Exponential Stabilization of Impulsive Switched Systems with Time Delays Using Guaranteed Cost Control

Abstract

This paper investigates the stabilization problem for impulsive switched systems with time delays. First, exponential stability criteria of the delayed impulsive switched systems are established by use of the Lyapunov-Krasovskii functional method. Based on these results, sufficient conditions for the existence of a guaranteed cost control are also given. Subject to these sufficient conditions, the closed-loop impulsive switched system under the guaranteed cost control law will be exponentially stable with a guaranteed cost value.

1. Introduction

During the evolution process of a variety of disciplines such as engineering, economics, and biology, there often exists an impressive characteristic that the system state changes rapidly due to state jumping and the coexistence of continuous dynamics and discrete events. These jumping effects and switching phenomena can be modeled as an impulsive switched system. The system stability problem is one of main problems for such impulsive switched systems.

Recently, there are many results on stability analysis of impulsive switched systems. For instance, the robust stabilization problem for a class of impulsive switched systems under the LQ guaranteed cost control is studied in [1]. In addition, sufficient conditions for the existence of a guaranteed cost control law are given as well. In [2], sufficient conditions, independent of time delays and impulsive switching intervals, are derived by using a Lyapunov-Krasovskii’s technique, ensuring asymptotical stability of impulsive switched systems with time invariant delays. Moreover, stability criteria problems for uncertain impulsive switched systems with input delay are studied by using the receding horizon method in [3]. Appropriate switching controllers are designed and linear matrix inequality conditions are derived to guarantee the closed-loop uncertain impulsive switched systems under a designed delayed controller. Other relevant references can be found in [4–9].

On the other hand, time delay has become a common phenomenon frequently occurred in science and engineering. Stability of time delay systems has received increasing attention among the applied mathematics and control community; for example, see [10–12] and the references therein. Most of these results are derived by use of either the Razumikhin method or the Lyapunov-Krasovskii method.

Since both impulses and time delays are ubiquitous in the real world, it is necessary to analyze their stability performance of dynamical systems involving impulses, time delays, and switchings. There already existed some effective results for the stability analysis of either impulsive switched systems or time delay systems mentioned above. However, there are few results on impulsive switched systems with time delays. This motivates our research in this paper. Our aim is to apply a set of improved Lyapunov functions to analyze the exponential stability performance of the impulsive switched systems with time delays, to derive sufficient conditions to ensure the exponential stability, and to design guaranteed cost controllers based on the linear matrix inequality approach. The designed guaranteed cost controller can ensure not only exponential stability of the closed-loop system but also the guaranteed cost performance.

The remaining sections of the paper proceed as follows. Section 2 provides necessary propositions and definitions. In Section 3, sufficient conditions for ensuring the closed-loop system exponentially stable are presented, and the designed guaranteed cost controller for impulsive switched systems with time delays are presented. Finally, conclusion remarks are presented in Section 4.

2. Problem Statement

Consider the following impulsive switched system with time delay:𝑥̇(𝑡)=𝐴_𝑖𝑘𝑥(𝑡)+𝐵_𝑖𝑘𝑥(𝑡−ℎ(𝑡))+𝐶_𝑖𝑘𝑢(𝑡)𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝐷_𝑘𝑥(𝑡)𝑡=𝑡_𝑘,𝑥(𝑡)=𝜑(𝑡)−1≤𝑡≤0,𝐴_𝑖𝑘=𝐴_𝑖𝑘+Δ𝐴_𝑖𝑘,𝐵_𝑖𝑘=𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘, (1a)(1b)(1c)(1d)where 𝑥(𝑡)∈𝑅^𝑛 is the state, 𝑢(𝑡)∈𝑅^𝑃 is the control input, 𝑛,𝑝∈𝑁. 𝐴_𝑖𝑘, 𝐵_𝑖𝑘, 𝐶_𝑖𝑘, and 𝐷_𝑘 are constant real matrices with appropriate dimensions. The delay function ℎ(𝑡) satisfies the condition 0≤ℎ₁≤ℎ(𝑡)≤ℎ₂. (2)Δ𝑥(𝑡)=𝑥(𝑡⁺)−𝑥(𝑡⁻), 𝑥(𝑡⁺)=lim_𝑣→0⁺𝑥(𝑡+𝑣), lim_𝑣→0⁺𝑥(𝑡−𝑣)=𝑥(𝑡⁻_𝑘)=𝑥(𝑡_𝑘), where 𝑖_𝑘∈{1,2,3,…,𝑚}, 𝑘∈𝑁, 𝑚∈𝑁. 𝑡_𝑘 is an impulsive switching point, 𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯<𝑡_∞. Δ𝐴_𝑖𝑘(⋅) and Δ𝐵_𝑖𝑘(⋅) are unknown real norm-bounded matrix valued functions, representing time-varying parameter uncertainties. The admissible uncertainties are assumed to be of the form Δ𝐴_𝑖𝑘(𝑡)Δ𝐵_𝑖𝑘(𝑡)=𝐸_𝑖𝑘Ψ_𝑖𝑘(𝑡)𝐻_𝑖𝑘𝐽_𝑖𝑘. (3)𝐸_𝑖𝑘, 𝐻_𝑖𝑘, 𝐽_𝑖𝑘 are known real constant matrices, Ψ_𝑖𝑘(𝑡) is an unknown real time-varying matrix satisfying Ψ^𝑇_𝑖𝑘(𝑡)Ψ_𝑖𝑘(𝑡)<𝐼, and 𝐼 is the identity matrix with appropriate dimension.

