On Boundedness and Attractiveness of Nonlinear Switched Delay Systems

Abstract

This paper is concerned with the boundedness and attractiveness of nonlinear switched delay systems whose subsystems have different equilibria. Some sufficient conditions which can guarantee the system’s boundedness are obtained. In addition, we work out the region where the solution will remain and furthermore the relationship between the initial function and the bounded region. Based on the new concept of attractor with switching laws, we show that the nonlinear switched delay system is attractive and then obtain the attractive region.

1. Introduction

A switched system is a collection of finite continuous variable systems (called subsystems) along with a discrete event governing the “switching” among them (called the switching law). In reality, there are many switched systems that occur naturally or by design, such as those in the fields of control, communication, computer, and signal processes. Indeed, these systems are always suited to describe practical dynamical behaviors with hybrid nature in engineering and technology [1, 2]. In the last decades, they have attracted considerable attention among control theorists, computer scientists, and practicing engineers in the study of switched systems and switching control design and hence rich theoretical results have been obtained (see [3–20] and the references therein).

Up to now, most of the studies on switched systems require that all subsystems share a common equilibrium. However, in many real world problems, the assumption that all subsystems share a common equilibrium may not hold and thus the assumption may limit the applicability of stability results. Recently, it is pointed out in [21, 22] that, when subsystems have different equilibria or no equilibrium, a switched system can still exhibit interesting behaviors under appropriate switching laws. Such behaviors are similar to those of a conventional bounded or stable system near an equilibrium point. In this paper, we introduce some boundedness and attractiveness notions to define such behaviors for switched delay systems. Such notions are extensions of the traditional boundedness and attractiveness concepts in [15, 16].

In this paper, we focus on the boundedness and attractiveness problems for a simple yet important class of nonlinear switched delay systems, which includes three main contributions. First, we propose the concept of attractor with the switching law and the notation of a function 𝜑∈𝐶_𝜏 in the neighbourhood of a subset Ω∈𝑅^𝑛 which is different from the conventional concept of the distance between two subsets of 𝑅^𝑛. Second, we propose sufficient conditions which guarantee that the switched delay system is bounded or attractive. Third, we explicitly construct a switching law and the region in which the solutions remain (boundedness) and explicitly construct the attractor for the switched delay systems.

