Delay-Dependent Finite-Time and 𝐿₂-Gain Analysis for Switched Systems with Time-Varying Delay

Abstract

For the switched systems, switching behavior always affects the finite-time stability (FTS) property, which was neglected by most previous studies. This paper is mainly concerned with the problem of delay-dependent finite-time and 𝐿₂-gain analysis for switched systems with time-varying delay. Several less conservative sufficient conditions related to finite-time stability and boundness of switched system with time-varying delays are proposed; the system trajectory stays within a special bound with the information of switching signal. At last, a numerical example is also given to illustrate the efficiency of the developed method.

1. Introduction

During the past decades, the systems with time delays have received much attention since they often fall in various practical systems, for instance, neural networks, networked control systems, engineering systems, biology, economics, and other fields. However, time delay is always frequently the major cause of oscillation and instability; the stability for systems with time delays has been devoted to a lot of effort. Nowadays, the stability of time delay is parted into two classes: delay-independent stability criteria and delay-dependent ones. It should be pointed out that delay-independent criteria turn out to be more conservative, especially for the small-size delays. The problem of delay-dependent stability analysis for time-delay systems has received considerable attention, and lots of important results have also been reported [1–12].

In recent years, there has been an increasing interest in analysis of hybrid and switched systems due to their importance both in theory and its applications. As an important class of hybrid systems, switched linear systems comprise a collection of linear subsystems described by differential equations as well as a switching law to specify the switching during these subsystems. A switched system is a type of hybrid system with a combination of the discrete and continuous dynamical systems. These systems are recognized as models for appearance which cannot be denoted by exclusively continuous or discrete processes. Recently, based on the Lyapunov functions and other analysis tools, the corresponding stability and stabilization for both linear and nonlinear and switched systems have been investigated and many good results have been obtained; for a recent survey on this issue and related matters, one can also refer to [13–26].

Most of the existing results have concentrated on Lyapunov-based asymptotic stability for the switched systems; the behavior of that is above the infinite time interval. However, in practical engineering, there exists the bad transient characteristics; the system is asymptotically stable without unusable. For the systems which work in a short time interval, for instance, control system, missile system, and communication network system, which mainly care about the behavior over a finite time interval. To deal with this problem, in 1961, Dorato presented the idea of finite-time stability in [27]. Over the past few years, many study efforts have been dedicated to the finite-time stability (FTS) of switched systems due to its wide applications. Comparing with the classical Lyapunov stability, which currently is the focus on a large and growing interdisciplinary area of research. To study the transient behavior of systems, FTS concerns the stability of a system over a finite interval of time and plays an important role. It is important to emphasize the disconnection between classical Lyapunov stability and finite-time stability. The problem about finite-time stabilization has been widely learned in the literature [28–44]. It is worth pointing out that there is a difference between finite-time stability and Lyapunov asymptotic stability, and they are also independent of each other.

Recently, some papers are found to be related to finite-time stability for switched systems. For example, based on the technique of average dwell time, the problem of finite-time boundedness for the switched linear system with time delays was investigated in [36]. The issue of finite-time stability and 𝐻_∞ stabilization for switched nonlinear discrete-time systems was developed in [37]. The problem of finite-time quantized 𝐻_∞ control problem for a class of discrete-time switched time-delay systems with time-varying exogenous disturbances was also discussed in [38]. But, to the best of the authors' knowledge, the finite-time 𝐿₂-gain problems for switched systems have not been completely studied. The previous research has some conservatism of stability criterion; it is natural to look for an alternative way to reduce the conservatism of corresponding stability criteria. This idea motivates to our study.

The main contribution of this paper is that we present a novel approach for finite-time stability of the given switched system. Moreover, several sufficient conditions ensuring the finite-time stability and boundness are proposed with different information we know about the switching signal. It shows the less conservative results when more information about the switching signal is available. By selecting the appropriate Lyapunov-Krasovskii functional, the sufficient conditions are obtained to guarantee finite-time stability of the systems and the closed-loop system trajectory stays in a special bound. The finite-time stability criteria can also be dealt with in the form of linear matrix inequalities and average dwell time. Finally, a numerical example is given to illustrate the effectiveness of the developed techniques.

2. Preliminaries

In this paper, the following switched systems are described as follows: 𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)𝑥(𝑡−𝜏(𝑡))+𝐺_𝜎(𝑡)𝜔(𝑡),𝑧(𝑡)=𝐶_𝜎(𝑡)𝑥(𝑡)+𝐷_𝜎(𝑡)𝜔(𝑡)𝑥(𝑡)(𝑡)[],,𝑡≥0,=𝜙,𝑡∈−ℎ,0 (1) where 𝑥(𝑡)∈ℛ^𝑛 is the state vector of the system, 𝑧(𝑡)∈ℛ^𝑚 is the controlled output, and 𝜔(𝑡)∈𝐿^𝑞₂[0,∞] is the external disturbance vector and satisfies the constraint: ^𝑇₀𝜔^⊺(𝑡)𝜔(𝑡)≤𝑑,𝑑≥0. (2)𝐴_𝜎(𝑡), 𝐵_𝜎(𝑡), 𝐺_𝜎(𝑡), 𝐶_𝜎(𝑡), and 𝐷_𝜎(𝑡) are the constant real matrices with appropriate dimension. 𝜏(𝑡) represents the mode-dependent time-varying state delay in the switched system and satisfies the following conditions: 0<𝜏(𝑡)≤ℎ<∞,𝜏̇(𝑡)≤𝜏, (3)(4) where ℎ is the upper bound of the time-varying delay 𝜏(𝑡) and 𝜏 is the variation rate of the time-varying delay 𝜏(𝑡). 𝜙(𝑡) is the differentiable vector-valued initial function on [−ℎ,0]. 𝜎(𝑡)∶[0,∞)→𝒩={1,2,…,𝑁} is the right continuous piecewise constant switching signal to be designed.

Corresponding to the switching signal 𝜎(𝑡), we get the follwing switching sequence: =𝑥₀;𝑖₀,𝑡₀,…,𝑖_𝑘,𝑡_𝑘,…,∣𝑖_𝑘∈𝒩,𝑘=0,1,…, (5) where 𝑡₀ is the initial time when 𝑡_𝑘∈[𝑡_𝑘,𝑡_𝑘+1), 𝑥(𝑡₀) is the initial state and 𝑖_𝑘th subsystem is active. Therefore, the trajectory 𝑥(𝑡) of the switched system (1) is called the trajectory of the 𝑖_𝑘th subsystem. As assumed before, we get rid of Zeno behavior for all types related to switching signal. Throughout this paper, we assumed that the state of the switched system (1) does not jump at the switching instants; that is, the trajectory 𝑥(𝑡) is continuous everywhere.

Firstly, we will give the definitions and lemmas about switched system (1), which plays an major role in the derivation of our results.

