tionx1

x1

Time t

Figure 1 The trajectories of 1199091

with

119860 = [

minus5 0

0 minus5

] 120582max (119860) = minus5 120583 (119860) = minus5

1198601= [

0 1

minus1 0

] 1205911119871= 0 120591

1= 1205911119867

= 01 120583 = 0

1198910(119905 119909) =

[

[

cos( 1199091

1 + 1199052)

0

]

]

12057501= 0 120575

02= 1

1198911(119905 119909 (119905 minus 1)) = [

002 sin (1199091(119905 minus 01) + 05)

0

]

12057511= 002 120575

12= 01

(74)

119909 = (1199091 1199092)119879

isin R2 Hence using (25) we obtain 1205751=

002 1205752= 11

The matrix

119875 (119905) = [

11990111

11990112

11990112

11990122

] (75)

can be computed as follows

11990111= 11990122= minus120573 + 119902

1(5 minus 120572)

11990112= radic1199021sdot radicminus10120573 + 2120572120573 + 119902

1(120572 minus 5)

2

minus 120598

(76)

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Next if 1205752satisfies

int

+infin

0

1205754

2(119904) 119889119904 lt +infin (52)

then 119868 = int+infin0

1199032(119904)119889119904 lt +infin

It follows that

119881 (119905 119909119905) le 119881 (0 119909

0) 119890minus2120572119905

+

11986812

2radic120572

forall119905 ge 0 (53)

Hence using (50) one gets

1003817100381710038171003817119909 (119905 120601)

1003817100381710038171003817le radic

119881 (0 1199090)

120573

119890minus120572119905

+ 1199032 forall119905 ge 0 (54)

with 1199032= radic119868122120573radic120572

In this case the solution converges to the ball 119861(0 1199032)

Remark that from (45) if we suppose that 1205752(119905) tends

to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr

+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909

0) as follows

(1) Estimate 1198811(0 1199090) as

1198811(0 1199090) le (119901 + 120573)

10038171003817100381710038171206011003817100381710038171003817

2 (55)

(2) Estimate 1198812(0 1199090) as

1198812(0 1199090) =

119898

sum

119894=1

12058121119894int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058122119894int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058123119894int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

+

119898

sum

119894=1

12058124119894int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

(56)

(i) Considering int119905119905minus120591119894119871

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119871

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119871

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119871

10038171003817100381710038171206011003817100381710038171003817

2

(57)

(ii) Considering int119905119905minus120591119894119867

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894119867

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894119867

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894119867

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(58)

(iii) Considering int119905119905minus120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus120591119894(119905)

1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus120591119894(119905)

1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(59)

(iv) Considering int119905119905minus]120591119894(119905)

1198902120572(119904minus119905)

119909(119904)2119889119904 ge 0

int

0

minus]120591119894(119905)1198902120572119904

119909 (119904)2

119889119904

le10038171003817100381710038171206011003817100381710038171003817

2

int

0

minus]120591119894(119905)1198902120572119904

119889119904 =

1

2120572

(1 minus 119890minus2120572]120591119894(119905)

)10038171003817100381710038171206011003817100381710038171003817

2

le ]120591119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(60)

1198812(0 1199090) le

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867)10038171003817100381710038171206011003817100381710038171003817

2

(61)

(3) Estimate 1198813(0 1199090) as

1198813(0 1199090)

=

119898

sum

119894=1

12058131119894120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

+

119898

sum

119894=1

12058132119894120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

(62)

The first member of 1198813(0 1199090) is

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867int

0

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867

2120572

(1205911198941198671198902120572120591119894119867

minus

1

2120572

(1 minus 119890minus2120572120591119894119867

))10038171003817100381710038171206011003817100381710038171003817

2

le 1198881198941198671205913

119894119867

10038171003817100381710038171206011003817100381710038171003817

2

(63)

with ((119909119890119909 + 119890minus119909 minus 1)1199092 ge 15) 119888119894119867= ((2120572120591

1198941198671198902120572120591119894119867

+

119890minus2120572120591119894119867

minus 1)(2120572120591119894119867)2

)

