research article robust h-infinity stabilization and...

Research ArticleRobust H-Infinity Stabilization and Resilient Filtering forDiscrete-Time Constrained Singular Piecewise-Affine Systems

Zhenhua Zhou Mao Wang and Qitian Yin

Space Control and Inertial Technology Research Center Harbin Institute of Technology Harbin 150080 China

Correspondence should be addressed to Zhenhua Zhou zhouzhenhuazzh163com

Received 7 June 2014 Accepted 12 November 2014

Academic Editor Asier Ibeas

Copyright copy 2015 Zhenhua Zhou et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with the problem of designing robust H-infinity output feedback controller and resilient filtering for aclass of discrete-time singular piecewise-affine systems with input saturation and state constraints Based on a singular piecewiseLyapunov function combined with S-procedure and some matrix inequality convexifying techniques the H-infinity stabilizationcondition is established and the resilient H-infinity filtering error dynamic system is investigated and meanwhile the domain ofattraction is well estimated Under energy bounded disturbance the input saturation disturbance tolerance condition is proposedthen the resilient H-infinity filter is designed in some restricted region It is shown that the controller gains and filter designparameters can be obtained by solving a family of LMIs parameterized by one or two scalar variables Meanwhile by using thecorresponding optimizationmethods the domain of attraction and the disturbance tolerance level is maximized and theH-infinityperformance 120574 is minimized Numerical examples are given to illustrate the effectiveness of the proposed design methods

1 Introduction

Piecewise-affine systems offer a good modeling frameworkfor hybrid systems involving nonlinear phenomena whichhave the characteristic of both continuous dynamics anddiscrete events with the nature of the model switching [1ndash5]Piecewise-affine systems which are composed of a partitionof the state space and local dynamics valid can describe arich class of practical circuits and control systems when somenonlinear components are encountered such as saturationdead-zone and relays [6ndash8] In fact many nonlinearitiesthat appear frequently in engineering systems either arepiecewise-affine or can be approximated as piecewise-affinefunctions which can be used to analyze smooth nonlinearsystems with arbitrary accuracy [9 10]

Robust stabilization problems of piecewise-affine systemswith norm-bounded time-varying parameters uncertaintieshave been extensively studied and various results have beenobtained on the analysis and controller synthesis [11] Tomention a few the problem of well-posedness was inves-tigated as a basic issue for piecewise-affine systems in theliterature [12] By presenting a number of algorithms the

authors in [6] tested the controllability and observabilityof piecewise-affine systems In [9 10] more attention waspaid by constructing a piecewise-affine Lyapunov functionon stability and optimal performance analysis for piecewise-affine system By the same Lyapunov functions as in [9]controller synthesis and state estimation of piecewise-affinesystem were considered in [13ndash16] By using a commonLyapunov function and a piecewise Lyapunov functionrespectively the authors in [17ndash23] investigated the analysisand control of systems that may involve multiple equilibriumpoints Using a similarmethod to that in [10ndash12] some resultshave also been reported in [24ndash26] where the piecewiseLyapunov function might be discontinuous across the regionboundaries Recently much more attention has been paidto the problem that the stabilization conditions can bedetermined by solving a set of linear matrix inequalities(LMIs) A number of results have been reported based onthe piecewise Lyapunov function [27ndash30] such as controllersynthesis state estimation output regulation and tracking ofpiecewise-affine system For a filtering error dynamic systemthe objective of filter designing is to estimate the unavailablestate variables During the past decades much more filtering

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 878120 16 pageshttpdxdoiorg1011552015878120

2 Mathematical Problems in Engineering

6

5

4

3

2

1

0

minus1

minus2

minus3

Time (step)

x1x2

x3

0 2 4 6 8 10 12 14 16 18 20

(H-infin) the states of the closed-loop system

Stat

ey(k)

Figure 1 State responses of the closed-loop system

schemes have been investigated such as Kalman filtering H-infinity filtering and reduced-order H-infinity filtering [31ndash34] Then the authors in [35] paid more attention to theproblemof resilientKalmanfilteringwith respect to estimatorgain perturbations And the resilient H-infinity filtering wasalso raised

For output feedback control systems an actuator withamplitude and rate limitations may be considered in mostreal-world applications that often suffer from the state con-straints For controller synthesis ignoring these constraintsmay degrade the performance and may even cause the insta-bility of closed-loop system [36 37] (Figure 1) On the otherhand output feedback control can be easily implementedwith low cost which is particularly very useful and morerealistic [38] In the literature [39] the low gain feedbackdesigns were investigated for linear systems with all of itsopen loop poles in the closed left-half plane Based on anauxiliary feedback matrix the authors in [40 41] studied thestabilization and 119871

2-gain control of piecewise-linear systems

with actuator saturation Recently the authors in [42 43]investigated state feedback H-infinity control and outputfeedback H-infinity control for piecewise-affine systemsrespectively

In this paper the H-infinity output feedback controland resilient filter design problems of singular piecewise-affine systems with input saturation and state constraints areconsidered Based on a singular piecewise Lyapunov functioncombined with S-procedure and some matrix inequalityconvexifying techniques the H-infinity stabilization con-dition is established and the resilient H-infinity filteringerror dynamic system is investigated Under energy boundeddisturbance the input saturation disturbance tolerance con-dition is proposed then the resilient H-infinity filter isdesigned in some restricted region The results are given interms of solutions to a set of linear matrix inequalities Mean-while by using the corresponding optimizationmethods thedomain of attraction and the disturbance tolerance level ismaximized and the H-infinity performance 120574 is minimized

According to the existing results the main contributions ofthis paper can be summarized as follows (1) by adding aresilient block to output feedback controller the robust sta-bilization of resilient H-infinity output feedback closed-loopcontrol systems is investigated which was not consideredin [44] (Figure 3) (2) for singular piecewise-affine systemsthe analysis and maximization of the disturbance tolerancecapability are considered (3) by investigating resilient H-infinity filter the robust stabilization of resilient filtering errordynamic system is considered for the first time

The paper is organized as follows In Section 2 modeldescription and some preliminaries are given Sufficientconditions for designing robust H-infinity output feedbackcontrollers are proposed in Section 3 firstly then a resilientH-infinity filter is given by using the analysis and synthesismethods described previously the resulting filtering errordynamic system is admissible with the resilient H-infinityfilter Two numerical examples are presented to illustrate theeffectiveness of the proposed approaches in Section 4 whichis followed by some conclusions finally

Notation 1 The notations used throughout the paper arestandard R119899 denotes the 119899-dimensional Euclidean spacewhile R119898times119899 refers to the set of all real matrices with 119898

rows and 119899 columns A119879 represents the transpose of thematrixA whileAminus1 denotes the inverse matrix ofA l

2[0infin)

refers to the space of square summable infinite sequenceswith the Euclidean norm sdot I is the identity matrix withappropriate dimensions For real symmetric matrices X andY the notation X ge Y (resp X gt Y) means that the matrixX minus Y is positive semidefinite (resp positive definite) Thenotation lowast is used to indicate the terms that can be inducedby symmetry [1 ℎ] denotes the set of 1 2 ℎ in which theelements are integers

2 Model Description andProblem Formulation

Consider a discrete-time singular piecewise-affine systemwith norm-bounded uncertainties and input saturationdescribed by the following dynamics

Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896)

z (119896) = F119894x (119896) + G

119894sat (u (119896)) +D

1198942w (119896)

(1)

x (119896) y (119896) isin R119894 119894 isin 119868 x (0) = x

0 (2)

where x(119896) isin R119899119909 is the system state vector u(119896) isin R119899119906 is thesystem control input y(119896) isin R119899119910 is the system measurementoutput vector z(119896) isin R119899119911 is the controlled output vector andw(119896) isin R119899119908 is an energy bounded disturbance input whichbelongs to l

2[0infin) and satisfies sum

infin

119905=0w119879(119905)w(119905) le 120573 lt infin

A119894 B119894 C119894 D1198941 D1198942 F119894 G119894 b119894 and E are known real constant

matrices with appropriate dimensions denoting the 119894th localmodel of the system and Eb

119894is the offset termThe index set

Mathematical Problems in Engineering 3

of cells is denoted by 119868 = 1 2 119873 The matrix E isin R119899119909times119899119909may be singular and rank(E) = 119903 le 119899

119909is assumed ΔA

119894and

Δb119894are real matrices representing parameter uncertainties of

the 119894th local model of the system which are assumed to benorm-bounded as

[ΔA119894 EΔb119894] = W1198941Δ119894(119905) [E1198941 E

1198942] 119894 isin 119868 (3)

whereW1198941 E1198941 andE

1198942are known real constantmatrices with

appropriate dimensionsΔ119894(119896) 119885

+rarr R1199041times1199042 is an unknown

real-valued time-varying matrix function with the Lebesguemeasurable elements satisfying

Δ119879

119894(119896) Δ119894(119896) le I

1199042

(4)

The parameter uncertainties are said to be admissible if (3)-(4) hold

Remark 1 In order to refrain unnecessarily complicatednotations in this paper we only consider the norm-boundeduncertainty parameters involving matrices A

119894and b

119894 Never-

theless the methods to be investigated in this paper can beeasily extended to the case when the uncertainty parametersalso emerge in the matrices B

119894D1198941D1198942 F119894 and G

119894 119894 isin 119868

In addition we consider that the system issubject to the input and state constraints wherethe saturated input is expressed by sat(u(119896)) =

[sat(u1(119896)) sat(u2(119896)) sdot sdot sdot sat(u

119898(119896))]119879 sat(u

119897) =

sgn(u119897)min|u

119897| u 119897 isin [1119898] with saturation level u

and the system state is bounded by the following condition

minusg le Lx le g (5)

where L isin R119899119903times119899119909 is a known real constant matrix and g isin R119899119903is a given constant vector

In the following we will introduce a new set

Ω = (119894 119895) | y (119896) isin R119894 y (119896 + 1) isin R

119895 119894 119895 isin 119868 (6)

which represents the index pairs denoting all possible transi-tions of the system state trajectories

Remark 2 For mixed logical dynamical (MLD) system [6]the reachability analysis can usually determine the set Ω Ifit is probable that the transitions happen between all regionsthenΩ = 119868 times 119868 = (119894 119895) | 119894 119895 isin 119868

It is assumed in this paper that the polyhedral regionR119894 119894 isin 119868 is slabs of the following form

R119894= y | 120572

119894le y le 120573

119894 y = C

119894x 119894 isin 119868 (7)

Each slab can be exactly described by a degenerateellipsoid

120576119894= x |

10038171003817100381710038171003817F119894x + 119891119894

10038171003817100381710038171003817le 1 119894 isin 119868 (8)

where F119894= 2C119894(120573119894minus 120572119894) 119891119894= minus(120573

119894+ 120572119894)(120573119894minus 120572119894) Then we

have the following relationship for each ellipsoid region

[x(119896)1

]

119879

[F119879119894F119894

F119879119894119891119894

lowast 119891119879

119894119891119894minus 1

] [x (119896)1

] le 0 119894 isin 119868 (9)

Let the partitioned regions be separated into two classes119868 = 119868

0cup 1198681 where 119868

0denotes the index set of regions with

119891119879

119894119891119894minus 1 le 0 which contains the origin and 119868

1denotes the

index set of regions otherwiseFor system (1) we introduce the following definitions

Definition 3 (see [45]) According to the discrete-timepiecewise-affine singular system (1) with u(119896) = 0 one hasthe following

(i) The system is said to be regular if det(zE minus A119894) is not

identically zero 119894 isin 119868(ii) The system is said to be causal if deg(det(zE minus A

119894)) =

rank(E) 119894 isin 119868(iii) The system is said to be stable if all roots of 120582(EA

119894) sub

Dint(0 1)(iv) The system is said to be admissible if it is regular

causal and stable(v) For the system there exists a grade 1 (infinite gener-

alised) eigenvector of the pair (EA119894) such that for any

nonzero vector ^1 satisfying E^1 = 0 and there existsa grade 119896 (infinite generalised) eigenvector of the pair(EA119894) such that for any nonzero vector ^119896 (119896 ge 2)

satisfying E^119896 = A119894^119896minus1

Definition 4 For the convenience of the subsequent descrip-tions one defines the sets 120576(P Υ) = x(119905) | x119879(119905)Px(119905) le

Υ Υ gt 0 and 120577(H 120591) = x(119905) | |H119897x(119905)| le 120591 120591 gt 0 as an

ellipsoid and a polyhedron where 119897 isin [1119898] respectively P isa positive definite matrix and H

119897is the 119897th row of the matrix

H isin R119899119906times119899119909 Also one introduces the following lemmas

Lemma 5 (see [46]) For all u isin R119899119906 and ^ isin R119899119906 such that|^119897| lt u 119897 isin [1 119898] one has sat(u) isin co w

119904u + w

119904^ 119904 isin

[1 2119898] where ldquocordquo denotes the convex hull Herew

119904is an 119899

119906times

119899119906diagonal matrix with elements either 1 or 0 On the other

hand w119904= I minus w

119904 There are 2119898 such matrices

Lemma 6 (see [46]) Let matrices M = M119879 S N and Δ(119905)

be real matrices of appropriate dimensions and the inequalityM + SΔ(119905)N + N119879Δ119879(119905)S119879 lt 0 holds for all Δ119879(119905)Δ(119905) le 119868 ifand only if for some positive scalar 120576 gt 0 such thatM + 120576SS119879 +120576minus1N119879N lt 0

(A) Resilient Output Feedback Control of the Discrete-TimeSingular Piecewise-Affine System In this paper we considera resilient output feedback controller which is described asfollows

u (119896) = (K119894+ ΔK119894) y (119896) K

119894isin R119899119906times119899119910

y (119896) isin R119894 119894 isin I

(10)

where ΔK119894

= W1198941Δ119894(119905)E1198943 119894 isin 119868 and E

1198943is known real

constant matrices with appropriate dimensionsIn this paper for all u isin R119899119906 and ^ isin R119899119906 such that |^

119897| lt

u 119897 isin [1 119898] we assume that x(119896) isin 120577(H119894 u) x(119896) isin R

119894


119894 isin 119868 according to Lemma 5 and (10) the saturated controlinput sat(u(119896)) can be written in the following form

sat (u (119896))

=

2119898

sum

119904=1

120588119894119904(119896) (W

119894119904K119894C119894+W119894119904ΔK119894C119894+ 2W

119894119904H119894) x (119896)

x (119896) isin R119894 119894 isin 119868

(11)

where 1205881198941(119870) ge 0 120588

1198942(119870) ge 0 120588

1198942119898(119870) ge 0 and

sum2119898

119904=1120588119894119904(119896) = 1

The closed-loop singular piecewise-affine control systemconsisting of system (1) and the saturated control input (11)can be described as

Ex (119896 + 1) = [A119894+ B119894W119894119904(119896)] x (119896)

+D1198941w (119896) + E (b

119894+ Δb119894)

z (119896) = [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

x (119896) isin R119894 119894 isin 119868

(12)

where A119894

= A119894+ ΔA

119894 W119894119904(119896) = sum

2119898

119904=1120588119894119904(119896)(W

119894119904K119894C119894+

W119894119904ΔK119894C119894+ 2W

119894119904H119894)x(119896)

