Applied Mathematics Letters 30 (2014) 12–18

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Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

An explicit criterion for finite-time stability of linearnonautonomous systems with delaysLe Van Hien ∗

Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:Received 24 September 2013Received in revised form 13 December 2013Accepted 13 December 2013

Keywords:Finite-time stabilityNonautonomous systemsTime-varying delayMetzler matrix

a b s t r a c t

In this paper, the problem of finite-time stability of linear nonautonomous systems withtime-varying delays is considered. Using a novel approach based on some techniques de-veloped for linear positive systems, we derive new explicit conditions in terms of matrixinequalities ensuring that the state trajectories of the system do not exceed a certainthreshold over a pre-specified finite time interval. These conditions are shown to be re-laxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate theeffectiveness of the obtained result.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The stability of time delay systems has been one of the most attractive research topics during the past decades [1–5].While the concept of Lyapunov stability, recognized as infinite time behavior, has beenwell investigated and developed, theconcept of finite-time stability (FTS) (or short-time stability) has been extensively studied in recent years (see, [6–14] andthe references therein).

Roughly speaking, a system is finite-time stable if, for a given bound on the initial condition, its state trajectories do notexceed a certain threshold during a pre-specified time interval [7]. It is noted that, a systemmay be finite-time stable but notLyapunov asymptotic stable, and vice versa [6,7,13] (see, also, Remark 2.1 in this paper). Although the Lyapunov asymptoticstability (LAS) has been successfully applied in many models, FTS is a useful concept to study in many practical systems inthe vivid world [8–10,12,13].

However, most of the existing results in the literature so far have been devoted to linear autonomous (i.e. time-invariant)systems. For linear time-invariant systems with constant delay, some finite-time stability conditions have been derivedin terms of feasible linear matrix inequalities based on the main approach is the Lyapunov–Krasovskii functional method[6,10–14]. There has been no result concerned with the FTS of nonautonomous systems (time-varying systems) with time-varying delays. Moreover, it should be noted that, the proposed conditions for FTS of time-varying systems based on theLyapunov functional approach have usually been derived in terms of Lyapunov or Riccati matrix differential equations [7–9]which lead to indefinitematrix inequalitieswith lack of efficient computational tools to solve them. Therefore, an alternativeapproachwhen dealingwith the FTS of time-varying systemswith delays is clearly needed, which hasmotivated our presentinvestigation.

In this paper, we consider the problem of FTS of linear nonautonomous systems with discrete and distributed time-varying delays. By utilizing some techniques developed for linear positive systems, we derive new explicit conditions in

∗ Tel.: +84 912494391.E-mail address: hienlv@hnue.edu.vn.

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L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18 13

terms of matrix inequalities ensuring that, for each given bound on the initial conditions, the state trajectories of the systemdo not exceed a certain threshold over a pre-specified finite time interval. Our conditions are derived in terms of sometypes of the Metzler matrix which can be easily verified. The novel feature of the result obtained in this paper is twofold.Firstly, the system considered in this paper is time-varying subjected to interval, nondifferentiable delays, whichmeans thatthe lower and the upper bounds for the time-varying delays are available but the delay functions are not necessary to bedifferentiable. This allows that the time-delays can be fast time-varying functions. Secondly, by a novel approach withoutusing the Lyapunov–Krasovskii functional method, we derive an explicit criterion for the FTS of the system in terms ofMetzler matrix inequalities which is intuitive and easy to verify.

Notations. For a given positive integer n, we denote n := {1, 2, . . . , n}. Rn denotes the n-dimensional space with the norm∥x∥∞ = maxi∈n |xi|. The set of realm× n-matrices is denoted by Rm×n. For u = (ui), v = (vi) in Rn, u ≥ v iff ui ≥ vi, ∀i ∈ n;u ≫ v iff ui > vi, ∀i ∈ n. We denote a vector e = (1 . . . 1)T ∈ Rn.