Mathematical Problems in Engineering 9

The second member of 1198813(0 1199090) is

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867119871

2120572

(120591119894119867119871

1198902120572120591119894119867

minus

1

2120572

[119890minus2120572120591119894119871

minus 119890minus2120572120591119894119867

])10038171003817100381710038171206011003817100381710038171003817

2

le

120591119894119867119871

(2120572)2(2120572120591119894119867119871

1198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

(

21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 1

(2120572)2

minus

21205721205911198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1

(2120572)2

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

119889119894119867119871

10038171003817100381710038171206011003817100381710038171003817

2

(64)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) and 119888

119894119871= (2120572120591

1198941198711198902120572120591119894119867

+

119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 Consider

1198813(0 1199090) le

119898

sum

119894=1

(120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

)10038171003817100381710038171206011003817100381710038171003817

2

(65)

Therefore from (55) (61) and (65) it follows that size

119881 (0 1199090)

le (119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+ 120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))10038171003817100381710038171206011003817100381710038171003817

2

(66)

Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575

(sdot)2char-

acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575

(sdot)2

increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin

Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575

(sdot)1are large enough then there is trade-off

between the values of 1205811and the maximum matrix measure

120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581

1is large then the constant 120598 is

requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)

Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes

4 Examples

Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591

1cos2(045119905)

1198911(119905 119909 (119905 minus 120591

1(119905)))

=[

[

minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575

2lowast cos( 119905

1 + 1199092

1

)

120575 cos (119905) 1199091(119905 minus 1205911(119905))

]

]

(67)where 120591

1 120575 gt 0 120575

2ge 0 will be chosen later and

119860 (119905) = [

119886 (119905) 1

minus1 119886 (119905)

] (68)

where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0

and 119902119894= 0 for (119894 = 1 119898) By computation we obtain

120583(119860) asymp minus81033741

Let120573 = 001 120572 = 1 120583 = 09 ] = 11

1205911119867

= 1205911 120591

1119871= 0 997904rArr 120591

1119867119871= 1205911119867

1205751= 12057511= 120575 120575

12= 1205752

1205811= 11989010

120581211

= 0 120581221

= 0 120581231

= 11989010

120581241

= 0 997904rArr 12058121= 11989010

120581311

=

1

1205912

1119867exp (2120572120591

1119867)

120581321

= 0

997904rArr 12058131=

1

exp (21205721205911119867)

(69)

Then120598 = 2 (119901 + 120573) 120575

01+ 2120572120573 + 120581

21

+ 1205813111989021205721205911119867

+ 1205811= 102 + 2119890

10

gt 0

(70)

10 Mathematical Problems in Engineering

Table 1

Delay bound (1205911119867) 2 4 6 8 10

Perturbation bound 1205751

3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245

We can verify that a solution 119875(119905) is given by

119875 (119905) =

[

[

[

[

119890sin(119905)

10

0

0

119890sin(119905)

10

]

]

]

]

(71)

We have 119901 = sup119905isinR+119875(119905) = 11989010

1205751lt

radic120581231119890minus21205721205911119867 (1 minus 120583) (120581

1minus 2120573120583 (119860))

(119901 + 120573)

1205811= 11989010

gt 2120573120583 (119860) = minus162067

120578 = 1205811minus 2120573120583 (119860) minus

(119901 + 120573)2

1205752

120581231119890minus21205721205911119867 (1 minus 120583)

gt 0

120578120577= (1 minus 120577

2

) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575

1 120577 lt 1

120574 =radic

119901 + 120573 + 1205812311205911119867

+ (1198881119867120591111986711989021205721205911119867

)

120573

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 = radic(119901 + 120573)

2

2120572120573120578120577

1205752=

1295910minus3

radic1 minus 1205772

1205752

(72)

If we take 120577 = 09 then 119903 = 297310minus31205752

We see that the perturbation bound 1205751in this example is

the same as in [5] if 1205752= 0 and is better than [4] as shown in

Table 1 The simulation of this example is shown in Figures 1and 2

Example 2 Consider the following second-order differentialsystem

1= minus5119909

1+ 1199092(119905 minus 01) + cos( 119909

1

1 + 1199052)

+ 002 sin (1199091(119905 minus 01) + 05)