(B) Resilient Filter for Discrete-Time Singular Piecewise-AffineSystems with Uncertain Parameters We consider a resilientfiltering for discrete-time singular piecewise-affine systemswith uncertain parameters as follows

x (119896 + 1) = A119891x (119896) + B

119891y (119896)

z (119896) = C119891x (119896) +D

119891y (119896) y (119896) isin R

119894 119894 isin 119868

u (119896) = (K119894+ ΔK119894) x (119896)

(13)

where A119891B119891C119891 andD

119891are the design parameters

According to (11) the saturated control input sat(u(119896))can be written in the following form

sat (u (119896)) =

2119898

sum

119904=1

120588119894119904(119896) (W

119894119904K119894+W119894119904ΔK119894+ 2W

119894119904H119894) x (119896)

x (119896) isin R119894 119894 isin 119868

(14)

Assume that x119879(119896) = [x119879(119896) x119879(119896)] the resilient filteringerror dynamic system consisting of system (1) 2 and thesaturated control input (14) can be described as

Ex (119896 + 1) = Ax (119896) + Bw (119896) + b119894

z (119896) = Cx (119896) +D1198942w (119896)

(15)

where C = [119865119894minusD119891C119894G119894W119894119904(119896) minus C

119891] A =

[A119894+ΔA119894B119894W119894119904(119896)

B119891C119894

A119891

] E = [ E 00 I ] W119894119904(119896) = sum

2119898

119904=1120588119894119904(119896)(W

119894119904K119894+

W119894119904ΔK119894+ 2W119894119904H119894)x(119896) B = [

D1198941

0 ] and z(119896) = z(119896) minus z(119896)b119894=

[E(b119894+Δb119894)

0 ]

3 Main Results

Theorem 7 If there exist matrices 0 lt P119894= P119879119894isin R119899119909times119899119909 K

119894isin

R119899119906times119899119910 119894 isin 119868 and positive scalar 120573 le 120573119872 120576119894119895 1205761015840

119894119895 12057610158401015840

119894119895 119894 isin 119868

120582119894119895 119894 isin 119868

1 (119894 119895) isin Ω such that the following LMIs hold

E119879P119894E ge 0 (16)

[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

lowast minus1205761015840

119894119895I1199042

0 0 0 0 E1198943C119894

lowast lowast minus120576119894119895I1199042

0 0 0 E1198941

lowast lowast lowast 120601 0 D1198942

Ξ

lowast lowast lowast lowast minusPminus1119895

+ Ψ D1198941

Λ

lowast lowast lowast lowast lowast minusI 0lowast lowast lowast lowast lowast lowast minusE119879P

119894E

]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]

(17)

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894

0lowast lowast minus120576

119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894


Ξ 0lowast lowast lowast lowast minusPminus1

119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894

lowast lowast lowast lowast lowast minusI 0 0lowast lowast lowast lowast lowast lowast Π + 120599 120582

119894119895F119879119894119891119894119891119879

119894

lowast lowast lowast lowast lowast lowast lowast 120582119894119895(119891119879

119894119891119894minus 119868)

]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

(18)


[[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119897 isin [1 119898] (19)

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903] (20)

whereQ119894119904= Q119894119904Pminus1119894

= (W119894119904K119894C119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894

Z119894119897is the 119897th row of matrix Z

119894 L119896is the 119896th row of matrix L

and g119896is the 119896th row of vector g then for any initial condition

x0starting from the region cup

119894isin119868(120576(P119894 120573) cap R

119894) the discrete-

time singular piecewise-affine system (1) can be asymptoticallystabilized by the controller (10) with K

119894= Y119894P119894Cminus1119894 Consider

120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

Ξ = F119894+ G119894Q119894119904+ G119894W119894119904H119894 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

Λ = A119894+ B119894Q119894119904+ B119894W119894119904H119894 120578 = 120582

119894119895F119879119894119891119894119891119879

119894

120580 = 120582119894119895(119891119879

119894119891119894minus 119868) Φ = 120582

minus1

119894119895Eb119894(Eb119894)119879

120603 = E1198942(F119879119894119891119894)

119879

120579 = Eb119894(F119879119894119891119894)

119879

120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)

119879

Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

(21)

Proof In this paper we consider the following singularpiecewise quadratic Lyapunov function

V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896) 119894 isin 119868 (22)

According to the Lyapunov function defined in (22)to make the closed-loop system (12) possess H-infinityperformance 120574 we know that it suffices to show the followinginequality

V (119896 + 1 x (119896 + 1)) minus V (119896 x (119896))

+ 120574minus2z119879 (119896) z (119896) minus w119879 (119896)w (119896) lt 0

(23)

In the case of 119894 isin 1198681 (119894 119895) isin Ω it follows from (23) that

[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896) + E(b

119894+ Δb119894)119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896) + E (b

119894+ Δb119894)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896) minus w (119896)

119879w (119896) lt 0

(24)

Equation (24) can be rewritten as follows with (119894 119895) isin Ω

for any nonzero w(119896) isin l2[0infin)

[

[

w(119896)

x(119896)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[

minusI 0 0lowast minusE119879P

119894E 0

lowast lowast 0]

]

times [

[

w (119896)

x (119896)1

]

]

lt 0

(25)

where A = A119894+ B119894W119894119904(119896) B = F

119894+ G119894W119894119904(119896)

From (25) we get

A119879P119895A minus E119879P

119894E + 120574minus2B119879B lt 0 (26)

It is easy to see the following inequality

A119879P119895A minus E119879P

119894E lt 0 (27)

Assume that (EA) is not causal We multiply (27) by thegrade 1 eigenvectors ^1 and its Hermitian ^1lowast respectivelyIn view of Definition 3 replacing A^1 by E^2 and noting thatE^1 = 0 it gives

^2lowastE119879P119894E^2 lt 0 (28)

which contradicts (16) Therefore (EA) is causal Thus theregularity of (EA) is implied As a result the closed-loopsystem (12) is said to be admissible

Then by taking into consideration the partition infor-mation (9) and applying the S-procedure we have that the


following inequality implies (25) with 120582119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[

[

w(119905)

x(119905)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[


119894E 0

lowast lowast 0]

]

+ 120582119894119895

[[

[

0 0 0lowast F119879119894F119894

F119879119894119891119894

lowast lowast 119891119879

119894119891119894minus 1

]]

]

times [

[

w (119905)

x (119905)1

]

]

lt 0 (119894 119895) isin Ω

(29)

For the matrix inequalities in (30) taking linear com-binations over 119904 and using the well-known Schur comple-ment it is easy to see that the following inequality implies(29)

[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0

lowast minusPminus1119895

D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894

lowast lowast minusI 0 0

lowast lowast lowast minusE119879P119894E + 120582119894119895F119879119894F119894

120582119894119895F119879119894119891119894

lowast lowast lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]]]]]]]

]

+

[[[[[[[[

[

0 0 0 0 0lowast 0 0 ΔA

119894EΔb119894

lowast lowast 0 0 0lowast lowast lowast 0 0lowast lowast lowast lowast 0

]]]]]]]]

]

lt 0 119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

(30)

where Q119894119904= W119894119904K119894C119894+ W119894119904H119894 On the other hand by using

the relations given in (3) the left-hand side (LHS) of (30) canbe easily rewritten as follows

LHS (30) =

[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+ Sym

[[[[[[

[

0W1198941

000

]]]]]]

]

Δ119894(119905) [0 0 0 E

1198941E1198942]

(31)


Remark 8 Firstly we consider the uncertainty terms appear-ing in the matrices A

119894and b

119894 according to the relations

ΔK119894

= W1198941Δ119894(119905)E1198943 119894 isin 119868 we can eliminate the uncer-

tainty terms appearing in G119894W119894119904ΔK119894C119894and B

119894W119894119904ΔK119894C119894by

the same means SymΓ is the shorthand notation forΓ + Γ119879

Thus based on Lemma 6 by introducing a set of positivescalar parameters 120576

119894119895gt 0 119894 isin 119868

1 (119894 119895) isin Ω it is easy to see

that the following inequality implies (31)

[[[[[[[[

[

minus120576119894119895I1199042

0 0 0 E1198941

E1198942

lowast minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0lowast lowast minusPminus1

119895+ 120576119894119895W1198941W1198791198941

D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894

lowast lowast lowast minusI 0 0lowast lowast lowast lowast minusE119879P

119894E + 120582119894119895F119879119894F119894

120582119894119895F119879119894119891119894

lowast lowast lowast lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]]]]]]

]

lt 0

(32)

For the matrix inequalities in (33) introduce two sets ofpositive scalar parameters 1205761015840

119894119895 12057610158401015840

119894119895

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 E1198941

E1198942

lowast lowast lowast minus1205742I + 1205761015840

119894119895120572120572119879 0 D

1198942Ξ 0


+ Ψ D1198941

Λ Eb119894

lowast lowast lowast lowast lowast minusI 0 0lowast lowast lowast lowast lowast lowast Π 120582

119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]]]]]

]

lt 0 (33)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 Ξ = F

119894+ G119894Q119894119904+

G119894W119894119904H119894 Λ = A

119894+ B119894Q119894119904+ B119894W119894119904H119894 Π = minusE119879P

119894E + 120582119894119895F119879119894F119894

and Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

Using Schur complement again it is seen that the aboveinequality is equivalent to the following inequality

[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ D1198941

Λ

lowast lowast lowast lowast lowast minusI 0lowast lowast lowast lowast lowast lowast Π

]]]]]]]]]]

]

+ (minus120582minus1

119894119895)

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

(119891119879

119894119891119894minus 1)minus1

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

119879

lt 0

(34)

By the well-known fact (119891119879

119894119891119894minus 1)minus1

= minus1 + 119891119879

119894times

(119891119894119891119894

119879minus 119868)minus1119891119894(matrix inversion lemma) and some simple

calculations the matrix inequality (34) can be rewritten asfollows


[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ + Φ D1198941

Λ + 120579

lowast lowast lowast lowast lowast minusI 0lowast lowast lowast lowast lowast lowast Π + 120599

]]]]]]]]]]]]]

]


119894119895)

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

(119891119879

119894119891119894minus 119868)minus1

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

119879

lt 0

(35)

where Φ = 120582minus1

119894119895Eb119894(Eb119894)119879 120603 = E

1198942(F119879119894119891119894)119879 120579 = Eb

119894(F119879119894119891119894)119879

and 120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)119879

By using the well-known Schur complement again it iseasy to see that the following inequality implies (35)

[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894

lowast lowast lowast lowast lowast minusI 0 0

lowast lowast lowast lowast lowast lowast Π + 120599 120582119894119895F119879119894119891119894119891119879

119894


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]

]

lt 0 (36)

where 120601 = minus1205742I + 120576

1015840

119894119895120572120572119879 on the other hand it follows that

b119894+ Δb119894= 0 it is easy to see the following inequality

[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896)

119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

minus w (119896)119879w (119896) lt 0 119894 isin 119868

0 (119894 119895) isin Ω

(37)

For thematrix inequalities in (38) taking linear combina-tions over 119904 and using the Schur complement one can obtain

from (37) that the inequality also holds for 119894 isin 1198680 (119894 119895) isin Ω

Consider

[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

lowast lowast minusI 0

lowast lowast lowast minusE119879P119894E

]]]]]]

]

lt 0

(38)

By a similar technique dealing with the uncertainties asin (30) it is easy to see that the matrix inequalities in (38)hold if and only if there exist three sets of positive scalars


120576119894119895 1205761015840

119894119895 and 120576

10158401015840

119894119895such that the matrix inequalities in (39) hold

Consider

[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]]]

]

lt 0

(39)

It is easy to see that the inequalities in (39) and (36) areequivalent to LMIs in (17) and (18)

Remark 9 In the case of b119894+ Δb119894= 0 119894 isin 119868

0 where 119868

0

denotes the index set of regions with 119891119879

119894119891119894minus 1 le 0 the

partition information (9) is not considered in the process ofTheorem 7 proof To make the results simple in Theorem 7we only investigate the system matrix A

119894and offset term Eb

119894

is time-variant In fact when matrices B119894 D1198941 D1198942 F119894 and

G119894are time-variant we can get the resilient H-infinity output

feedback controllers with the same courseIn addition by Schur complement the LMIs in (19) are

equivalent to Z119894119897Xminus1119894Z119879119894119897

le 1199062120573 119894 isin 119868 119897 isin [1 119898] which

are equivalent to H119894119897Pminus1119894H119879119894119897

le 1199062120573 119894 isin 119868 119897 isin [1 119898] It

is easy to see that H119894119897Pminus1119894H119879119894119897

le 1199062120573 119894 isin 119868 119897 isin [1 119898] are

equivalent to 120576(P119894 120573) sub 120577(H

119894 119906) 119894 isin 119868 (see [41]) Similarly

the LMIs in (20) are equivalent to 120576(P119894 120573) sub 120577(L

119896 g119896) 119894 isin 119868

119896 isin [1 119903]

Using inequalities (23) and noticing that ΔV(119896) = V(119896 +

1 x(119896 + 1)) minus V(119896 x(119896)) and V(0) = 0 one can obtain

V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896)

=

119905minus1

sum

119896=0

ΔV (119896) le

119905minus1

sum

119896=0

w119879 (119905)w (119905) le 120573

x (119896) isin R119894 119894 isin 119868 119905 ge 1

(40)

At the time 119896 it is seen from (40) that x(119896) isin 120576(P119894 120573) cap

R119894 x(119896) isin R

119894 119894 isin 119868 Using the conditions 120576(P

119894 120573) sub

120577(H119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub 120577(L

119896 g119896) 119894 isin 119868 119896 isin [1 119903]

we can obtain the relationships x(119896) isin 120577(H119894 119906) and minusg le

Lx(119896) le g Assume that at the next time 119896 + 1 the state ofthe closed-loop system (12) transits to x(119896+1) isin R

119895 119895 isin 119868 It

is clear from (40) that x(119896+1) isin 120576(P119895 120573)capR

119895Then it follows

from the conditions 120576(P119894 120573) sub 120577(H

119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub

120577(L119896 g119896) 119894 isin 119868 119896 isin [1 119903] that x(119896 + 1) isin 120577(H

119895 119906) and

minusg le Lx(119896 + 1) le g Using the above deductions recursivelyit can be concluded that x(119896) isin 120577(H

119894 119906) and minusg le Lx(119896) le g

always hold for x(119896) isin R119894 119894 isin 119868 119905 ge 0 and the state

trajectory of the closed-loop system (12) will remain in theregion cup

119894isin119868(120576(P119894 120573) capR

119894) This completes the proof

For a fixed scalar 120573 le 120573119872 the smallest H-infinity

performance 120574 can be measured by solving this optimizationproblem (min Pminus1

