# Exponential Stability for a Class of Switched Nonlinear Systems with Mixed Time-Varying Delays via an Average Dwell-Time Method

Authors: N. Yotha; T. Botmart; T. Mouktonglang
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

The problem of exponential stability for a class of switched nonlinear systems with discrete and distributed time-varying delays is studied. The constraint on the derivative of the time-varying delay is not required which allows the time delay to be a fast time-varying function. We study the stability properties of switched nonlinear systems consisting of both stable and unstable subsystems. Average dwell-time approached and improved piecewise Lyapunov functional combined with Leibniz-Newton are formulated. New delay-dependent sufficient conditions for the exponential stabilization of the switched systems are first established in terms of LMIs. A numerical example is also given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

The switched systems are an important class of hybrid systems. They are described by a family of continuous or discrete-time subsystems and a rule that orchestrates the switching between the subsystems. Recently, switched systems have attracted much attention due to the widespread application in control, chemical engineering processing [1], communication networks, traffic control [2, 3], and control of manufacturing systems [4–6]. A switched nonlinear system with time delay is called switched nonlinear delay system, where delay may be contained in the system state, control input, or switching signals. In [7–9], some stability properties of switched linear delay systems composed of both stable and unstable subsystems have been studied by using an average dwell-time approach and piecewise Lyapunov functions. It is shown that when the average dwell time is sufficiently large and the total activation time of the unstable subsystems is relatively small compared with that of the Hurwitz stable subsystems, global exponential stability is guaranteed. The concept of dwell time was extended to average dwell time by Hespanha and Morse [10] with switching among stable subsystems. Furthermore, [11] generalized the results to the case where stable and unstable subsystems coexist.The stability analysis of nonlinear time-delay systems has received increasing attention. Time-delay systems are frequently encountered in various areas such as chemical engineering systems, biological modeling, and economics. The stability analysis for nonlinear time-delay systems has been investigated extensively. Various approaches to such problems have been proposed, see [12–14] and the references therein. It is well known that the existences of time delay in a system may cause instability and oscillations system. Thus, the stability analysis of nonlinear time-delay systems has received considerable attention for the last few decades, see [15, 16].Recently, the exponential stability problems for time-delay systems have been studied by many researchers, see for examples [17–21]. The problems have been dealt with for various control areas such as exponential stabilization for linear time-delay systems [22], exponential stabilization for uncertain time-delay systems [21, 23]. Hien and Phat [23] presented exponential stability and stabilization conditions for a class of uncertain linear system with time-varying delay, based on an improved Lyapunov-Krasovskii functional combined with Leibniz-Newton formula. The robust stability conditions are derived in terms of LMIs, but the time-varying delays are required to be differentiable and the lower bound is restricted to zero. However, in most cases, these conditions are difficult to satisfy. Therefore, in this paper, we will employ some new techniques so that the above conditions can be removed.Stability analysis of nonlinear systems with distributed delays is of both practical and theoretical importance. For some systems, delay phenomena may not be simply considered as delays in the velocity terms and/or discrete delays in the states. Therefore, it is desirable to extend the system model to include distributed delays. Practical applications, modeled by systems with distributed delays, can be found in [24–26]. In [27], Gao et al. studied the problem of linear switching systems with discrete and continuous time delay. By constructing Lyapunov functional under a condition on the time delay, we show that it stabilizes the system for sufficiently small delays.In this paper, the problem of exponential stability for a class of switched nonlinear systems with discrete and distributed time-varying delays is studied. The discrete time delay is a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available. However, the delay function is not necessary to be differentiable. We study the stability properties of switched nonlinear systems consisting of both stable and unstable subsystems. Average dwell-time approached and improved piecewise Lyapunov functional combined with Leibniz-Newton's formula, and new delay-dependent sufficient conditions for the exponential stabilization of the switched systems are first established in terms of LMIs. A numerical example is also given to illustrate the effectiveness of the proposed method.The rest of this paper is organized as follows. In Section2, we give notations, definition, propositions, and lemma which will be used in the proof of the main results. Delay-dependent sufficient conditions for the exponential stability of switched nonlinear time-varying delay systems are presented in Section 3. A Numerical example illustrated the obtained results is given in Section 4. The paper ends with conclusions in Section 5 and cited references.


## 2. Preliminaries

The following notation will be used in this paper.ℝ+ denotes the set of all real nonnegative numbers. ℝn denotes the n-dimensional space and the vector norm ∥·∥. Mn×r denotes the space of all (n×r) matrices.We also letAT denote the transpose of matrix A:A is symmetric if A=AT. :I denotes the identity matrix. λ(A) denotes the set of all eigenvalues of A. λmax(A)=max{Reλ:λ∈λ(A)}:xt∶={x(t+s):s∈[-h,0]},∥xt∥=sups∈[-h,0]∥x(t+s)∥:C([0,t],ℝn) denotes the set of all ℝn-valued continuous functions on [0,t]. L2([0,t],ℝm) denotes the set of all the ℝm-valued square integrable functions on [0,t]: matrix A is called semipositive definite (A≥0) if 〈Ax,x〉≥0, for all x∈ℝn:A is positive definite (A>0) if 〈Ax,x〉>0 for all x≠0:A>B means A-B>0: the symmetric term in a matrix is denoted by *.Consider a nonlinear system with interval time-varying and distributed delays of the form(2.1)x˙(t)=Aσ(t)x(t)+Dσ(t)x(t-hσ(t)(t))+Cσ(t)∫t-kσ(t)(t)t‍x(s)ds+Eσ(t)fσ(t)(t,x(t),x(t-hσ(t)(t)),∫t-kσ(t)(t)t‍x(s)ds),x(t)=ϕ(t),t∈[-τ,0],τ=max{h2,k},
where x(t)∈ℝn is the state. Aσ(t),Cσ(t),Dσ(t),andEσ(t) are given matrices of appropriate dimensions. σ(t):[0,+∞)→M={1,2,…,m} is piecewise constant and left continuous and is called a switching signal and ϕ(t)∈C([-h2,0],ℝn) is the initial function with the norm
(2.2)∥ϕ∥=sups∈[-τ,0]∥ϕ(s)∥2+∥ϕ˙(s)∥2.
The time delays h(t) and k(t) denoted the discrete time-varying delay and distributed time-varying delay, respectively. They are assumed to satisfy the following conditions:
(2.3)0≤h1σ(t)≤hσ(t)(t)≤h2σ(t),0≤kσ(t)(t)≤kσ(t).
The nonlinear perturbation satisfies the following growth condition:
(2.4)∃ai,bi,ci>0,∥fi(t,x,y,z)∥≤ai∥x∥+bi∥y∥+ci∥z∥,∀(t,x,y,z)∈ℝ+×ℝn×ℝn×ℝn,i∈M.We introduce the following technical well-known propositions, which will be used in the proof of our results.Definition 2.1 (see [7]).
For any switching signalσ(t) and any T2>T1≥0, let Nσ(T1,T2) denote the number of switching of σ(t) over the interval (T1,T2). For given Ta>0,N0≥0, if the inequality Nσ(T1,T2)≤N0+((T2-T1)/Ta) holds, then Ta is call average dwell time and N0 is called chattering bound. As commonly used in the literature, for convenience, we choose N0=0 in this paper.Proposition 2.2 (Cauchy inequality).
For any symmetric positive definite matrixN∈Mn×n and x,y∈ℝn one has
(2.5)±2xTy≤xTNx+yTN-1y.Proposition 2.3 (see [28]).
For any symmetric positive definite matrixM>0, scalar γ>0 and vector function ω:[0,γ]→ℝn such that the integrations concerned are well defined, the following inequality holds:
(2.6)(∫0γ‍ω(s)ds)TM(∫0γ‍ω(s)ds)≤γ(∫0γ‍ωT(s)Mω(s)ds).Proposition 2.4 (see [28] (Schur complement lemma)).
Given constant symmetric matricesX,Y,Z with appropriate dimensions satisfying X=XT,Y=YT>0. Then X+ZTY-1Z<0 if and only if
(2.7)(XZTZ-Y)<0or(-YZZTX)<0.