Consider the following cost function: 𝐽=^∞_𝑘=0^𝑡𝑘+1_𝑡𝑘𝑥^𝑇(𝑡)𝑅_𝑖𝑘𝑥(𝑡)+𝑢^𝑇(𝑡)𝑄_𝑖𝑘𝑢(𝑡)𝑑𝑡, (4) where 𝑅_𝑖𝑘>0 and 𝑄_𝑖𝑘>0 are positive definite symmetric matrices.

Definition 1. Suppose that there exist a control law 𝑢(𝑡)=𝐹_𝑖𝑘𝑥(𝑡) and a positive constant 𝐽^∗ such that, for all admissible uncertainties, the closed-loop system ((1a), (1b), (1c), and (1d)) is exponentially stable and the cost function (4) satisfies 𝐽≤𝐽^∗, and then 𝑢(𝑡) is the exponential guaranteed cost control law for the impulsive switched system with time delay. Here, 𝐽^∗ is the guaranteed cost value.

In addition, the following lemmas are necessary to proceed.

Lemma 2 (see [13, Proposition 2.2]). Let 𝐴, 𝐷, 𝐸, and 𝐹 be real matrices of appropriate dimensions with ‖𝐹‖≤1. Then, for any matrix 𝑃=𝑃^𝑇>0 and scalar 𝜀>0, 𝜀𝐼−𝐸𝑃𝐸^𝑇()>0,𝐴+𝐷𝐹𝐸𝑃(𝐴+𝐷𝐹𝐸)^𝑇≤𝐴𝑃𝐴^𝑇+𝐴𝑃𝐸^𝑇𝜀𝐼−𝐸𝑃𝐸^𝑇−1𝐸𝑃𝐴^𝑇+𝜀𝐷𝐷^𝑇. (5)

Lemma 3 (Schur complement lemma (see [14])). Given constant matrices 𝑋, 𝑌, and 𝑍 with appropriate dimensions satisfying 𝑋=𝑋^𝑇 and 𝑌=𝑌^𝑇>0. then 𝑋+𝑍^𝑇𝑌⁻¹𝑍<0 if and only if 𝑋𝑍^𝑇𝑍−𝑌<0. (6)

Lemma 4 (integral inequality [15]). For any constant matrix 𝑍=𝑍^𝑇>0 and a scalar b such that the following integrations are well defined, 𝑏^𝑡_𝑡−𝑏𝑥(𝑠)^𝑇𝑍𝑥(𝑠)𝑑𝑠≥^𝑡_𝑡−𝑏𝑥(𝑠)𝑑𝑠^𝑇𝑍^𝑡_𝑡−𝑏𝑥(𝑠)𝑑𝑠. (7)

3. Main Results

First, we will present sufficient conditions for exponential stability of the impulsive switched system ((1a), (1b), (1c), and (1d)) without control input.

Theorem 5. Consider the time-delayed impulsive switched system ((1a), (1b), (1c), and (1d)) with the cost function (4). Suppose there exist 𝛼>0, 𝜀₁>0, 𝑃_𝑖𝑘=𝑃^𝑇_𝑖𝑘>0, such that the following conditions are satisfied: (i)𝑘₁₁0𝜆₂𝑆₁𝜆₃𝑆₂0∗𝑘₂₂𝜆₃𝑆₃𝜆₃𝑆₃0∗∗𝑘₃₃00∗∗∗𝑘₄₄0∗∗∗∗𝑘₅₅<0, (8)(ii)𝑃_𝑖𝑘−1(𝐼+𝐷_𝑘)^𝑇𝑃_𝑖𝑘𝑃_𝑖𝑘𝐼+𝐷_𝑘𝑃_𝑖𝑘>0, (9)
where, 𝑘₁₁=𝐴_𝑖𝑘^𝑇𝑃_𝑖𝑘+𝑃_𝑖𝑘𝐴_𝑖𝑘+𝑈₁+𝑈₂−𝜆₂𝑆₁−𝜆₃𝑆₂+2𝛼𝑃_𝑖𝑘+𝑅_𝑖𝑘,𝑘₂₂=𝐵_𝑖𝑘^𝑇𝑃_𝑖𝑘+𝑃_𝑖𝑘𝐵_𝑖𝑘−2𝜆₃𝑆₃,𝑘₃₃=−𝜆₂𝑈₁−𝜆₂𝑆₁−𝜆₃𝑆₃,𝑘₄₄=−𝜆₃𝑈₂−𝜆₃𝑆₂−𝜆₃𝑆₃,𝑘₅₅=ℎ²₁𝑆₁+ℎ²₂𝑆₂+ℎ₂−ℎ₁²𝑆₃. (10) Then, the impulsive switched system ((1a), (1b), (1c), and (1d)) is exponentially stable.