Definition 1. Switch system (1) is said to be finite-time bounded with respect to (𝑐₁,𝑐₂,𝑇,𝑆,𝑑), if condition (6) holds: max_{−𝜏≤𝑡0≤0}𝑥^⊺𝑡₀𝑆𝑥𝑡₀,𝑥̇^⊺𝑡₀𝑆𝑥̇𝑡₀≤𝑐₁⟹𝑥^⊺(𝑡)𝑆𝑥(𝑡)<𝑐₂,∀𝑡∈[0,𝑇],∀𝜔(𝑡)∶^𝑇₀𝜔^⊺(𝑠)𝜔(𝑠)𝑑𝑠≤𝑑, (6) where 𝑐₂>𝑐₁≥0 and 𝑆>0.

Definition 2. For 𝛾>0, 𝑑>0, 𝑇>0, 𝜂>0, Λ>0, and 𝑐₂>𝑐₁>0, system (3) is said to be finite-time stable with a weighted 𝐿₂ performance 𝛾 with respect to (𝑐₁,𝑐₂,𝑇,𝑆,𝑑), if the following condition holds: ^𝑇₀𝜆𝜂𝑠−ln¹𝑐₂Λ𝑐₁+𝑑𝛾²1/𝜂1−𝑒^𝜂𝑇𝑧^⊺(𝑠)𝑧(𝑠)𝑑𝑠≤𝛾²𝑒^{−𝜂𝑇𝑇}₀𝜔^⊺(𝑠)𝜔(𝑠)𝑑𝑠, (7) and under zero initial condition, it holds for all nonzero 𝜔: ∫^𝑇₀𝜔^⊺(𝑠)𝜔(𝑠)𝑑𝑠≤𝑑.

Definition 3 (see [21]). For any 𝑇₂>𝑇₁≥0, let 𝑁_𝜎(𝑇₁,𝑇₂) denote the switching number of 𝜎(𝑡) during (𝑇₁,𝑇₂). If 𝑁_𝜎(𝑇₁,𝑇₂)≤𝑁₀+(𝑇₂−𝑇₁)/𝜏_𝑎 holds for 𝑁₀≥0 and 𝜏_𝑎>0, then 𝑁₀ and 𝜏_𝑎 are called chattering bound and average dwell time, respectively. Here we assume 𝑁₀=0 for simplicity as supported by reference.

Lemma 4 (see [9]). Let 𝑓_𝑖∶ℛ^𝑚→ℛ(𝑖=1,2,…,𝑁) have positive values in an open subset 𝒟 of ℛ^𝑚. Then, the reciprocally convex combination of 𝑓_𝑖 over 𝒟 satisfies min_{𝛽𝑖∣𝛽𝑖>0,∑𝑖𝛽𝑖=1𝑖}1𝛽_𝑖𝑓_𝑖(𝑡)=_𝑖𝑓_𝑖(𝑡)+max_{𝑔𝑖,𝑗(𝑡)𝑖≠𝑗}𝑔_𝑖,𝑗(𝑡)𝑔𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜_𝑖,𝑗∶ℛ^𝑚⟶ℛ,𝑔_𝑗,𝑖(𝑡)=𝑔_𝑖,𝑗(𝑡),𝑓_𝑖(𝑡)𝑔_𝑖,𝑗(𝑡)𝑔_𝑖,𝑗(𝑡)𝑓_𝑗(𝑡)≥0. (8)

Lemma 5 (see [11]). For any constant matrix 𝑍∈ℛ^𝑛×𝑛, 𝑍=𝑍^⊺>0, scalars ℎ>0, such that the following integrations are well defined; then −ℎ^𝑡_𝑡−ℎ𝑥^⊺(𝑠)𝑍𝑥(𝑠)𝑑𝑠≤−^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠^⊺𝑍^𝑡_𝑡−ℎ𝑥(𝑠),−ℎ𝑑𝑠₂2⁰_−ℎ^𝑡_𝑡+𝜃𝑥^⊺(𝑠)𝑍𝑥(𝑠)𝑑𝑠𝑑𝜃≤−⁰_−ℎ^𝑡_𝑡+𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃^⊺𝑍⁰_−ℎ^𝑡_𝑡+𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃. (9)

Lemma 6 (Schur’s complement). Given constant matrices 𝑋, 𝑌, 𝑍, where 𝑋=𝑋^⊺ and 0<𝑌=𝑌^⊺, then 𝑋+𝑍^⊺𝑌⁻¹𝑍<0 if and only if 𝑋𝑍^⊺∗−𝑌<0𝑜𝑟−𝑌𝑍∗𝑋<0. (10)