## 3. Main Results

### 3.1. Stability of Switched Nonlinear Time-Varying Delay Systems

First, we present a delay-dependent exponential stabilizability analysis conditions for the given nonlinear time-varying delay systems (2.1). We consider the case when the stable and unstable subsystems coexist. We assume that the subsystem i(1≤i≤k) is stable, where the positive integer k satisfies 1≤k≤m and the subsystem j(k+1≤j≤m) is unstable. Let us set ηi=(h2i-h1i),i=1,2,…,m.For the system (2.1), given α>0, the following lemma provides a change estimation of Lyapunov-Krasovskii functional candidate:
(3.1)Vi(x(t))=Vi1+Vi2+Vi3+Vi4+Vi5+Vi6+Vi7,
where
(3.2)Vi1=xT(t)Yix(t),Vi2=∫t-h1it‍eα(s-t)xT(s)YiQiYix(s)ds,Vi3=∫t-h2it‍eα(s-t)xT(s)YiQiYix(s)ds,Vi4=h1i∫-h1i0‍∫t+st‍eα(τ-t)x˙T(τ)YiRiYix˙(τ)dτds,Vi5=h2i∫-h2i0‍∫t+st‍eα(τ-t)x˙T(τ)YiRiYix˙(τ)dτds,Vi6=ηi∫-h2i-h1i‍∫t+st‍eα(τ-t)x˙T(τ)YiUiYix˙(τ)dτds,Vi7=∫ki0‍∫t+st‍eα(τ-t)xT(τ)YiSiYix(τ)dτds.Lemma 3.1.
For a given constantα>0, suppose (2.3) holds. If there exist symmetric positive definite matrices Pi,Qi,Ri,Ui,Si such that the following LMI holds:
(3.3)Σ1i=Wi-[000-II000]T×e-αh2iUi[000-II000]<0,(3.4)Σ2i=Wi-[00I0-I000]T×e-αh2iUi[00I0-I000]<0,(3.5)Σ3i=[ΨikiCiPi2aiPiEiTkici2EiPi*-kie-αkiSi400**-2aiI0***-kici2e-αkiSi4]<0,(3.6)Wi=[W11iW12iW13iW14iW15i000*W22i00W25ikiCiPikici2EiPi0**W33i0W35i000***W44iW45i000****W55i002biPiEiT*****W66i00******W77i0*******-2biI],i=1,2,3,…,k,
where
(3.7)Ψi=-0.1e-αh1iRi-0.1e-αh2iRi,W11i=[Ai+αI]Pi+Pi[Ai+αI]T-0.9(e-αh1i+e-αh2i)Ri+2Qi+kiSi+(ai+bi)I,W12i=PiAiT,W13i=e-αh1iRi,W14i=e-αh2iRi,W15i=DiPi,W22i=(h1i2+h2i2)Ri+ηi2Ui-2Pi+(ai+bi)I,W25i=DiPi,W33i=-e-αh1i(Qi+Ri)-e-αh2iUi,W35i=e-αh2iUi,W44i=-e-αh2i(Qi+Ri+Ui),W45i=e-αh2iUi,W55i=-2e-αh2iUi,W66i=-kie-αkiSi4,W77i=-kici2e-αkiSi4.
Then, along the trajectory of system (2.1), one has
(3.8)Vi(x(t))≤e-α(t-t0)Vi(x(t0)),t≥t0≥0.Proof.
LetYi=Pi-1,y(t)=Yix(t). We consider the following Lyapunov-Krasovskii functional (3.1). By taking the derivative of Lyapunov-Krasovskii functional candidate (3.1) along the trajectory of the system (2.1) we have
(3.9)V˙i1=yT(t)[PiAiT+AiPi]y(t)+2yT(t)DiPiy(t-hi(t))+2yT(t)CiPi∫t-ki(t)t‍y(s)ds+2yT(t)Eifi(t,x(t),x(t-hi(t)),∫t-ki(t)t‍x(s)ds),V˙i2=yT(t)Qiy(t)-e-αh1iyT(t-h1i)Qiy(t-h1i)-αVi2,V˙i3=yT(t)Qiy(t)-e-αh2iyT(t-h2i)Qiy(t-h2i)-αVi3,V˙i4≤h1i2y˙T(t)Riy˙(t)-h1ie-αh1i∫t-h1it‍y˙T(s)Riy˙(s)ds-αVi4,V˙i5≤h2i2y˙T(t)Riy˙(t)-h2ie-αh2i∫t-h2it‍y˙T(s)Riy˙(s)ds-αVi5,V˙i6≤ηi2y˙T(t)Uiy˙(t)-ηie-αh2i∫t-h2it-h1i‍y˙T(s)Uiy˙(s)ds-αVi6,V˙i7≤kiyT(t)Siy(t)-e-αki∫t-ki(t)t‍yT(s)Siy(s)ds-αVi7.
Applying Proposition 2.3 and the Leibniz-Newton formula, we have
(3.10)-h1∫t-h1it‍y˙T(s)Riy˙(s)ds≤-[∫t-h1it‍y˙(s)ds]TRi[∫t-h1it‍y˙(s)ds]≤-[y(t)-y(t-h1i)]TRi[y(t)-y(t-h1i)]=-yT(t)Riy(t)+2yT(t)Riy(t-h1i)-yT(t-h1i)Riy(t-h1i),-h2∫t-h2it‍y˙T(s)Riy˙(s)ds≤-[∫t-h2it‍y˙(s)ds]TRi[∫t-h2it‍y˙(s)ds]≤-[y(t)-y(t-h2i)]TRi[y(t)-y(t-h2i)]=-yT(t)Riy(t)+2yT(t)Riy(t-h2i)-yT(t-h2i)Riy(t-h2i).
Note that
(3.11)-(h2i-h1i)∫t-h2it-h1i‍y˙T(s)Uiy˙(s)ds=-(h2i-h1i)∫t-h2it-hi(t)‍y˙T(s)Uiy˙(s)ds=-(h2i-h1i)∫t-hi(t)t-h1i‍y˙T(s)Uiy˙(s)ds=-(h2i-hi(t))∫t-h2it-hi(t)‍y˙T(s)Uiy˙(s)ds=-(hi(t)-h1i)∫t-h2it-hi(t)‍y˙T(s)Uiy˙(s)ds=-(hi(t)-h1i)∫t-hi(t)t-h1i‍y˙T(s)Uiy˙(s)ds=-(h2i-hi(t))∫t-hi(t)t-h1i‍y˙T(s)Uiy˙(s)ds.
Using Proposition 2.3 gives
(3.12)-(h2i-hi(t))∫t-h2it-hi(t)‍y˙T(s)Uiy˙(s)ds≤-[∫t-h2it-hi(t)‍y˙(s)ds]TUi[∫t-h2it-hi(t)‍y˙(s)ds]≤-[y(t-hi(t))-y(t-h2i)]TUi[y(t-hi(t))-y(t-h2i)],-(hi(t)-h1i)∫t-hi(t)t-h1i‍y˙T(s)Uiy˙(s)ds≤-[∫t-hi(t)t-h1i‍y˙(s)ds]TUi[∫t-hi(t)t-h1i‍y˙(s)ds]≤-[y(t-h1i)-y(t-hi(t))]TUi[y(t-h1i)-y(t-hi(t))].
Let βi=((h2i-hi(t))/(h2i-h1i))≤1. Then