Proof. When 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], consider the Lyapunov-Krasovskii function candidate 𝑉𝑡,𝑥_𝑡=⁴_𝑖=1𝑉_𝑖𝑡,𝑥_𝑡, (11) where 𝑉₁𝑡,𝑥_𝑡=𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝑥(𝑡),𝑉₂𝑡,𝑥_𝑡=^𝑡_𝑡−ℎ1𝑒^{2𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑈₁𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ2}𝑒^{2𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑈₂𝑥(𝑠)𝑉𝑑𝑠,₃𝑡,𝑥_𝑡=ℎ₁⁰_−ℎ1^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑠)}𝑥̇^𝑇(𝜏)𝑆₁𝑥̇(𝜏)𝑑𝜏𝑑𝑠+ℎ₂⁰_−ℎ2^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑠)}𝑥̇^𝑇(𝜏)𝑆₂𝑥̇(𝜏)𝑉𝑑𝜏𝑑𝑠,₄𝑡,𝑥_𝑡=ℎ₂−ℎ₁×^−ℎ1_−ℎ2^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑠)}𝑥̇^𝑇(𝜏)𝑆₃𝑥̇(𝜏)𝑑𝜏𝑑𝑠. (12) Let Ω=𝜆_max𝑃_𝑖𝑘+ℎ₁𝜆_max𝑈₁+ℎ₂𝜆_max𝑈₂+0.5ℎ³₁𝜆_max𝑆₁+0.5ℎ³₂𝜆_max𝑆₂+ℎ²₂−ℎ²₁ℎ₂−ℎ₁𝜆_max𝑆₃; (13) then, we can easily check that 𝜆‖𝑥(𝑡)‖²≤𝑉𝑡,𝑥_𝑡≤Ω‖𝑥(𝑡)‖²,𝑡∈𝑅⁺. (14) Taking the derivative of 𝑉_𝑖(𝑡,𝑥_𝑡), 𝑖=1,2,3,4, along the trajectory of the closed-loop system, we obtain 𝑉₁̇𝑡,𝑥_𝑡=[𝐴_𝑖𝑘𝑥(𝑡)+𝐵_𝑖𝑘𝑥(𝑡−ℎ(𝑡))]^𝑇𝑃_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝐴_𝑖𝑘𝑥(𝑡)+𝐵_𝑖𝑘𝑥(𝑡−ℎ(𝑡))=𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑘𝑃_𝑖𝑘+𝑃_𝑖𝑘𝐴_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡−ℎ(𝑡))𝐵^𝑇_𝑖𝑘𝑃_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝐵^𝑇_𝑖𝑘𝑥(𝑡−ℎ(𝑡)),𝑉₂̇𝑡,𝑥_𝑡=−2𝛼𝑒^{−2𝛼𝑡𝑡}_𝑡−ℎ1𝑒^2𝛼𝑠𝑥^𝑇(𝑠)𝑈₁𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ2}𝑒^2𝛼𝑠𝑥^𝑇(𝑠)𝑈₂𝑥(𝑠)𝑑𝑠+𝑒^−2𝛼𝑡𝑒^2𝛼𝑡𝑥^𝑇(𝑡)𝑈₁𝑥(𝑡)−𝑒^{2𝛼(𝑡−ℎ1)}𝑥^𝑇𝑡−ℎ₁×𝑈₁𝑥𝑡−ℎ₁+𝑒^2𝛼𝑡𝑥^𝑇(𝑡)𝑈₂𝑥(𝑡)−𝑒^{2𝛼(𝑡−ℎ2)}𝑥^𝑇𝑡−ℎ₂𝑈₁𝑥𝑡−ℎ₂=−2𝛼𝑉₂+𝑥^𝑇(𝑡)𝑈₁+𝑈₂𝑥(𝑡)−𝜆₂𝑥^𝑇𝑡−ℎ₁𝑈₁𝑥𝑡−ℎ₁−𝜆₃𝑥^𝑇𝑡−ℎ₂𝑈₁𝑥𝑡−ℎ₂,𝑉₃̇𝑡,𝑥_𝑡=−2𝛼𝑉₃+𝑒^−2𝛼𝑡𝑑ℎ𝑑𝑡¹⁰_−ℎ1^𝑡_𝑡+𝑠𝑒^2𝛼𝜏𝑥̇^𝑇(𝜏)𝑆₁𝑥̇(𝜏)𝑑𝜏𝑑𝑠+ℎ₂⁰_−ℎ2^𝑡_𝑡+𝑠𝑒^2𝛼𝜏𝑥̇^𝑇(𝜏)𝑆₂𝑥̇(𝜏)𝑑𝜏𝑑𝑠=−2𝛼𝑉₃+𝑥̇^𝑇(𝑡)ℎ²₁𝑆₁+ℎ²₂𝑆₂𝑥̇(𝑡)−ℎ₁⁰_−ℎ1𝑒^2𝛼𝑠𝑥̇^𝑇(𝑡+𝑠)𝑆₁𝑥̇(𝑡+𝑠)𝑑𝑠−ℎ₂⁰_−ℎ2𝑒^2𝛼𝑠𝑥̇^𝑇(𝑡+𝑠)𝑆₂𝑥̇(𝑡+𝑠)𝑑𝑠. (15)(16)(17) From Lemma 4, we have 𝑉₃̇𝑡,𝑥_𝑡≤−2𝛼𝑉₃+𝑥̇^𝑇(𝑡)ℎ²₁𝑆₁+ℎ²₂𝑆₂𝑥̇(𝑡)−𝑥^𝑇(𝑡)𝜆₂𝑆₁+𝜆₃𝑆₂𝑥(𝑡)+2𝜆₂𝑥^𝑇(𝑡)𝑆₁𝑥𝑡−ℎ₁−𝜆₂𝑥^𝑇𝑡−ℎ₁𝑆₁𝑥𝑡−ℎ₁+2𝜆₃𝑥^𝑇(𝑡)𝑆₂𝑥𝑡−ℎ₂−𝜆₃𝑥^𝑇𝑡−ℎ₂𝑆₂𝑥𝑡−ℎ₂,𝑉₄̇𝑡,𝑥_𝑡=−2𝛼𝑉₄+𝑒^−2𝛼𝑡ℎ₂−ℎ₁×^−ℎ1_−ℎ2𝑒^2𝛼𝑡𝑥̇^𝑇(𝑡)𝑆₃𝑥̇(𝑡)−𝑒^{2𝛼(𝑡+𝑠)}𝑥̇^𝑇(𝑡+𝑠)𝑆₃𝑥̇(𝑡+𝑠)𝑑𝑠=−2𝛼𝑉₄+ℎ₂−ℎ₁²𝑥̇^𝑇(𝑡)𝑆₃𝑥̇(𝑡)−ℎ₂−ℎ₁^−ℎ1_−ℎ2𝑒^2𝛼𝑠𝑥̇^𝑇(𝑡+𝑠)𝑆₃𝑥̇(𝑡+𝑠)𝑑𝑠≤−2𝛼𝑉₄+ℎ₂−ℎ₁²𝑥̇^𝑇(𝑡)𝑆₃𝑥̇(𝑡)−ℎ₂−ℎ₁𝑒^{−2𝛼ℎ2−ℎ}_−ℎ2𝑥̇^𝑇(𝑡+𝑠)𝑆₃𝑥̇()+𝑡+𝑠𝑑𝑠_{−ℎ1−ℎ}𝑥̇^𝑇(𝑡+𝑠)𝑆₃𝑥̇(𝑡+𝑠)𝑑𝑠≤−2𝛼𝑉₄+ℎ₂−ℎ₁²𝑥̇^𝑇(𝑡)𝑆₃𝑥̇(𝑡)−𝜆₃2𝑥^𝑇(𝑡−ℎ(𝑡))𝑆₃𝑥(𝑡−ℎ(𝑡))−2𝑥^𝑇(𝑡−ℎ(𝑡))𝑆₃𝑥𝑡−ℎ₂+𝑥^𝑇𝑡−ℎ₂𝑆₃𝑥𝑡−ℎ₂+𝑥^𝑇𝑡−ℎ₁𝑆₃𝑥𝑡−ℎ₁−2𝑥^𝑇(𝑡−ℎ(𝑡))𝑆₃𝑥𝑡−ℎ₁. (18)(19) Then, we