Robust Finite-Time 𝐻_∞ Control for Impulsive Switched Nonlinear Systems with State Delay

Abstract

This paper investigates robust finite-time 𝐻_∞ control for a class of impulsive switched nonlinear systems with time-delay. Firstly, using piecewise Lyapunov function, sufficient conditions ensuring finite-time boundedness of the impulsive switched system are derived. Then, finite-time 𝐻_∞ performance analysis for impulsive switched systems is developed, and a robust finite-time 𝐻_∞ state feedback controller is proposed to guarantee that the resulting closed-loop system is finite-time bounded with 𝐻_∞ disturbance attenuation. All the results are given in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to show the effectiveness of the proposed method.

1. Introduction

A switched system is a hybrid dynamical system consisting of a family of continuous-time or discrete-time subsystems and a switching law that orchestrates the switching between them [1]. In the last decades, in the stability analysis and stabilization for switched systems, lots of valuable results are established (see [2–5]). Most recently, on the basis of Lyapunov functions and other analysis tools, the stability problem of linear and nonlinear switched systems with time-delay has been further investigated (see [6–15]), and lots of valuable results are established for 𝐻_∞ control problems (see [16–22]).

It is well known that impulsive dynamical behaviors inevitably exist in some practical systems like physical, biological, engineering, and information science systems due to abrupt changes at certain instants during the dynamical process. Although hybrid system and switched system are important models for dealing with complex real systems, there is little work concerned with the above impulsive phenomena. Such a phenomenon can be modeled as an impulsive switched system, it is characteristic that their states change during the switching because of the occurrence of impulses [23].

In recent years, the impulsive switched systems have drawn more and more attention and many useful conclusions have been obtained. Multiple Krasovskii-Lyapunov function approach is employed to study the problem of ISS stability of a class of impulsive switched systems with time-delay in [24]. By the Lyapunov-Razumikhin technique, a delay-independent criterion of the exponential stability is established on the minimum dwell time in [25]. The problem of robust 𝐻_∞ stabilization of nonlinear impulsive switched system with time-delays is studied in [23].

Usually, the stability of a system is defined over an infinite-time interval. But in many practical systems, we focus on the dynamical behavior of a system over a fixed finite-time interval. Based on this, finite-time stability is first proposed by Dorato in 1961 [26]. Compared with the classical Lyapunov stability, finite-time stability is proposed for the study of the transient performance of the system, which is a totally different concept. The so-called finite-time stability means the boundedness of the state of a system over a fixed finite-time interval. Finite-time stability problems can be found in [27–32]. The finite-time stability of linear impulsive systems is analyzed in [33], the finite-time stability and stabilization of impulsive dynamic systems are carried out in [34–36]. The finite-time stability and stabilization of switched systems are investigated in [37].

Recently, robust finite-time control of switched systems is studied in [38, 39]. However, to the best of our knowledge, there are very few results on finite-time boundedness and robust 𝐻_∞ control of the impulsive switched systems, which motivates the present study. The paper is organized as follows. In Section 2, problem formulation and some necessary lemmas are given. In Section 3, based on the dwell time approach, finite-time boundedness and finite-time 𝐻_∞ performance for switched impulsive systems are addressed, and sufficient conditions for the existence of a robust finite-time 𝐻_∞ state feedback controller are proposed in terms of a set of matrix inequalities. Numerical examples are provided to show the effectiveness of the proposed approach in Section 4. Concluding remarks are given in Section 5.

Notations. The notations used in this paper are standard. The notation 𝑃>0 means that 𝑃 is a real positive definite matrix; diag{⋯} stands for a block-diagonal matrix; 𝜆_max(𝑃) and 𝜆_min(𝑃) denote the maximum and minimum eigenvalues of matrix 𝑃, respectively; ‖𝑥(𝑡)‖=𝑥^𝑇(𝑡)𝑥(𝑡) and ‖𝑥(𝑡)‖₂=(∫^∞₀‖𝑥(𝑡)‖²𝑑𝑡)^1/2.

2. Problem Formulation and Preliminaries

Consider the following impulsive switched system: 𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝑥(𝑡−ℎ)+𝐵_1𝜎(𝑡)𝑢₁(𝑡)+𝑓_𝜎(𝑡)(𝑥(𝑡))+𝐵_2𝜎(𝑡)𝑤(𝑡),𝑡≠𝑡_𝑘Δ𝑥=𝐸_𝜎(𝑡)𝑥(𝑡)+𝑢₂(𝑡),𝑡=𝑡_𝑘,𝑘=1,2,3,…𝑧(𝑡)=𝐶_𝜎(𝑡)𝑥(𝑡)+𝐷_𝜎(𝑡)𝑢₁(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈𝑡₀−ℎ,𝑡₀, (2.1a)(2.1b)(2.1c)(2.1d)where 𝑥(𝑡)∈𝑅^𝑛 is the state vector, 𝑧(𝑡)∈𝑅^𝑟 is the controlled output, 𝑤(𝑡)∈𝑅^𝑝 is the disturbance input which belongs to 𝐿₂[0,∞), 𝑢₁(𝑡)∈𝑅^𝑚, 𝑡≠𝑡_𝑘 is the switched control input, 𝑢₂(𝑡_𝑘)∈𝑅^𝑛 is the impulsive control input at 𝑡_𝑘, on the other hand, 𝑢₂(𝑡)=0, 𝑡≠𝑡_𝑘, 𝑘=1,2,3,…. 𝜎(𝑡)∶[𝑡₀,+∞)→𝑁={1,2,…,𝑁} is a switching signal. 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝜎(𝑡)=𝑖_𝑘, 𝑖_𝑘∈𝑁, 𝑘=0,1,2,3,…. Δ𝑥(𝑡)=𝑥(𝑡⁺)−𝑥(𝑡⁻),𝑥(𝑡⁺)=lim_ℎ→0⁺𝑥(𝑡+ℎ),𝑥(𝑡)=𝑥(𝑡⁻)=lim_ℎ→0⁺𝑥(𝑡−ℎ). 𝑡_𝑘, 𝑘=0,1,2,3,… are the impulsive jumping points or switching points. 𝑡₀ is the initial time, 𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯, and lim_𝑘→∞𝑡_𝑘=+∞. ℎ>0 is the time-delay which is a positive constant. 𝑓_𝑖(⋅)∶𝑅^𝑛→𝑅^𝑛,𝑖∈𝑁 is nonlinear vector-valued function. 𝜑(𝑡),𝑡∈[𝑡₀−ℎ,𝑡₀] is a continuous vector-valued initial function. 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐵_1𝑖, 𝑖∈𝑁 are uncertain real-valued matrices with appropriate dimensions, 𝐵_2𝑖, 𝐸_𝑖, 𝐶_𝑖, 𝐷_𝑖, 𝑖∈𝑁 are known real constant matrices with appropriate dimensions.

Assumption 2.1. For each 𝑖∈𝑁, 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐵_1𝑖 are uncertain real-valued matrices with appropriate dimensions. We assume that the uncertainties are of the form𝐴_𝑖=𝐴_𝑖+Δ𝐴_𝑖,𝐴_𝑑𝑖=𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖,𝐵_1𝑖=𝐵_1𝑖+Δ𝐵_1𝑖,Δ𝐴_𝑖Δ𝐴_𝑑𝑖Δ𝐵_1𝑖=𝐻_𝑖𝐹_𝑖(𝑡)𝐸_𝐴𝑖𝐸_𝐴𝑑𝑖𝐸_𝐵𝑖, (2.2a)(2.2b)where 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐵_1𝑖, 𝐻_𝑖, 𝐸_𝐴𝑖, 𝐸_𝐴𝑑𝑖, and 𝐸_𝐵𝑖 are known real-valued constant matrices with appropriate dimensions, 𝐹_𝑖(𝑡) is the uncertain matrix satisfying 𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼. (2.3)

Assumption 2.2. For each 𝑖∈𝑁, nonlinear vector-valued function 𝑓_𝑖 satisfies Lipschitz condition 𝑓_𝑖(𝑥(𝑡))≤𝑈_𝑖𝑥(𝑡), (2.4) where 𝑈_𝑖 is the Lipschitz constant matrix.

Assumption 2.3. For a given time constant 𝑇_𝑓>𝑡₀, the external disturbance 𝑤(𝑡) satisfies ^𝑇𝑓₀𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡≤𝑑². (2.5)

Assumption 2.4. For system (2.1a)–(2.1d), the impulsive jump matrices 𝐸_𝑖 satisfy that (𝐼+𝐸_𝑖) are invertible.

Definition 2.5 (see [32]). For a given time constant 𝑇_𝑓>𝑡₀, impulsive switched system (2.1a), (2.1b), (2.1c) and (2.1d) with 𝑢₁(𝑡)≡0, 𝑢₂(𝑡)≡0, and 𝑤(𝑡)≡0, is said to be finite-time stable with respect to (𝑐²₁,𝑐²₂,𝑇_𝑓,𝑅,𝜎(𝑡)) if the following inequality holds: sup_{𝑡0−ℎ≤𝜏≤𝑡0}𝑥^𝑇(𝜏)𝑅𝑥(𝜏)≤𝑐²₁⇒𝑥^𝑇(𝑡)𝑅𝑥(𝑡)<𝑐²₂,𝑡∈𝑡₀,𝑇_𝑓, (2.6) where 𝑐₂>𝑐₁>0, 𝑅 is a positive definite matrix, and 𝜎(𝑡) is a switching signal.