(3.13)-(h2i-hi(t))∫t-hi(t)t-h1‍y˙T(s)Uiy˙(s)ds=-βi∫t-hi(t)t-h1‍(h2i-h1i)y˙T(s)Uiy˙(s)ds≤-βi∫t-hi(t)t-h1‍(hi(t)-h1i)y˙T(s)Uiy˙(s)ds≤-βi[y(t-h1i)-y(t-hi(t))]TUi[y(t-h1i)-y(t-hi(t))]-(hi(t)-h1i)∫t-h2it-hi(t)‍y˙T(s)Uiy˙(s)ds=-(1-βi)∫t-h2it-hi(t)‍(h2i-h1i)y˙T(s)Uiy˙(s)ds≤-(1-βi)∫t-h2it-hi(t)‍(h2i-h(t))y˙T(s)Uiy˙(s)ds≤-(1-βi)[y(t-hi(t))-y(t-h2i)]T≤.×Ui[y(t-hi(t))-y(t-h2i)].
Therefore, from (3.12) and (3.13), we obtain
(3.14)-(h2i-h1i)∫t-h2it-h1i‍y˙T(s)Uiy˙(s)ds≤-[y(t-hi(t))-y(t-h2i)]TUi[y(t-hi(t))-y(t-h2i)]-[y(t-h1i)-y(t-hi(t))]TUi[y(t-h1i)-y(t-hi(t))]-βi[y(t-h1i)-y(t-hi(t))]TUi[y(t-h1i)-y(t-hi(t))]-(1-βi)[y(t-hi(t))-y(t-h2i)]TUi[y(t-hi(t))-y(t-h2i)].
By using the following identity relation
(3.15)Piy˙(t)-AiPiy(t)-DiPiy(t-hi(t))-CiPi∫t-ki(t)t‍y(s)ds-Eifi(t,x(t),x(t-hi(t)),∫t-ki(t)t‍x(s)ds)=0,
we have
(3.16)-2y˙T(t)Piy˙(t)+2y˙T(t)AiPiy(t)+2y˙T(t)DiPiy(t-hi(t))+2y˙T(t)CiPi∫t-ki(t)t‍y(s)ds+2y˙T(t)Eifi(t,x(t),x(t-hi(t)),∫t-ki(t)t‍x(s)ds)=0.
Applying Propositions 2.2 and 2.3 gives
(3.17)2yT(t)CiPi∫t-ki(t)t‍y(s)ds≤4kieαkiyT(t)CiPiSi-1PiCiTy(t)≤+14kie-αki(∫t-ki(t)t‍y(s)ds)TSi(∫t-ki(t)t‍y(s)ds)≤4kieαkiyT(t)CiPiSi-1PiCiTy(t)≤+14e-αki∫t-ki(t)t‍yT(s)Siy(s)ds,2y˙T(t)CiPi∫t-ki(t)t‍y(s)ds≤4kieαkiy˙T(t)CiPiSi-1PiCiTy˙(t)≤+14kie-αki(∫t-ki(t)t‍y(s)ds)TSi(∫t-ki(t)t‍y(s)ds)≤4kieαkiy˙T(t)CiPiSi-1PiCiTy˙(t)≤+14e-αki∫t-ki(t)t‍yT(s)Siy(s)ds.
By using Proposition 2.2 and condition (2.4), we have
(3.18)2yT(t)Eifi(t,x(t),x(t-h(t)),∫t-k(t)t‍x(s)ds)=≤2∥y(t)∥Ei∥fi(t,x(t),x(t-h(t)),∫t-k(t)t‍x(s)ds)∥=≤2∥y(t)∥Ei(ai∥x∥+bi∥x(t-hi(t))∥+ci∥∫t-ki(t)t‍x(s)ds∥)==2ai∥y(t)∥∥EiPiy(t)∥+2bi∥y(t)∥∥EiPiy(t-hi(t))∥+2ci∥y(t)∥∥EiPi∫t-ki(t)t‍y(s)ds∥=≤(ai+bi)yT(t)y(t)+aiyT(t)PiEiTEiPiy(t)+biyT(t-hi(t))PiEiTEiPiy(t-h(t))==+4kici2eαkiyT(t)EiPiSi-1PiEiTy(t)+14e-αki∫t-ki(t)t‍yT(s)Siy(s)ds,2y˙T(t)Eifi(t,x(t),x(t-h(t)),∫t-k(t)t‍x(s)ds)=≤(ai+bi)y˙T(t)y˙(t)+aiyT(t)PiEiTEiPiy(t)+biyT(t-hi(t))PiEiTEiPiy(t-hi(t))==+4kici2eαkiy˙T(t)EiPiSi-1PiEiTy˙(t)+14e-αki∫t-ki(t)t‍yT(s)Siy(s)ds.
Hence, according to (3.10)–(3.18), we have
(3.19)V˙i(x(t))+αVi(x(t))≤ζT(t)Wiζ(t)=-βi[y(t-h1i)-y(t-hi(t))]Te-αh2iUi[y(t-h1i)-y(t-hi(t))]=-(1-βi)[y(t-hi(t))-y(t-h2i)]Te-αh2iUi[y(t-hi(t))-y(t-h2i)]=ζT(t)[(1-βi)Σ1i+βiΣ2i]ζ(t)+yT(t)Σiy(t),
where Σ1i and Σ2i are defined in (3.3), (3.4), respectively, and
(3.20)Σi=Ψi+4kieαkiCiPiSi-1PiCiT+2aiPiEiTEiPi+4kici2eαkiEiPiSi-1PiEiT,ζ(t)=[y(t),y˙(t),y(t-h1i),y(t-h2i),y(t-h(t))].
Since 0≤βi≤1, (1-βi)Σ1i+βiΣ2i is a convex combination of Σ1i and Σ2i, therefore, (1-βi)Σ1i+βiΣ2i<0 is equivalent to Σ1i<0 and Σ2i<0. Applying Shur complement lemma, the inequalities Σi is equivalent to Σ3i<0. Thus, it follows from (3.3)–(3.5) and (3.19), we obtain
(3.21)V˙(x(t))+αV(x(t))≤0,t≥t0≥0.
Thus, by the above differential inequality, we have
(3.22)Vi(x(t))≤e-α(t-t0)Vi(x(t0)),t≥t0≥0.For the system (2.1) and given β>0, the following lemma provides a change estimate of Lyapunov-Krasovskii functional candidate:
(3.23)Vj(x(t))=Vj1+Vj2+Vj3+Vj4+Vj5+Vj6+Vj7,
where
(3.24)Vj1=xT(t)Yjx(t),Vj2=∫t-h1jt‍eβ(t-s)xT(s)YjQjYjx(s)ds,Vj3=∫t-h2jt‍eβ(t-s)xT(s)YjQjYjx(s)ds,Vj4=h1i∫-h1j0‍∫t+st‍eβ(t-τ)x˙T(τ)YjRjYjx˙(τ)dτds,Vj5=h2j∫-h2j0‍∫t+st‍eβ(t-τ)x˙T(τ)YjRjYjx˙(τ)dτds,Vj6=ηj∫-h2j-h1j‍∫t+st‍eβ(t-τ)x˙T(τ)YjUjYjx˙(τ)dτds,Vj7=∫kj0‍∫t+st‍eβ(t-τ)xT(τ)YjSjYjx(τ)dτds.Lemma 3.2.
For a given constantβ>0, suppose (2.3) holds. If there exist symmetric positive definite matrices Pj,Qj,Rj,Uj,Sj such that the following LMI holds:
(3.25)Σ1j=Wj-[000-II000]T×eβh2jUj[000-II000]<0,(3.26)Σ2j=Wj-[00I0-I000]T×eβh2jUj[00I0-I000]<0,(3.27)Σ3j=[ΨjkjCjPj2ajPjEjTkjcj2EjPj*-kjeβkjSj400**-2ajI0***-kjcj2eβkjSj4]<0,(3.28)Wj=[W11jW12jW13jW14jW15j000*W22j00W25jkjCjPjkjcj2EjPj0**W33j0W35j000***W44jW45j000****W55j002bjPjEjT*****W66j00******W77j0*******-2bjI],j=k+1,…,m,
where