add (15) to (19) together to yield 𝑉̇𝑡,𝑥_𝑡+2𝛼𝑉𝑡,𝑥_𝑡≤𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑘𝑃_𝑖𝑘+𝑃_𝑖𝑘𝐴_𝑖𝑘+𝑈₁+𝑈₂+2𝛼𝑃_𝑖𝑘−𝜆₂𝑆₁−𝜆₃𝑆₂+𝑅_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡−ℎ(𝑡))𝐵^𝑇_𝑖𝑘𝑃_𝑖𝑘+𝑃_𝑖𝑘𝐵^𝑇_𝑖𝑘−2𝜆₃𝑆₃×𝑥(𝑡−ℎ(𝑡))+𝑥^𝑇𝑡−ℎ₁−𝜆₂𝑈₁−𝜆₂𝑆₁−𝜆₃𝑆₃𝑥𝑡−ℎ₁+𝑥^𝑇𝑡−ℎ₂−𝜆₃𝑈₂−𝜆₃𝑆₂−𝜆₃𝑆₃𝑥𝑡−ℎ₂+2𝜆₃𝑥^𝑇(𝑡)𝑆₂𝑥𝑡−ℎ₂+2𝜆₂𝑥^𝑇(𝑡)𝑆₁𝑥𝑡−ℎ₁+2𝜆₃𝑥^𝑇(𝑡−ℎ(𝑡))𝑆₃𝑥𝑡−ℎ₂+2𝜆₃𝑥^𝑇(𝑡−ℎ(𝑡))𝑆₃𝑥𝑡−ℎ₁+𝑥̇^𝑇(𝑡)ℎ²₁𝑆₁+ℎ²₂𝑆₂+ℎ₂−ℎ₁²𝑆₃𝑥̇(𝑡)−𝑥^𝑇(𝑡)𝑅_𝑖𝑘𝑥(𝑡). (20) Let 𝜉^𝑇(𝑡)=[𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ(𝑡)),𝑥^𝑇(𝑡−ℎ₁),𝑥^𝑇(𝑡−ℎ₂),𝑥̇^𝑇(𝑡)], 𝑘𝑘=¹¹0𝜆₂𝑆₁𝜆₃𝑆₂0∗𝑘₂₂𝜆₃𝑆₃𝜆₃𝑆₃0∗∗𝑘₃₃00∗∗∗𝑘₄₄0∗∗∗∗𝑘₅₅. (21) We have 𝑉̇𝑡,𝑥_𝑡+2𝛼𝑉𝑡,𝑥_𝑡≤𝜉^𝑇(𝑡)𝑘𝜉(𝑡)−𝑥^𝑇(𝑡)𝑅_𝑖𝑘𝑥(𝑡). (22) Since 𝑘<0 from ((1a), (1b), (1c), and (1d)), it is clear that 𝑉̇(𝑡,𝑥_𝑡)+2𝛼𝑉(𝑡,𝑥_𝑡)<0 or, equivalently, 𝑉(𝑡,𝑥_𝑡)≤𝑉(0,𝑥₀)𝑒^−2𝛼𝑡, ∀𝑡∈𝑅⁺.
Taking (14) into account, we have 𝜆‖𝑥(𝑡)‖²≤𝑉𝑡,𝑥_𝑡≤𝑉0,𝑥₀𝑒^−2𝛼𝑡≤Ω𝑒^−2𝛼𝑡‖𝑥(0)‖², (23) and hence ‖𝑥(𝑡)‖≤Ω𝜆‖𝑥(0)‖𝑒^−𝛼𝑡, (24) which shows that the impulsive switched system ((1a), (1b), (1c), and (1d)) is exponentially stable when 𝑡∈(𝑡_𝑘,𝑡_𝑘+1].
Next, let us consider the case at the impulsive and switching time point 𝑡_𝑘,𝑡_𝑘=1,2,3,…, 𝑉𝑡⁺_𝑘−𝑉𝑡⁻_𝑘=𝑥𝑡⁺_𝑘^𝑇𝑃_𝑖𝑘𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘^𝑇𝑃_𝑖𝑘−1𝑥𝑡⁻_𝑘. (25) Noting that Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘), 𝑥(𝑡⁻_𝑘)=𝑥(𝑡_𝑘), we have 𝑉𝑡⁺_𝑘−𝑉𝑡⁻_𝑘=𝑥𝑡⁺_𝑘^𝑇𝑃_𝑖𝑘𝑥𝑡⁺_𝑘−𝑥𝑡_𝑘^𝑇𝑃_𝑖𝑘−1𝑥𝑡_𝑘=𝑥𝑡_𝑘^𝑇𝐼+𝐷_𝑘^𝑇𝑃_𝑖𝑘𝐼+𝐷_𝑘−𝑃_𝑖𝑘−1𝑥𝑡_𝑘. (26) From (ii), we know that ^{𝑃𝑖𝑘−1(𝐼+𝐷𝑘)𝑇𝑃𝑖𝑘}_{𝑃𝑖𝑘(𝐼+𝐷𝑘)𝑃𝑖𝑘}>0, which is equivalent to (𝐼+𝐷_𝑘)^𝑇𝑃_𝑖𝑘(𝐼+𝐷_𝑘)−𝑃_𝑖𝑘−1<0.
So, 𝑉(𝑡⁺_𝑘)−𝑉(𝑡⁻_𝑘)<0. Therefore, we can conclude that the impulsive switched system ((1a), (1b), (1c), and (1d)) is exponentially stable.
Theorem 5 develops sufficient conditions for exponential stability of the impulsive switched system ((1a), (1b), (1c), and (1d)) under the case that 𝑢(𝑡)=0. In the next theorem, we will design the guaranteed cost controller in the form of 𝑢(𝑡)=𝐹_𝑖𝑘𝑥(𝑡) and sufficient conditions for the existence of a guaranteed cost control are also presented as well.