2. Problem statement and preliminaries

Consider the following linear nonautonomous system with time-varying delays

x(t) = A(t)x(t) + D(t)x(t − τ(t)) + G(t) t

t−κ(t)x(s)ds, t ≥ 0,

x(t) = φ(t), t ∈ [−d, 0],(2.1)

where x(t) ∈ Rn is the state; A(t) = (aij(t)) ∈ Rn×n, D(t) = (dij(t)) ∈ Rn×n and G(t) = (gij(t)) ∈ Rn×n are the systemmatrices; τ(t), κ(t) are time-varying delays satisfying 0 ≤ τ ≤ τ(t) ≤ τ , 0 ≤ κ(t) ≤ κ , t ≥ 0; φ(t) = (φi(t)) ∈

C([−d, 0], Rn), where d = max{τ , κ}, is the initial condition. Let us denote |φi| = sup−d≤t≤0 |φi(t)| and ∥φ∥∞ = maxi∈n |φi|.

Definition 2.1. For given a time T > 0 and positive numbers r1 < r2, system (2.1) is said to be finite-time stablewith respectto (r1, r2, T ) if for any initial condition φ(t) ∈ C([−d, 0], Rn), ∥φ∥∞ ≤ r1 implies that ∥x(t, φ)∥∞ < r2 for all t ∈ [0, T ].

Remark 2.1. It should be noted that, FTS and LAS are independent concepts in the following sense: a system which is FTSmay be not LAS, and vice versa. This will be illustrated in the following example.

Example 2.1. Consider the following delay differential equations

x(t) = −1.2x(t) +t + 2t + 1

x(t − 1), t ≥ 0, (2.2)

x(t) = −0.8x(t) +t

t + 6x(t − 1), t ≥ 0. (2.3)

Eq. (2.2) is globally LAS. However, this equation is not FTS with respect to r1 = 1, r2 = 1.25 and T = 10. Conversely,Eq. (2.3) is FTS with respect to r1 = 1, r2 = 1.5 and T = 10 but (2.3) is not LAS, even every non-zero solution of (2.3) goesto infinity as time tends to infinity. The state trajectories of (2.2) and (2.3) with initial condition φ(t) = 1, t ∈ [−1, 0], arepresented in Figs. 1 and 2, respectively.

The main purpose of this paper is to find conditions for the stability of system (2.1) over a finite time interval [0, T ]. Byutilizing some techniques developed for positive systems, some new explicit conditions are derived in terms of the Metzlermatrix for the FTS of system (2.1).

3. Main results

Let A(t) = (aij(t)),D(t) = (dij(t)) and G(t) = (gij(t)) be given matrices with continuous elements. We make thefollowing assumptions which are usually used for time-varying systems (see, for example, [3]). For given T > 0, assumethat

A1. aii(t) ≤ aii, i ∈ n, |aij(t)| ≤ aij, i = j, i, j ∈ n, t ∈ [0, T ].A2. |dij(t)| ≤ dij, |gij(t)| ≤ g ij, t ∈ [0, T ], i, j ∈ n.

We denote A =aij, D =

dijand G =

g ij. For a nonnegative scalar γ , let us define the matrix Mγ = A −

γ I + e−γ τ D + κG.

14 L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18

Fig. 1. A state trajectory of Eq. (2.2).

Fig. 2. A state trajectory of Eq. (2.3).

We are now in a position to state our main result as follows.

Theorem 3.1. Let assumptions A1,A2 hold. Then for given 0 < r1 < r2 and T > 0, system (2.1) is finite-time stable with respectto (r1, r2, T ) if there exist γ ≥ 0, positive numbers λ1, λ2 and a vector ξ ∈ Rn satisfying the following conditions

Mγ ξ ≪ 0, (3.1a)

λ1e ≤ ξ ≤ λ2e, (3.1b)λ2

λ1<

r2r1

e−γ (T+d). (3.1c)

L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18 15

Remark 3.1. It is worth noting that, if condition (3.1a) is satisfied with γ = 0 then system (2.1) is exponentially stablein the sense of Lyapunov (see the Appendix). Then we obtain a similar result in [3]. Moreover, in this case, system (2.1) isfinite-time stable with respect to (r1, r2, T ) for any 0 < r1 < r2, T > 0.