(3.29)Ψj=-0.1eβh1jRj-0.1eβh2jRj,W11j=[Aj-βI]Pj+Pj[Aj-βI]T-0.9(eβh1j+eβh2j)Rj+2Qj+kjSj+(aj+bj)I,W12j=PjAjT,W13j=eβh1jRj,W14j=eβh2jRj,W15j=DjPj,W22j=(h1j2+h2j2)Rj+ηj2Uj-2Pj+(aj+bj)I,W25j=DjPj,W33j=-eβh1j(Qj+Rj)-eβh2jUj,W35j=eβh2jUj,W44j=-eβh2j(Qj+Rj+Uj),W45j=eβh2jUj,W55j=-2eβh2jUj,W66j=-kjeβkjSj4,W77j=-kjcj2eβkjSj4.
Then, along the trajectory of system (2.1), one has
(3.30)Vj(x(t))≤eβ(t-t0)Vj(x(t0)),t≥t0≥0.Proof.
By taking the derivative of Lyapunov-Krasovskii functional candidate (3.23) along the trajectory of the system (2.1), we are able to do similar estimation as we did for the Lemma 3.1. We have the following:
(3.31)V˙j(t)-βVj(t)≤0,t≥t0≥0.
Thus, by the above differential inequality, we have
(3.32)Vj(x(t))≤eβ(t-t0)Vj(x(t0)),t≥t0≥0.From Lemmas3.1 and 3.2, it is easy to show the following properties of the Lyapunov functional candidate (3.1) and (3.23).(i)
There exist scalarsε1>0, ε2>0, such that
(3.33)ε1∥x(t)∥2≤Vσ(t)(x(t))≤ε2∥x(t0)∥cl2,σ(t)∈M.(ii)
There exists a constant scalarμ≥1 such that
(3.34)Vi(x(t))≤μVj(x(t)),i,j∈M.(iii)
The Lyapunov functional candidate (3.1) and (3.23) whose derivative along the trajectory of the corresponding subnetwork satisfies
(3.35)V(x(t))≤{e-α(t-tk)Vi(x(tk)),ifi∈1,2,…,k,fort∈[tk,tk+1),eβ(t-tk)Vj(x(tk)),ifj∈k+1,k+2,…,m,fort∈[tk,tk+1).Now, for any piecewise constant switching signalσ(t) and any 0≤t0<t, we let T-(t0,t) (resp., T+(t0,t)) denote the total activation time of the stable subsystem (resp., the ones of unstable subsystem) during (t0,t). Then, we choose a scalar α*∈(0,α) arbitrarily to propose the following switching law:(
S
1
): determine the switching signal σ(t) so that
(3.36)inft≥t0T-(t0,t)T+(t0,t)≥β+α*α-α*
holds on time interval (t0,t). Meanwhile, we choose α*<α as the average dwell-time scheme: for any t>t0,
(3.37)Nσ(t0,t)≤N0+t-t0Ta,Ta≥Ta*=lnμα*.Theorem 3.3.
For a given constantα>0, β>0 and time-varying delay satisfying (2.3), suppose that the subsystem (1≤i≤k) of switched system (2.1) satisfies the conditions of Lemma 3.1, and the others satisfy the conditions of Lemma 3.2. Then the system (2.1) is exponentially stable for switching signal satisfying (3.36), (3.37) and the state decay estimate is given by
(3.38)∥x(t)∥≤c0biaie-λ(t-t0)∥x(t0)∥cl,
where
(3.39)c0=eN0lnμ,λ=12(α*-lnμTa),ℒ=min1≤i≤mλmin(Pi-1),ℳ=max1≤i≤mλmax(Pi-1)+max1≤i≤m2h2iλmax[Pi-1QiPi-1]+max1≤i≤m2h2i3λmax[Pi-1RiPi-1]+max1≤i≤mηi3λmax[Pi-1UiPi-1]+max1≤i≤mki2λmax[Pi-1SiPi-1],
and there exists μ≥1 such that
(3.40)Pi≤μPj,Qi≤μQj,Ri≤μRj,Ui≤μUj,Si≤μSj,∀i,j∈M.Proof.
Suppose thatt∈[tk,tk+1). For piecewise Lyapunov functional candidate (3.1) and (3.23) along trajectory of network system (2.1), we have
(3.41)V(x(t))≤{e-α(t-tk)Vi(x(tk)),ifi∈1,2,…,k,eβ(t-tk)Vj(x(tk)),ifj∈k+1,k+2,…,m.
SinceVσ(tk)(x(tk))≤μVσ(tk-)(x(tk-)) is true from (3.34) at the switching point tk, where tk-=limt→tkt and the relation k=Nσ(t0,t)≤N0+((t-t0)/Ta), we obtain
(3.42)V(x(t))≤eβT+(tk,t)-αT-(tk,t)Vσ(tk)(x(tk))≤eβT+(tk,t)-αT-(tk,t)μVσ(tk-)(x(tk-))≤eβT+(tk,t)-αT-(tk,t)μeβT+(tk-1,tk)-αT-(tk-1,tk)Vσ(tk-1)(x(tk-1))≤μeβT+(tk-1,t)-αT-(tk-1,t)Viσ(tk-1)(x(tk-1))⋮≤μkeβT+(t0,t)-αT-(t0,t)Vσ(t0)(x(t0))≤eβT+(t0,t)-αT-(t0,t)+(N0+((t-t0)/Ta))lnμVσ(t0)(x(t0)).
Under the switching law (S1) for any t0, t, we have
(3.43)βT+(t0,t)-αT-(t0,t)≤-α*(T+(t0,t)+T-(t0,t))=-α*(t-t0).
Thus,
(3.44)V(x(t))≤eN0lnμe-(α*-(lnμ/Ta))(t-t0)Vσ(t0)(x(t0))≤c0e-2λ(t-t0)Vσ(t0)(x(t0)),
where c0=eN0lnμ, λ=(1/2)(α*-(lnμ/Ta)). According to (3.33), we have
(3.45)ℒ∥x(t)∥2≤V(x(t)),Vσ(t0)(x(t0))≤ℳ∥x(t0)∥cl2.
Combining (3.44) and (3.45) leads to
(3.46)∥x(t)∥2≤1ℒV(x(t))≤ℳℒc0e-2λ(t-t0)∥x(t0)∥cl2.
Therefore,
(3.47)∥x(t)∥≤c0ℳℒe-λ(t-t0)∥x(t0)∥cl.
which means that solution to (2.1) is exponentially stable. The proof is thus completed.