3. Finite-Time Boundedness Analysis

Theorem 7. System (3) is said to be finite-time stability with respect to (𝑐₁,𝑐₂,𝑅,𝑑,𝑇) if there exist symmetric positive matrices 𝑃_𝑖, 𝑄_𝑠𝑖  𝑠=1,2, 𝑋_𝑘𝑖  (𝑘=1,2), 𝑌_𝑖 and matrices 𝑀_𝑖, 𝑁_𝑖, 𝑉_𝑖, scalars 𝛼≥0, 𝜇≥1, 𝜆_𝑙>0  (𝑙=1,2,…,8), 𝑑>0, ℎ>0, Λ>0, 𝜏>0, 𝑟(𝜏); such that ∀𝑖,𝑗∈𝒩, we have the following linear matrix inequalities:Ξ_1𝑖=Ξ_11𝑖Ξ_12𝑖𝑊_𝑖𝐴^⊺_𝑖𝑉^⊺_𝑖ℎ𝐴^⊺_𝑖𝑀^⊺_𝑖+𝑌_𝑖+𝑌^⊺_𝑖𝑌_𝑖+𝑌^⊺_𝑖𝑃_𝑖𝐺_𝑖∗Ξ_22𝑖−𝑊_𝑖+𝑋_2𝑖ℎ𝐵^⊺_𝑖𝑉^⊺_𝑖ℎ𝐵^⊺_𝑖𝑀^⊺_𝑖00∗∗−𝑄_1𝑖−𝑋_2𝑖ℎ0000∗∗∗Ξ_44𝑖−ℎ𝑀^⊺_𝑖0𝑉_𝑖𝐺_𝑖∗∗∗∗−ℎ𝑋_1𝑖−𝑌_𝑖−𝑌^⊺_𝑖0ℎ𝑀_𝑖𝐺_𝑖∗∗∗∗∗−𝑌_𝑖−𝑌^⊺_𝑖0∗∗∗∗∗∗−𝜂𝐻_𝑖Ξ<0,_2𝑖=Ξ_11𝑖Ξ_12𝑖𝑊_𝑖𝐴^⊺_𝑖𝑉^⊺_𝑖𝑌_𝑖+𝑌^⊺_𝑖ℎ𝐴^⊺_𝑖𝑁^⊺_𝑖+𝑌_𝑖+𝑌^⊺_𝑖𝑃_𝑖𝐺_𝑖∗Ξ_22𝑖−𝑊_𝑖+𝑋_2𝑖ℎ𝐵^⊺_𝑖𝑉^⊺_𝑖0ℎ𝐵^⊺_𝑖𝑁^⊺_𝑖0∗∗−𝑄_1𝑖−𝑋_2𝑖ℎ0000∗∗∗Ξ_44𝑖0−ℎ𝑁^⊺_𝑖𝑉_𝑖𝐺_𝑖∗∗∗∗−𝑌_𝑖−𝑌^⊺_𝑖00∗∗∗∗∗−ℎ𝑋_1𝑖−𝑌_𝑖−𝑌^⊺_𝑖ℎ𝑁_𝑖𝐺_𝑖∗∗∗∗∗∗−𝜂𝐻_𝑖<0, (11)𝑃_𝑖<𝜇𝑃_𝑗,𝑄_1𝑖<𝜇𝑄_1𝑗,𝑄_2𝑖<𝜇𝑄_2𝑗,𝑋_1𝑖<𝜇𝑋_1𝑗,𝑋_2𝑖<𝜇𝑋_2𝑗,𝑌_𝑖<𝜇𝑌_𝑗,𝜆₁𝑐₂𝑒^−𝜂𝑇>Λ𝑐₁+𝑑𝜆₈1−𝑒^−𝜂𝑇. (12)(13) with the average dwell time of the switching signal 𝜎 satisfying 𝜏_𝑎>𝜏^∗_𝑎=𝑇ln𝜇ln𝜆₁𝑐₂−lnΛ𝑐₁+𝑑𝜆₈1−𝑒^−𝜂𝑇,−𝜂𝑇 (14) where Ξ_11𝑖=𝛿𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐴^⊺_𝑖𝑃_𝑖+𝑒^𝛿ℎ𝑄_1𝑖+𝑄_2𝑖+ℎ𝑋_1𝑖−𝑋_2𝑖ℎ−2𝑌_𝑖−2𝑌^⊺_2𝑖,Ξ_12𝑖=𝑃_𝑖𝐵_𝑖+𝑋_2𝑖ℎ−𝑊_𝑖,Ξ_22𝑖=𝑟(𝜏)𝑄_1𝑖−𝑋_2𝑖ℎ+𝑊_𝑖+𝑊^⊺_𝑖−𝑋^⊺_2𝑖ℎ,Ξ_44𝑖=ℎ𝑋_2𝑖+𝑒^𝛿ℎ−𝛿ℎ−1𝛿²𝑌_𝑖−𝑉_𝑖−𝑉^⊺_𝑖,𝑟(𝜏)=−(1−𝜏)𝑒^𝛿ℎ−(),𝑖𝑓𝜏>1;1−𝜏,𝑖𝑓𝜏≤1. (15) Consider Λ=𝜆₂+ℎ𝑒^𝛿ℎ𝜆₃+𝜆₄+ℎ²𝑒^𝛿ℎ𝜆₅+𝜆₆+12𝜆₇ℎ³𝑒^𝛿ℎ,𝜆₁=min_𝑖∈𝒩𝜆_min𝑃_𝑖,𝜆₂=max_𝑖∈𝒩𝜆_max𝑃_𝑖,𝜆₃=max_𝑖∈𝒩𝜆_max𝑄_1𝑖,𝜆₄=max_𝑖∈𝒩𝜆_max𝑄_2𝑖,𝜆₅=max_𝑖∈𝒩𝜆_max𝑋_1𝑖,𝜆₆=max_𝑖∈𝒩𝜆_max𝑋_2𝑖,𝜆₇=max_𝑖∈𝒩𝜆_max𝑌_𝑖,𝜆₈=max_𝑖∈𝒩𝜆_max𝐻_𝑖. (16)(17)(18)(19)(20)