Proof. It is necessary to prove for the case γ > 0. Let ξ ∈ Rn satisfies (3.1a), (3.1b). Then we haveA − γ I + e−γ τ D + κG

ξ ≪ 0

and thus

nj=1

aij + dije−γ τ

+ κg ijξj ≤ γ ξi, ∀i ∈ n. (3.2)

In the following, we will use x(t) to denote the solution x(t, φ) if it does not make any confusion. At first, from (2.1) wehave

D+|xi(t)| = sgn(xi(t))xi(t)

≤ aii(t)|xi(t)| +

nj=1,j=i

|aij(t)∥xj(t)| +

nj=1

|dij(t)∥xj(t − τ(t))| +

nj=1

|gij(t)| t

t−κ(t)|xj(s)|ds

≤ aii|xi(t)| +

nj=1,j=i

aij|xj(t)| +

nj=1

dij|xj(t − τ(t))| +

nj=1

g ij

t

t−κ(t)|xj(s)|ds, ∀t ≥ 0, i ∈ n, (3.3)

where D+ denotes the Dini upper-right derivative.Next, let us consider the functions vi(t), i ∈ n, as follows

vi(t) =eγ d

λ1∥φ∥∞ξieγ t , t ≥ −d. (3.4)

It is noted that, for all t ≥ 0 and j ∈ n, we have

vj(t − τ(t)) =eγ d

λ1∥φ∥∞ξjeγ (t−τ(t))

≤eγ d

λ1∥φ∥∞ξjeγ te−γ τ

≤ e−γ τvj(t) (3.5)

and t

t−κ(t)vj(s)ds =

eγ d

λ1∥φ∥∞ξj

1 − e−γ κ(t)

γeγ t

≤ κvj(t) (3.6)

by using the fact that 1−e−γ κ

γ≤ limγ↓0+

1−e−γ κ

γ= κ for any κ ≥ 0.

Therefore, from (3.2), (3.4)–(3.6), we obtain

aiivi(t) +

nj=1,j=i

aijvj(t) +

nj=1

dijvj(t − τ(t)) +

nj=1

g ij

t

t−κ(t)vj(s)ds

≤ ηeγ taiiξi +

nj=1,j=i

aijξj +n

j=1

ξjdije−γ τ+

nj=1

κg ijξj

,

≤ ηeγ tn

j=1

aij + dije−γ τ

+ κg ijξj

≤ ηγ ξieγ t , ∀t ≥ 0, i ∈ n, (3.7)

where η =eγ d

λ1∥φ∥∞. Thus, it follows from (3.7) that

vi(t) ≥ aiivi(t) +

nj=1,j=i

aijvj(t) +

nj=1

dijvj(t − τ(t)) + g ij

t

t−κ(t)vj(s)ds, t ≥ 0. (3.8)

We will prove that |xi(t)| ≤ vi(t), ∀t ∈ [0, T ], i ∈ n. Let ρi(t) = |xi(t)| − vi(t), t ≥ −d. Noticing that, for t ∈ [−d, 0],we have |xi(t)| ≤ |φi| ≤ ηξieγ t

= vi(t), and hence, ρi(t) ≤ 0, for all t ∈ [−d, 0], i ∈ n. Assume that there exists an index

16 L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18

i ∈ n and t1 ∈ (0, T ] such that ρi(t1) = 0, ρi(t) > 0, t ∈ (t1, t1 + δ) for some δ > 0 and ρj(t) ≤ 0, ∀t ∈ [−d, t1], j ∈ n.Then D+ρi(t1) > 0. However, from (3.3) and (3.8), it follows for t ∈ [0, t1) that

D+ρi(t) ≤ aiiρi(t) +

nj=1,j=i

aijρj(t) +

nj=1

dijρj(t − τ(t)) + g ij

t

t−κ(t)ρj(s)ds ≤ aiiρi(t),

and therefore, D+ρi(t1) ≤ 0, which yields a contradiction. This shows that ρi(t) ≤ 0 for all t ≥ 0, i ∈ n, and thus, we readilyobtain |xi(t)| ≤

eγ d

λ1∥φ∥∞ξieγ t , ∀t ≥ 0, i ∈ n. Consequently,

∥x(t)∥∞ ≤eγ d

λ1∥φ∥∞∥ξ∥∞eγ t

≤λ2eγ d

λ1∥φ∥∞eγ t , t ∈ [0, T ]. (3.9)

If ∥φ∥∞ ≤ r1 then, by (3.9), ∥x(t)∥∞ ≤λ2λ1r1eγ (T+d) < r2, ∀t ∈ [0, T ]. This shows that system (2.1) is finite-time stable with

respect to (r1, r2, T ). The proof is completed. �

Remark 3.2. Since Mγ is a Metzler matrix (see [3,15] for more details), condition (3.1a) can be checked by various criteria.Condition (3.1b) can be relaxed in the following sense, if ξ ∈ Rn satisfies (3.1a) then ξ also satisfies (3.1b) with λ1 =

mini∈n ξi, λ2 = ∥ξ∥∞. However, in order to use the parameter searching method in Matlab, we impose condition (3.1b) forimproving FTS parameters r1, r2, T and the delay size d.