## 3.1. Stability of Switched Nonlinear Time-Varying Delay Systems

First, we present a delay-dependent exponential stabilizability analysis conditions for the given nonlinear time-varying delay systems (2.1). We consider the case when the stable and unstable subsystems coexist. We assume that the subsystem i(1≤i≤k) is stable, where the positive integer k satisfies 1≤k≤m and the subsystem j(k+1≤j≤m) is unstable. Let us set ηi=(h2i-h1i),i=1,2,…,m.For the system (2.1), given α>0, the following lemma provides a change estimation of Lyapunov-Krasovskii functional candidate:
(3.1)Vi(x(t))=Vi1+Vi2+Vi3+Vi4+Vi5+Vi6+Vi7,
where
(3.2)Vi1=xT(t)Yix(t),Vi2=∫t-h1it‍eα(s-t)xT(s)YiQiYix(s)ds,Vi3=∫t-h2it‍eα(s-t)xT(s)YiQiYix(s)ds,Vi4=h1i∫-h1i0‍∫t+st‍eα(τ-t)x˙T(τ)YiRiYix˙(τ)dτds,Vi5=h2i∫-h2i0‍∫t+st‍eα(τ-t)x˙T(τ)YiRiYix˙(τ)dτds,Vi6=ηi∫-h2i-h1i‍∫t+st‍eα(τ-t)x˙T(τ)YiUiYix˙(τ)dτds,Vi7=∫ki0‍∫t+st‍eα(τ-t)xT(τ)YiSiYix(τ)dτds.Lemma 3.1.
For a given constantα>0, suppose (2.3) holds. If there exist symmetric positive definite matrices Pi,Qi,Ri,Ui,Si such that the following LMI holds:
(3.3)Σ1i=Wi-[000-II000]T×e-αh2iUi[000-II000]<0,(3.4)Σ2i=Wi-[00I0-I000]T×e-αh2iUi[00I0-I000]<0,(3.5)Σ3i=[ΨikiCiPi2aiPiEiTkici2EiPi*-kie-αkiSi400**-2aiI0***-kici2e-αkiSi4]<0,(3.6)Wi=[W11iW12iW13iW14iW15i000*W22i00W25ikiCiPikici2EiPi0**W33i0W35i000***W44iW45i000****W55i002biPiEiT*****W66i00******W77i0*******-2biI],i=1,2,3,…,k,
where
(3.7)Ψi=-0.1e-αh1iRi-0.1e-αh2iRi,W11i=[Ai+αI]Pi+Pi[Ai+αI]T-0.9(e-αh1i+e-αh2i)Ri+2Qi+kiSi+(ai+bi)I,W12i=PiAiT,W13i=e-αh1iRi,W14i=e-αh2iRi,W15i=DiPi,W22i=(h1i2+h2i2)Ri+ηi2Ui-2Pi+(ai+bi)I,W25i=DiPi,W33i=-e-αh1i(Qi+Ri)-e-αh2iUi,W35i=e-αh2iUi,W44i=-e-αh2i(Qi+Ri+Ui),W45i=e-αh2iUi,W55i=-2e-αh2iUi,W66i=-kie-αkiSi4,W77i=-kici2e-αkiSi4.
Then, along the trajectory of system (2.1), one has
(3.8)Vi(x(t))≤e-α(t-t0)Vi(x(t0)),t≥t0≥0.Proof.
LetYi=Pi-1,y(t)=Yix(t). We consider the following Lyapunov-Krasovskii functional (3.1). By taking the derivative of Lyapunov-Krasovskii functional candidate (3.1) along the trajectory of the system (2.1) we have
(3.9)V˙i1=yT(t)[PiAiT+AiPi]y(t)+2yT(t)DiPiy(t-hi(t))+2yT(t)CiPi∫t-ki(t)t‍y(s)ds+2yT(t)Eifi(t,x(t),x(t-hi(t)),∫t-ki(t)t‍x(s)ds),V˙i2=yT(t)Qiy(t)-e-αh1iyT(t-h1i)Qiy(t-h1i)-αVi2,V˙i3=yT(t)Qiy(t)-e-αh2iyT(t-h2i)Qiy(t-h2i)-αVi3,V˙i4≤h1i2y˙T(t)Riy˙(t)-h1ie-αh1i∫t-h1it‍y˙T(s)Riy˙(s)ds-αVi4,V˙i5≤h2i2y˙T(t)Riy˙(t)-h2ie-αh2i∫t-h2it‍y˙T(s)Riy˙(s)ds-αVi5,V˙i6≤ηi2y˙T(t)Uiy˙(t)-ηie-αh2i∫t-h2it-h1i‍y˙T(s)Uiy˙(s)ds-αVi6,V˙i7≤kiyT(t)Siy(t)-e-αki∫t-ki(t)t‍yT(s)Siy(s)ds-αVi7.
Applying Proposition 2.3 and the Leibniz-Newton formula, we have
(3.10)-h1∫t-h1it‍y˙T(s)Riy˙(s)ds≤-[∫t-h1it‍y˙(s)ds]TRi[∫t-h1it‍y˙(s)ds]≤-[y(t)-y(t-h1i)]TRi[y(t)-y(t-h1i)]=-yT(t)Riy(t)+2yT(t)Riy(t-h1i)-yT(t-h1i)Riy(t-h1i),-h2∫t-h2it‍y˙T(s)Riy˙(s)ds≤-[∫t-h2it‍y˙(s)ds]TRi[∫t-h2it‍y˙(s)ds]≤-[y(t)-y(t-h2i)]TRi[y(t)-y(t-h2i)]=-yT(t)Riy(t)+2yT(t)Riy(t-h2i)-yT(t-h2i)Riy(t-h2i).
Note that
(3.11)-(h2i-h1i)∫t-h2it-h1i‍y˙T(s)Uiy˙(s)ds=-(h2i-h1i)∫t-h2it-hi(t)‍y˙T(s)Uiy˙(s)ds=-(h2i-h1i)∫t-hi(t)t-h1i‍y˙T(s)Uiy˙(s)ds=-(h2i-hi(t))∫t-h2it-hi(t)‍y˙T(s)Uiy˙(s)ds=-(hi(t)-h1i)∫t-h2it-hi(t)‍y˙T(s)Uiy˙(s)ds=-(hi(t)-h1i)∫t-hi(t)t-h1i‍y˙T(s)Uiy˙(s)ds=-(h2i-hi(t))∫t-hi(t)t-h1i‍y˙T(s)Uiy˙(s)ds.
Using Proposition 2.3 gives
(3.12)-(h2i-hi(t))∫t-h2it-hi(t)‍y˙T(s)Uiy˙(s)ds≤-[∫t-h2it-hi(t)‍y˙(s)ds]TUi[∫t-h2it-hi(t)‍y˙(s)ds]≤-[y(t-hi(t))-y(t-h2i)]TUi[y(t-hi(t))-y(t-h2i)],-(hi(t)-h1i)∫t-hi(t)t-h1i‍y˙T(s)Uiy˙(s)ds≤-[∫t-hi(t)t-h1i‍y˙(s)ds]TUi[∫t-hi(t)t-h1i‍y˙(s)ds]≤-[y(t-h1i)-y(t-hi(t))]TUi[y(t-h1i)-y(t-hi(t))].
Let βi=((h2i-hi(t))/(h2i-h1i))≤1. Then