Proof. Define 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠). We consider the following Lyapunov-Krasovskii functional: 𝑉_𝜎(𝑡)𝑥_𝑡,𝑡=⁴_𝑙=1𝑉_{𝑙𝜎(𝑡)}𝑥_𝑡,𝑉,𝑡_1𝜎(𝑡)𝑥_𝑡,𝑡=𝑥^⊺(𝑡)𝑒^𝛿𝑡𝑃_𝜎(𝑡)𝑥(𝑡),𝑉_2𝜎(𝑡)𝑥_𝑡,𝑡=^𝑡_𝑡−ℎ𝑒^{𝛿(𝑠+ℎ)}𝑥^⊺(𝑠)𝑄_1𝜎(𝑡)𝑥(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝜏}𝑒^{𝛿(𝑠+ℎ)}𝑥^⊺(𝑠)𝑄_2𝜎(𝑡)𝑥(𝑠)𝑉𝑑𝑠,_3𝜎(𝑡)𝑥_𝑡,𝑡=⁰_−ℎ^𝑡_𝑡+𝜃𝑒^{𝛿(𝑠−𝜃)}𝑥^⊺(𝑠)𝑋_1𝜎(𝑡)𝑥(𝑠)𝑑𝑠𝑑𝜃+⁰_−ℎ^𝑡_𝑡+𝜃𝑒^{𝛿(𝑠−𝜃)}𝑥̇^⊺(𝑠)𝑋_2𝜎(𝑡)𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,_4𝜎(𝑡)𝑥_𝑡,𝑡=⁰_−ℎ⁰_𝜃^𝑡_𝑡+𝜐𝑒^{𝛿(𝑠−𝜃)}𝑥̇^⊺(𝑠)𝑌_𝜎(𝑡)𝑥̇(𝑠)𝑑𝑠𝑑𝜐𝑑𝜃. (21)
Taking the time derivative of 𝑉_𝜎(𝑡)(𝑥_𝑡,𝑡) along the trajectory of the system (1), one has 𝑉̇_1𝑖𝑥_𝑡,𝑡=𝑥^⊺(𝑡)𝑒^𝛿𝑡𝛿𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐴^⊺_𝑖𝑃_𝑖𝑥(𝑡)+2𝑒^𝛿𝑡𝑥^⊺(𝑡)𝑃_𝑖𝐵_𝑖𝑥(𝑡−𝜏(𝑡))+2𝑒^𝛿𝑡𝑥^⊺(𝑡)𝑃_𝑖𝐺_𝑖𝜔(𝑡),𝑉̇_2𝑖𝑥_𝑡,𝑡=𝑒^𝛿𝑡𝑥^⊺(𝑡)𝑒^𝛿ℎ𝑄_1𝑖+𝑄_2𝑖𝑥(𝑡)−𝑒^𝛿𝑡𝑥^⊺(𝑡−ℎ)𝑄_1𝑖𝑥()−(𝜏̇(𝑡𝑒𝑡−ℎ1−))^𝛿𝑡𝑥^⊺(𝑡−𝜏(𝑡))𝑄_2𝑖𝑥(𝑡−𝜏(𝑡)). (22)
Since 0≤𝜏(𝑡)≤ℎ, (23) we define 𝑟(𝜏)=−(1−𝜏)𝑒^𝛿ℎ,if−(),𝜏>1;1−𝜏if𝜏≤1. (24)
Then, 𝑉̇_2𝑖𝑥_𝑡,𝑡=𝑒^𝛿𝑡𝑥^⊺(𝑡)𝑒^𝛿ℎ𝑄_1𝑖+𝑄_2𝑖𝑥(𝑡)−𝑒^𝛿𝑡𝑥^⊺(𝑡−ℎ)𝑄_1𝑖𝑥(𝑡−ℎ)+𝑒^𝛿𝑡𝑥^⊺(𝑡−𝜏(𝑡))𝑟(𝜏)𝑄_2𝑖𝑥((𝑡,𝑉̇𝑡−𝜏))_3𝑖𝑥_𝑡,𝑡=𝑒^𝛿𝑡𝑥^⊺(𝑡)ℎ𝑋_1𝑖𝑥(𝑡)+𝑒^𝛿𝑡𝑥̇^⊺(𝑡)ℎ𝑋_2𝑖𝑥̇(𝑡)−^𝑡_𝑡−ℎ𝑒^{𝛿(𝑠+ℎ)}𝑥^⊺(𝑠)𝑋_1𝑖𝑥(𝑠)−𝑑𝑠_{𝑡𝑡−ℎ}𝑒^{𝛿(𝑠+ℎ)}𝑥̇^⊺(𝑠)𝑋_2𝑖𝑥̇(𝑠)𝑑𝑠≤𝑒^𝛿𝑡𝑥^⊺(𝑡)ℎ𝑋_1𝑖𝑥(𝑡)+𝑒^𝛿𝑡𝑥̇^⊺(𝑡)ℎ𝑋_2𝑖𝑥̇(𝑡)−𝑒^𝛿𝑡𝑡_{(𝑡)𝑡−𝜏}𝑥^⊺(𝑠)𝑋_1𝑖𝑥(𝑠)𝑑𝑠−𝑒^{𝛿𝑡𝑡−𝜏(𝑡)}_𝑡−ℎ𝑥^⊺(𝑠)𝑋_1𝑖𝑥(𝑠)𝑑𝑠−𝑒^𝛿𝑡𝑡_𝑡−ℎ𝑥̇^⊺(𝑠)𝑋_2𝑖𝑥̇(𝑠)𝑑𝑠. (25)
By Lemma 5, one can obtain −^𝑡_{(𝑡)𝑡−𝜏}𝑥^⊺(𝑠)𝑋_1𝑖𝑥(𝑠)𝑑𝑠≤−𝜏(𝑡)𝑈^⊺₁𝑋_1𝑖𝑈₁,−^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑥^⊺(𝑠)𝑋_1𝑖𝑥(𝑠)𝑑𝑠≤−(ℎ−𝜏(𝑡))𝑈^⊺₂𝑋_1𝑖𝑈₂, (26) where 𝑈₁=1𝜏(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)𝑑𝑠,𝑈₂=1ℎ−𝜏(𝑡)^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑑𝑠,lim_{𝜏(𝑡)→0}1𝜏(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)𝑑𝑠=𝑥(𝑡),lim_{𝜏(𝑡)→ℎ}1ℎ−𝜏(𝑡)^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑑𝑠=𝑥(𝑡−ℎ). (27)
From the Leibniz-Newton formula, the following equation is true for any matrices 𝑀_𝑖, 𝑁_𝑖, and 𝑉_𝑖 with appropriate dimensions: 2𝜏(𝑡)𝑈^⊺₁𝑀_𝑖+2(ℎ−𝜏(𝑡))𝑈^⊺₂𝑁_𝑖+2𝑥̇^⊺(𝑡)𝑉_𝑖×−𝑥̇(𝑡)+𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝜏(𝑡))+𝐺_𝑖𝜔(𝑡)=0. (28)