Theorem 6. Consider the impulsive switched system with time delay ((1a), (1b), (1c), and (1d)) with the cost function (4) and suppose there exist 𝛼>0, 𝜀₁>0, 𝑃_𝑖𝑘=𝑃^𝑇_𝑖𝑘>0. If there exist symmetric positive definite matrices 𝑈₁, 𝑈₂, 𝑆₁, 𝑆₂, 𝑆₃, and 𝐾 such that the following conditions are satisfied:(i)𝜀₁𝐼𝐸^𝑇_𝑖𝑘𝑃_𝑖𝑘𝑃_𝑖𝑘𝐸_𝑖𝑘𝑃_𝑖𝑘>0, (27)(ii)Π₁Π₂∗Π₃<0, (28)(iii)𝑃_𝑖𝑘−1𝐼+𝐷_𝑘^𝑇𝑃_𝑖𝑘𝑃_𝑖𝑘𝐼+𝐷_𝑘𝑃_𝑖𝑘>0, (29)
where Π₁=𝑤₁₁0𝜆₂𝑆₁𝜆₃𝑆₂0∗𝑤₂₂𝜆₃𝑆₃𝜆₃𝑆₃0∗∗𝑤₃₃00∗∗∗𝑤₄₄0∗∗∗∗𝑤₅₅,Π₂=𝐴^𝑇_𝑖𝑘𝐸_𝑖𝑘𝐾𝐾(𝐶_𝑖⁻¹_𝑘)^𝑇𝐻^𝑇_𝑖𝑘00000𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘^𝑇000000000000000,Π₃=diag𝐸^𝑇_𝑖𝑘𝐸_𝑖𝑘−𝜀₁𝐼,−𝐼,−𝑄_𝑖⁻¹_𝑘1,−𝜀₁,𝑤𝐼,−𝐼₁₁=𝐴^𝑇_𝑖𝑘𝐴_𝑖𝑘+3𝑃²_𝑖𝑘+𝑈₁+𝑈₂−𝜆₂𝑆₁−𝜆₃𝑆₂+2𝛼𝑃_𝑖𝑘+𝑅_𝑖𝑘,𝑤₂₂=−2𝜆₃𝑆₃,𝑤₃₃=−𝜆₂𝑈₁−𝜆₂𝑆₁−𝜆₃𝑆₃,𝑤₄₄=−𝜆₃𝑈₂−𝜆₃𝑆₂−𝜆₃𝑆₃,𝑤₅₅=ℎ²₁𝑆₁+ℎ²₂𝑆₂+ℎ₂−ℎ₁²𝑆₃,𝜆₁=𝑒^{−2𝛼(ℎ1+ℎ2)},𝜆₂=𝑒^−2𝛼ℎ1,𝜆₃=𝑒^−2𝛼ℎ2,𝜆=𝜆_min𝑃_𝑖𝑘, (30) then the guaranteed cost control law 𝑢(𝑡)=𝐶_𝑖⁻¹_𝑘𝐾^𝑇𝑥(𝑡) can ensure that the impulsive switched system ((1a), (1b), (1c), and (1d)) is exponentially stable. Moreover, the guaranteed cost value is 𝐽^∗=𝑉0,𝑥₀. (31)