Remark 2.6. Equation (2.6) stands for the boundedness of the state of a system over a fixed finite-time interval (𝑡₀,𝑇_𝑓], when the initial state is bounded.

Definition 2.7 (see [40]). For a given time constant 𝑇_𝑓, impulsive switched system (2.1a)–(2.1d) with 𝑢₁(𝑡)≡0, 𝑢₂(𝑡)≡0, and 𝑤(𝑡) satisfying (2.5), is said to be finite-time bounded with respect to (𝑐²₁,𝑐²₂,𝑇_𝑓,𝑑²,𝑅,𝜎(𝑡)) if the condition (2.6) holds, where 𝑐₂>𝑐₁>0, 𝑅 is a positive definite matrix and 𝜎(𝑡) is a switching signal.

Definition 2.8. For any 𝑇₂>𝑇₁>0, let 𝑁_𝜎(𝑡)(𝑇₁,𝑇₂) denote the switching number of 𝜎(𝑡) on an interval (𝑇₁,𝑇₂). If 𝑁_𝜎(𝑡)(𝑇₁,𝑇₂)≤𝑁₀+(𝑇₂−𝑇₁)/𝜏_𝑎 holds for given 𝑁₀≥0, 𝜏_𝑎>0, then the constant 𝜏_𝑎 is called the average dwell time. In this paper we let 𝑁₀=0.

Definition 2.9. For a given time constant 𝑇_𝑓, impulsive switched system (2.1a)–(2.1d) with 𝑢₁(𝑡)≡0, 𝑢₂(𝑡)≡0 is said to have finite-time 𝐻_∞ performance with respect to (0,𝑐²₂,𝑇_𝑓,𝑑²,𝛾,𝑅,𝜎(𝑡)) if the system is finite-time bounded and the following inequality holds: ‖𝑧(𝑡)‖₂≤𝛾‖𝑤(𝑡)‖₂,∀𝑤(𝑡)∈𝐿₂[0,∞), (2.7) where 𝑐₂>0,𝛾>0, 𝑅 is a positive definite matrix and 𝜎(𝑡)is a switching signal.

Definition 2.10. For a given time constant 𝑇_𝑓, impulsive switched system (2.1a)–(2.1d) is said to be robust finite-time stabilization with 𝐻_∞ disturbance attenuation level 𝛾, if there exists a switched controller 𝑢₁(𝑡)=𝐾_𝜎(𝑡)𝑥(𝑡), 𝑡≠𝑡_𝑘 and an impulsive controller 𝑢₂(𝑡_𝑘)=𝐾_𝜎(𝑡)𝑥(𝑡_𝑘),𝑡=𝑡_𝑘, where 𝑡∈(𝑡₀,𝑇_𝑓] such that(i) the corresponding closed-loop system is finite-time bounded with respect to (0,𝑐²₂,𝑇_𝑓,𝑑²,𝑅,𝜎(𝑡));(ii) under zero initial condition, inequality (2.7) holds for any 𝑤(𝑡) satisfying (2.5).

Lemma 2.11. Let 𝑈, 𝑉, 𝑊, and 𝑋 be real matrices of appropriate dimensions with 𝑋 satisfying 𝑋=𝑋^𝑇, then for all 𝑉^𝑇𝑉≤𝐼, 𝑋+𝑈𝑉𝑊+𝑊^𝑇𝑉^𝑇𝑈^𝑇<0, (2.8) if and only if there exists a scalar 𝜀>0 such that 𝑋+𝜀𝑈𝑈^𝑇+𝜀⁻¹𝑊^𝑇𝑊<0. (2.9)

3. Main Results

3.1. Finite-Time Boundedness Analysis

In this subsection, we focus on the finite-time boundedness of the following impulsive switched system:𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝑥(𝑡−ℎ)+𝑓_𝜎(𝑡)(𝑥(𝑡))+𝐵_2𝜎(𝑡)𝑤(𝑡),𝑡≠𝑡_𝑘Δ𝑥=𝐸_𝜎(𝑡)𝑥(𝑡),𝑡=𝑡_𝑘,𝑘=1,2,3,…𝑥(𝑡)=𝜑(𝑡),𝑡∈𝑡₀−ℎ,𝑡₀. (3.1a)(3.1b)(3.1c)

Before proceeding to Lemma 3.2, we first introduce a function 𝑣(𝑡). For given positive definite matrices 𝑄_𝑖𝑘, 𝑖_𝑘∈𝑁, by Assumption 2.4, there exists a real number 𝜌_𝑖𝑘≥1, 𝜌^∗=max{𝜌_𝑖𝑘,𝑖_𝑘∈𝑁} such that 𝑄_𝑖𝑘−1≤𝜌_𝑖𝑘𝐼+𝐸_𝑖𝑘−1^𝑇𝑄_𝑖𝑘𝐼+𝐸_𝑖𝑘−1. (3.2) Furthermore, we define the following function 𝑣_𝑘(𝑡)=𝜌_𝑖𝑘−𝑡−𝑡_𝑘²𝑡_𝑘+1−𝑡_𝑘²𝜌_𝑖𝑘−1,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (3.3) Finally, a piecewise continuous function 𝑣(𝑡) is as follows: 𝑣(𝑡)=𝑣_𝑘(𝑡),𝑡∈𝑡_𝑘,𝑡_𝑘+1. (3.4) Consider the function 𝑣(𝑡), for each interval (𝑡_𝑘,𝑡_𝑘+1], 𝑣(𝑡⁺_𝑘)=𝜌_𝑖𝑘, 𝑣(𝑡_𝑘+1)=1, and 𝑣(𝑡) is monotonically nonincreasing and bounded function, 𝑣(𝑡_𝑘+1)≤𝑣(𝑡)≤𝑣(𝑡⁺_𝑘).

Remark 3.1. Note that the previous works require the condition 𝑄_𝑖𝑘−1≤(𝐼+𝐸_𝑖𝑘−1)^𝑇𝑄_𝑖𝑘(𝐼+𝐸_𝑖𝑘−1) (see [23, 41]), which can be obtained by setting 𝜌_𝑖𝑘=1 in (3.2). Thus, the proposed approach may provide more relaxed conditions.

Lemma 3.2. Consider the following Lyapunov functional candidate: 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃_𝜎(𝑡)𝑥(𝑡)+^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)𝑥(𝑠)𝑑𝑠 (3.5) for system (3.1a), (3.1b), and (3.1c), where 𝑃_𝑖 and 𝑄_𝑖, 𝑖∈𝑁 are symmetric positive definite matrices with appropriate dimensions.
The following inequality is derived: 𝑉̇(𝑡)≤2𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝑥̇(𝑡)+𝛼^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)𝑥(𝑠)(𝑡)𝑥𝑑𝑠+𝑣^𝑇(𝑡)𝑄_𝑖𝑘𝑥(𝑡)−𝑣(𝑡−ℎ)𝑥^𝑇(𝑡−ℎ)𝑄_{𝑖𝑘−𝑚}𝑥(𝑡−ℎ)𝑒^𝛼ℎ𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑡−ℎ∈𝑡_𝑘−𝑚,𝑡_{𝑘−𝑚+1},𝑚∈{0,1,2,3,…}. (3.6)