From Lemma 4, it yields −^𝑡_𝑡−ℎ𝑥̇^⊺(𝑠)𝑋_2𝑖𝑥̇(𝑠)𝑑𝑠=−^𝑡_{(𝑡)𝑡−𝜏}𝑥̇^⊺𝑋_2𝑖𝑥̇(𝑠)−𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}𝑥̇^⊺𝑋_2𝑖𝑥̇(𝑠)ℎ𝑑𝑠≤−ℎ−𝜏(𝑡)^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑥̇𝑑𝑠^⊺𝑋_2𝑖ℎ^{𝑡−𝜏(𝑡)}_𝑡−ℎ̇−ℎ𝑥𝑑𝑠𝜏(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑥̇𝑑𝑠^⊺𝑋_2𝑖ℎ^𝑡_{𝑡−𝜏(𝑡)}̇𝑥(𝑡)((𝑡𝑥((𝑡()𝑥𝑑𝑠≤−−𝑥𝑡−𝜏))𝑡−𝜏))−𝑥𝑡−ℎ^⊺×𝑋_2𝑖ℎ𝑊_𝑖∗𝑋_2𝑖ℎ𝑥(𝑡)((𝑡𝑥((𝑡()𝑉̇−𝑥𝑡−𝜏))𝑡−𝜏))−𝑥𝑡−ℎ_4𝑖𝑥_𝑡,𝑡=𝑒^𝛿𝑡𝑥̇^⊺(𝑡)𝑌_𝑖𝑥̇(𝑡)⁰_−ℎ⁰_𝜃𝑒^−𝛿𝜐𝑑𝜐𝑑𝜃−𝑒^𝛿𝑡0_−ℎ^𝑡_𝑡+𝜃𝑥̇^⊺(𝑠)𝑌_𝑖𝑥̇(𝑠)𝑑𝑠𝑑𝜃=𝑒^𝛿𝑡𝑥̇^⊺(𝑡)𝑒^𝛿ℎ−𝛿ℎ−1𝛿²𝑌_𝑖𝑥̇(𝑡)−𝑒^𝛿𝑡0_−ℎ^𝑡_𝑡+𝜃𝑥̇^⊺(𝑠)𝑌_𝑖𝑥̇(𝑠)𝑑𝑠𝑑𝜃. (29) By using Lemma 5, we have −⁰_−ℎ^𝑡_𝑡+𝜃𝑥̇(𝑠)𝑌_𝑖𝑥̇(𝑠)𝑑𝑠𝑑𝜃=−⁰_{(𝑡)−𝜏}^𝑡_𝑡+𝜃𝑥̇(𝑠)𝑌_𝑖𝑥̇(𝑠)−𝑑𝑠𝑑𝜃_{−𝜏(𝑡)−ℎ𝑡𝑡+𝜃}𝑥̇(𝑠)𝑌_𝑖𝑥̇(𝑠)2𝑑𝑠𝑑𝜃≤−𝜏²(𝑡)⁰_{(𝑡)−𝜏}^𝑡_𝑡+𝜃𝑥̇(𝑠)𝑑𝑠𝑑𝜃^⊺×𝑌_𝑖⁰_{(𝑡)−𝜏}^𝑡_𝑡+𝜃𝑥̇(𝑠)−2𝑑𝑠𝑑𝜃(ℎ−𝜏(𝑡))^{2−𝜏(𝑡)−ℎ𝑡𝑡+𝜃}𝑥̇(𝑠)𝑑𝑠𝑑𝜃^⊺×𝑌_𝑖^{−𝜏(𝑡)}_−ℎ^𝑡_𝑡+𝜃𝑥̇(𝑠)2𝑑𝑠𝑑𝜃=−𝜏²(𝑡)𝜏(𝑡)𝑥(𝑡)−^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)𝑑𝑠^⊺×𝑌_𝑖𝜏(𝑡)𝑥(𝑡)−^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)−2𝑑𝑠(ℎ−𝜏(𝑡))²(ℎ−𝜏(𝑡)𝑥(𝑡))−^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑑𝑠^⊺×𝑌_𝑖(ℎ−𝜏(𝑡)𝑥(𝑡))−^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑥(𝑡)−1𝑑𝑠=−2𝜏(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)𝑑𝑠^⊺×𝑌_𝑖𝑥(𝑡)−1𝜏(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)𝑥(𝑡)−1𝑑𝑠−2ℎ−𝜏(𝑡)^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑑𝑠^⊺×𝑌_𝑖𝑥(𝑡)−1ℎ−𝜏(𝑡)^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑥(𝑡)𝑑𝑠=−2−𝑈₁^⊺𝑌_𝑖𝑥(𝑡)−𝑈₁−2𝑥(𝑡)−𝑈₂^⊺𝑌_𝑖𝑥(𝑡)−𝑈₂. (30)
Therefore, for a given 𝜂>0 and from (21)–(30), one can obtain that 𝑉̇_𝑖𝑥_𝑡,𝑡−𝜂𝜔^⊺𝐻_𝑖𝜔(𝑡)≤𝑒^𝛿𝑡𝜉^⊺(𝑡)Ξ_𝑖𝜉(𝑡), (31)
where𝜉^⊺(𝑡)=𝑥^⊺(𝑡),𝑥^⊺(𝑡−𝜏(𝑡)),𝑥^⊺(𝑡−ℎ),𝑥̇^⊺(𝑡),𝑈^⊺₁,𝑈^⊺₂,𝜔^⊺(𝑡).Ξ_𝑖=Ξ_11𝑖Ξ_12𝑖𝑊_𝑖𝐴^⊺_𝑖𝑉^⊺_𝑖𝜏(𝑡)𝐴^⊺_𝑖𝑀^⊺_𝑖+𝑌_𝑖+𝑌^⊺_𝑖(ℎ−𝜏(𝑡))𝐴^⊺_𝑖𝑁^⊺_𝑖+𝑌_𝑖+𝑌^⊺_𝑖𝑃_𝑖𝐺_𝑖∗Ξ_22𝑖−𝑊_𝑖+𝑋_2𝑖ℎ𝐵^⊺_𝑖𝑉^⊺_𝑖𝜏(𝑡)𝐵^⊺_𝑖𝑀^⊺_𝑖(ℎ−𝜏(𝑡))𝐵^⊺_𝑖𝑁^⊺_𝑖0∗∗−𝑄_1𝑖−𝑋_2𝑖ℎ0000∗∗∗Ξ_44𝑖−𝜏(𝑡)𝑀^⊺_𝑖−(ℎ−𝜏(𝑡))𝑁^⊺_𝑖𝑉_𝑖𝐺_𝑖∗∗∗∗−𝜏(𝑡)𝑋_1𝑖−𝑌_𝑖−𝑌^⊺_𝑖0𝜏(𝑡)𝑀_𝑖𝐺_𝑖∗∗∗∗∗−(ℎ−𝜏(𝑡))𝑋_1𝑖−𝑌_𝑖−𝑌^⊺_𝑖(ℎ−𝜏(𝑡))𝑁_𝑖𝐺_𝑖∗∗∗∗∗∗−𝜂𝐻_𝑖. (32)
The LMIs (11) lead to 𝜏(𝑡)→ℎ and to 𝜏(𝑡)→0, respectively. It is easy to see that Ξ_1𝑖 results from Ξ_𝑖∣𝜏(𝑡)=ℎ and Ξ_2𝑖 results from Ξ_𝑖∣𝜏(𝑡)=0. Thus, we obtain 𝑉̇_𝑖𝑥_𝑡,𝑡−𝜂𝑉_𝑖𝑥_𝑡,𝑡<𝜂𝜔^⊺(𝑡)𝐻_𝑖𝜔(𝑡). (33)
Notice that 𝑑𝑒𝑑𝑡^−𝜂𝑡𝑉_𝑖𝑥_𝑡,𝑡<𝜂𝑒^−𝜂𝑡𝜔^⊺(𝑡)𝐻_𝑖𝜔(𝑡). (34)
Integrating (34) from 𝑡_𝑘 to 𝑡, we can get that 𝑉_𝑖𝑥_𝑡,𝑡<𝑒^{𝜂(𝑡−𝑡𝑘)}𝑉_𝑖𝑥_𝑡𝑘,𝑡_𝑘+𝜂^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠. (35)
Note that (12) and 𝜇≥1, it yields 𝑉_{𝜎(𝑡𝑘)}𝑥_𝑡𝑘,𝑡_𝑘≤𝜇𝑉_{𝜎(𝑡𝑘−1)}𝑥_𝑡𝑘,𝑡_𝑘. (36)
Then, we can easily have 𝑉_{𝜎(𝑡𝑘−1)}𝑥_𝑡𝑘,𝑡_𝑘<𝑒^{𝜂(𝑡𝑘−𝑡𝑘−1)}𝑉_{𝜎(𝑡𝑘−1)}𝑥_𝑡𝑘−1,𝑡_𝑘−1+𝜂^𝑡𝑘_𝑡𝑘−1𝑒^{𝜂(𝑡𝑘−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠. (37)
Thus, from (35)–(37), it yields 𝑉_𝜎(𝑡)𝑥_𝑡,𝑡≤𝑒^{𝜂(𝑡−𝑡𝑘)}𝑉_{𝜎(𝑡𝑘)}𝑥_𝑡𝑘,𝑡_𝑘+𝜂^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠≤𝜇𝑒^{𝜂(𝑡−𝑡𝑘)}𝑉_{𝜎(𝑡𝑘−1)}𝑥_𝑡𝑘,𝑡_𝑘+𝜂^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠≤𝜇𝑒^{𝜂(𝑡−𝑡𝑘−1)}𝑉_{𝜎(𝑡𝑘−1)}𝑥_𝑡𝑘−1,𝑡_𝑘−1+𝜂𝜇^𝑡𝑘_𝑡𝑘−1𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠+𝜂^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠≤𝜇²𝑒^{𝜂(𝑡−𝑡𝑘−2)}𝑉_{𝜎(𝑡𝑘−2)}𝑥_𝑡𝑘−2,𝑡_𝑘−2+𝜂𝜇^{2𝑡𝑘−1}_𝑡𝑘−2𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠+𝜂𝜇^𝑡𝑘_𝑡𝑘−1𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠+𝜂^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠≤⋯≤𝜇^{𝑁𝜎(0,𝑡)}𝑒^𝜂𝑡𝑉_𝜎(0)𝑥₀,0+𝜂𝜇^{𝑁𝜎(0,𝑡)𝑡1}₀𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠+𝜂𝜇^{𝑁𝜎(𝑡1,𝑡)𝑡2}_𝑡1𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠+⋯+𝜂^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠=𝜇^{𝑁𝜎(0,𝑡)}𝑒^𝜂𝑡𝑉_𝜎(0)𝑥₀,0+𝜂^𝑡₀𝑒^{𝜂(𝑡−𝑠)}𝜇^{𝑁𝜎(𝑠,𝑡)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠≤𝜇^{𝑁𝜎(0,𝑡)}𝑒^𝜂𝑇𝑉_𝜎(0)𝑥₀,0+𝜂𝜇^{𝑁𝜎(0,𝑡)}𝑑𝜆_max𝐻_𝑖𝑒^𝛼𝑇𝑡₀𝑒^−𝜂𝑠𝑑𝑠≤𝜇^{𝑁𝜎(0,𝑇)}𝑒^𝜂𝑇×𝑉_𝜎(0)𝑥₀,0+𝑑𝜆_max𝐻_𝑖𝜂^𝑇₀𝑒^−𝜂𝑠𝑑𝑠≤𝜇^{𝑇/𝜏𝑎}𝑒^𝜂𝑇𝑉_𝜎(0)𝑥₀,0+𝑑𝜆_max𝐻_𝑖1−𝑒^−𝜂𝑇≤𝜇^{𝑁𝜎(0,𝑇)}𝑒^𝜂𝑇×𝑉_𝜎(0)𝑥₀,0+𝑑𝜆_max𝐻_𝑖𝜂^𝑇₀𝑒^−𝜂𝑠𝑑𝑠=𝜇^{𝑇/𝜏𝑎}𝑒^𝜂𝑇𝑉_𝜎(0)𝑥₀,0+𝑑𝜆₈1−𝑒^−𝜂𝑇. (38)
Define 𝑃_𝑖=𝑅^−1/2𝑃_𝑖𝑅^−1/2, 𝑄_𝑠𝑖=𝑅^−1/2𝑄_𝑠𝑖𝑅^−1/2(𝑠=1,2), 𝑋_𝑙𝑖=𝑅^−1/2𝑋_𝑙𝑖𝑅^−1/2(𝑙=1,2),𝑌_𝑖=𝑅^−1/2𝑌_𝑖𝑅^−1/2.
Note that 𝑉_𝜎(0)𝑥₀,0=max_𝑖∈𝒩𝜆_max𝑃_𝑖𝑥^⊺(0)(0)+𝑅𝑥max_𝑖∈𝒩𝜆_max𝑄_1𝑖+max_𝑖∈𝒩𝜆_max𝑄_2𝑖𝑒^𝛿ℎ×⁰_−ℎ𝑒^𝛿𝑠𝑥^⊺(𝑠)𝑅𝑥(𝑠)𝑑𝑠+max_𝑖∈𝒩𝜆_max𝑋_1𝑖𝑒^𝛿ℎ0_−ℎ⁰_𝜃𝑒^−𝛿𝜃𝑥^⊺(𝑠)𝑅𝑥(𝑠)𝑑𝑠𝑑𝜃+max_𝑖∈𝒩𝜆_max𝑋_2𝑖𝑒^𝛿ℎ0_−ℎ⁰_𝜃𝑒^−𝛿𝜃𝑥̇^⊺(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠𝑑𝜃+max_𝑖∈𝒩𝜆_max𝑋_2𝑖𝑒^𝛿ℎ×⁰_−ℎ⁰_𝜃⁰_𝜐𝑒^−𝛿𝜃𝑥̇^⊺(𝑠)𝑅𝑥̇(𝑠)≤𝑑𝑠𝑑𝜃𝑑𝜐max_𝑖∈𝒩𝜆_max𝑃_𝑖+ℎ𝑒^𝛿ℎmax_𝑖∈𝒩𝜆_max𝑄_1𝑖+max_𝑖∈𝒩𝜆_max𝑄_2𝑖+ℎ²𝑒^𝛿ℎmax_𝑖∈𝒩𝜆_max𝑋_1𝑖+max_𝑖∈𝒩𝜆_max𝑋_2𝑖+12ℎ³𝑒^𝛿ℎmax_𝑖∈𝒩𝜆_max𝑌_𝑖×sup_{−ℎ≤𝑠≤0}{𝑥^⊺(𝑠)𝑅𝑥(𝑠),𝑥̇^⊺(𝑠)𝑅𝑥̇(𝑠≤𝜆)}₂+ℎ𝑒^𝛿ℎ𝜆₃+𝜆₄+ℎ²𝑒^𝛿ℎ𝜆₅+𝜆₆+12𝜆₇ℎ³𝑒^𝛿ℎ×sup_{−ℎ≤𝑠≤0}{𝑥^⊺(𝑠)𝑅𝑥(𝑠),𝑥̇^⊺(𝑠)𝑅𝑥̇(𝑠≤𝜆)}₂+ℎ𝑒^𝛿ℎ𝜆₃+𝜆₄+ℎ²𝑒^𝛿ℎ𝜆₅+𝜆₆+12𝜆₇ℎ³𝑒^𝛿ℎ𝑐₁=Λ𝑐₁. (39)
Thus, 𝑉_𝜎(𝑡)𝑥_𝑡,𝑡≤𝜇^{𝑇/𝜏𝑎}𝑒^𝛼𝑇Λ𝑐₁+𝑑𝜆₈1−𝑒^−𝜂𝑇=𝑒^{(𝛼+ln𝜇/𝜏𝑎)𝑇}Λ𝑐₁+𝑑𝜆₈1−𝑒^−𝜂𝑇. (40)
On the other hand, 𝑉_𝜎(𝑡)𝑥_𝑡,𝑡≥𝜆_min𝑃_𝑖𝑥^⊺(𝑡)𝑅𝑥(𝑡)=𝜆₁𝑥^⊺(𝑡)𝑅𝑥(𝑡). (41)
From (40) and (41), one obtains 𝑥^⊺(𝑡)𝑅𝑥(𝑡)≤Λ𝑐₁+𝑑𝜆₈1−𝑒^−𝜂𝑇𝜆₁𝑒^{(𝜂+ln𝜇/𝜏𝑎)𝑇}. (42)
When 𝜇=1, which is the trivial case, from (13), 𝑥^⊺(𝑡)𝑅𝑥(𝑡)<𝑐₂𝑒^−𝜂𝑇𝑒^𝜂𝑇=𝑐₂. When 𝜇≥1, from (13), ln(𝜆₁𝑐₂)−ln[Λ𝑐₁+𝑑𝜆₈(1−𝑒^−𝜂𝑇)]−𝛼𝑇>0, we have 𝑇𝜏_𝑎<ln𝜆₁𝑐₂−lnΛ𝑐₁+𝑑𝜆₈1−𝑒^−𝜂𝑇−𝜂𝑇=𝜆ln𝜇ln₁𝑐₂𝑒^−𝜂𝑇/Λ𝑐₁+𝑑𝜆₈1−𝑒^−𝜂𝑇.ln𝜇 (43)
Substituting (43) into (42) yields 𝑥^⊺(𝑡)𝑅𝑥(𝑡)<𝑐₂. (44) The proof is completed.