Proof. When 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], choose the Lyapunov-Krasovskii function candidate to be similar to Theorem 5𝑉𝑡,𝑥_𝑡=⁴_𝑖=1𝑉_𝑖𝑡,𝑥_𝑡, (32) where 𝑉₁𝑡,𝑥_𝑡=𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝑥(𝑡),𝑉₂𝑡,𝑥_𝑡=^𝑡_𝑡−ℎ1𝑒^{2𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑈₁𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ2}𝑒^{2𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑈₂𝑥(𝑠)𝑉𝑑𝑠,₃𝑡,𝑥_𝑡=ℎ₁⁰_−ℎ1^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑠)}𝑥̇^𝑇(𝜏)𝑆₁𝑥̇(𝜏)𝑑𝜏𝑑𝑠+ℎ₂⁰_−ℎ2^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑠)}𝑥̇^𝑇(𝜏)𝑆₂𝑥̇(𝜏)𝑉𝑑𝜏𝑑𝑠,₄𝑡,𝑥_𝑡=ℎ₂−ℎ₁×^−ℎ1_−ℎ2^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑠)}𝑥̇^𝑇(𝜏)𝑆₃𝑥̇(𝜏)𝑑𝜏𝑑𝑠. (33) Taking the derivative of 𝑉_𝑖(𝑡,𝑥_𝑡), 𝑖=1,2,3,4, along the trajectory of the closed-loop system, we obtain 𝑉₁̇𝑡,𝑥_𝑡=𝐴_𝑖𝑘𝑥(𝑡)+𝐵_𝑖𝑘𝑥(𝑡−ℎ(𝑡))+𝐶_𝑖𝑘𝑢(𝑡)^𝑇𝑃_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝐴_𝑖𝑘𝑥(𝑡)+𝐵_𝑖𝑘𝑥(𝑡−ℎ(𝑡))+𝐶_𝑖𝑘𝑢(𝑡). (34) Put (1d) and 𝑢(𝑡)=𝐹_𝑖𝑘𝑥(𝑡) into it and then we can get 𝑉₁̇𝑡,𝑥_𝑡=𝑥^𝑇(𝑡)𝐴_𝑖𝑘+Δ𝐴_𝑖𝑘^𝑇𝑃_𝑖𝑘+𝑃_𝑖𝑘𝐴_𝑖𝑘+Δ𝐴_𝑖𝑘+𝐹^𝑇_𝑖𝑘𝐶^𝑇_𝑖𝑘𝑃_𝑖𝑘+𝑃_𝑖𝑘𝐶_𝑖𝑘𝐹_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡−ℎ(𝑡))𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘^𝑇𝑃_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘𝑥(𝑡−ℎ(𝑡)). (35) Next, it follows from Lemma 2 that 𝐴_𝑖𝑘+Δ𝐴_𝑖𝑘^𝑇𝑃_𝑖𝑘+𝑃_𝑖𝑘𝐴_𝑖𝑘+Δ𝐴_𝑖𝑘≤𝐴_𝑖𝑘+Δ𝐴_𝑖𝑘^𝑇𝐴_𝑖𝑘+Δ𝐴_𝑖𝑘+𝑃²_𝑖𝑘≤𝐴^𝑇_𝑖𝑘𝐴_𝑖𝑘+𝐴^𝑇_𝑖𝑘𝐸_𝑖𝑘𝜀₁𝐼−𝐸^𝑇_𝑖𝑘𝐸_𝑖𝑘⁻¹𝐸^𝑇_𝑖𝑘𝐴_𝑖𝑘+𝜀₁𝐻^𝑇_𝑖𝑘𝐻_𝑖𝑘+𝑃²_𝑖𝑘,𝑥^𝑇(𝑡−ℎ(𝑡))𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘^𝑇𝑃_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝑘𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘𝑥(𝑡−ℎ(𝑡))≤𝑥^𝑇(𝑡)𝑃²_𝑖𝑘𝑥(𝑡)+𝑥^𝑇((𝑡×𝐵𝑡−ℎ))_𝑖𝑘+Δ𝐵_𝑖𝑘^𝑇𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘𝑥(𝑡−ℎ(𝑡)). (36) Substituting (36) into (35) yields 𝑉₁̇𝑡,𝑥_𝑡≤𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑘𝐴_𝑖𝑘+𝐴^𝑇_𝑖𝑘𝐸_𝑖𝑘𝜀₁𝐼−𝐸^𝑇_𝑖𝑘𝐸_𝑖𝑘⁻¹𝐸^𝑇_𝑖𝑘𝐴_𝑖𝑘+𝜀₁𝐻^𝑇_𝑖𝑘𝐻_𝑖𝑘+3𝑃²_𝑖𝑘+𝐹^𝑇_𝑖𝑘𝐶^𝑇_𝑖𝑘𝐶_𝑖𝑘𝐹_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡−ℎ(𝑡))𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘^𝑇𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘×𝑥(𝑡−ℎ(𝑡)). (37) The derivatives of 𝑉₂(𝑡,𝑥_𝑡), 𝑉₃(𝑡,𝑥_𝑡), and 𝑉₄(𝑡,𝑥_𝑡) are the same as those in Theorem 5. Then, we can put (37), (16), (18), and (19) together to obtain 𝑉̇𝑡,𝑥_𝑡+2𝛼𝑉𝑡,𝑥_𝑡≤𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑘𝐴_𝑖𝑘+𝐴^𝑇_𝑖𝑘𝐸_𝑖𝑘𝜀₁𝐼−𝐸^𝑇_𝑖𝑘𝐸_𝑖𝑘⁻¹𝐸^𝑇_𝑖𝑘𝐴_𝑖𝑘+𝜀₁𝐻^𝑇_𝑖𝑘𝐻_𝑖𝑘+3𝑃²_𝑖𝑘+𝐹^𝑇_𝑖𝑘𝐶^𝑇_𝑖𝑘𝐶_𝑖𝑘𝐹_𝑖𝑘+𝑈₁+𝑈₂−𝜆₂𝑆₁−𝜆₃𝑆₂+2𝛼𝑃_𝑖𝑘+𝑅_𝑖𝑘+𝐹^𝑇_𝑖𝑘𝑄_𝑖𝑘𝐹_𝑖𝑘𝑥(𝑡)+𝑥^𝑇(𝑡−ℎ(𝑡))𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘^𝑇𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘−2𝜆₃𝑆₃×𝑥(𝑡−ℎ(𝑡))+𝑥^𝑇𝑡−ℎ₁−𝜆₂𝑈₁−𝜆₂𝑆₁−𝜆₃𝑆₃𝑥𝑡−ℎ₁+𝑥^𝑇𝑡−ℎ₂−𝜆₃𝑈₂−𝜆₃𝑆₂−𝜆₃𝑆₃𝑥𝑡−ℎ₂+2𝜆₃𝑥^𝑇(𝑡)𝑆₂𝑥𝑡−ℎ₂+2𝜆₂𝑥^𝑇(𝑡)𝑆₁𝑥𝑡−ℎ₁+2𝜆₃𝑥^𝑇(𝑡−ℎ(𝑡))𝑆₃𝑥𝑡−ℎ₂+2𝜆₃𝑥^𝑇(𝑡−ℎ(𝑡))𝑆₃𝑥𝑡−ℎ₁+𝑥̇^𝑇(𝑡)ℎ²₁𝑆₁+ℎ²₂𝑆₂+ℎ₂−ℎ₁²𝑆₃𝑥̇(𝑡)−𝑥^𝑇(𝑡)𝑅_𝑖𝑘+𝐹^𝑇_𝑖𝑘𝑄_𝑖𝑘𝐹_𝑖𝑘𝑥(𝑡). (38) Let 𝑤𝑤=¹¹0𝜆₂𝑆₁𝜆₃𝑆₂0∗𝑤₂₂𝜆₃𝑆₃𝜆₃𝑆₃0∗∗𝑤₃₃00∗∗∗𝑤₄₄0∗∗∗∗𝑤₅₅,𝑤₁₁=𝐴^𝑇_𝑖𝑘𝐴_𝑖𝑘+𝐴^𝑇_𝑖𝑘𝐸_𝑖𝑘𝜀₁𝐼−𝐸^𝑇_𝑖𝑘𝐸_𝑖𝑘⁻¹𝐸^𝑇_𝑖𝑘𝐴_𝑖𝑘+𝜀₁𝐻^𝑇_𝑖𝑘𝐻_𝑖𝑘+3𝑃²_𝑖𝑘+𝐹^𝑇_𝑖𝑘𝐶^𝑇_𝑖𝑘𝐶_𝑖𝑘𝐹_𝑖𝑘+𝑈₁+𝑈₂−𝜆₂𝑆₁−𝜆₃𝑆₂+2𝛼𝑃_𝑖𝑘+𝑅_𝑖𝑘+𝐹^𝑇_𝑖𝑘𝑄_𝑖𝑘𝐹_𝑖𝑘,𝑤₂₂=𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘^𝑇𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘−2𝜆₃𝑆₃,𝑤₃₃=−𝜆₂𝑈₁−𝜆₂𝑆₁−𝜆₃𝑆₃,𝑤₄₄=−𝜆₃𝑈₂−𝜆₃𝑆₂−𝜆₃𝑆₃,𝑤₅₅=ℎ²₁𝑆₁+ℎ²₂𝑆₂+ℎ₂−ℎ₁²𝑆₃. (39) Then, we have 𝑉̇𝑡,𝑥_𝑡+2𝛼𝑉𝑡,𝑥_𝑡≤𝜉^𝑇(𝑡)𝑤𝜉(𝑡)−𝑥^𝑇(𝑡)𝑅_𝑖𝑘+𝐹^𝑇_𝑖𝑘𝑄_𝑖𝑘𝐹_𝑖𝑘𝑥(𝑡). (40) By Lemma 3, it proves that 𝑤<0 holds if 𝑤₁𝑤₂∗𝑤₃<0, (41) where 𝑤₁=𝑤₁₁0𝜆₂𝑆₁𝜆₃𝑆₂0∗𝑤₂₂𝜆₃𝑆₃𝜆₃𝑆₃0∗∗𝑤₃₃00∗∗∗𝑤₄₄0∗∗∗∗𝑤₅₅,𝑤₂=𝐴^𝑇_𝑖𝑘𝐸_𝑖𝑘𝐹^𝑇_𝑖𝑘𝐶^𝑇_𝑖𝑘𝐹^𝑇_𝑖𝑘𝐻^𝑇_𝑖𝑘00000𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘^𝑇000000000000000,𝑤₃=diag𝐸^𝑇_𝑖𝑘𝐸_𝑖𝑘−𝜀₁𝐼,−𝐼,−𝑄_𝑖⁻¹_𝑘1,−𝜀₁,𝑤𝐼,−𝐼₁₁=𝐴^𝑇_𝑖𝑘𝐴_𝑖𝑘+3𝑃²_𝑖𝑘+𝑈₁+𝑈₂−𝜆₂𝑆₁−𝜆₃𝑆₂+2𝛼𝑃_𝑖𝑘+𝑅_𝑖𝑘,𝑤₂₂=−2𝜆₃𝑆₃. (42) Letting 