(3.13)-(h2i-hi(t))∫t-hi(t)t-h1‍y˙T(s)Uiy˙(s)ds=-βi∫t-hi(t)t-h1‍(h2i-h1i)y˙T(s)Uiy˙(s)ds≤-βi∫t-hi(t)t-h1‍(hi(t)-h1i)y˙T(s)Uiy˙(s)ds≤-βi[y(t-h1i)-y(t-hi(t))]TUi[y(t-h1i)-y(t-hi(t))]-(hi(t)-h1i)∫t-h2it-hi(t)‍y˙T(s)Uiy˙(s)ds=-(1-βi)∫t-h2it-hi(t)‍(h2i-h1i)y˙T(s)Uiy˙(s)ds≤-(1-βi)∫t-h2it-hi(t)‍(h2i-h(t))y˙T(s)Uiy˙(s)ds≤-(1-βi)[y(t-hi(t))-y(t-h2i)]T≤.×Ui[y(t-hi(t))-y(t-h2i)].
Therefore, from (3.12) and (3.13), we obtain
(3.14)-(h2i-h1i)∫t-h2it-h1i‍y˙T(s)Uiy˙(s)ds≤-[y(t-hi(t))-y(t-h2i)]TUi[y(t-hi(t))-y(t-h2i)]-[y(t-h1i)-y(t-hi(t))]TUi[y(t-h1i)-y(t-hi(t))]-βi[y(t-h1i)-y(t-hi(t))]TUi[y(t-h1i)-y(t-hi(t))]-(1-βi)[y(t-hi(t))-y(t-h2i)]TUi[y(t-hi(t))-y(t-h2i)].
By using the following identity relation
(3.15)Piy˙(t)-AiPiy(t)-DiPiy(t-hi(t))-CiPi∫t-ki(t)t‍y(s)ds-Eifi(t,x(t),x(t-hi(t)),∫t-ki(t)t‍x(s)ds)=0,
we have
(3.16)-2y˙T(t)Piy˙(t)+2y˙T(t)AiPiy(t)+2y˙T(t)DiPiy(t-hi(t))+2y˙T(t)CiPi∫t-ki(t)t‍y(s)ds+2y˙T(t)Eifi(t,x(t),x(t-hi(t)),∫t-ki(t)t‍x(s)ds)=0.
Applying Propositions 2.2 and 2.3 gives
(3.17)2yT(t)CiPi∫t-ki(t)t‍y(s)ds≤4kieαkiyT(t)CiPiSi-1PiCiTy(t)≤+14kie-αki(∫t-ki(t)t‍y(s)ds)TSi(∫t-ki(t)t‍y(s)ds)≤4kieαkiyT(t)CiPiSi-1PiCiTy(t)≤+14e-αki∫t-ki(t)t‍yT(s)Siy(s)ds,2y˙T(t)CiPi∫t-ki(t)t‍y(s)ds≤4kieαkiy˙T(t)CiPiSi-1PiCiTy˙(t)≤+14kie-αki(∫t-ki(t)t‍y(s)ds)TSi(∫t-ki(t)t‍y(s)ds)≤4kieαkiy˙T(t)CiPiSi-1PiCiTy˙(t)≤+14e-αki∫t-ki(t)t‍yT(s)Siy(s)ds.
By using Proposition 2.2 and condition (2.4), we have
(3.18)2yT(t)Eifi(t,x(t),x(t-h(t)),∫t-k(t)t‍x(s)ds)=≤2∥y(t)∥Ei∥fi(t,x(t),x(t-h(t)),∫t-k(t)t‍x(s)ds)∥=≤2∥y(t)∥Ei(ai∥x∥+bi∥x(t-hi(t))∥+ci∥∫t-ki(t)t‍x(s)ds∥)==2ai∥y(t)∥∥EiPiy(t)∥+2bi∥y(t)∥∥EiPiy(t-hi(t))∥+2ci∥y(t)∥∥EiPi∫t-ki(t)t‍y(s)ds∥=≤(ai+bi)yT(t)y(t)+aiyT(t)PiEiTEiPiy(t)+biyT(t-hi(t))PiEiTEiPiy(t-h(t))==+4kici2eαkiyT(t)EiPiSi-1PiEiTy(t)+14e-αki∫t-ki(t)t‍yT(s)Siy(s)ds,2y˙T(t)Eifi(t,x(t),x(t-h(t)),∫t-k(t)t‍x(s)ds)=≤(ai+bi)y˙T(t)y˙(t)+aiyT(t)PiEiTEiPiy(t)+biyT(t-hi(t))PiEiTEiPiy(t-hi(t))==+4kici2eαkiy˙T(t)EiPiSi-1PiEiTy˙(t)+14e-αki∫t-ki(t)t‍yT(s)Siy(s)ds.
Hence, according to (3.10)–(3.18), we have
(3.19)V˙i(x(t))+αVi(x(t))≤ζT(t)Wiζ(t)=-βi[y(t-h1i)-y(t-hi(t))]Te-αh2iUi[y(t-h1i)-y(t-hi(t))]=-(1-βi)[y(t-hi(t))-y(t-h2i)]Te-αh2iUi[y(t-hi(t))-y(t-h2i)]=ζT(t)[(1-βi)Σ1i+βiΣ2i]ζ(t)+yT(t)Σiy(t),
where Σ1i and Σ2i are defined in (3.3), (3.4), respectively, and
(3.20)Σi=Ψi+4kieαkiCiPiSi-1PiCiT+2aiPiEiTEiPi+4kici2eαkiEiPiSi-1PiEiT,ζ(t)=[y(t),y˙(t),y(t-h1i),y(t-h2i),y(t-h(t))].
Since 0≤βi≤1, (1-βi)Σ1i+βiΣ2i is a convex combination of Σ1i and Σ2i, therefore, (1-βi)Σ1i+βiΣ2i<0 is equivalent to Σ1i<0 and Σ2i<0. Applying Shur complement lemma, the inequalities Σi is equivalent to Σ3i<0. Thus, it follows from (3.3)–(3.5) and (3.19), we obtain
(3.21)V˙(x(t))+αV(x(t))≤0,t≥t0≥0.
Thus, by the above differential inequality, we have
(3.22)Vi(x(t))≤e-α(t-t0)Vi(x(t0)),t≥t0≥0.For the system (2.1) and given β>0, the following lemma provides a change estimate of Lyapunov-Krasovskii functional candidate:
(3.23)Vj(x(t))=Vj1+Vj2+Vj3+Vj4+Vj5+Vj6+Vj7,
where
(3.24)Vj1=xT(t)Yjx(t),Vj2=∫t-h1jt‍eβ(t-s)xT(s)YjQjYjx(s)ds,Vj3=∫t-h2jt‍eβ(t-s)xT(s)YjQjYjx(s)ds,Vj4=h1i∫-h1j0‍∫t+st‍eβ(t-τ)x˙T(τ)YjRjYjx˙(τ)dτds,Vj5=h2j∫-h2j0‍∫t+st‍eβ(t-τ)x˙T(τ)YjRjYjx˙(τ)dτds,Vj6=ηj∫-h2j-h1j‍∫t+st‍eβ(t-τ)x˙T(τ)YjUjYjx˙(τ)dτds,Vj7=∫kj0‍∫t+st‍eβ(t-τ)xT(τ)YjSjYjx(τ)dτds.Lemma 3.2.
For a given constantβ>0, suppose (2.3) holds. If there exist symmetric positive definite matrices Pj,Qj,Rj,Uj,Sj such that the following LMI holds:
(3.25)Σ1j=Wj-[000-II000]T×eβh2jUj[000-II000]<0,(3.26)Σ2j=Wj-[00I0-I000]T×eβh2jUj[00I0-I000]<0,(3.27)Σ3j=[ΨjkjCjPj2ajPjEjTkjcj2EjPj*-kjeβkjSj400**-2ajI0***-kjcj2eβkjSj4]<0,(3.28)Wj=[W11jW12jW13jW14jW15j000*W22j00W25jkjCjPjkjcj2EjPj0**W33j0W35j000***W44jW45j000****W55j002bjPjEjT*****W66j00******W77j0*******-2bjI],j=k+1,…,m,
where