2= minus1199091(119905 minus 01) minus 5119909

2

(73)

0 001 002 003 004 0050

001

002

003

004

005

006

007

008

009

Solu

tionx1

x1

Time t

Figure 1 The trajectories of 1199091

with

119860 = [

minus5 0

0 minus5

] 120582max (119860) = minus5 120583 (119860) = minus5

1198601= [

0 1

minus1 0

] 1205911119871= 0 120591

1= 1205911119867

= 01 120583 = 0

1198910(119905 119909) =

[

[

cos( 1199091

1 + 1199052)

0

]

]

12057501= 0 120575

02= 1

1198911(119905 119909 (119905 minus 1)) = [

002 sin (1199091(119905 minus 01) + 05)

0

]

12057511= 002 120575

12= 01

(74)

119909 = (1199091 1199092)119879

isin R2 Hence using (25) we obtain 1205751=

002 1205752= 11

The matrix

119875 (119905) = [

11990111

11990112

11990112

11990122

] (75)

can be computed as follows

11990111= 11990122= minus120573 + 119902

1(5 minus 120572)

11990112= radic1199021sdot radicminus10120573 + 2120572120573 + 119902

1(120572 minus 5)

2

minus 120598

(76)

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

The second member of 1198813(0 1199090) is

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894(1199051)

1198902120572(119904+120591119894119867)

119909 (119904)2

119889119904 1198891199051

le10038171003817100381710038171206011003817100381710038171003817

2

120591119894119867119871

int

minus120591119894119871

minus120591119894119867

int

0

1199051minus120591119894119867

1198902120572(119904+120591119894119867)

119889119904 1198891199051

le

120591119894119867119871

2120572

(120591119894119867119871

1198902120572120591119894119867

minus

1

2120572

[119890minus2120572120591119894119871

minus 119890minus2120572120591119894119867

])10038171003817100381710038171206011003817100381710038171003817

2

le

120591119894119867119871

(2120572)2(2120572120591119894119867119871

1198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 119890minus2120572120591119894119871

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

(

21205721205911198941198671198902120572120591119894119867

+ 119890minus2120572120591119894119867

minus 1

(2120572)2

minus

21205721205911198941198711198902120572120591119894119867

+ 119890minus2120572120591119894119871

minus 1

(2120572)2

)10038171003817100381710038171206011003817100381710038171003817

2

le 120591119894119867119871

119889119894119867119871

10038171003817100381710038171206011003817100381710038171003817

2

(64)

with 119889119894119867119871

= (1198881198941198671205912

119894119867minus 1198881198941198711205912

119894119871) and 119888

119894119871= (2120572120591

1198941198711198902120572120591119894119867

+

119890minus2120572120591119894119871

minus 1)(2120572120591119894119871)2 Consider

1198813(0 1199090) le

119898

sum

119894=1

(120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

)10038171003817100381710038171206011003817100381710038171003817

2

(65)

Therefore from (55) (61) and (65) it follows that size

119881 (0 1199090)

le (119901 + 120573 +

119898

sum

119894=1

(12058121119894120591119894119871+ (12058122119894+ 12058123119894+ 12058124119894]) 120591119894119867

+ 120581311198941198881198941198671205913

119894119867+ 12058132119894120591119894119867119871

119889119894119867119871

))10038171003817100381710038171206011003817100381710038171003817

2

(66)

Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575

(sdot)2char-

acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575

(sdot)2

increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin

Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575

(sdot)1are large enough then there is trade-off

between the values of 1205811and the maximum matrix measure

120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581

1is large then the constant 120598 is

requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)

Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes

4 Examples

Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591

1cos2(045119905)

1198911(119905 119909 (119905 minus 120591

1(119905)))

=[

[

minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575

2lowast cos( 119905

1 + 1199092

1

)

120575 cos (119905) 1199091(119905 minus 1205911(119905))

]

]

(67)where 120591

1 120575 gt 0 120575

2ge 0 will be chosen later and

119860 (119905) = [

119886 (119905) 1

minus1 119886 (119905)