Proof. (i) When 𝑡_𝑘+ℎ≥𝑡_𝑘+1, 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃_𝜎(𝑡)𝑥(𝑡)+^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)𝑥(𝑠)𝑑𝑠=𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝑥(𝑡)+^{𝑡𝑘−𝑚+1}_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_{𝑖𝑘−𝑚}𝑥(𝑠)+𝑑𝑠_{𝑡𝑘−𝑚+2𝑡𝑘−𝑚+1}𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_{𝑖𝑘−𝑚+1}𝑥(𝑠)𝑑𝑠⋯+^𝑡_𝑡𝑘𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝑖𝑘𝑥(𝑠)𝑉̇(𝑡)𝑑𝑠,=2𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝑥̇(𝑡)+𝛼^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)𝑥(𝑠)𝑡𝑑𝑠+𝑣_{𝑘−𝑚+1}𝑥^𝑇𝑡_{𝑘−𝑚+1}𝑒^{𝛼(𝑡−𝑡𝑘−𝑚+1)}𝑄_{𝑖𝑘−𝑚}𝑥𝑡_{𝑘−𝑚+1}−𝑣(𝑡−ℎ)𝑥^𝑇(𝑡−ℎ)𝑒^𝛼ℎ𝑄_{𝑖𝑘−𝑚}𝑥()𝑡𝑡−ℎ+𝑣_{𝑘−𝑚+2}𝑥^𝑇𝑡_{𝑘−𝑚+2}𝑒^{𝛼(𝑡−𝑡𝑘−𝑚+2)}𝑄_{𝑖𝑘−𝑚+1}𝑥𝑡_{𝑘−𝑚+2}−𝑣𝑡⁺_{𝑘−𝑚+1}𝑥^𝑇𝑡⁺_{𝑘−𝑚+1}𝑒^{𝛼(𝑡−𝑡𝑘−𝑚+1)}𝑄_{𝑖𝑘−𝑚+1}𝑥𝑡⁺_{𝑘−𝑚+1}⋯+𝑣𝑡_𝑘𝑥^𝑇𝑡_𝑘𝑒^{𝛼(𝑡−𝑡𝑘)}𝑄_𝑖𝑘−1𝑥𝑡_𝑘−𝑣𝑡⁺_𝑘−1𝑥^𝑇𝑡⁺_𝑘−1𝑒^{𝛼(𝑡−𝑡𝑘)}𝑄_𝑖𝑘−1𝑥𝑡⁺_𝑘−1+𝑣(𝑡)𝑥^𝑇(𝑡)𝑄_𝑖𝑘𝑥(𝑡)−𝑣𝑡⁺_𝑘𝑥^𝑇𝑡⁺_𝑘𝑒^{𝛼(𝑡−𝑡𝑘)}𝑄_𝑖𝑘𝑥𝑡⁺_𝑘,𝑉̇(𝑡)=2𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝑥̇(𝑡)+𝛼^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)𝑥(𝑠)(𝑡)𝑥𝑑𝑠+𝑣^𝑇(𝑡)𝑄_𝑖𝑘𝑥(𝑡)−𝑣(𝑡−ℎ)𝑥^𝑇(𝑡−ℎ)𝑒^𝛼ℎ𝑄_{𝑖𝑘−𝑚}𝑥(𝑡−ℎ)+𝑥^𝑇𝑡_{𝑘−𝑚+1}𝑒^{𝛼(𝑡−𝑡𝑘−𝑚+1)}𝑄_{𝑖𝑘−𝑚}−𝜌_{𝑖𝑘−𝑚+1}𝐼+𝐸_{𝑖𝑘−𝑚}^𝑇𝑄_{𝑖𝑘−𝑚+1}𝐼+𝐸_{𝑖𝑘−𝑚}𝑥𝑡_{𝑘−𝑚+1}…+𝑥^𝑇𝑡_𝑘𝑒^{𝛼(𝑡−𝑡𝑘)}𝑄_𝑖𝑘−1−𝜌_𝑖𝑘𝐼+𝐸_𝑖𝑘−1^𝑇𝑄_𝑖𝑘𝐼+𝐸_𝑖𝑘−1𝑥𝑡_𝑘. (3.7) From (3.2), we can obtain that 𝑄_{𝑖𝑘−𝑚}−𝜌_{𝑖𝑘−𝑚+1}𝐼+𝐸_{𝑖𝑘−𝑚}^𝑇𝑄_{𝑖𝑘−𝑚+1}𝐼+𝐸_{𝑖𝑘−𝑚}⋮𝑄≤0_𝑖𝑘−1−𝜌_𝑖𝑘𝐼+𝐸_𝑖𝑘−1^𝑇𝑄_𝑖𝑘𝐼+𝐸_𝑖𝑘−1≤0. (3.8) Combining (3.7) and (3.8), (3.6) is obtained.(ii) When 𝑡_𝑘+ℎ<𝑡_𝑘+1,(1)𝑡∈(𝑡_𝑘,𝑡_𝑘+ℎ], the proof is similar to the proof line in the situation (i).(2)𝑡∈(𝑡_𝑘+ℎ,𝑡_𝑘+1], 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃_𝜎(𝑡)𝑥(𝑡)+^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)𝑥(𝑠)𝑑𝑠=𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝑥(𝑡)+^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝑖𝑘𝑥(𝑠)𝑑𝑠. (3.9)The proof for this situation is omitted.
The proof is completed.

Lemma 3.3. Consider the following Lyapunov function: 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃_𝜎(𝑡)𝑥(𝑡)+^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)𝑥(𝑠)𝑑𝑠 (3.10) for system (3.1a), (3.1b), and (3.1c), where 𝑃_𝑖 and 𝑄_𝑖,𝑖∈{1,2,…,𝑁} are symmetric positive definite matrices with appropriate dimensions. Under the condition −𝑒^𝛼ℎ𝜌^∗𝑃_𝑗𝐼+𝐸^𝑇_𝑗𝐸^𝑇_𝑖∗−𝑃_𝑖⁻¹0∗∗−𝑒^−𝛼ℎ𝜌^∗−1𝑄_𝑖⁻¹<0,∀𝑖,𝑗∈𝑁, (3.11) we have 𝑉𝑡⁺_𝑘<𝑒^𝛼ℎ𝜌^∗𝑉𝑡_𝑘, (3.12) where 𝜌^∗=max{𝜌_𝑖𝑘,𝑖_𝑘∈𝑁}.

Proof. Without loss of generality, let 𝜎(𝑡⁺_𝑘)=𝑖, 𝜎(𝑡_𝑘)=𝑗. Then, we have 𝑉𝑡⁺_𝑘=𝑥^𝑇𝑡⁺_𝑘𝑃_𝜎(𝑡⁺_𝑘)𝑥𝑡⁺_𝑘+^𝑡+𝑘_{𝑡+𝑘−ℎ}𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡+𝑘−𝑠)}𝑄_𝜎(𝑠)𝑥(𝑠)𝑑𝑠≤𝑥^𝑇𝑡⁺_𝑘𝑃_𝜎(𝑡⁺_𝑘)𝑥𝑡⁺_𝑘+𝑒^𝛼ℎ𝜌^∗𝑡+𝑘_{𝑡+𝑘−ℎ}𝑥^𝑇(𝑠)𝑄_𝜎(𝑠)𝑥(𝑠)𝑑𝑠≤𝑥^𝑇𝑡_𝑘𝐼+𝐸_𝑗^𝑇𝑃_𝑖𝐼+𝐸_𝑗𝑥𝑡_𝑘+𝑒^𝛼ℎ𝜌^∗𝑥^𝑇𝑡_𝑘𝐸^𝑇_𝑗𝑄_𝑖𝐸_𝑗𝑥𝑡_𝑘+𝑒^𝛼ℎ𝜌^∗𝑡𝑘_{𝑡𝑘−ℎ}𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡𝑘−𝑠)}𝑄_𝜎(𝑠)𝑥(𝑠)𝑑𝑠,𝑉𝑡_𝑘=𝑥^𝑇𝑡_𝑘𝑃_𝑗𝑥𝑡_𝑘+^𝑡𝑘_{𝑡𝑘−ℎ}𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡𝑘−𝑠)}𝑄_𝜎(𝑠)𝑥(𝑠)𝑑𝑠. (3.13)(3.14) Combining (3.13) with (3.14), we have 𝑉𝑡⁺_𝑘−𝑒^𝛼ℎ𝜌^∗𝑉𝑡_𝑘≤𝑥^𝑇𝑡_𝑘𝐼+𝐸_𝑗^𝑇𝑃_𝑖𝐼+𝐸_𝑗𝑥𝑡_𝑘+𝑒^𝛼ℎ𝜌^∗𝑥^𝑇𝑡_𝑘𝐸^𝑇_𝑗𝑄_𝑖𝐸_𝑗𝑥𝑡_𝑘−𝑒^𝛼ℎ𝜌^∗𝑥^𝑇𝑡_𝑘𝑃_𝑗𝑥𝑡_𝑘=𝑥^𝑇𝑡_𝑘Σ_𝑖𝑗𝑥𝑡_𝑘, (3.15) where _𝑖𝑗=𝐼+𝐸_𝑗^𝑇𝑃_𝑖𝐼+𝐸_𝑗+𝑒^𝛼ℎ𝜌^∗𝐸^𝑇_𝑗𝑄_𝑖𝐸_𝑗−𝑒^𝛼ℎ𝜌^∗𝑃_𝑗. (3.16) Using Schur complement, (3.11) is equivalent to Σ_𝑖𝑗<0or𝑉𝑡⁺_𝑘−𝑒^𝛼ℎ𝜌^∗𝑉𝑡_𝑘<0. (3.17) The proof is completed.