Remark 8. It should be mentioned that to reduce the conservatism, −∫⁰_{−𝜏(𝑡)}∫^𝑡_𝑡+𝜃𝑥̇^⊺(𝑠)𝑍_𝑖𝑥̇𝑑𝑠𝑑𝜃 and −∫^{−𝜏(𝑡)}_−ℎ∫^𝑡_𝑡+𝜃𝑥̇^⊺(𝑠)𝑍_𝑖𝑥̇𝑑𝑠𝑑𝜃 are bounded with −2[𝑥(𝑡)−𝑈₁]^⊺𝑍_𝑖[𝑥(𝑡)−𝑈₁] and −2[𝑥(𝑡)−𝑈₂]^⊺ 𝑍_𝑖[𝑥(𝑡)−𝑈₂], respectively.

Remark 9. One can clearly see that criteria given in Theorem 7 are delay-dependent. It is well known that the delay-dependent criteria are less conservative than delay-independent criteria when the delay is small or belongs to a given interval. Interval time-varying delay is a time delay that varies in an interval in which the lower bound is not restricted to zero.

Remark 10. In Theorem 7, the free weight matrix technology is employed, which reduces the conservativeness of the stability condition.

4. Finite-Time Weighted 𝐿₂-Gain Analysis

Theorem 11. System (3) is finite-time bounded with respect to (𝑐₁,𝑐₂,𝑅,𝑑,𝑇) if there exist symmetric positive matrices 𝑃_𝑖, 𝑄_𝑠𝑖  𝑠=1,2, 𝑋_𝑘𝑖(𝑘=1,2), 𝑌_𝑖 and matrices 𝑀_𝑖, 𝑁_𝑖, 𝑉_𝑖, scalars 𝛼≥0, 𝜇≥1, 𝜆_𝑙>0(𝑙=1,2,…,7), 𝑑>0, ℎ>0, Λ>0, 𝛾>0, 𝜏>0, 𝑟(𝜏), such that ∀𝑖,𝑗∈𝒩; we have the following linear matrix inequalities:Σ_1𝑖=Ξ_11𝑖Ξ_12𝑖𝑊_𝑖𝐴^⊺_𝑖𝑉^⊺_𝑖ℎ𝐴^⊺_𝑖𝑀^⊺_𝑖+𝑌_𝑖+𝑌^⊺_𝑖𝑌_𝑖+𝑌^⊺_𝑖𝑃_𝑖𝐺_𝑖𝐶^⊺_𝑖∗Ξ_22𝑖−𝑊_𝑖+𝑋_2𝑖ℎ𝐵^⊺_𝑖𝑉^⊺_𝑖ℎ𝐵^⊺_𝑖𝑀^⊺_𝑖000∗∗−𝑄_1𝑖−𝑋_2𝑖ℎ00000∗∗∗Ξ_44𝑖−ℎ𝑀^⊺_𝑖0𝑉_𝑖𝐺_𝑖0∗∗∗∗−ℎ𝑋_1𝑖−𝑌_𝑖−𝑌^⊺_𝑖0ℎ𝑀_𝑖𝐺_𝑖0∗∗∗∗∗−𝑌_𝑖−𝑌^⊺_𝑖00∗∗∗∗∗∗−𝛾²𝐼𝐷^⊺_𝑖Σ∗∗∗∗∗∗∗−𝐼<0,_2𝑖=Ξ_11𝑖Ξ_12𝑖𝑊_𝑖𝐴^⊺_𝑖𝑉^⊺_𝑖𝑌_𝑖+𝑌^⊺_𝑖ℎ𝐴^⊺_𝑖𝑁^⊺_𝑖+𝑌_𝑖+𝑌^⊺_𝑖𝑃_𝑖𝐺_𝑖𝐶^⊺_𝑖∗Ξ_22𝑖−𝑊_𝑖+𝑋_2𝑖ℎ𝐵^⊺_𝑖𝑉^⊺_𝑖0ℎ𝐵^⊺_𝑖𝑁^⊺_𝑖00∗∗−𝑄_1𝑖−𝑋_2𝑖ℎ00000∗∗∗Ξ_44𝑖0−ℎ𝑁^⊺_𝑖𝑉_𝑖𝐺_𝑖0∗∗∗∗−𝑌_𝑖−𝑌^⊺_𝑖000∗∗∗∗∗−ℎ𝑋_1𝑖−𝑌_𝑖−𝑌^⊺_𝑖ℎ𝑁_𝑖𝐺_𝑖0∗∗∗∗∗∗−𝛾²𝐼𝐷^⊺_𝑖∗∗∗∗∗∗∗−𝐼<0, (45)𝑃_𝑖<𝜇𝑃_𝑗,𝑄_1𝑖<𝜇𝑄_1𝑗,𝑄_2𝑖<𝜇𝑄_2𝑗,𝑋_1𝑖<𝜇𝑋_1𝑗,𝑋_2𝑖<𝜇𝑋_2𝑗,𝑌_𝑖<𝜇𝑌_𝑗,𝜆₁𝑐₂𝑒^−𝜂𝑇>Λ𝑐₁+𝑑𝛾²1𝜂1−𝑒^−𝜂𝑇, (46) with the average dwell time of the switching signal 𝜎 satisfying 𝜏_𝑎>𝜏^∗_𝑎=𝑇ln𝜇ln𝜆₁𝑐₂−lnΛ𝑐₁+𝑑𝛾²1/𝜂1−𝑒^−𝜂𝑇.−𝜂𝑇 (47)