2. Problem Statement

Consider the nonlinear switched delay system 𝑥̇=𝐴_𝛾𝑥+𝑓_𝛾(𝑥(𝑡−𝜏)),𝛾∈𝕀≜{1,20,…,𝑁}, (1) where 𝑥∈𝑅^𝑛, 𝑓_𝛾∈𝐶(𝑅^𝑛,𝑅^𝑛), and 𝐴_𝛾∈𝑅^𝑛×𝑛, 𝜏≥0. It is assumed that(H1) there exist 𝛼>0,𝑀≥1 such that ‖𝑒^𝐴𝛾𝑡‖≤𝑀𝑒^−𝛼𝑡; (H2)there exists 𝐿, 0<𝐿<𝛼/𝑀, such that ‖𝑓_𝛾(𝑥)−𝑓_𝛾(𝑦)‖≤𝐿‖𝑥−𝑦‖,𝑥,𝑦∈ℝ^𝑛, 𝛾∈𝕀;(H3)𝐴_𝛾𝑥+𝑓_𝛾(𝑥)=0 has only one solution 𝑥^𝛾∈ℝ^𝑛 and 𝑥^𝛾≠𝑥^𝑙 for 𝛾≠𝑙, 𝛾,𝑙∈𝕀. Let Γ={𝑥_𝛾(𝑡;0,𝑥^𝑗),𝛾,𝑗∈𝕀,𝑡∈𝑅⁺}, where 𝑥_𝛾𝑡;0,𝑥^𝑗=𝑒^𝐴𝛾𝑡𝑥^𝑗+^𝑡₀𝑒^{𝐴𝛾(𝑡−𝑢)}𝑓_𝛾𝑥_𝛾(𝑢−𝜏)𝑑𝑢 (2) and denote that 𝐷=max{‖𝑥^𝛾−𝑥^𝑗‖,𝛾≠𝑗,𝛾,𝑗∈𝕀}, 𝐶_𝜏≜𝐶([−𝜏,0],𝑅^𝑛), and for 𝜑∈𝐶_𝜏, define 𝜑_𝜏=sup_{𝜃∈[−𝜏,0]}𝜑(𝜃). (3) Let 𝑥_𝛾(𝑡;𝑡₀,𝜑)≜𝑥_𝛾(𝑡) be the solution of the initial value problem 𝑥̇_𝛾(𝑡)=𝐴_𝛾𝑥_𝛾+𝑓_𝛾𝑥_𝛾(,𝑥𝑡−𝜏)_𝛾(𝜃)=𝜑(𝜃),𝜃∈[−𝜏,0]. (4) If 𝜑≡𝑥^𝑗 especially, denote that 𝑥_𝛾(𝑡;𝑡₀,𝑥^𝑗)≜𝑥^𝑗_𝛾(𝑡),𝑗∈𝕀, and let 𝑥(𝑡;𝑡₀,𝜑,𝜎)≜𝑥(𝑡) be the solution of (1) with the switching law 𝜎 as 𝑥̇(𝑡)=𝐴_𝛾𝑥+𝑓_𝛾(𝑥(𝑥(𝜃)(𝜃)[],𝑡−𝜏)),𝛾∈𝕀,=𝜑,𝜃∈−𝜏,0 (5) where 𝑥_𝑡≜𝑥(𝑡+𝜃),𝜃∈[−𝜏,0]. Since all the subsystems are autonomous, we can take 𝑡₀=0 without loss of generality. Let 𝜎=((𝑡₁,𝑖₁),…,(𝑡_𝑘,𝑖_𝑘),…) be the switching law of the switched system (1), where 𝑖_𝑘∈𝕀 and 0=𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯; that is, under the switching law 𝜎, the subsystem 𝑖_𝑘 is active during the time interval [𝑡_𝑘−1,𝑡_𝑘). Suppose that 𝑖_𝑘≠𝑖_𝑘+1 holds, which means that a subsystem will be definitely switched to a different subsystem when the switching triggers.

3. Boundedness

In this section, we will study the boundedness problem of the switched delay system (1).

Definition 1. Suppose that 𝛿>0, Ω⊂𝑅^𝑛, and then a neighborhood of Ω in 𝐶_𝜏 is defined as 𝐵_𝛿(Ω)=𝜑∣𝜑∈𝐶_𝜏,sup_{𝜃∈[−𝜏,0]}𝜌𝜑(𝜃),Ω≤𝛿, (6) where the distance 𝜌 between 𝑥∈𝑅^𝑛 and Ω is defined as 𝜌(𝑥,Ω)=inf𝑥−𝑦∣𝑦∈Ω. (7)

Definition 2. A nonnegative number 𝜇 is called the dwell time of the switching law 𝜎=𝑡₁,𝑖₁,…,𝑡_𝑘,𝑖_𝑘,…, (8) if 𝜇=inf𝑡_𝑘+1−𝑡_𝑘,𝑘=1,2,… (9) holds.