Theorem 3.4. 𝑅 is a positive definite matrix. Let 𝑃_𝑖=𝑅^−1/2𝑃_𝑖𝑅^−1/2, 𝑄_𝑖=𝑅^−1/2𝑄_𝑖𝑅^−1/2, For??all 𝑖∈𝑁, if there exist positive scalars 𝜌_𝑖≥1, 𝑖∈𝑁, 𝜌^∗=max{𝜌_𝑖,𝑖∈𝑁}, 𝛼, 𝜆₁, 𝜆₂, 𝜆₃ and symmetric positive matrices 𝑃_𝑖,𝑃_𝑗,𝑄_𝑖,𝑇_𝑖, 𝑖,𝑗∈𝑁 such that 1𝜌^∗𝐼+𝐸_𝑖⁻¹𝑄_𝑗𝐼+𝐸_𝑖^−𝑇−𝑄_𝑖≤0,∀𝑖,𝑗∈𝑁𝑃_𝑖𝐴^𝑇_𝑖+𝐴_𝑖𝑃_𝑖−𝛼𝑃_𝑖+𝐼𝐴_𝑑𝑖𝑄_𝑖𝐵_2𝑖𝑃_𝑖∗−𝑒^𝛼ℎ𝑄_𝑗00∗∗−𝑇_𝑖0∗∗∗−𝜌^∗−1𝑄_𝑖+𝑈^𝑇_𝑖𝑘𝑈_𝑖𝑘<0,∀𝑖,𝑗∈𝑁−𝑒^𝛼ℎ𝜌^∗𝑃_𝑗𝑃_𝑗𝐼+𝐸^𝑇_𝑗𝑃_𝑗𝐸^𝑇_𝑗∗−𝑃_𝑖0∗∗−𝑒^−𝛼ℎ𝜌^∗−1𝑄_𝑖<0,∀𝑖,𝑗∈𝑁𝜆₁𝑅⁻¹<𝑃_𝑖<𝑅⁻¹,𝜆₂𝑅⁻¹<𝑄_𝑖,𝑇_𝑖<𝜆₃𝐼,∀𝑖∈𝑁−𝑐²₂𝑒^{−𝛼𝑇𝑓}+𝑑²𝜆₃𝑐₁𝑐₁∗−𝜆₁01∗∗−𝜌^∗ℎ𝑒^−𝛼ℎ𝜆₂<0 (3.18)(3.19)(3.20)(3.21)(3.22) hold, under the average dwell time scheme 𝜏_𝑎>𝜏^∗_𝑎=𝑇_𝑓𝛼ℎ+ln𝜌^∗ln𝑐²₂𝑒^{−𝛼𝑇𝑓}−ln1/𝜆₁+ℎ𝜌^∗𝑒^𝛼ℎ/𝜆₂𝑐²₁+𝑑²𝜆₃, (3.23) system (3.1a)–(3.1c) is finite-time bounded with respect to (𝑐²₁,𝑐²₂,𝑇_𝑓,𝑑²,𝑅,𝜎(𝑡)).

Proof. Assuming that when 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝜎(𝑡)=𝑖_𝑘, 𝑖_𝑘∈𝑁, 𝑘=0,1,2,3,….
Choose the following Lyapunov functional candidate: 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃_𝜎(𝑡)⁻¹𝑥(𝑡)+^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)⁻¹𝑥(𝑠)𝑑𝑠. (3.24)
When 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], according to (3.18) and Lemma 3.2, we have 𝑉̇(𝑡)≤2𝑥^𝑇(𝑡)𝑃_𝑖⁻¹_𝑘𝑥̇(𝑡)+𝛼^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)⁻¹𝑥(𝑠)(𝑡)𝑥𝑑𝑠+𝑣^𝑇(𝑡)𝑄_𝑖⁻¹_𝑘𝑥(𝑡)−𝑣(𝑡−ℎ)𝑥^𝑇(𝑡−ℎ)𝑄_𝑖⁻¹_𝑘−𝑚𝑥(𝑡−ℎ)𝑒^𝛼ℎ,𝑉̇(𝑥(𝑡))−𝛼𝑉(𝑥(𝑡))−𝑤^𝑇(𝑡)𝑇_𝑖𝑘𝑤(𝑡)≤2𝑥^𝑇(𝑡)𝑃_𝑖⁻¹_𝑘𝑥̇(𝑡)+𝛼^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)⁻¹𝑥(𝑠)(𝑡)𝑥𝑑𝑠+𝑣^𝑇(𝑡)𝑄_𝑖⁻¹_𝑘𝑥(𝑡)−𝑣(𝑡−ℎ)𝑥^𝑇(𝑡−ℎ)𝑄_𝑖⁻¹_𝑘−𝑚𝑥(𝑡−ℎ)𝑒^𝛼ℎ−𝛼𝑥^𝑇(𝑡)𝑃_𝑖⁻¹_𝑘𝑥(𝑡)−𝛼^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)⁻¹𝑥(𝑠)𝑑𝑠−𝑤^𝑇(𝑡)𝑇_𝑖𝑘𝑤(𝑡)≤2𝑥^𝑇(𝑡)𝑃_𝑖⁻¹_𝑘𝑥̇(𝑡)+𝜌^∗𝑥^𝑇(𝑡)𝑄_𝑖⁻¹_𝑘𝑥(𝑡)−𝑥^𝑇(𝑡−ℎ)𝑄_𝑖⁻¹_𝑘−𝑚𝑥(𝑡−ℎ)𝑒^𝛼ℎ−𝛼𝑥^𝑇(𝑡)𝑃_𝑖⁻¹_𝑘𝑥(𝑡)−𝑤^𝑇(𝑡)𝑇_𝑖𝑘𝑤(𝑡). (3.25) According to (3.1a)–(3.1c), and (3.25), Assumption 2.2, and the fallowing inequality: 2𝑥^𝑇(𝑡)𝑃_𝑖⁻¹_𝑘𝑓(𝑥(𝑡))≤𝑓^𝑇_𝑖𝑘(𝑥(𝑡))𝑓_𝑖𝑘(𝑥(𝑡))+𝑥^𝑇(𝑡)𝑃_𝑖⁻¹_𝑘𝑃_𝑖⁻¹_𝑘𝑥(𝑡)≤𝑥^𝑇(𝑡)𝑈^𝑇_𝑖𝑘𝑈_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖⁻¹_𝑘𝑃_𝑖⁻¹_𝑘𝑥(𝑡), (3.26) we have 𝑉̇(𝑥(𝑡))−𝛼𝑉(𝑥(𝑡))−𝑤^𝑇(𝑡)𝑇_𝑖𝑘𝑤(𝑡)≤𝑋^𝑇(𝑡)Ξ_𝑘𝑋(𝑡), (3.27) where 𝑋^𝑇(𝑡)=(𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ)𝑤^𝑇(𝑡)), Ξ_𝑘=Δ_𝑘𝑃_𝑖⁻¹_𝑘𝐴_𝑑𝑖𝑘𝑃_𝑖⁻¹_𝑘𝐵_2𝑖𝑘∗−𝑒^𝛼ℎ𝑄_𝑖⁻¹_𝑘−𝑚0∗∗−𝑇_𝑖𝑘,Δ_𝑘=𝐴^𝑇_𝑖𝑘𝑃_𝑖⁻¹_𝑘+𝑃_𝑖⁻¹_𝑘𝐴_𝑖𝑘+𝜌^∗𝑄_𝑖⁻¹_𝑘−𝛼𝑃_𝑖⁻¹_𝑘+𝑈^𝑇_𝑖𝑘𝑈_𝑖𝑘+𝑃_𝑖⁻¹_𝑘𝑃_𝑖⁻¹_𝑘. (3.28) Using Schur complement, we obtain from (3.19) that 𝑂_𝑖𝑃_𝑖⁻¹𝐴_𝑑𝑖𝑃_𝑖⁻¹𝐵_2𝑖∗−𝑒^𝛼ℎ𝑄_𝑗⁻¹0∗∗−𝑇_𝑖<0, (3.29) where 𝑂_𝑖=𝐴^𝑇_𝑖𝑃_𝑖⁻¹+𝑃_𝑖⁻¹𝐴_𝑖+𝜌^∗𝑄_𝑖⁻¹−𝛼𝑃_𝑖⁻¹+𝑈^𝑇_𝑖𝑈_𝑖+𝑃_𝑖⁻¹𝑃_𝑖⁻¹. (3.30) Noticing that the above inequality holds for all 𝑖,𝑗∈𝑁, then we have Ξ_𝑘<0 for 𝑖_𝑘,𝑖_𝑘−1∈𝑁. Thus, 𝑉̇(𝑥(𝑡))−𝛼𝑉(𝑥(𝑡))−𝑤^𝑇(𝑡)𝑇_𝑖𝑘𝑤(𝑡)<0. (3.31) When 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], according to Lemma 3.3, we can obtain (3.12) from condition (3.20).
Combining (3.31) and (3.12), we can obtain that 𝑉(𝑡)<𝑒^{𝛼(𝑡−𝑡𝑘)}𝑉𝑡⁺_𝑘+^𝑡_𝑡𝑘𝑒^{𝛼(𝑡−𝑠)}𝑤^𝑇(𝑠)𝑇_𝑖𝑘𝑤(𝑠)𝑑𝑠<𝑒^{𝛼(𝑡−𝑡𝑘)}𝑒^𝛼ℎ𝜌^∗𝑉𝑡_𝑘+^𝑡_𝑡𝑘𝑒^{𝛼(𝑡−𝑠)}𝑤^𝑇(𝑠)𝑇_𝑖𝑘𝑤(𝑠)𝑑𝑠<𝑒^{𝛼(𝑡−𝑡𝑘)}𝑒^𝛼ℎ𝜌^∗𝑒^{𝛼(𝑡𝑘−𝑡𝑘−1)}𝑉𝑡⁺_𝑘−1+^𝑡𝑘_𝑡𝑘−1𝑒^{𝛼(𝑡𝑘−𝑠)}𝑤^𝑇(𝑠)𝑇_𝑖𝑘−1𝑤(𝑠)𝑑𝑠+^𝑡_𝑡𝑘𝑒^{𝛼(𝑡−𝑠)}𝑤^𝑇(𝑠)𝑇_𝑖𝑘𝑤(𝑠)𝑑𝑠<⋯<𝑒^{𝛼(𝑡−𝑡0)}𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑡0,𝑡)}𝑉𝑡₀+𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑡0,𝑡)𝑡1}_𝑡0𝑒^{𝛼(𝑡−𝑠)}𝑤^𝑇(𝑠)𝑇_𝑖0𝑤(𝑠)+𝑒𝑑𝑠_𝛼ℎ𝜌^{∗𝑁𝜎(𝑡1,𝑡)𝑡2}_𝑡1𝑒^{𝛼(𝑡−𝑠)}𝑤^𝑇(𝑠)𝑇_𝑖1𝑤(𝑠)𝑑𝑠+⋯+𝑒^𝛼ℎ𝜌^∗𝑡𝑘_𝑡𝑘−1𝑒^{𝛼(𝑡−𝑠)}𝑤^𝑇(𝑠)𝑇_𝑖𝑘−1𝑤(𝑠)𝑑𝑠+^𝑡_𝑡𝑘𝑒^{𝛼(𝑡−𝑠)}𝑤^𝑇(𝑠)𝑇_𝑖𝑘𝑤(𝑠)𝑑𝑠=𝑒^{𝛼(𝑡−𝑡0)}𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑡0,𝑡)}𝑉𝑡₀+^𝑡_𝑡0𝑒^{𝛼(𝑡−𝑠)}𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑠,𝑡)}𝑤^𝑇(𝑠)𝑇_𝑖𝑘𝑤(𝑠)𝑑𝑠<𝑒^𝛼𝑡𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑡0,𝑡)}𝑉𝑡₀+𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑡0,𝑡)}𝑒^𝛼𝑡𝑡_𝑡0𝑤^𝑇(𝑠)𝑇_𝑖𝑘𝑤(𝑠)𝑑𝑠<𝑒^𝛼𝑇𝑓𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑡0,𝑡)}𝑉𝑡₀+^𝑇𝑓_𝑡0𝑤^𝑇(𝑠)𝑇_𝑖𝑘𝑤(𝑠)𝑑𝑠<𝑒^𝛼𝑇𝑓𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑡0,𝑡)}𝑉𝑡₀+𝜆_max𝑇_𝑖𝑘𝑑². (3.32) Noticing that 𝑁_𝜎(𝑡₀,𝑇_𝑓)<𝑇_𝑓/𝜏_𝑎 and according to (3.21), we have 𝑉(𝑡)<𝑒^{(𝛼+𝛼ℎ/𝜏𝑎)𝑇𝑓}𝜌^{∗𝑇𝑓/𝜏𝛼}𝑉𝑡₀+𝜆₃𝑑²,𝑉(𝑡)≥𝑥^𝑇(𝑡)𝑃_𝑖⁻¹_𝑘𝑥(𝑡)=𝑥^𝑇(𝑡)𝑅^1/2𝑃_𝑖⁻¹_𝑘𝑅^1/2𝑥(𝑡)≥𝜆_min𝑃_𝑖⁻¹_𝑘𝑥^𝑇(𝑡)(𝑡)=1𝑅𝑥𝜆_max𝑃_𝑖𝑘𝑥^𝑇(𝑡)𝑅𝑥(𝑡). (3.33) Because 𝜆₁𝑅⁻¹<𝑃_𝑖<𝑅⁻¹, we have 𝑉(𝑡)>𝑥^𝑇(𝑡)𝑅𝑥(𝑡). (3.34) According to the Lyapunov function that we have chosen, we have 𝑉𝑡₀=𝑥^𝑇𝑡₀𝑃_𝑖⁻¹𝑥𝑡₀+^𝑡0_𝑡0−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{−𝛼(𝑡0−𝑠)}𝑄_𝑖⁻¹𝑥(𝑠)𝑑𝑠≤max_𝑖∈𝑁𝜆_max𝑃_𝑖⁻¹𝑥^𝑇𝑡₀𝑅𝑥𝑡₀+ℎ𝑒^𝛼ℎ𝜌^∗max_𝑖∈𝑁𝜆_max𝑄_𝑖⁻¹sup_{𝑡0−ℎ≤𝜃≤𝑡0}𝑥^𝑇(𝜃)(𝜃)≤1𝑅𝑥min_𝑖∈𝑁𝜆_min𝑃_𝑖+𝜌^∗ℎ𝑒^𝛼ℎmin_𝑖∈𝑁𝜆_min𝑄_𝑖sup_{𝑡0−ℎ≤𝜃≤𝑡0}𝑥^𝑇(𝜃)𝑅𝑥(𝜃). (3.35) According to (3.21), the following inequality is derived: 𝑉𝑡₀<1𝜆₁+𝜌^∗ℎ𝑒^𝛼ℎ𝜆₂𝑐²₁. (3.36) Combining (3.33), (3.34), and (3.36), we can obtain that 𝑥^𝑇(𝑡)𝑅𝑥(𝑡)<𝑉(𝑡)<𝑒^{(𝛼+𝛼ℎ/𝜏𝑎)𝑇𝑓}𝜌^{∗𝑇𝑓/𝜏𝛼}1𝜆₁+𝜌^∗ℎ𝑒^𝛼ℎ𝜆₂𝑐²₁+𝜆₃𝑑². (3.37) Using Schur complement, (3.22) is equivalent to 1𝜆₁+𝜌^∗ℎ𝑒^𝛼ℎ𝜆₂𝑐²₁+𝜆₃𝑑²<𝑐²₂𝑒^{−𝛼𝑇𝑓}. (3.38) From (3.38), we can obtain that 𝜏_𝑎>0.
Substituting (3.23) into (3.37) leads to 𝑥^𝑇(𝑡)𝑅𝑥(𝑡)<𝑐²₂. (3.39) Thus, system (3.1a)–(3.1c) is finite-time bounded with respect to (𝑐²₁,𝑐²₂,𝑇_𝑓,𝑑²,𝑅,𝜎(𝑡)).
The proof is completed.