] (68)

where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0

and 119902119894= 0 for (119894 = 1 119898) By computation we obtain

120583(119860) asymp minus81033741

Let120573 = 001 120572 = 1 120583 = 09 ] = 11

1205911119867

= 1205911 120591

1119871= 0 997904rArr 120591

1119867119871= 1205911119867

1205751= 12057511= 120575 120575

12= 1205752

1205811= 11989010

120581211

= 0 120581221

= 0 120581231

= 11989010

120581241

= 0 997904rArr 12058121= 11989010

120581311

=

1

1205912

1119867exp (2120572120591

1119867)

120581321

= 0

997904rArr 12058131=

1

exp (21205721205911119867)

(69)

Then120598 = 2 (119901 + 120573) 120575

01+ 2120572120573 + 120581

21

+ 1205813111989021205721205911119867

+ 1205811= 102 + 2119890

10

gt 0

(70)

10 Mathematical Problems in Engineering

Table 1

Delay bound (1205911119867) 2 4 6 8 10

Perturbation bound 1205751

3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245

We can verify that a solution 119875(119905) is given by

119875 (119905) =

[

[

[

[

119890sin(119905)

10

0

0

119890sin(119905)

10

]

]

]

]

(71)

We have 119901 = sup119905isinR+119875(119905) = 11989010

1205751lt

radic120581231119890minus21205721205911119867 (1 minus 120583) (120581

1minus 2120573120583 (119860))

(119901 + 120573)

1205811= 11989010

gt 2120573120583 (119860) = minus162067

120578 = 1205811minus 2120573120583 (119860) minus

(119901 + 120573)2

1205752

120581231119890minus21205721205911119867 (1 minus 120583)

gt 0

120578120577= (1 minus 120577

2

) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575

1 120577 lt 1

120574 =radic

119901 + 120573 + 1205812311205911119867

+ (1198881119867120591111986711989021205721205911119867

)

120573

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 = radic(119901 + 120573)

2

2120572120573120578120577

1205752=

1295910minus3

radic1 minus 1205772

1205752

(72)

If we take 120577 = 09 then 119903 = 297310minus31205752

We see that the perturbation bound 1205751in this example is

the same as in [5] if 1205752= 0 and is better than [4] as shown in

Table 1 The simulation of this example is shown in Figures 1and 2

Example 2 Consider the following second-order differentialsystem

1= minus5119909

1+ 1199092(119905 minus 01) + cos( 119909

1

1 + 1199052)

+ 002 sin (1199091(119905 minus 01) + 05)

2= minus1199091(119905 minus 01) minus 5119909

2

(73)

0 001 002 003 004 0050

001

002

003

004

005

006

007

008

009

Solu

tionx1

x1

Time t

Figure 1 The trajectories of 1199091

with

119860 = [

minus5 0

0 minus5

] 120582max (119860) = minus5 120583 (119860) = minus5

1198601= [

0 1

minus1 0

] 1205911119871= 0 120591

1= 1205911119867

= 01 120583 = 0

1198910(119905 119909) =

[

[

cos( 1199091

1 + 1199052)

0

]

]

12057501= 0 120575

02= 1

1198911(119905 119909 (119905 minus 1)) = [

002 sin (1199091(119905 minus 01) + 05)

0

]

12057511= 002 120575

12= 01

(74)

119909 = (1199091 1199092)119879

isin R2 Hence using (25) we obtain 1205751=

002 1205752= 11

The matrix

119875 (119905) = [

11990111

11990112

11990112

11990122

] (75)

can be computed as follows

11990111= 11990122= minus120573 + 119902

1(5 minus 120572)

11990112= radic1199021sdot radicminus10120573 + 2120572120573 + 119902

1(120572 minus 5)

2

minus 120598

(76)

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 1

Delay bound (1205911119867) 2 4 6 8 10

Perturbation bound 1205751

3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245

We can verify that a solution 119875(119905) is given by

119875 (119905) =

[

[

[

[

119890sin(119905)

10

0

0

119890sin(119905)

10

]

]

]

]

(71)

We have 119901 = sup119905isinR+119875(119905) = 11989010

1205751lt

radic120581231119890minus21205721205911119867 (1 minus 120583) (120581

1minus 2120573120583 (119860))

(119901 + 120573)