𝐹^𝑇_𝑖𝑘𝐶^𝑇_𝑖𝑘=𝐾, we get 𝐹_𝑖𝑘=𝐶_𝑖⁻¹_𝑘𝐾^𝑇. Then, we have 𝑤₁𝑤₂∗𝑤₃=Π₁Π₂∗Π₃Π<0,₁=𝑤₁,Π₂=𝐴^𝑇_𝑖𝑘𝐸_𝑖𝑘𝐾𝐾(𝐶_𝑖⁻¹_𝑘)^𝑇𝐻^𝑇_𝑖𝑘00000𝐵_𝑖𝑘+Δ𝐵_𝑖𝑘^𝑇000000000000000Π₃=𝑤₃. (43) In addition, since ^Π1Π2_∗Π3<0 (see Theorem 6 (ii)), ^𝑤1𝑤2_∗𝑤3<0, then we plug it into (40) and obtain 𝑉̇𝑡,𝑥_𝑡+2𝛼𝑉𝑡,𝑥_𝑡<0. (44) Now, we can easily conclude that the impulsive switched system ((1a), (1b), (1c), and (1d)) is exponentially stable. Looking back to (40), we can get 𝑉̇𝑡,𝑥_𝑡≤−𝑥^𝑇(𝑡)𝑅_𝑖𝑘+𝐹^𝑇_𝑖𝑘𝑄_𝑖𝑘𝐹_𝑖𝑘𝑥(𝑡),𝐽=^∞_𝑘=0^𝑡𝑘+1_𝑡𝑘𝑥^𝑇(𝑡)𝑅_𝑖𝑘𝑥(𝑡)+𝑢^𝑇(𝑡)𝑄_𝑖𝑘𝑢(𝑡)𝑑𝑡≤−lim^𝑘_{𝑘→∞𝑗=0}^𝑡𝑗+1_𝑡𝑗𝑑𝑉𝑡,𝑥_𝑡𝑑𝑡𝑑𝑡=−lim_𝑘→∞−𝑉0,𝑥₀+𝑉𝑡⁻₁,𝑥_𝑡⁻₁−𝑉𝑡⁺₁,𝑥_𝑡⁺₁+𝑉𝑡⁻₂,𝑥_𝑡⁻₂−𝑉𝑡⁺₂,𝑥_𝑡⁺₂+⋯−𝑉𝑡⁻_𝑘,𝑥_𝑡⁻_𝑘+𝑉𝑡⁺_𝑘,𝑥_𝑡⁺_𝑘−𝑉𝑡⁻_𝑘+1,𝑥_𝑡⁻_𝑘+1=lim_𝑘→∞𝑉0,𝑥₀+^𝑘_𝑗=1𝑉𝑡⁺_𝑗,𝑥_𝑡⁺_𝑗−𝑉𝑡⁻_𝑗,𝑥_𝑡⁻_𝑗+𝑉𝑡⁻_𝑘+1,𝑥_𝑡⁻_𝑘+1=𝑉0,𝑥₀+lim^𝑘_{𝑘→∞𝑗=1}𝑥𝑡_𝑗^𝑇𝐼+𝐷_𝑗^𝑇𝑃_𝑖𝑗𝐼+𝐷_𝑗−𝑃_𝑖𝑗−1𝑥𝑡_𝑗+lim_𝑘→∞𝑉𝑡⁻_𝑘+1,𝑥_𝑡⁻_𝑘+1. (45)(46) We know that lim_𝑘→∞𝑉𝑡⁻_𝑘+1,𝑥_𝑡⁻_𝑘+1=0,lim^𝑘_{𝑘→∞𝑗=1}𝑥𝑡_𝑗^𝑇𝐼+𝐷_𝑗^𝑇𝑃_𝑖𝑗𝐼+𝐷_𝑗−𝑃_𝑖𝑗−1𝑥𝑡_𝑗<0. (47) Thus, we can get 𝐽<𝑉0,𝑥₀=𝐽^∗. (48) This completes the proof.

4. Conclusion

The exponential stabilization problem for impulsive switched systems with time delays is considered in this paper. Sufficient conditions for exponential stability of the impulsive switched systems without control input are derived via the Lyapunov-Krasovskii functional method. Moreover, sufficient conditions for the exponential stability of the impulsive switched systems under the designed guaranteed cost control law are established and the corresponding guaranteed cost value can be obtained as well.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (nos. 11171079, 61290325, and 61321003), HUST Startup Research Fund, and HUST Independent Innovation Research Fund (GF and Natural Science).