119894Y119894Z119894)120574 so that the LMIs in (17)ndash(20)

hold

Theorem 10 For given positive scalars 120574 and 120573 le 120573119872 if there

exist matrices 0 lt H1

= H1198791

isin R119899119909times119899119909 0 lt H3

= H1198793

isin

R119899119909times119899119909 H2isin R119899119909times119899119909 0 lt P

119894= P119879119894

isin R2119899119909times2119899119909 K119894isin R119899119906times119899119910

A119891isin R119899119909times119899119909 B

119891isin R119899119909times119899119909 C

119891isin R119899119909times119899119909 and D

119891isin R119899119909times119899119909 and

positive scalars 120576119894119895 1205761015840

119894119895 12057610158401015840

119894119895 119894 isin 119868 120582

119894119895lt 0 119894 isin 119868

1 (119894 119895) isin Ω

such that the following LMIs hold

E119879P119894E ge 0 (41)

[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941

0

lowast lowast lowast 120601 0 0 D1198942

F119894minusD119891C119894G119894Q119894119904+ G119894W119894119904H119894minus C119891

lowast lowast lowast lowast 120590 H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894

lowast lowast lowast lowast lowast H3

0 B119891C119894

A119891

lowast lowast lowast lowast lowast lowast minusI 0 0lowast lowast lowast lowast lowast lowast lowast 119869

11198692

lowast lowast lowast lowast lowast lowast lowast lowast 1198693

]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]


[[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942I1198791

120582minus1

119894119895E1198942I1198792

E1198942


F119894minusD119891C119894

alefsym 0lowast lowast lowast lowast weierp H

2D1198941

A119894+ 120582minus1

119894119895Eb119894I1198791

I Eb119894


0 B119891C119894

A119891

0lowast lowast lowast lowast lowast lowast minusI 0 0 0lowast lowast lowast lowast lowast lowast lowast 119869

1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681

lowast lowast lowast lowast lowast lowast lowast lowast 1198693+ 120582minus1

1198941198951198682119868119879

21198682

lowast lowast lowast lowast lowast lowast lowast lowast lowast 120582119894119895(119891119879

119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]

]

ge 0 119894 isin 119868 119897 isin [1 119898]

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903]

(42)

where Q119894119904= Q119894119904Pminus1119894

= (W119894119904K119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894




1199090starting from the region cup

119894isin119868(120576(P119894 120573) cap R


time singular piecewise-affine system (1) can be asymptoticallystabilized by the resilient H-infinity filter 2 with K

119894= Y119894P119894

Consider

weierp = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879+ 120582minus1

119894119895Eb119894(Eb119894)119879

I = B119894Q119894119904+ B119894W119894119904H119894+ 120582minus1

119894119895Eb119894119868119879

2

alefsym = G119894Q119894119904+ G119894W119894119904H119894minus C119891

120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

120601 = minus1205742I + 1205761015840

119894119895120572120572119879 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879

120575 = 120582minus1

119894119895E1198942E1198791198942

120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [

1198681

1198682

]

minusE119879P119894E + 120582119894119895F119879119894F119894= [

1198691

1198692

lowast 1198693

]

(43)

Proof From the proof of Theorem 7 we also consider thesingular piecewise quadratic Lyapunov function defined in(22) to make the resilient filtering error dynamic system (15)possess H-infinity performance 120574 we know that it suffices toshow the following inequality

119881 (119896 + 1 x (119896 + 1)) minus 119881 (119896 x (119896))


(44)

where z(119896) = z(119896) minus z(119896) for any nonzero w(119896) isin l2[0infin)

and (44) can be rewritten in the following inequality

x (119896)119879A119879 + w (119896)119879 B119879 + b

119894

119879

times P119895Ax (119896) + Bw (119896) + b

119894

minus x119879 (119896)E119879P119894Ex (119896) minus w (119896)

119879w (119896) + 120574minus2

times x (119896)119879 [F119894 minusD119891C119894G119894W119894119904(119896) minus C

119891]119879

+ w (119896)119879D1198791198942

times [F119894minusD119891C119894G119894W119894119904(119896) minus C

119891] x (119896) +D

1198942w (119896) lt 0

(45)

By the system state-space equation it is easy to see thefollowing inequality


[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894

lowast A119879P119895A minus E119879P

119894E + 120574minus2C119879C A119879P

119895b119894

lowast lowast b119894

119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)

From the proof of Theorem 7 it is easy to see that (EA) iscausal Thus the regularity of (EA) is implied As a resultthe closed-loop system (15) is said to be admissible

Remark 11 According to the proof of Theorem 7 we use thesame method to get the regularity of (EA) We also multiply(27) by the grade 1 eigenvectors ^1 and its Hermitian ^1lowastrespectively In view of Definition 3 replacingA^1 byE^2 andnoting that E^1 = 0 as a result the closed-loop system (15) issaid to be admissible

Then by taking into consideration the partition infor-mation (9) and applying the S-procedure we have that thefollowing inequality implies (46) with 120582

119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]

P119895(lowast) + 120574minus2

[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0

lowast minusE119879P119894E + 120582119894119895F119879119894F119894

120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)

By using the well-known Schur complement it is easy tosee that the following inequality implies (48)

[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)

On the other hand by using the relations given in (3) (49)can be easily rewritten as follows

[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D

1198942F119894minusD119891C119894G119894W119894119904(119896) minus C

1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0

lowast lowast lowast minusI 0 0 0

lowast lowast lowast lowast 1198691

1198692

1198681

lowast lowast lowast lowast lowast 1198693

1198682

lowast lowast lowast lowast lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)

where minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]


It is easy to see that the following inequality implies (50)

[[[[[[[[[[

[

minus1205742I 0 0 D

1198942F119894minusD119891C119894G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894minus C119891

0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891

0lowast lowast lowast minusI 0 0 0lowast lowast lowast lowast 119869

11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)

From the proof of Theorem 7 based on Lemma 6 byintroducing three sets of positive scalar parameters 120576

119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868

1 (119894 119895) isin Ω it is easy to see that the

following inequality implies (51)

[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942

On the other hand it follows that b119894+ Δb119894= 0 it is easy

to see the following inequality

[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941

0lowast lowast lowast 120601 0 0 D

1198942F119894minusD119891C119894G119894Q119894119904+ G119894W119894119904H119894minus C119891


A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879


On the other hand from the proof ofTheorem 7 it is clearthat all trajectories of system (1) starting from the origin willremain inside the region cup

119894isin119868(120576(P119894 120573) cap R

119894) This completes

the proof



119894Y119894Z119894)120574 so that the LMIs in 10 hold

4 Numerical Examples

Example 1 To illustrate the analysis and synthesis methodsdescribed in the previous sections we consider the stabiliza-tion problem for system (1) with the following data

A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)

Saturation level u = 6 and system state is bounded by (5)with L = 119868 and g = [30 30 30]

119879 It is assumed that

R1= x (119905) | 5 le [1 0 0] x (119905) le 30

R2= x (119905) | minus30 le [1 0 0] x (119905) le minus5

(55)

Let the coefficient matrices of ellipsoid be given as

F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9

Let the interference signal be given as w(119896) = 119890minus5119896

With these parameters solving LMIs we found that theyare feasible and the solution is as follows

(A) Resilient Output Feedback Control of the Discrete-TimeSingular Piecewise-Affine System By applying Theorem 7directly one can obtain

K1= [minus308973 minus170674 minus205996]

K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)

The H-infinity performance 120574 = 69025

From the above two sets of Lyapunov matrices Figure 2presents the estimated domains of attraction of system (12)which is bounded by the solid line

(B) Resilient Filter for Discrete-Time Singular Piecewise-AffineSystems with Uncertain Parameters Consider

1198701= [52373 60612 15916]

1198702= [minus39530 minus61848 minus08335]

(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25

minus10 minus5 0 5 10

x 2

x1

Figure 2 Estimated domains of attraction in Example 1

6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)


and the H-infinity performance 120574 = 1456232 Consider

119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)

Example 2 Now we consider a practical physical exampleto illustrate the analysis and synthesis methods described in

the previous sections we consider the stabilization problemfor system (1) with the tunnel diode circuit [47] Somemore details on the dynamics of the tunnel diode circuit inExample 2 are given in [47] For this system it is assumedthat the controller is subject to saturation constraint withu = 6 and the system state is bounded by (5) with L = 119868 andg = [30 30 30]

119879 x1(119896) can be exactly described by capacitor

voltage x2(119896) can be exactly described by inductive current

and x3(119896) can be exactly described by the current which

flows through tunnel diode The tunnel diode circuit can beabstracted as a system which is described by the followingdynamics

Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)

The data we need is given previously where 120573119872

= 56567Now we also need some data 120573 = 50 le 120573

119872

The gain matrices for resilient H-infinity output feedbackcontroller are obtained

K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)

The H-infinity performance 120574 = 75744The design parameters are obtained

A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259

minus80976 minus43324 minus44279

21836 52533 52859

]

]

(62)

5 Conclusions

In this paper we have proposed new LMI conditions forthe problem of designing robust H-infinity output feedbackcontroller and resilient filtering for a class of discrete-time singular piecewise-affine systems with input saturationand state constraints involving norm-bounded time-varyingparameters uncertainties Based on a singular piecewiseLyapunov function combined with S-procedure and somematrix inequality convexifying techniques the controllergains and the filter design parameters have been obtainedby solving a family of LMIs Meanwhile by presentingthe corresponding optimization methods the domain ofattraction and the disturbance tolerance level is maximized


and the H-infinity performance 120574 is minimized Simulationexamples are presented to demonstrate the effectiveness andpracticability of the proposed approaches

As for further studies the piecewise-affine controller willbe used to investigate a similar robust stabilization problemof continuous-time singular piecewise-affine systems Inaddition resilient H-infinity guaranteed cost controller willbe investigated for singular piecewise-affine systems withinput saturation and state constraints

Conflict of Interests

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

Acknowledgment

This research is sponsored by the National Natural ScienceFoundation of China (Grant no 61004038)

References

[1] S Mirzazad-Barijough and J-W Lee ldquoStability and transientperformance of discrete-time piecewise affine systemsrdquo IEEETransactions on Automatic Control vol 57 no 4 pp 936ndash9492012

[2] S Mirzazad-Barijough and J W Lee ldquoRobust stability andperformance analysis of discrete-time piecewise affine systemswith disturbancesrdquo in Proceedings of the IEEE 51st AnnualConference on Decision and Control (CDC rsquo12) pp 4229ndash42342012

[3] M Rubagotti L Zaccarian and A Bemporad ldquoStability analy-sis of discrete-time piecewise-affine systems over non-invariantdomainsrdquo in Proceedings of the 51st IEEE Conference on Decisionand Control (CDC rsquo12) pp 4235ndash4240 Maui Hawaii USADecember 2012

[4] N Eghbal N Pariz and A Karimpour ldquoDiscontinuous piece-wise quadratic Lyapunov functions for planar piecewise affinesystemsrdquo Journal ofMathematical Analysis andApplications vol399 no 2 pp 586ndash593 2013

[5] M Rubagotti S Trimboli and A Bemporad ldquoStability andinvariance analysis of uncertain discrete-time piecewise affinesystemsrdquo IEEE Transactions on Automatic Control vol 58 no9 pp 2359ndash2365 2013

[6] A Bemporad G Ferrari-Trecate andMMorari ldquoObservabilityand controllability of piecewise affine and hybrid systemsrdquo IEEETransactions on Automatic Control vol 45 no 10 pp 1864ndash1876 2000

[7] W P Heemels M K Camlibel and J M Schumacher ldquoOnthe dynamic analysis of piecewise-linear networksrdquo IEEETrans-actions on Circuits and Systems I Fundamental Theory andApplications vol 49 no 3 pp 315ndash327 2002

[8] J Imura ldquoClassification and stabilizability analysis of bimodalpiecewise affine systemsrdquo International Journal of Robust andNonlinear Control vol 12 no 10 pp 897ndash926 2002

[9] M Johansson and A Rantzer ldquoComputation of piecewisequadratic Lyapunov functions for hybrid systemsrdquo IEEE Trans-actions on Automatic Control vol 43 no 4 pp 555ndash559 1998

[10] A Rantzer and M Johansson ldquoPiecewise linear quadraticoptimal controlrdquo IEEE Transactions on Automatic Control vol45 no 4 pp 629ndash637 2000

[11] M Johansson Piecewise Linear Control SystemsmdashA Computa-tional Approach vol 284 2003

[12] J Imura and A van der Schaft ldquoCharacterization of well-posedness of piecewise-linear systemsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1600ndash1619 2000

[13] G Feng ldquoController design and analysis of uncertain piecewise-linear systemsrdquo IEEE Transactions on Circuits and Systems IFundamental Theory and Applications vol 49 no 2 pp 224ndash232 2002

[14] Y Zhu D Q Li and G Feng ldquoH-infinity controller synthesisof uncertain piecewise continuous-time linear systemsrdquo IEEProceedings-Control Theory and Applications vol 152 pp 513ndash519 2005

[15] J X Zhang and W S Tang ldquoOutput feedback H(infinity)control for uncertain piecewise linear systemsrdquo Journal ofDynamical and Control Systems vol 14 no 1 pp 121ndash144 2008

[16] J Zhang and W Tang ldquoOutput feedback optimal guaranteedcost control of uncertain piecewise linear systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 19 no 5 pp569ndash590 2009

[17] G Bianchini S Paoletti and A Vicino ldquoConvex relaxationsfor 119871

2-gain analysis of piecewise affinepolynomial systemsrdquo

International Journal of Control vol 86 no 7 pp 1207ndash12132013

[18] M di Bernardo U Montanaro and S Santini ldquoHybrid modelreference adaptive control of piecewise affine systemsrdquo IEEETransactions on Automatic Control vol 58 no 2 pp 304ndash3162013

[19] W P M H Heemels M Lazar N Van De Wouw andA Pavlov ldquoObserver-based control of discrete-time piecewiseaffine systems exploiting continuity twicerdquo in Proceedings of the47th IEEE Conference on Decision and Control (CDC rsquo08) pp4675ndash4680 Cancun Mexico December 2008

[20] G Pola and M D Di Benedetto ldquoSymbolic models and controlof discrete-time piecewise affine systems an approximate simu-lation approachrdquo IEEE Transactions on Automatic Control vol59 no 1 pp 175ndash180 2014

[21] L Rodrigues and J P How ldquoObserver-based control ofpiecewise-affine systemsrdquo International Journal of Control vol76 no 5 pp 459ndash477 2003

[22] A Hassibi and S Boyd ldquoQuadratic stabilization and control ofpiecewise-linear systemsrdquo in Proceedings of the American Con-trol Conference (ACC rsquo98) vol 6 pp 3659ndash3664 PhiladelphiaPa USA June 1998

[23] A L Juloski W P Heemels and S Weiland ldquoObserver designfor a class of piecewise linear systemsrdquo International Journal ofRobust andNonlinear Control vol 17 no 15 pp 1387ndash1404 2007

[24] S Pettersson and B Lennartson ldquoHybrid system stabilityand robustness verification using linear matrix inequalitiesrdquoInternational Journal of Control vol 75 no 16-17 pp 1335ndash13552002