Corollary 3.5. 𝑅 is a positive definite matrix, let 𝑤(𝑡)≡0, 𝑃_𝑖=𝑅^−1/2𝑃_𝑖𝑅^−1/2, 𝑄_𝑖=𝑅^−1/2𝑄_𝑖𝑅^−1/2 for??all ?𝑖∈𝑁. If there exist positive scalars 𝜌_𝑖≥1, 𝑖∈𝑁, 𝜌^∗=max{𝜌_𝑖,𝑖∈𝑁}, 𝛼,𝜆₁,𝜆₂ and symmetric positive matrices 𝑃_𝑖, 𝑃_𝑗, 𝑄_𝑖 for??all𝑖,𝑗∈𝑁 with appropriate dimensions such that 1𝜌^∗𝐼+𝐸_𝑖⁻¹𝑄_𝑗𝐼+𝐸_𝑖^−𝑇−𝑄_𝑖≤0,∀𝑖,𝑗∈𝑁𝑃_𝑖𝐴^𝑇_𝑖+𝐴_𝑖𝑃_𝑖−𝛼𝑃_𝑖+𝐼𝐴_𝑑𝑖𝑄_𝑖𝑃_𝑖∗−𝑒^𝛼ℎ𝑄_𝑗0∗∗−𝜌^∗−1𝑄_𝑖+𝑈^𝑇_𝑖𝑘𝑈_𝑖𝑘<0,∀𝑖,𝑗∈𝑁−𝑒^𝛼ℎ𝜌^∗𝑃_𝑗𝑃_𝑗𝐼+𝐸^𝑇_𝑗𝑃_𝑗𝐸^𝑇_𝑗∗−𝑃_𝑖0∗∗−𝑒^−𝛼ℎ𝜌^∗−1𝑄_𝑖<0,∀𝑖,𝑗∈𝑁𝜆₁𝑅⁻¹<𝑃_𝑖<𝑅⁻¹,𝜆₂𝑅⁻¹<𝑄_𝑖,∀𝑖∈𝑁−𝑐²₂𝑒^{−𝛼𝑇𝑓}𝑐₁𝑐₁∗−𝜆₁01∗∗−𝜌^∗ℎ𝑒^−𝛼ℎ𝜆₂<0 (3.40) hold with average dwell time 𝜏_𝑎>𝜏^∗_𝑎=𝑇_𝑓𝛼ℎ+ln𝜌^∗ln𝑐²₂𝑒^{−𝛼𝑇𝑓}−ln1/𝜆₁+𝜌^∗ℎ𝑒^𝛼ℎ/𝜆₂𝑐²₁. (3.41) System (3.1a)–(3.1c) with 𝑤(𝑡)≡0 is finite-time stable with respect to (𝑐²₁,𝑐²₂,𝑇_𝑓,𝑅,𝜎(𝑡)).