Lemma 3. Let 𝜇 be the dwell time of the switching law 𝜎=((𝑡₁,𝑖₁),…,(𝑡_𝑘,𝑖_𝑘),…), and 𝛿>0 is a constant. If 𝜇≥max{𝜏,(ln(𝑀(𝛿+𝐷)/𝛿))/(𝛼−𝑀𝐿)}, then 𝜑∈𝐵_𝛿(𝑥^𝑖0) implies that 𝑥_𝑡𝑘∈𝐵_𝛿(𝑥^𝑖𝑘), where 𝑥_𝑡𝑘=𝑥_𝑡𝑘(𝜃)=𝑥𝑡_𝑘+𝜃,0,𝜑,𝜎,𝜃∈[−𝜏,0]. (10)

Proof. Suppose that 𝜑∈𝐵_𝛿(𝑥^𝑖0) and the switching law is defined as 𝜎=((𝑡₁,𝑖₁),…,(𝑡_𝑘,𝑖_𝑘),…). Then, we will use induction to prove the result. When 𝑚=1, 𝑡∈[0,𝑡₁], we will prove that sup_{𝜃∈[−𝜏,0]}𝑥𝑡₁+𝜃,0,𝜑,𝜎−𝑥^𝑖1≤𝛿. (11) In fact, 𝑥𝑡,0,𝜑,𝜎−𝑥^𝑖1=𝑥_𝑖1(𝑡)−𝑥^𝑖1=𝑒^{𝐴𝑖1𝑡}𝜑(0)+^𝑡₀𝑒^{𝐴𝑖1(𝑡−𝑢)}𝑓_𝑖1𝑥_𝑖1(𝑢−𝜏)𝑑𝑢−𝑒^{𝐴𝑖1𝑡}𝑥^𝑖1−^𝑡₀𝑒^{𝐴𝑖1(𝑡−𝑢)}𝑓_𝑖1𝑥^𝑖1≤𝑒𝑑𝑢_{𝐴𝑖1𝑡}𝜑(0)−𝑥^𝑖1+^𝑡₀𝑒^{𝐴𝑖1(𝑡−𝑢)}𝑓_𝑖1𝑥_𝑖1(𝑢−𝜏)−𝑓_𝑖1𝑥^𝑖1𝑑𝑢≤𝑀𝑒^−𝛼𝑡𝜑(0)−𝑥^𝑖1+𝑀𝐿^𝑡₀𝑒^{−𝛼(𝑡−𝑢)}𝑥_𝑖1(𝑢−𝜏)−𝑥^𝑖1𝑑𝑢. (12) Then, 𝑒^𝛼𝑡𝑥_𝑖1(𝑡)−𝑥^𝑖1≤𝑀𝜑(0)−𝑥^𝑖1+𝑀𝐿^𝑡₀𝑒^𝛼𝑢𝑥_𝑖1(𝑢−𝜏)−𝑥^𝑖1𝜑(0)𝑑𝑢≤𝑀−𝑥^𝑖1+𝑀𝐿^𝑡₀𝑒^𝛼𝑢𝑥_𝑖1(𝑢−𝜏)−𝑥^𝑖1𝜑(0)𝑑𝑢≤𝑀−𝑥^𝑖0+𝑥^𝑖0−𝑥^𝑖1+𝑀𝐿^𝑡₀𝑒^𝛼𝑢𝑥_𝑖1(𝑢−𝜏)−𝑥^𝑖1𝐷𝑑𝑢≤𝑀𝛿++𝑀𝐿^𝑡₀𝑒^𝛼𝑢𝑥_𝑖1𝑢−𝑥^𝑖1_𝜏𝑑𝑢. (13) Furthermore, since the right hand side of the inequality is increasing function, we have 𝑒^𝛼𝑡𝑥_𝑖1𝑡−𝑥^𝑖1_𝜏≤𝑀𝛿+𝐷+𝑀𝐿^𝑡₀𝑒^𝛼𝑢𝑥_𝑖1𝑢−𝑥^𝑖1_𝜏𝑑𝑢, (14) and then, by applying the Bellman-Gronwall inequality to (14), we have 𝑒^𝛼𝑡𝑥_𝑖1𝑡−𝑥^𝑖1_𝜏≤𝑀𝛿+𝐷𝑒^𝑀𝐿𝑡, (15) and thus 𝑥_𝑖1𝑡−𝑥^𝑖1_𝜏≤𝑀𝛿+𝐷𝑒^{−(𝛼−𝑀𝐿)𝑡}. (16) Let 