[25] L Gao and Y Wu ldquoA LMI-based approach for robust stabi-lization of uncertain linear continuous time singular Markovswitched systems with time delaysrdquo in Proceedings of the 25thChinese Control Conference (CCC rsquo06) pp 844ndash849 IEEEHarbin China August 2006

[26] G Zong Y Tian and H Sun ldquoStabilization for a class ofpiecewise continuous-time switched control systemsrdquo in Pro-ceedings of the IEEE 7th World Congress on Intelligent Controland Automation (WCICA rsquo08) vol 1ndash23 pp 7226ndash7229 June2008


[27] G Feng ldquoObserver-based output feedback controller design ofpiecewise discrete-time linear systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 50 no 3 pp 448ndash451 2003

[28] H Lin and P J Antsaklis ldquoHybrid state feedback stabilizationwith 119897

2performance for disrete-time switched linear systemsrdquo


[29] J Xu L H Xie and G Feng ldquoFeedback control design fordiscrete-time piecewise affine systemsrdquo in Proceedings of theInternational Conference on Control and Automation (ICCArsquo05) vol 1 pp 425ndash430 IEEE June 2005

[30] T Zhang and G Feng ldquoOutput tracking of piecewise-linearsystems via error feedback regulator with application to syn-chronization of nonlinear Chuarsquos circuitrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 54 no 8 pp 1852ndash1863 2007

[31] Y L Wei M Wang and J B Qiu ldquoNew approach to delay-dependent 119867

infinfiltering for discrete-time Markovian jump

systems with time-varying delay and incomplete transitiondescriptionsrdquo IET Control Theory amp Applications vol 7 no 5pp 684ndash696 2013

[32] P Shi E-K Boukas and R K Agarwal ldquoKalman filtering forcontinuous-time uncertain systems with Markovian jumpingparametersrdquo IEEE Transactions on Automatic Control vol 44no 8 pp 1592ndash1597 1999

[33] C-H Lien J-D Chen C-T Lee R-S Chen and C-D YangldquoRobust 119867

infinfilter design for discrete-time switched systems

with interval time-varying delay and linear fractional pertur-bations LMI optimization approachrdquoApplied Mathematics andComputation vol 219 no 24 pp 11395ndash11407 2013

[34] L Y Chung C H Lien K W Yu and J D Chen ldquoRobust119867infin

filtering for discrete switched systems with interval time-varying delayrdquo Signal Processing vol 94 no 1 pp 661ndash669 2014

[35] G-H Yang and J L Wang ldquoRobust nonfragile Kalman filteringfor uncertain linear systems with estimator gain uncertaintyrdquoIEEE Transactions on Automatic Control vol 46 no 2 pp 343ndash348 2001

[36] G Song and Z Wang ldquoA delay partitioning approach to outputfeedback control for uncertain discrete time-delay systems withactuator saturationrdquo Nonlinear Dynamics vol 74 no 1-2 pp189ndash202 2013

[37] S Huang Z Xiang and H R Karimi ldquoRobust 1198972-gain control

for 2D nonlinear stochastic systems with time-varying delaysand actuator saturationrdquo Journal of the Franklin Institute vol350 no 7 pp 1865ndash1885 2013

[38] J Dong and G H Yang ldquoStatic output feedback controlsynthesis for linear systems with time-invariant parametricuncertaintiesrdquo IEEE Transactions on Automatic Control vol 52no 10 pp 1930ndash1936 2007

[39] B Zhou Z Lin and G-R Duan ldquoA parametric Lyapunovequation approach to low gain feedback design for discrete-timesystemsrdquo Automatica vol 45 no 1 pp 238ndash244 2009

[40] T Hu Z Lin and BM Chen ldquoAnalysis and design for discrete-time linear systems subject to actuator saturationrdquo Systems andControl Letters vol 45 no 2 pp 97ndash112 2002

[41] T Hu Z Lin and B M Chen ldquoAn analysis and designmethod for linear systems subject to actuator saturation anddisturbancerdquo Automatica vol 38 no 2 pp 351ndash359 2002

[42] J Qiu T Zhang G Feng and H Liu ldquoPiecewise affine model-based 119867

infinstatic output feedback control of constrained non-

linear processesrdquo IET Control Theory amp Applications vol 4 no11 pp 2315ndash2330 2010

[43] Y Gao Z Liu and H Chen ldquoRobust 119867infin

control for con-strained discrete-time piecewise affine systems with time-varying parametric uncertaintiesrdquo IET Control Theory amp Appli-cations vol 3 no 8 pp 1132ndash1144 2009

[44] Y Chen Q Zhou and S Fei ldquoRobust stabilization and 1198972-gain

control of uncertain discrete-time constrained piecewise-affinesystemsrdquoNonlinear Dynamics vol 75 no 1-2 pp 127ndash140 2014

[45] K L Hsiung and L Lee ldquoLyapunov inequality and bounded reallemma for discrete-time descriptor systemsrdquo IEE ProceedingsmdashControl Theory and Applications vol 146 no 4 pp 327ndash3311999

[46] J Qiu G Feng and H Gao ldquoApproaches to robust 119867infin

staticoutput feedback control of discrete-time piecewise-affine sys-tems with norm-bounded uncertaintiesrdquo International Journalof Robust and Nonlinear Control vol 21 no 7 pp 790ndash814 2011

[47] L Rodrigues and S Boyd ldquoPiecewise-affine state feedbackfor piecewise-affine slab systems using convex optimizationrdquoSystems amp Control Letters vol 54 no 9 pp 835ndash853 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


6

5

4

3

2

1

0

minus1

minus2

minus3

Time (step)

x1x2

x3

0 2 4 6 8 10 12 14 16 18 20


Stat

ey(k)


schemes have been investigated such as Kalman filtering H-infinity filtering and reduced-order H-infinity filtering [31ndash34] Then the authors in [35] paid more attention to theproblemof resilientKalmanfilteringwith respect to estimatorgain perturbations And the resilient H-infinity filtering wasalso raised

For output feedback control systems an actuator withamplitude and rate limitations may be considered in mostreal-world applications that often suffer from the state con-straints For controller synthesis ignoring these constraintsmay degrade the performance and may even cause the insta-bility of closed-loop system [36 37] (Figure 1) On the otherhand output feedback control can be easily implementedwith low cost which is particularly very useful and morerealistic [38] In the literature [39] the low gain feedbackdesigns were investigated for linear systems with all of itsopen loop poles in the closed left-half plane Based on anauxiliary feedback matrix the authors in [40 41] studied thestabilization and 119871

2-gain control of piecewise-linear systems

with actuator saturation Recently the authors in [42 43]investigated state feedback H-infinity control and outputfeedback H-infinity control for piecewise-affine systemsrespectively

In this paper the H-infinity output feedback controland resilient filter design problems of singular piecewise-affine systems with input saturation and state constraints areconsidered Based on a singular piecewise Lyapunov functioncombined with S-procedure and some matrix inequalityconvexifying techniques the H-infinity stabilization con-dition is established and the resilient H-infinity filteringerror dynamic system is investigated Under energy boundeddisturbance the input saturation disturbance tolerance con-dition is proposed then the resilient H-infinity filter isdesigned in some restricted region The results are given interms of solutions to a set of linear matrix inequalities Mean-while by using the corresponding optimizationmethods thedomain of attraction and the disturbance tolerance level ismaximized and the H-infinity performance 120574 is minimized

According to the existing results the main contributions ofthis paper can be summarized as follows (1) by adding aresilient block to output feedback controller the robust sta-bilization of resilient H-infinity output feedback closed-loopcontrol systems is investigated which was not consideredin [44] (Figure 3) (2) for singular piecewise-affine systemsthe analysis and maximization of the disturbance tolerancecapability are considered (3) by investigating resilient H-infinity filter the robust stabilization of resilient filtering errordynamic system is considered for the first time

The paper is organized as follows In Section 2 modeldescription and some preliminaries are given Sufficientconditions for designing robust H-infinity output feedbackcontrollers are proposed in Section 3 firstly then a resilientH-infinity filter is given by using the analysis and synthesismethods described previously the resulting filtering errordynamic system is admissible with the resilient H-infinityfilter Two numerical examples are presented to illustrate theeffectiveness of the proposed approaches in Section 4 whichis followed by some conclusions finally

Notation 1 The notations used throughout the paper arestandard R119899 denotes the 119899-dimensional Euclidean spacewhile R119898times119899 refers to the set of all real matrices with 119898

rows and 119899 columns A119879 represents the transpose of thematrixA whileAminus1 denotes the inverse matrix ofA l

2[0infin)

refers to the space of square summable infinite sequenceswith the Euclidean norm sdot I is the identity matrix withappropriate dimensions For real symmetric matrices X andY the notation X ge Y (resp X gt Y) means that the matrixX minus Y is positive semidefinite (resp positive definite) Thenotation lowast is used to indicate the terms that can be inducedby symmetry [1 ℎ] denotes the set of 1 2 ℎ in which theelements are integers

2 Model Description andProblem Formulation

Consider a discrete-time singular piecewise-affine systemwith norm-bounded uncertainties and input saturationdescribed by the following dynamics

Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896)

z (119896) = F119894x (119896) + G

119894sat (u (119896)) +D

1198942w (119896)

(1)

x (119896) y (119896) isin R119894 119894 isin 119868 x (0) = x

0 (2)

where x(119896) isin R119899119909 is the system state vector u(119896) isin R119899119906 is thesystem control input y(119896) isin R119899119910 is the system measurementoutput vector z(119896) isin R119899119911 is the controlled output vector andw(119896) isin R119899119908 is an energy bounded disturbance input whichbelongs to l

2[0infin) and satisfies sum

infin

119905=0w119879(119905)w(119905) le 120573 lt infin

A119894 B119894 C119894 D1198941 D1198942 F119894 G119894 b119894 and E are known real constant

matrices with appropriate dimensions denoting the 119894th localmodel of the system and Eb

119894is the offset termThe index set




119894and



[ΔA119894 EΔb119894] = W1198941Δ119894(119905) [E1198941 E

1198942] 119894 isin 119868 (3)






Δ119879

119894(119896) Δ119894(119896) le I

1199042

(4)



119894and b

119894 Never-


119894D1198941D1198942 F119894 and G

119894 119894 isin 119868



119898(119896))]119879 sat(u

119897) =

sgn(u119897)min|u






Ω = (119894 119895) | y (119896) isin R119894 y (119896 + 1) isin R

119895 119894 119895 isin 119868 (6)




R119894= y | 120572

119894le y le 120573

119894 y = C

119894x 119894 isin 119868 (7)


120576119894= x |

10038171003817100381710038171003817F119894x + 119891119894

10038171003817100381710038171003817le 1 119894 isin 119868 (8)


119894+ 120572119894)(120573119894minus 120572119894) Then we


[x(119896)1

]

119879

[F119879119894F119894

F119879119894119891119894

lowast 119891119879

119894119891119894minus 1

] [x (119896)1

] le 0 119894 isin 119868 (9)


0cup 1198681 where 119868


119891119879


1denotes the





119894)) =


119894) sub












119904u + w

119904^ 119904 isin


119904is an 119899

119906times







u (119896) = (K119894+ ΔK119894) y (119896) K

119894isin R119899119906times119899119910

y (119896) isin R119894 119894 isin I

(10)

where ΔK119894

= W1198941Δ119894(119905)E1198943 119894 isin 119868 and E



119897| lt


119894



sat (u (119896))

=

2119898

sum

119904=1

120588119894119904(119896) (W

119894119904K119894C119894+W119894119904ΔK119894C119894+ 2W

119894119904H119894) x (119896)

x (119896) isin R119894 119894 isin 119868

(11)

where 1205881198941(119870) ge 0 120588

1198942(119870) ge 0 120588

1198942119898(119870) ge 0 and

sum2119898

119904=1120588119894119904(119896) = 1


Ex (119896 + 1) = [A119894+ B119894W119894119904(119896)] x (119896)

+D1198941w (119896) + E (b

119894+ Δb119894)

z (119896) = [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

x (119896) isin R119894 119894 isin 119868

(12)

where A119894

= A119894+ ΔA

119894 W119894119904(119896) = sum

2119898

119904=1120588119894119904(119896)(W

119894119904K119894C119894+

W119894119904ΔK119894C119894+ 2W

119894119904H119894)x(119896)


x (119896 + 1) = A119891x (119896) + B

119891y (119896)

z (119896) = C119891x (119896) +D

119891y (119896) y (119896) isin R

119894 119894 isin 119868

u (119896) = (K119894+ ΔK119894) x (119896)

(13)

where A119891B119891C119891 andD



sat (u (119896)) =

2119898

sum

119904=1

120588119894119904(119896) (W

119894119904K119894+W119894119904ΔK119894+ 2W

119894119904H119894) x (119896)

x (119896) isin R119894 119894 isin 119868

(14)


Ex (119896 + 1) = Ax (119896) + Bw (119896) + b119894

z (119896) = Cx (119896) +D1198942w (119896)

(15)


119891] A =

[A119894+ΔA119894B119894W119894119904(119896)

B119891C119894

A119891

] E = [ E 00 I ] W119894119904(119896) = sum

2119898

119904=1120588119894119904(119896)(W

119894119904K119894+

W119894119904ΔK119894+ 2W119894119904H119894)x(119896) B = [

D1198941

0 ] and z(119896) = z(119896) minus z(119896)b119894=

[E(b119894+Δb119894)

0 ]

3 Main Results


119894isin


119894119895 12057610158401015840

119894119895 119894 isin 119868

120582119894119895 119894 isin 119868


E119879P119894E ge 0 (16)

[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]

(17)

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894


119894119895F119879119894119891119894119891119879

119894


119894119891119894minus 119868)

]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

(18)


[[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119897 isin [1 119898] (19)

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903] (20)

whereQ119894119904= Q119894119904Pminus1119894

= (W119894119904K119894C119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R




120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

Ξ = F119894+ G119894Q119894119904+ G119894W119894119904H119894 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

Λ = A119894+ B119894Q119894119904+ B119894W119894119904H119894 120578 = 120582

119894119895F119879119894119891119894119891119879

119894

120580 = 120582119894119895(119891119879

119894119891119894minus 119868) Φ = 120582

minus1

119894119895Eb119894(Eb119894)119879

120603 = E1198942(F119879119894119891119894)

119879

120579 = Eb119894(F119879119894119891119894)

119879

120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)

119879

Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

(21)


V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896) 119894 isin 119868 (22)


V (119896 + 1 x (119896 + 1)) minus V (119896 x (119896))


(23)


[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896) + E(b

119894+ Δb119894)119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896) + E (b

119894+ Δb119894)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896) minus w (119896)

119879w (119896) lt 0

(24)



[

[

w(119896)

x(119896)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[


119894E 0

lowast lowast 0]

]

times [

[

w (119896)

x (119896)1

]

]

lt 0

(25)

where A = A119894+ B119894W119894119904(119896) B = F

119894+ G119894W119894119904(119896)

From (25) we get

A119879P119895A minus E119879P

119894E + 120574minus2B119879B lt 0 (26)


A119879P119895A minus E119879P

119894E lt 0 (27)