3.2. 𝐻_∞ Performance Analysis

In this subsection, 𝐻_∞ performance of the following system is investigated:𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝑥(𝑡−ℎ)+𝑓_𝜎(𝑡)(𝑥(𝑡))+𝐵_2𝜎(𝑡)𝑤(𝑡),𝑡≠𝑡_𝑘Δ𝑥=𝐸_𝜎(𝑡)𝑥(𝑡),𝑡=𝑡_𝑘,𝑘=1,2,3,…𝑧(𝑡)=𝐶_𝜎(𝑡)𝑥(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈𝑡₀−ℎ,𝑡₀ (3.42a)(3.42b)(3.42c)(3.42d)

Theorem 3.6. 𝑅 is a positive definite matrix. Let 𝑃_𝑖=𝑅^−1/2𝑃_𝑖𝑅^−1/2, 𝑄_𝑖=𝑅^−1/2𝑄_𝑖𝑅^−1/2 for all𝑖∈𝑁. Suppose that there exist positive scalars 𝜌_𝑖≥1,𝑖∈𝑁, 𝜌^∗=max{𝜌_𝑖,𝑖∈𝑁}, 𝛼,𝛾,𝜀 and symmetric positive matrices 𝑃_𝑖,𝑃_𝑗,𝑄_𝑖 for all𝑖,𝑗∈𝑁 such that 1𝜌^∗𝐼+𝐸_𝑖⁻¹𝑄_𝑗𝐼+𝐸_𝑖^−𝑇−𝑄_𝑖≤0,∀𝑖,𝑗∈𝑁𝑃_𝑖𝐴^𝑇_𝑖+𝐴_𝑖𝑃_𝑖−𝛼𝑃_𝑖+𝐼𝐴_𝑑𝑖𝑄_𝑖𝐵_2𝑖𝑃_𝑖𝑃_𝑖𝐶^𝑇_𝑖∗−𝑒^𝛼ℎ𝑄_𝑗000∗∗−𝛾²𝜌00∗∗∗−^∗−1𝑄_𝑖+𝑈^𝑇_𝑖𝑘𝑈_𝑖𝑘0∗∗∗∗−𝐼<0,∀𝑖,𝑗∈𝑁−𝑒^𝛼ℎ𝜌^∗𝑃_𝑗𝑃_𝑗𝐼+𝐸^𝑇_𝑗𝑃_𝑗𝐸^𝑇_𝑗∗−𝑃_𝑖0∗∗−𝑒^−𝛼ℎ𝜌^∗−1𝑄_𝑖<0,∀𝑖,𝑗∈𝑁𝑃_𝑖<𝑅⁻¹,∀𝑖∈𝑁−𝑐²₂+𝑒^𝛼𝑇𝑓𝛾²𝑑²<0 (3.43)(3.44)(3.45)(3.46)(3.47) hold with average dwell time 𝜏_𝑎>𝜏^∗_𝑎𝑇=max^𝑓𝛼ℎ+ln𝜌^∗ln𝑐²₂−ln𝑒^𝛼𝑇𝑓𝛾²𝑑²,ℎ𝜀. (3.48) Then, system (3.42a)–(3.42d) is finite-time bounded and has 𝐻_∞ performance with respect to (0,𝑐²₂,𝑇_𝑓,𝑑²,𝛾,𝑅,𝜎(𝑡)), where 𝛾²=𝑒^{(1+𝜀)𝛼𝑇𝑓}(𝜌^∗)^{𝜀𝑇𝑓/ℎ}𝛾².

Proof. When 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝜎(𝑡)=𝑖_𝑘, 𝑖_𝑘∈𝑁, 𝑘=0,1,2,3,…. Choose the following Lyapunov functional candidate for system (3.42a)–(3.42d) 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃_𝜎(𝑡)⁻¹𝑥(𝑡)+^𝑡_𝑡−ℎ𝑣(𝑠)𝑥^𝑇(𝑠)𝑒^{𝛼(𝑡−𝑠)}𝑄_𝜎(𝑠)⁻¹𝑥(𝑠)𝑑𝑠. (3.49) When 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑉̇(𝑥(𝑡))−𝛼𝑉(𝑥(𝑡))+𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)≤𝑋^𝑇(𝑡)Ψ_𝑘𝑋(𝑡), (3.50) where 𝑋^𝑇(𝑡)=(𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ)𝑤^𝑇(𝑡)), Ψ_𝑘=Δ_𝑘𝑃_𝑖⁻¹_𝑘𝐴_𝑑𝑖𝑘𝑃_𝑖⁻¹_𝑘𝐵_2𝑖𝑘∗−𝑒^𝛼ℎ𝑄_𝑖⁻¹_𝑘−𝑚0∗∗−𝛾²𝐼,Δ_𝑘=𝐴^𝑇_𝑖𝑘𝑃_𝑖⁻¹_𝑘+𝑃_𝑖⁻¹_𝑘𝐴_𝑖𝑘+𝜌^∗𝑄_𝑖⁻¹_𝑘−𝛼𝑃_𝑖⁻¹_𝑘+𝑈^𝑇_𝑖𝑘𝑈_𝑖𝑘+𝑃_𝑖⁻¹_𝑘𝑃_𝑖⁻¹_𝑘+𝐶^𝑇_𝑖𝑘𝐶_𝑖𝑘. (3.51) Using Schur complement, we obtain from (3.44) that 𝐸_𝑖𝑃_𝑖⁻¹𝐴_𝑑𝑖𝑃_𝑖⁻¹𝐵_2𝑖∗−𝑒^𝛼ℎ𝑄_𝑗⁻¹0∗∗−𝛾²𝐼<0, (3.52) where 𝐸_𝑖=𝐴^𝑇_𝑖𝑃_𝑖⁻¹+𝑃_𝑖⁻¹𝐴_𝑖+𝜌^∗𝑄_𝑖⁻¹−𝛼𝑃_𝑖⁻¹+𝑈^𝑇_𝑖𝑈_𝑖+𝑃_𝑖⁻¹𝑃_𝑖⁻¹+𝐶^𝑇_𝑖𝐶_𝑖.
Noticing that the above inequality holds for??all 𝑖,𝑗∈𝑁, then we have Ψ_𝑘<0, for 𝑖_𝑘,𝑖_𝑘−𝑚∈𝑁.
Thus, 𝑉̇(𝑥(𝑡))−𝛼𝑉(𝑥(𝑡))+𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)<0, (3.53)
Let 𝛾²𝑤^𝑇(𝑠)𝑤(𝑠)−𝑧^𝑇(𝑠)𝑧(𝑠)=Δ(𝑠), from (3.32), we have 𝑉(𝑡)<𝑒^{𝛼(𝑡−𝑡0)}𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑡0,𝑡)}𝑉𝑡₀+^𝑡_𝑡0𝑒^{𝛼(𝑡−𝑠)}𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑠,𝑡)}Δ(𝑠)𝑑𝑠. (3.54) Under zero initial condition, we have 0<^𝑡_𝑡0𝑒^{𝛼(𝑡−𝑠)}𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑠,𝑡)}Δ(𝑠)𝑑𝑠, (3.55) that is, ^𝑡_𝑡0𝑒^{𝛼(𝑡−𝑠)}𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑠,𝑡)}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠<^𝑡_𝑡0𝑒^{𝛼(𝑡−𝑠)}𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑠,𝑡)}𝛾²𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠. (3.56) Noticing that ^𝑡_𝑡0𝑒^{𝛼(𝑡−𝑠)}𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑠,𝑡)}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠>^𝑡_𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠. (3.57) Then, we have ^𝑡_𝑡0𝑒^{𝛼(𝑡−𝑠)}𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑠,𝑡)}𝛾²𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠<𝑒^𝛼𝑡𝑒^𝛼ℎ𝜌^{∗𝑁𝜎(𝑡0,𝑡)𝑡}_𝑡0𝛾²𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠. (3.58) Let 𝑡=𝑇_𝑓, because 𝜏_𝑎>ℎ/𝜀, we have ^𝑇𝑓_𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠<𝑒^{(1+𝜀)𝛼𝑇𝑓}𝜌∗^{𝜀𝑇𝑓/ℎ}𝛾^2𝑇𝑓_𝑡0𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠, (3.59) then ^𝑇𝑓_𝑡0𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠<𝛾^2𝑇𝑓_𝑡0𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠. (3.60) Thus, system (3.42a)–(3.42d) is finite-time bounded and has 𝐻_∞ performance with respect to (0,𝑐²₂,𝑇_𝑓,𝑑²,𝛾,𝑅,𝜎(𝑡)), where 𝛾²=𝑒^{(1+𝜀)𝛼𝑇𝑓}(𝜌^∗)^{𝜀𝑇𝑓/ℎ}𝛾².
The proof is completed.

Remark 3.7. When 𝜌^∗=1, Theorem 3.6 degenerates to the result of [41], which cannot guarantee the finite-time boundedness of the addressed system if 𝜌^∗>1.