(3.29)Ψj=-0.1eβh1jRj-0.1eβh2jRj,W11j=[Aj-βI]Pj+Pj[Aj-βI]T-0.9(eβh1j+eβh2j)Rj+2Qj+kjSj+(aj+bj)I,W12j=PjAjT,W13j=eβh1jRj,W14j=eβh2jRj,W15j=DjPj,W22j=(h1j2+h2j2)Rj+ηj2Uj-2Pj+(aj+bj)I,W25j=DjPj,W33j=-eβh1j(Qj+Rj)-eβh2jUj,W35j=eβh2jUj,W44j=-eβh2j(Qj+Rj+Uj),W45j=eβh2jUj,W55j=-2eβh2jUj,W66j=-kjeβkjSj4,W77j=-kjcj2eβkjSj4.
Then, along the trajectory of system (2.1), one has
(3.30)Vj(x(t))≤eβ(t-t0)Vj(x(t0)),t≥t0≥0.Proof.
By taking the derivative of Lyapunov-Krasovskii functional candidate (3.23) along the trajectory of the system (2.1), we are able to do similar estimation as we did for the Lemma 3.1. We have the following:
(3.31)V˙j(t)-βVj(t)≤0,t≥t0≥0.
Thus, by the above differential inequality, we have
(3.32)Vj(x(t))≤eβ(t-t0)Vj(x(t0)),t≥t0≥0.From Lemmas3.1 and 3.2, it is easy to show the following properties of the Lyapunov functional candidate (3.1) and (3.23).(i)
There exist scalarsε1>0, ε2>0, such that
(3.33)ε1∥x(t)∥2≤Vσ(t)(x(t))≤ε2∥x(t0)∥cl2,σ(t)∈M.(ii)
There exists a constant scalarμ≥1 such that
(3.34)Vi(x(t))≤μVj(x(t)),i,j∈M.(iii)
The Lyapunov functional candidate (3.1) and (3.23) whose derivative along the trajectory of the corresponding subnetwork satisfies
(3.35)V(x(t))≤{e-α(t-tk)Vi(x(tk)),ifi∈1,2,…,k,fort∈[tk,tk+1),eβ(t-tk)Vj(x(tk)),ifj∈k+1,k+2,…,m,fort∈[tk,tk+1).Now, for any piecewise constant switching signalσ(t) and any 0≤t0<t, we let T-(t0,t) (resp., T+(t0,t)) denote the total activation time of the stable subsystem (resp., the ones of unstable subsystem) during (t0,t). Then, we choose a scalar α*∈(0,α) arbitrarily to propose the following switching law:(
S
1
): determine the switching signal σ(t) so that
(3.36)inft≥t0T-(t0,t)T+(t0,t)≥β+α*α-α*
holds on time interval (t0,t). Meanwhile, we choose α*<α as the average dwell-time scheme: for any t>t0,
(3.37)Nσ(t0,t)≤N0+t-t0Ta,Ta≥Ta*=lnμα*.Theorem 3.3.
For a given constantα>0, β>0 and time-varying delay satisfying (2.3), suppose that the subsystem (1≤i≤k) of switched system (2.1) satisfies the conditions of Lemma 3.1, and the others satisfy the conditions of Lemma 3.2. Then the system (2.1) is exponentially stable for switching signal satisfying (3.36), (3.37) and the state decay estimate is given by
(3.38)∥x(t)∥≤c0biaie-λ(t-t0)∥x(t0)∥cl,
where
(3.39)c0=eN0lnμ,λ=12(α*-lnμTa),ℒ=min1≤i≤mλmin(Pi-1),ℳ=max1≤i≤mλmax(Pi-1)+max1≤i≤m2h2iλmax[Pi-1QiPi-1]+max1≤i≤m2h2i3λmax[Pi-1RiPi-1]+max1≤i≤mηi3λmax[Pi-1UiPi-1]+max1≤i≤mki2λmax[Pi-1SiPi-1],
and there exists μ≥1 such that
(3.40)Pi≤μPj,Qi≤μQj,Ri≤μRj,Ui≤μUj,Si≤μSj,∀i,j∈M.Proof.
Suppose thatt∈[tk,tk+1). For piecewise Lyapunov functional candidate (3.1) and (3.23) along trajectory of network system (2.1), we have
(3.41)V(x(t))≤{e-α(t-tk)Vi(x(tk)),ifi∈1,2,…,k,eβ(t-tk)Vj(x(tk)),ifj∈k+1,k+2,…,m.
SinceVσ(tk)(x(tk))≤μVσ(tk-)(x(tk-)) is true from (3.34) at the switching point tk, where tk-=limt→tkt and the relation k=Nσ(t0,t)≤N0+((t-t0)/Ta), we obtain
(3.42)V(x(t))≤eβT+(tk,t)-αT-(tk,t)Vσ(tk)(x(tk))≤eβT+(tk,t)-αT-(tk,t)μVσ(tk-)(x(tk-))≤eβT+(tk,t)-αT-(tk,t)μeβT+(tk-1,tk)-αT-(tk-1,tk)Vσ(tk-1)(x(tk-1))≤μeβT+(tk-1,t)-αT-(tk-1,t)Viσ(tk-1)(x(tk-1))⋮≤μkeβT+(t0,t)-αT-(t0,t)Vσ(t0)(x(t0))≤eβT+(t0,t)-αT-(t0,t)+(N0+((t-t0)/Ta))lnμVσ(t0)(x(t0)).
Under the switching law (S1) for any t0, t, we have
(3.43)βT+(t0,t)-αT-(t0,t)≤-α*(T+(t0,t)+T-(t0,t))=-α*(t-t0).
Thus,
(3.44)V(x(t))≤eN0lnμe-(α*-(lnμ/Ta))(t-t0)Vσ(t0)(x(t0))≤c0e-2λ(t-t0)Vσ(t0)(x(t0)),
where c0=eN0lnμ, λ=(1/2)(α*-(lnμ/Ta)). According to (3.33), we have
(3.45)ℒ∥x(t)∥2≤V(x(t)),Vσ(t0)(x(t0))≤ℳ∥x(t0)∥cl2.
Combining (3.44) and (3.45) leads to
(3.46)∥x(t)∥2≤1ℒV(x(t))≤ℳℒc0e-2λ(t-t0)∥x(t0)∥cl2.
Therefore,
(3.47)∥x(t)∥≤c0ℳℒe-λ(t-t0)∥x(t0)∥cl.
which means that solution to (2.1) is exponentially stable. The proof is thus completed.