^2lowastE119879P119894E^2 lt 0 (28)





1 (119894 119895) isin

Ω

[

[

w(119905)

x(119905)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[


119894E 0

lowast lowast 0]

]

+ 120582119894119895

[[

[

0 0 0lowast F119879119894F119894

F119879119894119891119894


119894119891119894minus 1

]]

]

times [

[

w (119905)

x (119905)1

]

]

lt 0 (119894 119895) isin Ω

(29)


[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+

[[[[[[[[

[

0 0 0 0 0lowast 0 0 ΔA

119894EΔb119894


]]]]]]]]

]


119898]

(30)



LHS (30) =

[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+ Sym

[[[[[[

[

0W1198941

000

]]]]]]

]

Δ119894(119905) [0 0 0 E

1198941E1198942]

(31)



119894and b


ΔK119894



119894W119894119904ΔK119894C119894by



119894119895gt 0 119894 isin 119868



[[[[[[[[

[

minus120576119894119895I1199042

0 0 0 E1198941

E1198942


1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


119895+ 120576119894119895W1198941W1198791198941

D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894


119894E + 120582119894119895F119879119894F119894

120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]

]

lt 0

(32)


119894119895 12057610158401015840

119894119895

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 E1198941

E1198942


119894119895120572120572119879 0 D

1198942Ξ 0


+ Ψ D1198941

Λ Eb119894


119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]]]]]

]

lt 0 (33)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 Ξ = F

119894+ G119894Q119894119904+

G119894W119894119904H119894 Λ = A

119894+ B119894Q119894119904+ B119894W119894119904H119894 Π = minusE119879P

119894E + 120582119894119895F119879119894F119894

and Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879


[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ D1198941

Λ


]]]]]]]]]]

]


119894119895)

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

(119891119879

119894119891119894minus 1)minus1

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

119879

lt 0

(34)


119894119891119894minus 1)minus1

= minus1 + 119891119879

119894times

(119891119894119891119894




[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ + Φ D1198941

Λ + 120579


]]]]]]]]]]]]]

]


119894119895)

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

(119891119879

119894119891119894minus 119868)minus1

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

119879

lt 0

(35)


119894119895Eb119894(Eb119894)119879 120603 = E

1198942(F119879119894119891119894)119879 120579 = Eb

119894(F119879119894119891119894)119879

and 120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)119879


[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894



119894


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]

]

lt 0 (36)

where 120601 = minus1205742I + 120576

1015840



[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896)

119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

minus w (119896)119879w (119896) lt 0 119894 isin 119868

0 (119894 119895) isin Ω

(37)



Consider

[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894



]]]]]]

]

lt 0

(38)



120576119894119895 1205761015840

119894119895 and 120576

10158401015840


Consider

[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]]]

]

lt 0

(39)



0 where 119868

0


119894119891119894minus 1 le 0 the



119894







le 1199062120573 119894 isin 119868 119897 isin [1 119898] It


le 1199062120573 119894 isin 119868 119897 isin [1 119898] are




119896 g119896) 119894 isin 119868

119896 isin [1 119903]



V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896)

=

119905minus1

sum

119896=0

ΔV (119896) le

119905minus1

sum

119896=0

w119879 (119905)w (119905) le 120573

x (119896) isin R119894 119894 isin 119868 119905 ge 1

(40)


R119894 x(119896) isin R


119894 120573) sub

120577(H119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub 120577(L

119896 g119896) 119894 isin 119868 119896 isin [1 119903]



119895 119895 isin 119868 It




119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub


119895 119906) and





119894isin119868(120576(P119894 120573) capR





hold



= H1198791

isin R119899119909times119899119909 0 lt H3

= H1198793

isin


119894= P119879119894


A119891isin R119899119909times119899119909 B

119891isin R119899119909times119899119909 C

119891isin R119899119909times119899119909 and D

119891isin R119899119909times119899119909 and


119894119895 12057610158401015840

119894119895 119894 isin 119868 120582

119894119895lt 0 119894 isin 119868

1 (119894 119895) isin Ω


E119879P119894E ge 0 (41)

[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941

0




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]


[[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942I1198791

120582minus1

119894119895E1198942I1198792

E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894I1198791

I Eb119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]

]

ge 0 119894 isin 119868 119897 isin [1 119898]

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903]

(42)

where Q119894119904= Q119894119904Pminus1119894

= (W119894119904K119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R



119894= Y119894P119894

Consider

weierp = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879+ 120582minus1

119894119895Eb119894(Eb119894)119879

I = B119894Q119894119904+ B119894W119894119904H119894+ 120582minus1

119894119895Eb119894119868119879

2


120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

120601 = minus1205742I + 1205761015840

119894119895120572120572119879 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879

120575 = 120582minus1

119894119895E1198942E1198791198942

120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [

1198681

1198682

]

minusE119879P119894E + 120582119894119895F119879119894F119894= [

1198691

1198692

lowast 1198693

]

(43)


119881 (119896 + 1 x (119896 + 1)) minus 119881 (119896 x (119896))


(44)



x (119896)119879A119879 + w (119896)119879 B119879 + b

119894

119879

times P119895Ax (119896) + Bw (119896) + b

119894


119879w (119896) + 120574minus2


119891]119879

+ w (119896)119879D1198791198942


119891] x (119896) +D

1198942w (119896) lt 0

(45)



[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894


119894E + 120574minus2C119879C A119879P

119895b119894


119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)




119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]


[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0


120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)


[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)


[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D


1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0



1198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)


= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]



[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894and



[ΔA119894 EΔb119894] = W1198941Δ119894(119905) [E1198941 E

1198942] 119894 isin 119868 (3)






Δ119879

119894(119896) Δ119894(119896) le I

1199042

(4)



119894and b

119894 Never-


119894D1198941D1198942 F119894 and G

119894 119894 isin 119868



119898(119896))]119879 sat(u

119897) =

sgn(u119897)min|u






Ω = (119894 119895) | y (119896) isin R119894 y (119896 + 1) isin R

119895 119894 119895 isin 119868 (6)




R119894= y | 120572

119894le y le 120573

119894 y = C

119894x 119894 isin 119868 (7)


120576119894= x |

10038171003817100381710038171003817F119894x + 119891119894

10038171003817100381710038171003817le 1 119894 isin 119868 (8)


119894+ 120572119894)(120573119894minus 120572119894) Then we


[x(119896)1

]

119879

[F119879119894F119894

F119879119894119891119894

lowast 119891119879

119894119891119894minus 1

] [x (119896)1

] le 0 119894 isin 119868 (9)


0cup 1198681 where 119868


119891119879


1denotes the





119894)) =


119894) sub












119904u + w

119904^ 119904 isin


119904is an 119899

119906times







u (119896) = (K119894+ ΔK119894) y (119896) K

119894isin R119899119906times119899119910

y (119896) isin R119894 119894 isin I

(10)

where ΔK119894

= W1198941Δ119894(119905)E1198943 119894 isin 119868 and E



119897| lt


119894



sat (u (119896))

=

2119898

sum

119904=1

120588119894119904(119896) (W

119894119904K119894C119894+W119894119904ΔK119894C119894+ 2W

119894119904H119894) x (119896)

x (119896) isin R119894 119894 isin 119868

(11)

where 1205881198941(119870) ge 0 120588

1198942(119870) ge 0 120588

1198942119898(119870) ge 0 and

sum2119898

119904=1120588119894119904(119896) = 1


Ex (119896 + 1) = [A119894+ B119894W119894119904(119896)] x (119896)

+D1198941w (119896) + E (b

119894+ Δb119894)

z (119896) = [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

x (119896) isin R119894 119894 isin 119868

(12)

where A119894

= A119894+ ΔA

119894 W119894119904(119896) = sum

2119898

119904=1120588119894119904(119896)(W

119894119904K119894C119894+

W119894119904ΔK119894C119894+ 2W

119894119904H119894)x(119896)


x (119896 + 1) = A119891x (119896) + B

119891y (119896)

z (119896) = C119891x (119896) +D

119891y (119896) y (119896) isin R

119894 119894 isin 119868

u (119896) = (K119894+ ΔK119894) x (119896)

(13)

where A119891B119891C119891 andD



sat (u (119896)) =

2119898

sum

119904=1

120588119894119904(119896) (W

119894119904K119894+W119894119904ΔK119894+ 2W

119894119904H119894) x (119896)

x (119896) isin R119894 119894 isin 119868

(14)


Ex (119896 + 1) = Ax (119896) + Bw (119896) + b119894

z (119896) = Cx (119896) +D1198942w (119896)

(15)


119891] A =

[A119894+ΔA119894B119894W119894119904(119896)

B119891C119894

A119891

] E = [ E 00 I ] W119894119904(119896) = sum

2119898

119904=1120588119894119904(119896)(W

119894119904K119894+

W119894119904ΔK119894+ 2W119894119904H119894)x(119896) B = [

D1198941

0 ] and z(119896) = z(119896) minus z(119896)b119894=

[E(b119894+Δb119894)

0 ]

3 Main Results


119894isin


119894119895 12057610158401015840

119894119895 119894 isin 119868

120582119894119895 119894 isin 119868


E119879P119894E ge 0 (16)

[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]

(17)

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894


119894119895F119879119894119891119894119891119879

119894


119894119891119894minus 119868)

]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

(18)


[[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119897 isin [1 119898] (19)

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903] (20)

whereQ119894119904= Q119894119904Pminus1119894

= (W119894119904K119894C119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R




120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

Ξ = F119894+ G119894Q119894119904+ G119894W119894119904H119894 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

Λ = A119894+ B119894Q119894119904+ B119894W119894119904H119894 120578 = 120582

119894119895F119879119894119891119894119891119879

119894

120580 = 120582119894119895(119891119879

119894119891119894minus 119868) Φ = 120582

minus1

119894119895Eb119894(Eb119894)119879

120603 = E1198942(F119879119894119891119894)

119879

120579 = Eb119894(F119879119894119891119894)

119879

120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)

119879

Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

(21)


V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896) 119894 isin 119868 (22)


V (119896 + 1 x (119896 + 1)) minus V (119896 x (119896))


(23)


[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896) + E(b

119894+ Δb119894)119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896) + E (b

119894+ Δb119894)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896) minus w (119896)

119879w (119896) lt 0

(24)



[

[

w(119896)

x(119896)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[


119894E 0

lowast lowast 0]

]

times [

[

w (119896)

x (119896)1

]

]

lt 0

(25)

where A = A119894+ B119894W119894119904(119896) B = F

119894+ G119894W119894119904(119896)

From (25) we get

A119879P119895A minus E119879P

119894E + 120574minus2B119879B lt 0 (26)


A119879P119895A minus E119879P

119894E lt 0 (27)


^2lowastE119879P119894E^2 lt 0 (28)





1 (119894 119895) isin

Ω

[

[

w(119905)

x(119905)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[


119894E 0

lowast lowast 0]

]

+ 120582119894119895

[[

[

0 0 0lowast F119879119894F119894

F119879119894119891119894


119894119891119894minus 1

]]

]

times [

[

w (119905)

x (119905)1

]

]

lt 0 (119894 119895) isin Ω

(29)


[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+

[[[[[[[[

[

0 0 0 0 0lowast 0 0 ΔA

119894EΔb119894


]]]]]]]]

]


119898]

(30)



LHS (30) =

[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+ Sym

[[[[[[

[

0W1198941

000

]]]]]]

]

Δ119894(119905) [0 0 0 E

1198941E1198942]

(31)



119894and b


ΔK119894



119894W119894119904ΔK119894C119894by



119894119895gt 0 119894 isin 119868



[[[[[[[[

[

minus120576119894119895I1199042

0 0 0 E1198941

E1198942


1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


119895+ 120576119894119895W1198941W1198791198941

D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894


119894E + 120582119894119895F119879119894F119894

120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]

]

lt 0

(32)


119894119895 12057610158401015840

119894119895

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 E1198941

E1198942


119894119895120572120572119879 0 D

1198942Ξ 0


+ Ψ D1198941

Λ Eb119894


119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]]]]]

]

lt 0 (33)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 Ξ = F

119894+ G119894Q119894119904+

G119894W119894119904H119894 Λ = A

119894+ B119894Q119894119904+ B119894W119894119904H119894 Π = minusE119879P

119894E + 120582119894119895F119879119894F119894

and Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879


[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ D1198941

Λ


]]]]]]]]]]

]


119894119895)

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

(119891119879

119894119891119894minus 1)minus1

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

119879

lt 0

(34)


119894119891119894minus 1)minus1

= minus1 + 119891119879

119894times

(119891119894119891119894




[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ + Φ D1198941

Λ + 120579


]]]]]]]]]]]]]

]


119894119895)

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

(119891119879

119894119891119894minus 119868)minus1

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

119879

lt 0

(35)


119894119895Eb119894(Eb119894)119879 120603 = E

1198942(F119879119894119891119894)119879 120579 = Eb

119894(F119879119894119891119894)119879

and 120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)119879


[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894



119894


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]

]

lt 0 (36)

where 120601 = minus1205742I + 120576

1015840



[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896)

119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

minus w (119896)119879w (119896) lt 0 119894 isin 119868

0 (119894 119895) isin Ω

(37)



Consider

[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894



]]]]]]

]

lt 0

(38)



120576119894119895 1205761015840

119894119895 and 120576

10158401015840


Consider

[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]]]

]

lt 0

(39)



0 where 119868

0


119894119891119894minus 1 le 0 the



119894







le 1199062120573 119894 isin 119868 119897 isin [1 119898] It


le 1199062120573 119894 isin 119868 119897 isin [1 119898] are




119896 g119896) 119894 isin 119868

119896 isin [1 119903]



V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896)

=

119905minus1

sum

119896=0

ΔV (119896) le

119905minus1

sum

119896=0

w119879 (119905)w (119905) le 120573

x (119896) isin R119894 119894 isin 119868 119905 ge 1

(40)


R119894 x(119896) isin R


119894 120573) sub

120577(H119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub 120577(L

119896 g119896) 119894 isin 119868 119896 isin [1 119903]



119895 119895 isin 119868 It




119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub


119895 119906) and





119894isin119868(120576(P119894 120573) capR





hold



= H1198791

isin R119899119909times119899119909 0 lt H3

= H1198793

isin


119894= P119879119894


A119891isin R119899119909times119899119909 B

119891isin R119899119909times119899119909 C

119891isin R119899119909times119899119909 and D

119891isin R119899119909times119899119909 and


119894119895 12057610158401015840

119894119895 119894 isin 119868 120582

119894119895lt 0 119894 isin 119868

1 (119894 119895) isin Ω


E119879P119894E ge 0 (41)

[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941

0




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]


[[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942I1198791

120582minus1

119894119895E1198942I1198792

E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894I1198791

I Eb119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]

]

ge 0 119894 isin 119868 119897 isin [1 119898]

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903]

(42)

where Q119894119904= Q119894119904Pminus1119894

= (W119894119904K119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R



119894= Y119894P119894

Consider

weierp = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879+ 120582minus1

119894119895Eb119894(Eb119894)119879

I = B119894Q119894119904+ B119894W119894119904H119894+ 120582minus1

119894119895Eb119894119868119879

2


120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

120601 = minus1205742I + 1205761015840

119894119895120572120572119879 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879