1205811= 11989010

gt 2120573120583 (119860) = minus162067

120578 = 1205811minus 2120573120583 (119860) minus

(119901 + 120573)2

1205752

120581231119890minus21205721205911119867 (1 minus 120583)

gt 0

120578120577= (1 minus 120577

2

) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575

1 120577 lt 1

120574 =radic

119901 + 120573 + 1205812311205911119867

+ (1198881119867120591111986711989021205721205911119867

)

120573

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 = radic(119901 + 120573)

2

2120572120573120578120577

1205752=

1295910minus3

radic1 minus 1205772

1205752

(72)

If we take 120577 = 09 then 119903 = 297310minus31205752

We see that the perturbation bound 1205751in this example is

the same as in [5] if 1205752= 0 and is better than [4] as shown in

Table 1 The simulation of this example is shown in Figures 1and 2

Example 2 Consider the following second-order differentialsystem

1= minus5119909

1+ 1199092(119905 minus 01) + cos( 119909

1

1 + 1199052)

+ 002 sin (1199091(119905 minus 01) + 05)

2= minus1199091(119905 minus 01) minus 5119909

2

(73)

0 001 002 003 004 0050

001

002

003

004

005

006

007

008

009

Solu

tionx1

x1

Time t

Figure 1 The trajectories of 1199091

with

119860 = [

minus5 0

0 minus5

] 120582max (119860) = minus5 120583 (119860) = minus5

1198601= [

0 1

minus1 0

] 1205911119871= 0 120591

1= 1205911119867

= 01 120583 = 0

1198910(119905 119909) =

[

[

cos( 1199091

1 + 1199052)

0

]

]

12057501= 0 120575

02= 1

1198911(119905 119909 (119905 minus 1)) = [

002 sin (1199091(119905 minus 01) + 05)

0

]

12057511= 002 120575

12= 01

(74)

119909 = (1199091 1199092)119879

isin R2 Hence using (25) we obtain 1205751=

002 1205752= 11

The matrix

119875 (119905) = [

11990111

11990112

11990112

11990122

] (75)

can be computed as follows

11990111= 11990122= minus120573 + 119902

1(5 minus 120572)

11990112= radic1199021sdot radicminus10120573 + 2120572120573 + 119902

1(120572 minus 5)

2

minus 120598

(76)

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 001 002 003 004 005

1

0

01

02

03

04

05

06

07

08

09

Solu

tionx2

x2

Time t

Figure 2 The trajectories of 1199092

such that the constant

120572 lt 5 1199021isin ]

2120573

(5 minus 120572)

120581231119890minus21205721205911119867

[

0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)

2

120572 = 1 120573 = 0001 1199021= 00010489

1205811= 002 gt minus10120573 = minus001

120581211

= 0 120581221

= 0

120581241

= 0 997904rArr 12058121= 120581231

= 0001282

120581311

= 016374 120581321

= 0 997904rArr 12058131= 00016374

11990111= 11990122= 000319

11990112= 000125 997904rArr 119901 = 000445

1205751= 002 lt 00206

120578 = 68310minus4

gt 0

120574 = radic

119901 + 120573 + 1205812311205911119867

+ 12058131111988811198671205913

1119867

120573

= 2416

with 1198881119867

=

2120572120591111986711989021205721205911119867

+ 119890minus21205721205911119867

minus 1

(21205721205911119867)2

119903 =

(119901 + 120573)119872

radic2120572120573120578

= 513 with 119872 = 1205752

(77)

in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted

in Figure 3 The evolution of states 1199091and 119909

2is given It is

shown in Figure 1 that the time-delay perturbed system is

0 1 2 3 4 5minus02

0

02

04

06

08

1

12

x1

x2

Solu

tion x

Time t

Figure 3 Convergence of solutions

globally uniformly practically exponentially stable toward aneighborhood of the origin

5 Conclusion

Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

References

[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987

[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989

[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994

[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997

[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008

[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008

[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005

[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002

[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006

[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003

[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008

[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004

[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009

[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008

[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008

[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008

[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete

Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008

[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005

[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008

[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008

[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005

[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009

[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009

[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990

[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001

[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990

[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of