Proof. Choose the same Lyapunov-Krasovskii functional as in Theorem 7; after some mathematical manipulation and Schur complement, we can get 𝑉̇_𝜎(𝑡)𝑥_𝑡,𝑡+𝑒^⊺(𝑡)𝑒(𝑡)−𝛾²𝜔^⊺(𝑡)𝜔(𝑡)=𝜉^⊺(𝑡)Σ_𝑠𝑖𝜉(𝑡),(𝑠=1,2). (48)
Define 𝐽(𝑡)=𝑒^⊺(𝑡)𝑒(𝑡)−𝛾²𝜔^⊺(𝑡)𝜔(𝑡). (49)
From (45), we obtain 𝑉̇_𝜎(𝑡)𝑥_𝑡,𝑡−𝜂𝑉_𝜎(𝑡)𝑥_𝑡,𝑡+𝐽(𝑡)<0. (50)
When 𝑡∈[𝑡_𝑘,𝑡_𝑘+1], where 𝑡_𝑘 is the 𝑘th switching instant, 𝑉_𝜎(𝑡)𝑥_𝑡,𝑡<𝑒^{𝜂(𝑡−𝑡𝑘)}𝑉_{𝜎(𝑡𝑘)}𝑥_𝑡𝑘,𝑡_𝑘−^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝐽(𝑠)𝑑𝑠. (51)
Notice that 𝑥(𝑡_𝑘)=𝑥(𝑡⁻_𝑘); then one obtains 𝑉_{𝜎(𝑡𝑘)}𝑥𝑡_𝑘,𝑡_𝑘≤𝜇𝑉_𝜎(𝑡⁻_𝑘)𝑥𝑡_𝑘,𝑡_𝑘. (52)
For any 𝑡∈[0,𝑇], one has 𝑉_𝜎(𝑡)𝑥_𝑡,𝑡≤𝑒^{𝜂(𝑡−𝑡𝑘)}𝑉_{𝜎(𝑡𝑘)}𝑥_𝑡𝑘,𝑡_𝑘+𝜂^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝐽(𝑠)𝑑𝑠≤𝜇𝑒^{𝜂(𝑡−𝑡𝑘−1)}𝑉_{𝜎(𝑡𝑘−1)}𝑥_𝑡𝑘−1,𝑡_𝑘−1+𝜂𝜇^𝑡𝑘_𝑡𝑘−1𝑒^{𝜂(𝑡−𝑠)}𝐽(𝑠)𝑑𝑠+𝜂^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝐽(𝑠)𝑑𝑠+𝜂𝜇^𝑡𝑘_𝑡𝑘−1𝑒^{𝜂(𝑡−𝑠)}𝜔^⊺(𝑠)𝐻_𝑖𝜔(𝑠)𝑑𝑠+𝜂^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝐽(𝑠)𝑑𝑠≤⋯≤𝜇^{𝑁𝜎(0,𝑡)}𝑒^𝜂𝑡𝑉_𝜎(0)𝑥₀,0+𝜂𝜇^{𝑁𝜎(0,𝑡)𝑡1}₀𝑒^{𝜂(𝑡−𝑠)}𝐽(𝑠)𝑑𝑠+𝜂𝜇^{𝑁𝜎(𝑡1,𝑡)𝑡2}_𝑡1𝑒^{𝜂(𝑡−𝑠)}𝐽(𝑠)𝑑𝑠+⋯+𝜂^𝑡_𝑡𝑘𝑒^{𝜂(𝑡−𝑠)}𝐽(𝑠)𝑑𝑠≤𝜇^{𝑁𝜎(0,𝑇)}𝑒^𝜂𝑇𝑉_𝜎(0)𝑥₀,0+𝜂^𝑇₀𝑒^{𝜂(𝑇−𝑠)}𝜇^{𝑁𝜎(𝑠,𝑇)}𝐽(𝑠)𝑑𝑠. (53)
Under zero initial condition, we have ^𝑇₀𝑒^−𝜂𝑠𝜇^{𝑁𝜎(𝑠,𝑇)}𝐽(𝑠)𝑑𝑠<0, (54) which implies that ^𝑇₀𝑒^−𝛼𝑠𝜇^{𝑁𝜎(𝑠,𝑇)}𝑧^⊺(𝑠)𝑧(𝑠)<𝑑𝑠_𝑇0𝜇^{𝑁𝜎(𝑠,𝑇)}𝛾²𝜔^⊺(𝑠)𝜔(𝑠)𝑑𝑠. (55)
Multiplying both sides of (55) by 𝜇^{−𝑁𝜎(0,𝑇)} yields ^𝑇₀𝑒^−𝛼𝑠𝜇^{−𝑁𝜎(0,𝑠)}𝑧^⊺(𝑠)𝑧(𝑠)<𝑑𝑠_𝑇0𝜇^{−𝑁𝜎(0,𝑠)}𝛾²𝜔^⊺(𝑠)𝜔(𝑠)𝑑𝑠. (56)
It is easy to deduce from (47) that 𝑁_𝜎()≤𝑠0,𝑠𝜏_𝑎≤ln𝜆₁𝑐₂/Λ𝑐₁+𝑑𝛾²1/𝜂1−𝑒^𝜂𝑇−𝜂𝑠.ln𝜇 (57)
Since 𝜇≥1, we have ^𝑇₀𝜇^{ln((𝜂𝑠−ln(𝜆1𝑐2/(Λ𝑐1+𝑑𝛾2(1/𝜂)(1−𝑒𝜂𝑇))))/ln𝜇)}×𝑧^⊺(𝑠)𝑧(𝑠)𝑑𝑠≤^𝑇₀𝜇^{−𝑁𝜎(0,𝑠)}𝑒^−𝜂𝑠𝑧^⊺(𝑠)𝑧(𝑠)≤𝑑𝑠_𝑇0𝜇^{−𝑁𝜎(0,𝑠)}𝑒^−𝜂𝑠𝛾²𝜔^⊺(𝑠)𝜔(𝑠)𝑑𝑠≤𝑒^{−𝜂𝑇𝑇}₀𝛾²𝜔^⊺(𝑠)𝜔(𝑠)𝑑𝑠. (58)
Therefore, we can obtain ^𝑇₀𝜆𝜂𝑠−ln¹𝑐₂Λ𝑐₁+𝑑𝛾²1/𝜂1−𝑒^𝜂𝑇𝑧^⊺(𝑠)𝑧(𝑠)𝑑𝑠≤𝛾²𝑒^−𝜂𝑇∫^𝑇₀𝜔^⊺(𝑠)𝜔(𝑠)𝑑𝑠. (59)
According to Definition 2, we know that Theorem 11 holds. This completes the proof.

Remark 12. It can be seen that Theorem 11 presents a more general result than [33–42] which contain the decay 𝜂; such a condition is derived by advantage of the simultaneous usage of the Lyapunov functional and the state variable transformation.

Remark 13. In many real applications, the minimum value of 𝛾²_min is of interest. In Theorem 11, with a fixed 𝜇 and 𝜂, 𝛾_min can be obtained through following procedure: min𝛾²s.t.LMIs((37)−(39)). (60)

5. Illustrative Example

Example 1. Consider the system (1) with the following data: 𝐴₁=−1.71.701.3−10.70.71−0.6,𝐴₂=,𝐵1−100.70−0.61.70−1.7₁=1.5−1.70.1−1.31−0.3−0.710.6,𝐵₂=,𝐺−100.11.3−0.10.61.50.11.8₁=𝐺₂=(𝑡)=00,𝜔(𝑡)=(𝑡)()(()([].100010001,𝑥0.70.03sin0.02cos2𝑡0.015sin𝑡+1+cos𝑡−2)),𝑡∈−𝜏,0 (61) The values of 𝑐₁, 𝑐₂, 𝑇, 𝑑 and matrix 𝑅 are given as follows: 𝑐₁=0.5,𝑇=10,𝑅=𝐼,𝑑=0.01,𝛿=0.042,𝜂=0.075. (62)

Through Theorem 7, when 𝑐₂=50, we can see that the admissible maximum ℎ computed by [36] is 0.2, and the maximal value of 𝜏 in this paper is 0.262. It is obvious that our method shows less conservatism result than that in [36]. Moreover, when 𝜏=0.2, the optimal bound with minimum value of 𝑐_2min relies on the parameter 𝜂. One can obtain clearly that Theorem 7 in our paper can indeed provide much smaller admissible 𝑐_2min than the stability criteria in [36] which shows the less conservative result in this paper.

6. Conclusions

In this paper, we have examined the problems of delay-dependent finite-time and 𝐿₂-gain analysis for switched systems with time-varying delay by allowing new Lyapunov-Krasovskii functional and average dwell time. A numerical example has also been given to demonstrate the effectiveness of the proposed approach. It is possible to extend the main results obtained in this paper to real systems which contain some practical constrains, for example, the failures of controllers. Details will be reported in our follow-up work in the future.

Acknowledgment

This work was supported by the Fundamental Research Funds for the Central Universities (103.1.2E022050205).