Remark 3.3. When the finite-time stability parameters r1, r2, T are pre-specified, one seeks to find themaximum allowabledelay (MAD) dmax. The following optimization problem can be formulated to find MAD

min γ > 0, λ > 0 subject toMγ ξ ≪ 0e ≤ ξ ≤ λe.

Then the MAD can be determined by

dmax =1γ

ln

r2λr1

− T .

4. An illustrative example

Consider the following system

x(t) = A(t)x(t) + D(t)x(t − τ(t)) + G(t) t

t−κ(t)x(s)ds, t ≥ 0, (4.1)

where

A(t) =

−4 |cos t|

sin2 2t −4

, D(t) =

sin2 t

cos 2√t

0 cos2 t

, G(t) =

sin√t 0

sin2 3t |cos 2t|

,

and τ(t) = |sin 4t|, κ(t) = |cos t|.System (4.1) satisfies A1, A2 for all T > 0 and we have A =

−4 11 −4

, D =

1 10 1

, G =

1 01 1

. It should be noted

that system (4.1) does not satisfy the Lyapunov stability conditions proposed in [3]. More precisely, in this case the matrixM = A + D + κG is not invertible, and hence does not satisfy conditions of Theorem 2.5 in [3]. However, Mγ = M − γ Isatisfies (3.1a) for any γ > 0 and the domain of the solution ξ ∈ R2 of (3.1a) is defined by 2

2+γξ1 < ξ2 <

2+γ

2 ξ1. Let ustake r1 = 1, r2 = 1.25 and γ = 0.01 then system (4.1) is finite-time stable with respect to (r1, r2, T ) for any finite time0 < T < Tmax = 21.3144.

Remark 4.1. Since the matrix Mγ satisfies (3.1a) for any γ > 0, limγ↓0+1γln r2

r1= +∞ for all 0 < r1 < r2, then for given

T > 0, there exists γ > 0 such that 1 <r2r1e−γ (T+d). Therefore, system (4.1) is finite-time stable with respect to (r1, r2, T )

for any 0 < r1 < r2, T > 0. However, system (4.1) is not LAS as shown in Fig. 3.

5. Conclusion

This paper has dealt with the problem of finite-time stability for a class of linear nonautonomous systems with time-varying delays. An explicit criterion for the finite-time stability of the system has been proposed in terms of matrixinequalities for a type of Metzler matrix. The effectiveness of the proposed conditions has been illustrated by a numericalexample.

L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18 17

Fig. 3. A state trajectory of system (4.1).

Acknowledgments

The authors would like to thank the editors and anonymous reviewers for their constructive comments. This workwas supported by the NAFOSTED of Vietnam (101.01-2011.51) and the Ministry of Education and Training of Vietnam(B2013.17.42).

Appendix. Proof for the claim in Remark 3.1

Let ξ ∈ Rn, ξ ≫ 0, such that (A + D + κG)ξ ≪ 0. Then there exists η > 0 such thatn

j=1

aij + dij + κg ij

ξj ≤ −η, ∀i ∈ n.

Therefore, there exists γ ∗ > 0 satisfying

γ ξi +

nj=1

eγ τ

− 1dij +

eγ κ

− 1γ

− κ

g ij

ξj − η ≤ 0, ∀i ∈ n, γ ∈ (0, γ ∗

].

By the same arguments used in (3.3)–(3.9), we obtain

|xi(t)| ≤1

ξmin∥φ∥∞ξie−γ t , ∀i ∈ n, t ≥ 0,

where ξmin = mini∈n ξi. Therefore, ∥x(t)∥∞ ≤∥ξ∥∞

ξmin∥φ∥∞e−γ t , t ≥ 0, which shows that system (2.1) is Lyapunov

exponentially stable. The proof is completed.

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