120575 = 120582minus1

119894119895E1198942E1198791198942

120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [

1198681

1198682

]

minusE119879P119894E + 120582119894119895F119879119894F119894= [

1198691

1198692

lowast 1198693

]

(43)


119881 (119896 + 1 x (119896 + 1)) minus 119881 (119896 x (119896))


(44)



x (119896)119879A119879 + w (119896)119879 B119879 + b

119894

119879

times P119895Ax (119896) + Bw (119896) + b

119894


119879w (119896) + 120574minus2


119891]119879

+ w (119896)119879D1198791198942


119891] x (119896) +D

1198942w (119896) lt 0

(45)



[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894


119894E + 120574minus2C119879C A119879P

119895b119894


119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)




119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]


[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0


120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)


[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)


[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D


1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0



1198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)


= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]



[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sat (u (119896))

=

2119898

sum

119904=1

120588119894119904(119896) (W

119894119904K119894C119894+W119894119904ΔK119894C119894+ 2W

119894119904H119894) x (119896)

x (119896) isin R119894 119894 isin 119868

(11)

where 1205881198941(119870) ge 0 120588

1198942(119870) ge 0 120588

1198942119898(119870) ge 0 and

sum2119898

119904=1120588119894119904(119896) = 1


Ex (119896 + 1) = [A119894+ B119894W119894119904(119896)] x (119896)

+D1198941w (119896) + E (b

119894+ Δb119894)

z (119896) = [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

x (119896) isin R119894 119894 isin 119868

(12)

where A119894

= A119894+ ΔA

119894 W119894119904(119896) = sum

2119898

119904=1120588119894119904(119896)(W

119894119904K119894C119894+

W119894119904ΔK119894C119894+ 2W

119894119904H119894)x(119896)


x (119896 + 1) = A119891x (119896) + B

119891y (119896)

z (119896) = C119891x (119896) +D

119891y (119896) y (119896) isin R

119894 119894 isin 119868

u (119896) = (K119894+ ΔK119894) x (119896)

(13)

where A119891B119891C119891 andD



sat (u (119896)) =

2119898

sum

119904=1

120588119894119904(119896) (W

119894119904K119894+W119894119904ΔK119894+ 2W

119894119904H119894) x (119896)

x (119896) isin R119894 119894 isin 119868

(14)


Ex (119896 + 1) = Ax (119896) + Bw (119896) + b119894

z (119896) = Cx (119896) +D1198942w (119896)

(15)


119891] A =

[A119894+ΔA119894B119894W119894119904(119896)

B119891C119894

A119891

] E = [ E 00 I ] W119894119904(119896) = sum

2119898

119904=1120588119894119904(119896)(W

119894119904K119894+

W119894119904ΔK119894+ 2W119894119904H119894)x(119896) B = [

D1198941

0 ] and z(119896) = z(119896) minus z(119896)b119894=

[E(b119894+Δb119894)

0 ]

3 Main Results


119894isin


119894119895 12057610158401015840

119894119895 119894 isin 119868

120582119894119895 119894 isin 119868


E119879P119894E ge 0 (16)

[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]

(17)

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894


119894119895F119879119894119891119894119891119879

119894


119894119891119894minus 119868)

]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

(18)


[[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119897 isin [1 119898] (19)

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903] (20)

whereQ119894119904= Q119894119904Pminus1119894

= (W119894119904K119894C119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R




120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

Ξ = F119894+ G119894Q119894119904+ G119894W119894119904H119894 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

Λ = A119894+ B119894Q119894119904+ B119894W119894119904H119894 120578 = 120582

119894119895F119879119894119891119894119891119879

119894

120580 = 120582119894119895(119891119879

119894119891119894minus 119868) Φ = 120582

minus1

119894119895Eb119894(Eb119894)119879

120603 = E1198942(F119879119894119891119894)

119879

120579 = Eb119894(F119879119894119891119894)

119879

120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)

119879

Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

(21)


V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896) 119894 isin 119868 (22)


V (119896 + 1 x (119896 + 1)) minus V (119896 x (119896))


(23)


[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896) + E(b

119894+ Δb119894)119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896) + E (b

119894+ Δb119894)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896) minus w (119896)

119879w (119896) lt 0

(24)



[

[

w(119896)

x(119896)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[


119894E 0

lowast lowast 0]

]

times [

[

w (119896)

x (119896)1

]

]

lt 0

(25)

where A = A119894+ B119894W119894119904(119896) B = F

119894+ G119894W119894119904(119896)

From (25) we get

A119879P119895A minus E119879P

119894E + 120574minus2B119879B lt 0 (26)


A119879P119895A minus E119879P

119894E lt 0 (27)


^2lowastE119879P119894E^2 lt 0 (28)





1 (119894 119895) isin

Ω

[

[

w(119905)

x(119905)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[


119894E 0

lowast lowast 0]

]

+ 120582119894119895

[[

[

0 0 0lowast F119879119894F119894

F119879119894119891119894


119894119891119894minus 1

]]

]

times [

[

w (119905)

x (119905)1

]

]

lt 0 (119894 119895) isin Ω

(29)


[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+

[[[[[[[[

[

0 0 0 0 0lowast 0 0 ΔA

119894EΔb119894


]]]]]]]]

]


119898]

(30)



LHS (30) =

[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+ Sym

[[[[[[

[

0W1198941

000

]]]]]]

]

Δ119894(119905) [0 0 0 E

1198941E1198942]

(31)



119894and b


ΔK119894



119894W119894119904ΔK119894C119894by



119894119895gt 0 119894 isin 119868



[[[[[[[[

[

minus120576119894119895I1199042

0 0 0 E1198941

E1198942


1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


119895+ 120576119894119895W1198941W1198791198941

D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894


119894E + 120582119894119895F119879119894F119894

120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]

]

lt 0

(32)


119894119895 12057610158401015840

119894119895

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 E1198941

E1198942


119894119895120572120572119879 0 D

1198942Ξ 0


+ Ψ D1198941

Λ Eb119894


119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]]]]]

]

lt 0 (33)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 Ξ = F

119894+ G119894Q119894119904+

G119894W119894119904H119894 Λ = A

119894+ B119894Q119894119904+ B119894W119894119904H119894 Π = minusE119879P

119894E + 120582119894119895F119879119894F119894

and Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879


[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ D1198941

Λ


]]]]]]]]]]

]


119894119895)

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

(119891119879

119894119891119894minus 1)minus1

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

119879

lt 0

(34)


119894119891119894minus 1)minus1

= minus1 + 119891119879

119894times

(119891119894119891119894




[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ + Φ D1198941

Λ + 120579


]]]]]]]]]]]]]

]


119894119895)

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

(119891119879

119894119891119894minus 119868)minus1

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

119879

lt 0

(35)


119894119895Eb119894(Eb119894)119879 120603 = E

1198942(F119879119894119891119894)119879 120579 = Eb

119894(F119879119894119891119894)119879

and 120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)119879


[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894



119894


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]

]

lt 0 (36)

where 120601 = minus1205742I + 120576

1015840



[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896)

119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

minus w (119896)119879w (119896) lt 0 119894 isin 119868

0 (119894 119895) isin Ω

(37)



Consider

[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894



]]]]]]

]

lt 0

(38)



120576119894119895 1205761015840

119894119895 and 120576

10158401015840


Consider

[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]]]

]

lt 0

(39)



0 where 119868

0


119894119891119894minus 1 le 0 the



119894







le 1199062120573 119894 isin 119868 119897 isin [1 119898] It


le 1199062120573 119894 isin 119868 119897 isin [1 119898] are




119896 g119896) 119894 isin 119868

119896 isin [1 119903]



V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896)

=

119905minus1

sum

119896=0

ΔV (119896) le

119905minus1

sum

119896=0

w119879 (119905)w (119905) le 120573

x (119896) isin R119894 119894 isin 119868 119905 ge 1

(40)


R119894 x(119896) isin R


119894 120573) sub

120577(H119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub 120577(L

119896 g119896) 119894 isin 119868 119896 isin [1 119903]



119895 119895 isin 119868 It




119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub


119895 119906) and





119894isin119868(120576(P119894 120573) capR





hold



= H1198791

isin R119899119909times119899119909 0 lt H3

= H1198793

isin


119894= P119879119894


A119891isin R119899119909times119899119909 B

119891isin R119899119909times119899119909 C

119891isin R119899119909times119899119909 and D

119891isin R119899119909times119899119909 and


119894119895 12057610158401015840

119894119895 119894 isin 119868 120582

119894119895lt 0 119894 isin 119868

1 (119894 119895) isin Ω


E119879P119894E ge 0 (41)

[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941

0




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]


[[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942I1198791

120582minus1

119894119895E1198942I1198792

E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894I1198791

I Eb119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]

]

ge 0 119894 isin 119868 119897 isin [1 119898]

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903]

(42)

where Q119894119904= Q119894119904Pminus1119894

= (W119894119904K119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R



119894= Y119894P119894

Consider

weierp = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879+ 120582minus1

119894119895Eb119894(Eb119894)119879

I = B119894Q119894119904+ B119894W119894119904H119894+ 120582minus1

119894119895Eb119894119868119879

2


120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

120601 = minus1205742I + 1205761015840

119894119895120572120572119879 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879

120575 = 120582minus1

119894119895E1198942E1198791198942

120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [

1198681

1198682

]

minusE119879P119894E + 120582119894119895F119879119894F119894= [

1198691

1198692

lowast 1198693

]

(43)


119881 (119896 + 1 x (119896 + 1)) minus 119881 (119896 x (119896))


(44)



x (119896)119879A119879 + w (119896)119879 B119879 + b

119894

119879

times P119895Ax (119896) + Bw (119896) + b

119894


119879w (119896) + 120574minus2


119891]119879

+ w (119896)119879D1198791198942


119891] x (119896) +D

1198942w (119896) lt 0

(45)



[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894


119894E + 120574minus2C119879C A119879P

119895b119894


119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)




119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]


[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0


120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)


[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)


[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D


1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0



1198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)


= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]



[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119897 isin [1 119898] (19)

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903] (20)

whereQ119894119904= Q119894119904Pminus1119894

= (W119894119904K119894C119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R




120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

Ξ = F119894+ G119894Q119894119904+ G119894W119894119904H119894 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

Λ = A119894+ B119894Q119894119904+ B119894W119894119904H119894 120578 = 120582

119894119895F119879119894119891119894119891119879

119894

120580 = 120582119894119895(119891119879

119894119891119894minus 119868) Φ = 120582

minus1

119894119895Eb119894(Eb119894)119879

120603 = E1198942(F119879119894119891119894)

119879

120579 = Eb119894(F119879119894119891119894)

119879

120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)

119879

Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

(21)


V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896) 119894 isin 119868 (22)


V (119896 + 1 x (119896 + 1)) minus V (119896 x (119896))


(23)


[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896) + E(b

119894+ Δb119894)119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896) + E (b

119894+ Δb119894)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896) minus w (119896)

119879w (119896) lt 0

(24)



[

[

w(119896)

x(119896)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[


119894E 0

lowast lowast 0]

]

times [

[

w (119896)

x (119896)1

]

]

lt 0

(25)

where A = A119894+ B119894W119894119904(119896) B = F

119894+ G119894W119894119904(119896)

From (25) we get

A119879P119895A minus E119879P

119894E + 120574minus2B119879B lt 0 (26)


A119879P119895A minus E119879P

119894E lt 0 (27)


^2lowastE119879P119894E^2 lt 0 (28)





1 (119894 119895) isin

Ω

[

[

w(119905)

x(119905)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[


119894E 0

lowast lowast 0]

]

+ 120582119894119895

[[

[

0 0 0lowast F119879119894F119894

F119879119894119891119894


119894119891119894minus 1

]]

]

times [

[

w (119905)

x (119905)1

]

]

lt 0 (119894 119895) isin Ω

(29)


[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+

[[[[[[[[

[

0 0 0 0 0lowast 0 0 ΔA

119894EΔb119894


]]]]]]]]

]


119898]

(30)



LHS (30) =

[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+ Sym

[[[[[[

[

0W1198941

000

]]]]]]

]

Δ119894(119905) [0 0 0 E

1198941E1198942]

(31)



119894and b


ΔK119894



119894W119894119904ΔK119894C119894by



119894119895gt 0 119894 isin 119868



[[[[[[[[

[

minus120576119894119895I1199042

0 0 0 E1198941

E1198942


1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


119895+ 120576119894119895W1198941W1198791198941

D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894


119894E + 120582119894119895F119879119894F119894

120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]

]

lt 0

(32)


119894119895 12057610158401015840

119894119895

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 E1198941

E1198942


119894119895120572120572119879 0 D

1198942Ξ 0


+ Ψ D1198941

Λ Eb119894


119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]]]]]

]

lt 0 (33)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 Ξ = F

119894+ G119894Q119894119904+

G119894W119894119904H119894 Λ = A

119894+ B119894Q119894119904+ B119894W119894119904H119894 Π = minusE119879P

119894E + 120582119894119895F119879119894F119894

and Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879


[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ D1198941

Λ


]]]]]]]]]]

]


119894119895)

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

(119891119879

119894119891119894minus 1)minus1

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

119879

lt 0

(34)


119894119891119894minus 1)minus1

= minus1 + 119891119879

119894times

(119891119894119891119894




[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ + Φ D1198941

Λ + 120579


]]]]]]]]]]]]]

]


119894119895)

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

(119891119879

119894119891119894minus 119868)minus1

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

119879

lt 0

(35)


119894119895Eb119894(Eb119894)119879 120603 = E

1198942(F119879119894119891119894)119879 120579 = Eb

119894(F119879119894119891119894)119879

and 120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)119879


[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894



119894


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]

]

lt 0 (36)

where 120601 = minus1205742I + 120576

1015840



[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896)

119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

minus w (119896)119879w (119896) lt 0 119894 isin 119868

0 (119894 119895) isin Ω

(37)



Consider

[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894



]]]]]]

]

lt 0

(38)



120576119894119895 1205761015840

119894119895 and 120576

10158401015840


Consider

[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]]]

]

lt 0

(39)



0 where 119868

0


119894119891119894minus 1 le 0 the



119894







le 1199062120573 119894 isin 119868 119897 isin [1 119898] It


le 1199062120573 119894 isin 119868 119897 isin [1 119898] are




119896 g119896) 119894 isin 119868

119896 isin [1 119903]



V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896)

=

119905minus1

sum

119896=0

ΔV (119896) le

119905minus1

sum

119896=0

w119879 (119905)w (119905) le 120573

x (119896) isin R119894 119894 isin 119868 119905 ge 1

(40)


R119894 x(119896) isin R


119894 120573) sub

120577(H119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub 120577(L

119896 g119896) 119894 isin 119868 119896 isin [1 119903]