3.3. Robust Finite-Time 𝐻_∞ Control

Consider system (2.1a)–(2.1d), under the switching controller 𝑢₁(𝑡)=𝐾_𝜎(𝑡)𝑥(𝑡), 𝑡≠𝑡_𝑘 and impulsive controller 𝑢₂(𝑡_𝑘)=𝐾_𝜎(𝑡)𝑥(𝑡_𝑘),? ?𝑡=𝑡_𝑘, the corresponding closed-loop system is given by𝑥̇(𝑡)=𝐴_𝜎(𝑡)+𝐵_1𝜎(𝑡)𝐾_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝑥(𝑡−ℎ)+𝑓_𝜎(𝑡)(𝑥(𝑡))+𝐵_2𝜎(𝑡)𝑤(𝑡),𝑡≠𝑡_𝑘Δ𝑥=𝐸_𝜎(𝑡)+𝐾_𝜎(𝑡)𝑥(𝑡),𝑡=𝑡_𝑘,𝑘=1,2,3,…𝑧(𝑡)=𝐶_𝜎(𝑡)+𝐷_𝜎(𝑡)𝐾_𝜎(𝑡)𝑥(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈𝑡₀−ℎ,𝑡₀. (3.61a)(3.61b)(3.61c)(3.61d)

Theorem 3.8. Consider impulsive switched system (2.1a)–(2.1d), let 𝑃_𝑖=𝑅^−1/2𝑃_𝑖𝑅^−1/2, 𝑄_𝑖=𝑅^−1/2𝑄_𝑖𝑅^−1/2 for??all???𝑖∈𝑁. If there exist positive scalars 𝜌_𝑖≥1, ??𝑖∈𝑁,𝜌^∗=max{𝜌_𝑖,𝑖∈𝑁}𝛼,𝛾,𝜀,𝛿_𝑖 and positive definite symmetric matrices 𝑃_𝑖,𝑄_𝑖, and matrices 𝑌_𝑖, ??𝑖∈𝑁, with appropriate dimensions, such that the following inequalities hold 1𝜌^∗𝐼+𝐸_𝑖⁻¹𝑄_𝑗𝐼+𝐸_𝑖^−𝑇−𝑄_𝑖≤0,∀𝑖,𝑗∈𝑁Γ_𝑖𝐴_𝑑𝑖𝑄_𝑖𝐵_2𝑖𝑃_𝑖𝑃_𝑖𝐶^𝑇_𝑖+𝑌^𝑇_𝑖𝐷^𝑇_𝑖𝑌^𝑇_𝑖𝐸^𝑇_𝐵𝑖+𝑃_𝑖𝐸^𝑇_𝐴𝑖∗−𝑒^𝛼ℎ𝑄_𝑗000𝑄_𝑖𝐸^𝑇_𝐴𝑑𝑖∗∗−𝛾²𝜌000∗∗∗−^∗−1𝑄_𝑖+𝑈^𝑇_𝑖𝑘𝑈_𝑖𝑘00∗∗∗∗−𝐼0∗∗∗∗∗−𝛿_𝑖<0,∀𝑖,𝑗∈𝑁, (3.62)(3.63) where Γ_𝑖=𝑃_𝑖𝐴^𝑇_𝑖+𝑌^𝑇_𝑖𝐵^𝑇_1𝑖+𝐴_𝑖𝑃_𝑖+𝐵_1𝑖𝑌_𝑖−𝛼𝑃_𝑖+𝐼+𝛿_𝑖𝐻_𝑖𝐻^𝑇_𝑖,−𝑒^𝛼ℎ𝜌^∗𝑃_𝑗𝑃_𝑗∗−𝑃_𝑖<0,∀𝑖,𝑗∈𝑁𝑃_𝑖<𝑅⁻¹,𝑖∈𝑁,−𝑐²₂+𝑒^𝛼𝑇𝑓𝛾²𝑑²<0. (3.64)(3.65)(3.66) Then, under the controller 𝐾_𝑖=𝑌_𝑖𝑃_𝑖⁻¹, 𝐾_𝑖=−𝐸_𝑖, and the following average dwell time scheme 𝜏_𝑎>𝜏^∗_𝑎𝑇=max^𝑓𝛼ℎ+ln𝜌^∗ln𝑐²₂−ln𝑒^𝛼𝑇𝑓𝛾²𝑑²,ℎ𝜀, (3.67) the corresponding closed-loop system is finite-time bounded with 𝐻_∞ performance with respect to (0,𝑐²₂,𝑇_𝑓,𝑑²,𝛾,𝑅,𝜎(𝑡)) and 𝛾²=𝑒^{(1+𝜀)𝛼𝑇𝑓}(𝜌^∗)^{𝜀𝑇𝑓/ℎ}𝛾².

Proof. According to Assumption 2.1, we have 𝐴_𝑖+𝐵_1𝑖𝐾_𝑖=𝐴_𝑖+𝐵_1𝑖𝐾_𝑖+𝐻_𝑖𝐹_𝑖𝐸_𝐴𝑖+𝐸_𝐵𝑖𝐾_𝑖,𝐴_𝑑𝑖=𝐴_𝑑𝑖+𝐻_𝑖𝐹_𝑖𝐸_𝐴𝑑𝑖. (3.68) Now replacing 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐶_𝑖 in the left side of (3.44) with 𝐴_𝑖+𝐵_1𝑖𝐾_𝑖,𝐴_𝑑𝑖, 𝐶_𝑖+𝐷_𝑖𝐾_𝑖, we can obtain that Θ_𝑖𝑗=Ω_𝑖𝐴_𝑑𝑖+𝐻_𝑖𝐹_𝑖𝐸_𝐴𝑑𝑖𝑄_𝑖𝐵_2𝑖𝑃_𝑖𝑃_𝑖𝐶_𝑖+𝐷_𝑖𝐾_𝑖^𝑇∗−𝑒^𝛼ℎ𝑄_𝑗000∗∗−𝛾²𝜌00∗∗∗−^∗−1𝑄_𝑖+𝑈^𝑇_𝑖𝑘𝑈_𝑖𝑘0,∗∗∗∗−𝐼 (3.69) where Ω_𝑖=𝐴_𝑖+𝐵_1𝑖𝐾_𝑖+𝐻_𝑖𝐹_𝑖𝐸_𝐴𝑖+𝐸_𝐵𝑖𝐾_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐵_1𝑖𝐾_𝑖+𝐻_𝑖𝐹_𝑖𝐸_𝐴𝑖+𝐸_𝐵𝑖𝐾_𝑖^𝑇−𝛼𝑃_𝑖+𝐼. (3.70) From (3.69), we know that Θ_𝑖𝑗=Π_1𝑖𝑗+Π_2𝑖𝑗, (3.71) where Π_1𝑖𝑗=Υ_1𝑖𝐴_𝑑𝑖𝑄_𝑖𝐵_2𝑖𝑃_𝑖𝑃_𝑖𝐶_𝑖+𝐷_𝑖𝐾_𝑖^𝑇∗−𝑒^𝛼ℎ𝑄_𝑗000∗∗−𝛾²𝜌00∗∗∗−^∗−1𝑄_𝑖+𝑈^𝑇_𝑖𝑘𝑈_𝑖𝑘0,Π∗∗∗∗−𝐼_2𝑖𝑗=Υ_2𝑖𝐻_𝑖𝐹_𝑖𝐸_𝐴𝑑𝑖𝑄_𝑖,000∗0000∗∗000∗∗∗00∗∗∗∗0 (3.72) with Υ_1𝑖=𝑃_𝑖𝐴_𝑖+𝐵_1𝑖𝐾_𝑖^𝑇+𝐴_𝑖+𝐵_1𝑖𝐾𝑃_𝑖−𝛼𝑃_𝑖Υ+𝐼,_2𝑖=𝑃_𝑖𝐸_𝐴𝑖+𝐸_𝐵𝑖𝐾_𝑖^𝑇𝐹^𝑇_𝑖𝐻^𝑇_𝑖+𝐻_𝑖𝐹_𝑖𝐸_𝐴𝑖+𝐸_𝐵𝑖𝐾_𝑖𝑃_𝑖, (3.73) let 𝑌_𝑖=𝐾_𝑖𝑃_𝑖, then Υ_1𝑖=𝑃_𝑖𝐴^𝑇_𝑖+𝑌^𝑇_𝑖𝐵^𝑇_1𝑖+𝐴_𝑖𝑃_𝑖+𝐵_1𝑖𝑌_𝑖−𝛼𝑃_𝑖Υ+𝐼,_2𝑖=𝑌^𝑇_𝑖𝐸^𝑇_𝐵𝑖+𝑃_𝑖𝐸^𝑇_𝐴𝑖𝐹^𝑇_𝑖𝐻^𝑇_𝑖+𝐻_𝑖𝐹_𝑖𝐸_𝐴𝑖𝑃_𝑖+𝐸_𝐵𝑖𝑌_𝑖. (3.74) From Lemma 2.11, we can obtain that Θ_𝑖𝑗=Π_1𝑖𝑗+𝐻_𝑖0000𝐹_𝑖𝐸_𝐴𝑖𝑃_𝑖+𝐸_𝐵𝑖𝑌_𝑖𝐸_𝐴𝑑𝑖𝑄_𝑖+𝑌000_𝑇𝑖𝐸^𝑇_𝐵𝑖+𝑃_𝑖𝐸^𝑇_𝐴𝑖𝑄_𝑖𝐸^𝑇_𝐴𝑑𝑖000𝐹_𝑖𝐻^𝑇_𝑖0000≤Π_1𝑖𝑗+𝛿_𝑖𝐻_𝑖0000𝐻^𝑇_𝑖+10000𝛿_𝑖𝑌^𝑇_𝑖𝐸^𝑇_𝐵𝑖+𝑃_𝑖𝐸^𝑇_𝐴𝑖𝑄_𝑖𝐸^𝑇_𝐴𝑑𝑖000𝐸_𝐴𝑖𝑃_𝑖+𝐸_𝐵𝑖𝑌_𝑖𝐸_𝐴𝑑𝑖𝑄_𝑖000. (3.75) Using Schur complement lemma, we get from (3.63) that Θ_𝑖𝑗<0. (3.76) Now we choose 𝐾_𝑖=−𝐸_𝑖, and replacing 𝐸_𝑖 in (3.45) with 𝐸_𝑖+𝐾_𝑖, we know that −𝑒^𝛼ℎ𝜌^∗𝑃_𝑗𝑃_𝑗0∗−𝑃_𝑖0∗∗−𝑒^−𝛼ℎ𝜌^∗−1𝑄_𝑖<0, (3.77) by (3.64), we know that the condition(3.45) hold.
Then, system (2.1a)–(2.1d) is robust finite-time bounded with 𝐻_∞ performance with respect to (0,𝑐²₂,𝑇_𝑓,𝑑²,𝛾,𝑅,𝜎(𝑡)), and 𝛾²=𝑒^{(1+𝜀)𝛼𝑇𝑓}(𝜌^∗)^{𝜀𝑇𝑓/ℎ}𝛾².
The proof is completed.