## 4. Numerical Example

In this section, we now provide an example to show the effectiveness of the result in Theorem3.3.Example 4.1.
Consider the nonlinear systems with interval time-varying and distributed delays (2.1) with the following parameters:
(4.1)x˙(t)=Aσ(t)x(t)+Dσ(t)x(t-hσ(t)(t))+Cσ(t)∫t-kσ(t)(t)t‍x(s)ds+Eσ(t)fσ(t)(t,x(t),x(t-hσ(t)(t)),∫t-kσ(t)(t)t‍x(s)ds),
where
(4.2)A1=[-20.10-1],D1=[0.100.3-0.5],C1=[0.1200.2],E1=[0.1000.1],A2=[-10.10.10.1],D2=[0.200.1-1.3],C2=[0.250.200.1],E2=[0.1000.1],f1(t,x1(t),x1(t-h(t)),∫t-k1(t)t‍x1(s)ds)≤0.1∥x1(t)∥+0.1∥x1(t-h(t))∥+0.2∥∫t-k1(t)t‍x1(s)ds∥,f2(t,x2(t),x2(t-h(t)),∫t-k2(t)t‍x2(s)ds)≤0.1∥x2(t)∥+0.05∥x2(t-h(t))∥+0.1∥∫t-k2(t)t‍x2(s)ds∥.Givenh11=0.1, h12=0.2, h21=0.15, h22=0.25, k1=0.22, k2=0.2, α=0.25, and β=0.2, it is found that LMIs (3.3), (3.4), (3.5), (3.25), (3.26), and (3.27) of Lemmas 3.1 and 3.2 have feasible solutions. Thus, we can get μ=1.35. We now choose α*=0.155<α. Then, the switching law (S1) will require inft≥t0T-(t0,t)/T+(t0,t)≥((β+α*)/(α-α*))=4.2632 and the average dwell time is computed as Ta≥Ta*=lnμ/α*=3.6472.We leth1(t)=0.1+0.05|cost|, k1(t)=0.1|sint|, h2(t)=0.1+0.05|sint|, k2(t)=0.1|cost| and ϕ(t)=[-cost,cost], for all t∈[-0.25,0]. Figure 1 shows the trajectories of solutions x1(t) and x2(t) of the interval time-varying and distributed delays system.Figure 1
The trajectories ofx1(t) and x2(t) of the interval time-varying delay system (4.1).


## 5. Conclusions

In this paper, we have investigated the exponential stability for a class of switched nonlinear systems with discrete and distributed time-varying delays. The interval time-varying delay function is not necessary to be differentiable which allows time-delay function to be a fast time-varying function. By using the average dwell-time approached and improved piecewise Lyapunov-Krasovskii functionals which are constructed based on and combined with Leibniz-Newton’s formula, for the cases that subsystems are stable and unstable, some new delay-dependent stability criteria are derived by a set of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness of our theoretical results.



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*Source: 528259-2012-09-29.xml*