119895 119895 isin 119868 It




119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub


119895 119906) and





119894isin119868(120576(P119894 120573) capR





hold



= H1198791

isin R119899119909times119899119909 0 lt H3

= H1198793

isin


119894= P119879119894


A119891isin R119899119909times119899119909 B

119891isin R119899119909times119899119909 C

119891isin R119899119909times119899119909 and D

119891isin R119899119909times119899119909 and


119894119895 12057610158401015840

119894119895 119894 isin 119868 120582

119894119895lt 0 119894 isin 119868

1 (119894 119895) isin Ω


E119879P119894E ge 0 (41)

[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941

0




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]


[[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942I1198791

120582minus1

119894119895E1198942I1198792

E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894I1198791

I Eb119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]

]

ge 0 119894 isin 119868 119897 isin [1 119898]

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903]

(42)

where Q119894119904= Q119894119904Pminus1119894

= (W119894119904K119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R



119894= Y119894P119894

Consider

weierp = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879+ 120582minus1

119894119895Eb119894(Eb119894)119879

I = B119894Q119894119904+ B119894W119894119904H119894+ 120582minus1

119894119895Eb119894119868119879

2


120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

120601 = minus1205742I + 1205761015840

119894119895120572120572119879 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879

120575 = 120582minus1

119894119895E1198942E1198791198942

120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [

1198681

1198682

]

minusE119879P119894E + 120582119894119895F119879119894F119894= [

1198691

1198692

lowast 1198693

]

(43)


119881 (119896 + 1 x (119896 + 1)) minus 119881 (119896 x (119896))


(44)



x (119896)119879A119879 + w (119896)119879 B119879 + b

119894

119879

times P119895Ax (119896) + Bw (119896) + b

119894


119879w (119896) + 120574minus2


119891]119879

+ w (119896)119879D1198791198942


119891] x (119896) +D

1198942w (119896) lt 0

(45)



[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894


119894E + 120574minus2C119879C A119879P

119895b119894


119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)




119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]


[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0


120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)


[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)


[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D


1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0



1198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)


= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]



[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 (119894 119895) isin

Ω

[

[

w(119905)

x(119905)1

]

]

119879

[[

[

D1198791198941

A119879

(E (b119894+ Δb119894))119879

]]

]

P119895(lowast)

+ 120574minus2 [

[

[

D1198791198942

B119879

0

]]

]

(lowast)

+ [

[


119894E 0

lowast lowast 0]

]

+ 120582119894119895

[[

[

0 0 0lowast F119879119894F119894

F119879119894119891119894


119894119891119894minus 1

]]

]

times [

[

w (119905)

x (119905)1

]

]

lt 0 (119894 119895) isin Ω

(29)


[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+

[[[[[[[[

[

0 0 0 0 0lowast 0 0 ΔA

119894EΔb119894


]]]]]]]]

]


119898]

(30)



LHS (30) =

[[[[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894

0


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

+ Sym

[[[[[[

[

0W1198941

000

]]]]]]

]

Δ119894(119905) [0 0 0 E

1198941E1198942]

(31)



119894and b


ΔK119894



119894W119894119904ΔK119894C119894by



119894119895gt 0 119894 isin 119868



[[[[[[[[

[

minus120576119894119895I1199042

0 0 0 E1198941

E1198942


1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


119895+ 120576119894119895W1198941W1198791198941

D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894


119894E + 120582119894119895F119879119894F119894

120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]

]

lt 0

(32)


119894119895 12057610158401015840

119894119895

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 E1198941

E1198942


119894119895120572120572119879 0 D

1198942Ξ 0


+ Ψ D1198941

Λ Eb119894


119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]]]]]

]

lt 0 (33)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 Ξ = F

119894+ G119894Q119894119904+

G119894W119894119904H119894 Λ = A

119894+ B119894Q119894119904+ B119894W119894119904H119894 Π = minusE119879P

119894E + 120582119894119895F119879119894F119894

and Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879


[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ D1198941

Λ


]]]]]]]]]]

]


119894119895)

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

(119891119879

119894119891119894minus 1)minus1

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

119879

lt 0

(34)


119894119891119894minus 1)minus1

= minus1 + 119891119879

119894times

(119891119894119891119894




[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ + Φ D1198941

Λ + 120579


]]]]]]]]]]]]]

]


119894119895)

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

(119891119879

119894119891119894minus 119868)minus1

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

119879

lt 0

(35)


119894119895Eb119894(Eb119894)119879 120603 = E

1198942(F119879119894119891119894)119879 120579 = Eb

119894(F119879119894119891119894)119879

and 120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)119879


[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894



119894


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]

]

lt 0 (36)

where 120601 = minus1205742I + 120576

1015840



[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896)

119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

minus w (119896)119879w (119896) lt 0 119894 isin 119868

0 (119894 119895) isin Ω

(37)



Consider

[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894



]]]]]]

]

lt 0

(38)



120576119894119895 1205761015840

119894119895 and 120576

10158401015840


Consider

[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]]]

]

lt 0

(39)



0 where 119868

0


119894119891119894minus 1 le 0 the



119894







le 1199062120573 119894 isin 119868 119897 isin [1 119898] It


le 1199062120573 119894 isin 119868 119897 isin [1 119898] are




119896 g119896) 119894 isin 119868

119896 isin [1 119903]



V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896)

=

119905minus1

sum

119896=0

ΔV (119896) le

119905minus1

sum

119896=0

w119879 (119905)w (119905) le 120573

x (119896) isin R119894 119894 isin 119868 119905 ge 1

(40)


R119894 x(119896) isin R


119894 120573) sub

120577(H119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub 120577(L

119896 g119896) 119894 isin 119868 119896 isin [1 119903]



119895 119895 isin 119868 It




119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub


119895 119906) and





119894isin119868(120576(P119894 120573) capR





hold



= H1198791

isin R119899119909times119899119909 0 lt H3

= H1198793

isin


119894= P119879119894


A119891isin R119899119909times119899119909 B

119891isin R119899119909times119899119909 C

119891isin R119899119909times119899119909 and D

119891isin R119899119909times119899119909 and


119894119895 12057610158401015840

119894119895 119894 isin 119868 120582

119894119895lt 0 119894 isin 119868

1 (119894 119895) isin Ω


E119879P119894E ge 0 (41)

[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941

0




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]


[[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942I1198791

120582minus1

119894119895E1198942I1198792

E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894I1198791

I Eb119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]

]

ge 0 119894 isin 119868 119897 isin [1 119898]

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903]

(42)

where Q119894119904= Q119894119904Pminus1119894

= (W119894119904K119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R



119894= Y119894P119894

Consider

weierp = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879+ 120582minus1

119894119895Eb119894(Eb119894)119879

I = B119894Q119894119904+ B119894W119894119904H119894+ 120582minus1

119894119895Eb119894119868119879

2


120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

120601 = minus1205742I + 1205761015840

119894119895120572120572119879 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879

120575 = 120582minus1

119894119895E1198942E1198791198942

120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [

1198681

1198682

]

minusE119879P119894E + 120582119894119895F119879119894F119894= [

1198691

1198692

lowast 1198693

]

(43)


119881 (119896 + 1 x (119896 + 1)) minus 119881 (119896 x (119896))


(44)



x (119896)119879A119879 + w (119896)119879 B119879 + b

119894

119879

times P119895Ax (119896) + Bw (119896) + b

119894


119879w (119896) + 120574minus2


119891]119879

+ w (119896)119879D1198791198942


119891] x (119896) +D

1198942w (119896) lt 0

(45)



[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894


119894E + 120574minus2C119879C A119879P

119895b119894


119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)




119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]


[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0


120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)


[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)


[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D


1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0



1198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)


= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]



[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894and b


ΔK119894



119894W119894119904ΔK119894C119894by



119894119895gt 0 119894 isin 119868



[[[[[[[[

[

minus120576119894119895I1199042

0 0 0 E1198941

E1198942


1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


119895+ 120576119894119895W1198941W1198791198941

D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894

Eb119894


119894E + 120582119894119895F119879119894F119894

120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]

]

lt 0

(32)


119894119895 12057610158401015840

119894119895

[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 E1198941

E1198942


119894119895120572120572119879 0 D

1198942Ξ 0


+ Ψ D1198941

Λ Eb119894


119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]]]]]

]

lt 0 (33)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 Ξ = F

119894+ G119894Q119894119904+

G119894W119894119904H119894 Λ = A

119894+ B119894Q119894119904+ B119894W119894119904H119894 Π = minusE119879P

119894E + 120582119894119895F119879119894F119894

and Ψ = 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879


[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ D1198941

Λ


]]]]]]]]]]

]


119894119895)

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

(119891119879

119894119891119894minus 1)minus1

[[[[[[[[[

[

00E1198942

0Eb119894

0120582119894119895F119879119894119891119894

]]]]]]]]]

]

119879

lt 0

(34)


119894119891119894minus 1)minus1

= minus1 + 119891119879

119894times

(119891119894119891119894




[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ + Φ D1198941

Λ + 120579


]]]]]]]]]]]]]

]


119894119895)

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

(119891119879

119894119891119894minus 119868)minus1

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

119879

lt 0

(35)


119894119895Eb119894(Eb119894)119879 120603 = E

1198942(F119879119894119891119894)119879 120579 = Eb

119894(F119879119894119891119894)119879

and 120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)119879


[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894



119894


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]

]

lt 0 (36)

where 120601 = minus1205742I + 120576

1015840



[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896)

119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

minus w (119896)119879w (119896) lt 0 119894 isin 119868

0 (119894 119895) isin Ω

(37)



Consider

[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894



]]]]]]

]

lt 0

(38)



120576119894119895 1205761015840

119894119895 and 120576

10158401015840


Consider

[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]]]

]

lt 0

(39)



0 where 119868

0


119894119891119894minus 1 le 0 the



119894







le 1199062120573 119894 isin 119868 119897 isin [1 119898] It


le 1199062120573 119894 isin 119868 119897 isin [1 119898] are




119896 g119896) 119894 isin 119868

119896 isin [1 119903]



V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896)

=

119905minus1

sum

119896=0

ΔV (119896) le

119905minus1

sum

119896=0

w119879 (119905)w (119905) le 120573

x (119896) isin R119894 119894 isin 119868 119905 ge 1

(40)


R119894 x(119896) isin R


119894 120573) sub

120577(H119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub 120577(L

119896 g119896) 119894 isin 119868 119896 isin [1 119903]



119895 119895 isin 119868 It




119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub


119895 119906) and





119894isin119868(120576(P119894 120573) capR





hold



= H1198791

isin R119899119909times119899119909 0 lt H3

= H1198793

isin


119894= P119879119894


A119891isin R119899119909times119899119909 B

119891isin R119899119909times119899119909 C

119891isin R119899119909times119899119909 and D

119891isin R119899119909times119899119909 and


119894119895 12057610158401015840

119894119895 119894 isin 119868 120582

119894119895lt 0 119894 isin 119868

1 (119894 119895) isin Ω


E119879P119894E ge 0 (41)

[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941

0




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]


[[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942I1198791

120582minus1

119894119895E1198942I1198792

E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894I1198791

I Eb119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]

]

ge 0 119894 isin 119868 119897 isin [1 119898]

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903]

(42)

where Q119894119904= Q119894119904Pminus1119894

= (W119894119904K119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R



119894= Y119894P119894

Consider

weierp = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879+ 120582minus1

119894119895Eb119894(Eb119894)119879

I = B119894Q119894119904+ B119894W119894119904H119894+ 120582minus1

119894119895Eb119894119868119879

2


120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

120601 = minus1205742I + 1205761015840

119894119895120572120572119879 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879

120575 = 120582minus1

119894119895E1198942E1198791198942

120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [

1198681

1198682

]

minusE119879P119894E + 120582119894119895F119879119894F119894= [

1198691

1198692

lowast 1198693

]

(43)


119881 (119896 + 1 x (119896 + 1)) minus 119881 (119896 x (119896))


(44)



x (119896)119879A119879 + w (119896)119879 B119879 + b

119894

119879

times P119895Ax (119896) + Bw (119896) + b

119894


119879w (119896) + 120574minus2


119891]119879

+ w (119896)119879D1198791198942


119891] x (119896) +D

1198942w (119896) lt 0

(45)



[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894


119894E + 120574minus2C119879C A119879P

119895b119894


119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)




119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]


[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0


120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)


[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)


[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D


1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0



1198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)


= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]



[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603


119894119895120572120572119879 0 D

1198942Ξ


+ Ψ + Φ D1198941

Λ + 120579


]]]]]]]]]]]]]

]


119894119895)

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

(119891119879

119894119891119894minus 119868)minus1

[[[[[[[[[[[[[

[

00

E1198942119891119879

119894

0Eb119894119891119879

119894

0

120582119894119895F119879119894119891119894119891119879

119894

]]]]]]]]]]]]]

]

119879

lt 0

(35)


119894119895Eb119894(Eb119894)119879 120603 = E

1198942(F119879119894119891119894)119879 120579 = Eb

119894(F119879119894119891119894)119879

and 120599 = 120582119894119895F119879119894119891119894(F119879119894119891119894)119879


[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 E1198943C119894


119894119895I1199042

+ 120582minus1

119894119895E1198942E1198791198942

0 120582minus1

119894119895E1198942(Eb119894)119879 0 E

1198941+ 120603 E

1198942119891119879

119894



119895+ Ψ + Φ D

1198941Λ + 120579 Eb

119894119891119879

119894



119894


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]

]

lt 0 (36)

where 120601 = minus1205742I + 120576

1015840



[A119894+ B119894W119894119904(119896)]x(119896) +D

1198941w(119896)

119879

times P119895[A119894+ B119894W119894119904(119896)] x (119896) +D

1198941w (119896)

minus x (119896)119879 E119879P119894Ex (119896)

+ 120574minus2

[F119894+ G119894W119894119904(119896)]x(119896) +D

1198942w(119896)

119879

times [F119894+ G119894W119894119904(119896)] x (119896) +D

1198942w (119896)

minus w (119896)119879w (119896) lt 0 119894 isin 119868

0 (119894 119895) isin Ω

(37)



Consider

[[[[[[

[

minus1205742I 0 D

1198942F119894+ G119894Q119894119904+ G119894W119894119904H119894+ G119894W119894119904ΔK119894C119894


D1198941

A119894+ B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894C119894



]]]]]]

]

lt 0

(38)



120576119894119895 1205761015840

119894119895 and 120576

10158401015840


Consider

[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]]]

]

lt 0

(39)



0 where 119868

0


119894119891119894minus 1 le 0 the



119894







le 1199062120573 119894 isin 119868 119897 isin [1 119898] It


le 1199062120573 119894 isin 119868 119897 isin [1 119898] are




119896 g119896) 119894 isin 119868

119896 isin [1 119903]



V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896)

=

119905minus1

sum

119896=0

ΔV (119896) le

119905minus1

sum

119896=0

w119879 (119905)w (119905) le 120573

x (119896) isin R119894 119894 isin 119868 119905 ge 1

(40)


R119894 x(119896) isin R


119894 120573) sub

120577(H119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub 120577(L

119896 g119896) 119894 isin 119868 119896 isin [1 119903]