Remark 3.9. In order to eliminate the impulsive jump, we design an impulsive feedback controller 𝐾_𝑖=−𝐸_𝑖, 𝑡=𝑡_𝑘. Then the system becomes a switched system with continuous states.

4. Numerical Examples

In this section, we present two examples to illustrate the effectiveness of the proposed approach.

Example 4.1. Consider system (2.1a)–(2.1d) with the following parameters.

Subsystem 1
𝐴₁=−812−7,𝐴_𝑑1=−1.30.10.2−1,𝐻₁=−0.10.30.20.4,𝐸_𝐴𝑑1=,𝐸0.20.10.3−0.2₁=0.43000.15,𝑈₁=0.1000.1,𝐵₁₁=,𝐸3−304_𝐵1=0.20.10−0.3,𝐵₂₁=,𝐶1−121₁=1102,𝐷₁=0.8000.2,𝐸_𝐴1=,−0.3−0.10.2−0.1 (4.1)𝑓₁(𝑥(𝑡))=0.1sin𝑥(𝑡), where ‖𝑓₁(𝑥(𝑡))‖<‖𝑈₁𝑥(𝑡)‖.

Subsystem 2
𝐴₂=−721−6,𝐴_𝑑2=−1.20.10.3−1.1,𝐻₂=−0.10.2−0.2−0.1,𝐸_𝐴𝑑2=,𝐸−0.30.10.2−0.3₂=0.15000.4,𝑈₂=0.2000.18,𝐵₁₂=4−116,𝐸_𝐵2=,𝐵−0.30.100.2₂₂=−1020.8,𝐶₂=2103,𝐷₂=0.801−1,𝐸_𝐴2=,0.10.30.2−0.2 (4.2)𝑓₂(𝑥(𝑡))=0.18cos𝑥(𝑡), where ‖𝑓₂(𝑥(𝑡))‖<‖𝑈₂𝑥(𝑡)‖.
Choosing 𝑇_𝑓=12,ℎ=0.2,𝑑²=10,𝑅=𝐼,??𝛼=0.1,𝐶²₂=2,𝜀=0.1,??𝛾²=0.5441,??𝜌^∗=1, solving the LMIs in (3.62)–(3.66) leads to 𝑄₁=,𝑄1.3506−0.1265−0.12650.7891²=0.50420.05250.05250.3221,𝑌₁=,𝑌0.0234−0.35770.16310.2680₂=,𝑃−0.0001−0.52210.11090.0371¹=,𝑃0.98870.00110.00110.9921²=,𝐾0.9995−0.0001−0.00011.0006₁=0.0241−0.36050.16470.2699,𝐾₂=,𝑄−0.0001−0.52180.11090.0371₁₋₁−𝐼+𝐸₁^𝑇𝑄₁⁻¹𝐼+𝐸₁𝑄≤0,₂₋₁−𝐼+𝐸₂^𝑇𝑄₁⁻¹𝐼+𝐸₂𝑄≤0,₂₋₁−𝐼+𝐸₂^𝑇𝑄₂⁻¹𝐼+𝐸₂𝑄≤0,₁₋₁−𝐼+𝐸₁^𝑇𝑄₂⁻¹𝐼+𝐸₁≤0, (4.3)𝜏_𝑎>𝜏^∗_𝑎=1.2049, we choose 𝜏_𝑎=2, 𝛾²=𝑒^{(1+𝜀)𝛼𝑇𝑓}(𝜌^∗)^{𝜀𝛼𝑇𝑓}𝛾²=2.0368, then the system is finite-time bounded according to [41, Theorem 3].

Example 4.2. Consider system (2.1a)–(2.1d) with the following parameters.

Subsystem 1
𝐴₁=−812−7,𝐴_𝑑1=−1.30.10.2−1,𝐻₁=−0.10.30.20.4,𝐸_𝐴𝑑1=,𝐸0.20.10.3−0.2₁=−0.100−0.1,𝑈₁=0.01000.01,𝐵₁₁=3−304,𝐸_𝐵1=,𝐵0.20.10−0.3₂₁=1−121,𝐶₁=−310−2,𝐷₁=8002,𝐸_𝐴1=,−3−0.10.2−1 (4.4)𝑓₁(𝑥(𝑡))=0.01sin𝑥(𝑡).

Subsystem 2
𝐴₂=−721−6,𝐴_𝑑2=−1.20.10.3−1.1,𝐻₂=−0.10.2−0.2−0.1,𝐸_𝐴𝑑2=,𝐸−0.30.10.2−0.3₂=−0.100−0.1,𝑈₂=0.02000.08,𝐵₁₂=4−116,𝐸_𝐵2=,𝐵−0.30.100.2₂₂=−1020.8,𝐶₂=−210−3,𝐷₂=8018,𝐸_𝐴2=,−10.30.2−2 (4.5)𝑓₂(𝑥(𝑡))=0.02cos𝑥(𝑡).(1) Let ℎ=0.2,𝑇_𝑓=12,𝑑²=10, 𝑅=𝐼,?𝛼=0.001,??𝐶²₂=21,??𝜌^∗=1.3,𝛾²=0.9344. By solving the LMIs in (3.62)–(3.66), we can get 𝑄₁=,𝑄0.42520.03870.03871.2272²=0.43520.04700.04701.2369,𝑌₁=,𝑌0.0866−0.4834−0.08630.5554₂=,𝑃0.1064−0.2575−0.12600.2934¹=,𝑃0.46060.04180.04180.9965²=,𝐾0.5364−0.0611−0.06110.9884₁=0.2329−0.4949−0.23890.5673,𝐾₂=,0.1699−0.2500−0.20240.2844 (4.6)?and 𝜏_𝑎>𝜏^∗_𝑎=3.8340. We choose 𝜏_𝑎=4, 𝜀=0.05, 𝛾²=0.9464, the initial condition 𝑥(𝑡)=0, 𝑡∈[−ℎ,0], the switching signal is shown in Figure 1, and state trajectories of the closed-loop system are shown in Figure 2.

Figure 1 

Switching signal.

Figure 2 

State trajectories of the closed-loop system.

We can see from Figure 2 that the states of the system are continuous due to the feedback 𝐾_𝑖 in impulsive instants.(2) Let ℎ=0.2, 𝑇_𝑓=12, 𝑑²=10, 𝑅=𝐼,??and? ?𝛼=0.001. By solving the LMIs of [41, Theorem 3], we can get 𝑄₁=,𝑄0.40150.03590.03591.0563²=0.52240.11040.11041.0717,𝑌₁=,𝑌0.1245−0.6523−0.09980.5952₂=,𝑃0.1279−0.2380−0.10060.2699¹=,𝑃0.55770.00990.00990.9992²=,𝐾0.55770.00990.00990.9992₁=0.2349−0.6552−0.18960.5976,𝐾₂=,𝑄0.2336−0.2405−0.18520.2720₁₋₁−𝐼+𝐸₁^𝑇𝑄₁⁻¹𝐼+𝐸₁𝑄>0,₂₋₁−𝐼+𝐸₂^𝑇𝑄₁⁻¹𝐼+𝐸₂𝑄>0,₂₋₁−𝐼+𝐸₂^𝑇𝑄₂⁻¹𝐼+𝐸₂𝑄>0,₁₋₁−𝐼+𝐸₁^𝑇𝑄₂⁻¹𝐼+𝐸₁>0. (4.7)Obviously, the above inequalities do not satisfy the conditions of [41, Theorem 3]. Thus, we cannot draw the conclusion that the closed-loop system is finite-time bounded from Theorem 3 in [41].

5. Conclusions

This paper has investigated robust finite-time 𝐻_∞ control for a class of impulsive switched nonlinear systems with time-delay. Based on piecewise Lyapunov function, sufficient conditions which guarantee finite-time boundedness of the impulsive switched system are derived. Then, a feedback control scheme consisting of an impulsive feedback controller and a switching controller is proposed, and the proposed control strategy can guarantee that the closed-loop system is finite-time bounded with 𝐻_∞ disturbance attenuation level. Finally, the results are illustrated by means of two numerical examples.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant nos. 61074023, 60974027, and 61273120.