119895 119895 isin 119868 It




119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub


119895 119906) and





119894isin119868(120576(P119894 120573) capR





hold



= H1198791

isin R119899119909times119899119909 0 lt H3

= H1198793

isin


119894= P119879119894


A119891isin R119899119909times119899119909 B

119891isin R119899119909times119899119909 C

119891isin R119899119909times119899119909 and D

119891isin R119899119909times119899119909 and


119894119895 12057610158401015840

119894119895 119894 isin 119868 120582

119894119895lt 0 119894 isin 119868

1 (119894 119895) isin Ω


E119879P119894E ge 0 (41)

[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941

0




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]


[[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942I1198791

120582minus1

119894119895E1198942I1198792

E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894I1198791

I Eb119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]

]

ge 0 119894 isin 119868 119897 isin [1 119898]

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903]

(42)

where Q119894119904= Q119894119904Pminus1119894

= (W119894119904K119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R



119894= Y119894P119894

Consider

weierp = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879+ 120582minus1

119894119895Eb119894(Eb119894)119879

I = B119894Q119894119904+ B119894W119894119904H119894+ 120582minus1

119894119895Eb119894119868119879

2


120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

120601 = minus1205742I + 1205761015840

119894119895120572120572119879 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879

120575 = 120582minus1

119894119895E1198942E1198791198942

120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [

1198681

1198682

]

minusE119879P119894E + 120582119894119895F119879119894F119894= [

1198691

1198692

lowast 1198693

]

(43)


119881 (119896 + 1 x (119896 + 1)) minus 119881 (119896 x (119896))


(44)



x (119896)119879A119879 + w (119896)119879 B119879 + b

119894

119879

times P119895Ax (119896) + Bw (119896) + b

119894


119879w (119896) + 120574minus2


119891]119879

+ w (119896)119879D1198791198942


119891] x (119896) +D

1198942w (119896) lt 0

(45)



[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894


119894E + 120574minus2C119879C A119879P

119895b119894


119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)




119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]


[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0


120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)


[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)


[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D


1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0



1198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)


= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]



[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120576119894119895 1205761015840

119894119895 and 120576

10158401015840


Consider

[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 E1198943C119894


119894119895I1199042

0 0 0 0 E1198943C119894


0 0 0 E1198941


Ξ


+ Ψ D1198941

Λ


119894E

]]]]]]]]]]]]

]

lt 0

(39)



0 where 119868

0


119894119891119894minus 1 le 0 the



119894







le 1199062120573 119894 isin 119868 119897 isin [1 119898] It


le 1199062120573 119894 isin 119868 119897 isin [1 119898] are




119896 g119896) 119894 isin 119868

119896 isin [1 119903]



V (119896 x (119896)) = x119879 (119896)E119879P119894Ex (119896)

=

119905minus1

sum

119896=0

ΔV (119896) le

119905minus1

sum

119896=0

w119879 (119905)w (119905) le 120573

x (119896) isin R119894 119894 isin 119868 119905 ge 1

(40)


R119894 x(119896) isin R


119894 120573) sub

120577(H119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub 120577(L

119896 g119896) 119894 isin 119868 119896 isin [1 119903]



119895 119895 isin 119868 It




119894 119906) 119894 isin 119868 and 120576(P

119894 120573) sub


119895 119906) and





119894isin119868(120576(P119894 120573) capR





hold



= H1198791

isin R119899119909times119899119909 0 lt H3

= H1198793

isin


119894= P119879119894


A119891isin R119899119909times119899119909 B

119891isin R119899119909times119899119909 C

119891isin R119899119909times119899119909 and D

119891isin R119899119909times119899119909 and


119894119895 12057610158401015840

119894119895 119894 isin 119868 120582

119894119895lt 0 119894 isin 119868

1 (119894 119895) isin Ω


E119879P119894E ge 0 (41)

[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941

0




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198680 (119894 119895) isin Ω 119904 isin [1 2

119898]


[[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942I1198791

120582minus1

119894119895E1198942I1198792

E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894I1198791

I Eb119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]

]

ge 0 119894 isin 119868 119897 isin [1 119898]

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903]

(42)

where Q119894119904= Q119894119904Pminus1119894

= (W119894119904K119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R



119894= Y119894P119894

Consider

weierp = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879+ 120582minus1

119894119895Eb119894(Eb119894)119879

I = B119894Q119894119904+ B119894W119894119904H119894+ 120582minus1

119894119895Eb119894119868119879

2


120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

120601 = minus1205742I + 1205761015840

119894119895120572120572119879 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879

120575 = 120582minus1

119894119895E1198942E1198791198942

120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [

1198681

1198682

]

minusE119879P119894E + 120582119894119895F119879119894F119894= [

1198691

1198692

lowast 1198693

]

(43)


119881 (119896 + 1 x (119896 + 1)) minus 119881 (119896 x (119896))


(44)



x (119896)119879A119879 + w (119896)119879 B119879 + b

119894

119879

times P119895Ax (119896) + Bw (119896) + b

119894


119879w (119896) + 120574minus2


119891]119879

+ w (119896)119879D1198791198942


119891] x (119896) +D

1198942w (119896) lt 0

(45)



[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894


119894E + 120574minus2C119879C A119879P

119895b119894


119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)




119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]


[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0


120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)


[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)


[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D


1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0



1198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)


= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]



[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[[[[[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942I1198791

120582minus1

119894119895E1198942I1198792

E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894I1198791

I Eb119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]]]]]

]

lt 0

119894 isin 1198681 (119894 119895) isin Ω 119904 isin [1 2

119898]

[

[

1 Z119894119897

Z119879119894119897

(1199062

120573)Pminus1119894

]

]

ge 0 119894 isin 119868 119897 isin [1 119898]

[[

[

1 L119896Pminus1119894

Pminus1119894L119879119896

(g119896

2

120573)Pminus1119894

]]

]

ge 0 119894 isin 119868 119896 isin [1 119903]

(42)

where Q119894119904= Q119894119904Pminus1119894

= (W119894119904K119894+W119894119904H119894)Pminus1119894

= W119894119904Y119894+W119894119904Z119894





119894isin119868(120576(P119894 120573) cap R



119894= Y119894P119894

Consider

weierp = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879+ 120582minus1

119894119895Eb119894(Eb119894)119879

I = B119894Q119894119904+ B119894W119894119904H119894+ 120582minus1

119894119895Eb119894119868119879

2


120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941

120601 = minus1205742I + 1205761015840

119894119895120572120572119879 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879

120575 = 120582minus1

119894119895E1198942E1198791198942

120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879

minusPminus1119895

= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [

1198681

1198682

]

minusE119879P119894E + 120582119894119895F119879119894F119894= [

1198691

1198692

lowast 1198693

]

(43)


119881 (119896 + 1 x (119896 + 1)) minus 119881 (119896 x (119896))


(44)



x (119896)119879A119879 + w (119896)119879 B119879 + b

119894

119879

times P119895Ax (119896) + Bw (119896) + b

119894


119879w (119896) + 120574minus2


119891]119879

+ w (119896)119879D1198791198942


119891] x (119896) +D

1198942w (119896) lt 0

(45)



[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894


119894E + 120574minus2C119879C A119879P

119895b119894


119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)




119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]


[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0


120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)


[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)


[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D


1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0



1198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)


= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]



[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879P119895B minus 119868 + 120574

minus2D1198791198942D1198942

B119879P119895A + 120574minus2D1198791198942C B119879P

119895b119894


119894E + 120574minus2C119879C A119879P

119895b119894


119879

P119895b119894

]]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0

(46)

From (46) we get

A119879P119895A minus E119879P

119894E + 120574minus2C119879C lt 0 (47)




119894119895lt 0 119894 isin 119868

1 (119894 119895) isin

Ω

[[

[

w(119896)

x(119896)1

]]

]

119879

times

[[[[

[

B119879

A119879

b119894

119879

]]]]

]


[[[

[

D1198791198942

C119879

0

]]]

]

(lowast)

+[[[

[

minusI 0 0


120582119894119895F119879119894119891119894

lowast lowast 120582119894119895(119891119879

119894119891119894minus 1)

]]]

]

times[[

[

w (119896)

x (119896)1

]]

]

lt 0 (119894 119895) isin Ω

(48)


[[[[[[[[[

[

minus1205742I 0 D

1198942C 0


B A b119894



120582119894119895F119879119894119891119894


119894119891119894minus 1)

]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(49)


[[[[[[[[[[[[[[[[[

[

minus1205742I 0 0 D


1198910

lowast H1

H2

D1198941

A119894+ ΔA119894

B119894W119894119904(119896) E (b

119894+ Δb119894)

lowast lowast H3

0 B119891C119894

A119891

0



1198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(50)


= [H1H2

lowast H3

] 120582119894119895F119879119894119891119894= [1198681

1198682

] minusE119879P119894E + 120582

119894119895F119879119894F119894=

[11986911198692

lowast 1198693

]



[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[[[[[[[[[[

[

minus1205742I 0 0 D


0lowast H

1H2D1198941

A119894

B119894Q119894119904+ B119894W119894119904H119894+ B119894W119894119904ΔK119894

Eb119894

lowast lowast H3

0 B119891C119894

A119891


11198692

1198681


1198682


119894119891119894minus 1)

]]]]]]]]]]

]

+ sym

[[[[[[[[[

[

0W1198941

00000

]]]]]]]]]

]

[0 0 0 0 E1198941

0 E1198942]

lt 0 (119894 119895) isin Ω

(51)


119894119895gt 0

1205761015840

119894119895gt 0 and 120576

10158401015840

119894119895gt 0 119894 isin 119868



[[[[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943

0lowast minus120576

1015840

119894119895I1199042

0 0 0 0 0 0 E1198943


119894119895I1199042

+ 120575 0 120594 0 0 E1198941+ 120582minus1

119894119895E1198942119868119879

1120582minus1

119894119895E1198942119868119879

2E1198942


F119894minusD119891C119894


2D1198941

A119894+ 120582minus1

119894119895Eb119894119868119879

1I Eb

119894


0 B119891C119894

A119891


1+ 120582minus1

1198941198951198681119868119879

11198692+ 120582minus1

1198941198951198681119868119879

21198681


1198941198951198682119868119879

21198682


119894119891119894minus 119868)

]]]]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω

(52)

where 120572 = G119894W119894119904W1198941 120593 = B

119894W119894119904W1198941 weierp = H

1+ 120576119894119895W1198941W1198791198941+

12057610158401015840

119894119895120593120593119879+120582minus1

119894119895Eb119894(Eb119894)119879I = B

119894Q119894119904+B119894W119894119904H119894+120582minus1

119894119895Eb119894119868119879

2alefsym =

G119894Q119894119904+ G119894W119894119904H119894minus C119891 120594 = 120582

minus1

119894119895E1198942(Eb119894)119879 120601 = minus120574

2I + 1205761015840

119894119895120572120572119879

and 120575 = 120582minus1

119894119895E1198942E1198791198942



[[[[[[[[[[[[[[

[

minus12057610158401015840

119894119895I1199042

0 0 0 0 0 0 0 E1198943


119894119895I1199042

0 0 0 0 0 0 E1198943


0 0 0 0 E1198941




A119894

B119894Q119894119904+ B119894W119894119904H119894


0 B119891C119894

A119891


11198692


]]]]]]]]]]]]]]

]

lt 0 (119894 119895) isin Ω (53)

where 120590 = H1+ 120576119894119895W1198941W1198791198941+ 12057610158401015840

119894119895120593120593119879



119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894isin119868(120576(P119894 120573) cap R


the proof






A1= [

[

01432 02312 04232

02123 06765 03454

10001 20001 02281

]

]

A2= [

[

02343 03343 01565

10000 05232 06456

12131 05456 08564

]

]

B1= [

[

minus03

02

10

]

]

B2= [

[

01

02

05

]

]

G1= [

[

minus05

04

10

]

]

G2= [

[

minus01

02

10

]

]

D11

= [

[

11 23 10

13 10 29

20 02 04

]

]

D12

= [

[

01 13 30

12 22 03

05 41 43

]

]

D21

= [

[

10 38 11

30 19 23

10 02 04

]

]

D22

= [

[

17 01 12

13 30 20

20 02 04

]

]

C1= [

[

1 2 1

2 1 2

1 4 3

]

]

C2= [

[

2 1 3

3 1 2

4 2 1

]

]

E = [

[

1 1 1

0 0 0

0 0 0

]

]

b1= [

[

005

005

100

]

]

b2= [

[

100

minus100

050

]

]

W11

= [

[

000

003

100

]

]

W21

= [

[

000

002

200

]

]

E11

= [

[

000

002

100

]

]

119879

E12

= 003 E21

= [

[

0

001

3

]

]

119879

E22

= 004 E13

= [

[

040

002

100

]

]

119879

E23

= [

[

050

002

100

]

]

119879

(54)



R1= x (119905) | 5 le [1 0 0] x (119905) le 30


(55)


F1= 2 times

C1198791

(1205731minus 1205721) 119891

1= minus

1205731+ 1205721

1205731minus 1205721

F2= 2 times

C1198792

(1205732minus 1205722) 119891

2= minus

1205732+ 1205722

1205732minus 1205722

(56)

where 1205721= 3 120573

1= 10 120572

2= 4 and 120573

2= 9





K2= [189530 151858 101835]

P1= [

[

30269 minus15503 25663

minus15503 48114 minus71203

25663 minus71203 08454

]

]

P2= [

[

50239 minus45333 105123

minus45333 54514 minus11113

105123 minus11113 78674

]

]

(57)




1198701= [52373 60612 15916]


(58)


25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus25


x 2

x1


6

5

4

3

2

1

0

minus1

minus20 2 4 6 8 10 12 14 16 18 20

x1x2

x3

Time (step)


Stat

ey(k)



119860119891= [

[

77703 minus80816 00444

minus66017 1688 05420

70078 minus10010 16448

]

]

119861119891= [

[

32033 70812 minus04566

00821 50721 minus71028

53240 minus41035 52551

]

]

119862119891= [

[

71427 29229 08533

minus13315 143578 minus46118

minus84661 43829 12413

]

]

119863119891= [

[

minus00336 minus00403 00559

80976 40324 minus08279

minus01886 62133 82879

]

]

(59)







Ex (119896 + 1) = (A119894+ ΔA119894) x (119896) + B

119894sat (u (119896))

+D1198941w (119896) + E (b

119894+ Δb119894)

y (119896) = C119894x (119896) 119894 = 1 2

(60)



119872


K1= [minus142233 03412 115316]

K2= [309430 minus62848 minus452235]

(61)


A119891= [

[

145503 24416 35644

minus46417 214488 45420

90898 minus77010 66468

]

]

B119891= [

[

23233 50662 minus74666

00121 10121 minus231228

13240 341335 252651

]

]

C119891= [

[

11421 minus19229 38233

minus63615 133528 minus126138

minus44561 13869 212113

]

]

D119891= [

[

minus70666 20223 10259


21836 52533 52859

]

]

(62)

5 Conclusions







Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Acknowledgment


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article robust h-infinity stabilization and...

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