𝑡=𝑡₁; we get 𝑥_𝑡1−𝑥^𝑖1_𝜏=𝑥_𝑖1𝑡1−𝑥^𝑖1_𝜏≤𝑀𝛿+𝐷𝑒^{−(𝛼−𝑀𝐿)𝑡1}≤𝛿, (17) and (11) is proved. Suppose that, when 𝑚=𝑘, the result is right; that is, 𝑥_𝑡𝑘∈𝐵_𝛿(𝑥^𝑖𝑘), and then we will prove that 𝑥_𝑡𝑘+1∈𝐵_𝛿(𝑥^𝑖𝑘+1); that is, sup_{𝜃∈[−𝜏,0]}𝑥𝑡_𝑘+1+𝜃,0,𝜑,𝜎−𝑥^𝑖𝑘+1≤𝛿. (18) In fact, when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1], 𝑥(𝑡,0,𝜑,𝜎)=𝑥_𝑖𝑘+1(𝑡−𝑡_𝑘,0,𝑥_𝑡𝑘), and thus 𝑥𝑡,0,𝜑,𝜎−𝑥^𝑖𝑘+1=𝑥_𝑖𝑘+1𝑡−𝑡_𝑘,0,𝑥_𝑡𝑘−𝑥^𝑖𝑘+1=𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘)}𝑥_𝑡𝑘(0)+^{𝑡−𝑡𝑘}₀𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘−𝑢)}𝑓_𝑖𝑘+1𝑥_𝑖𝑘+1(𝑢−𝜏)𝑑𝑢−𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘)}𝑥^𝑖𝑘+1−^{𝑡−𝑡𝑘}₀𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘−𝑢)}𝑓_𝑖𝑘+1𝑥^𝑖𝑘+1≤𝑒𝑑𝑢_{𝐴𝑖𝑘+1(𝑡−𝑡𝑘)}𝑥_𝑡𝑘(0)−𝑥^𝑖𝑘+1+^{𝑡−𝑡𝑘}₀𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘−𝑢)}×𝑓_𝑖𝑘+1𝑥_𝑖𝑘+1(𝑢−𝜏)−𝑓_𝑖𝑘+1𝑥^𝑖𝑘+1𝑑𝑢≤𝑀𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑥_𝑡𝑘(0)−𝑥^𝑖𝑘+1+𝑀𝐿^{𝑡−𝑡𝑘}₀𝑒^{−𝛼(𝑡−𝑡𝑘−𝑢)}𝑥_𝑖𝑘+1(𝑢−𝜏)−𝑥^𝑖𝑘+1𝑑𝑢, (19) and then we have 𝑒^{𝛼(𝑡−𝑡𝑘)}𝑥_𝑖𝑘+1𝑡−𝑡_𝑘,0,𝜑−𝑥^𝑖𝑘+1≤𝑀𝑥_𝑡𝑘(0)−𝑥^𝑖𝑘+1+𝑀𝐿^{𝑡−𝑡𝑘}₀𝑒^𝛼𝑢𝑥_𝑖𝑘+1(𝑢−𝜏)−𝑥^𝑖𝑘+1𝑥𝑑𝑢≤𝑀_𝑡𝑘(0)−𝑥^𝑖𝑘+𝑥^𝑖𝑘−𝑥^𝑖𝑘+1+𝑀𝐿^{𝑡−𝑡𝑘}₀𝑒^𝛼𝑢𝑥_𝑖𝑘+1(𝑢−𝜏)−𝑥^𝑖𝑘+1𝐷𝑑𝑢≤𝑀𝛿++𝑀𝐿^{𝑡−𝑡𝑘}₀𝑒^𝛼𝑢𝑥_{𝑖𝑘+1𝑢}−𝑥^𝑖𝑘+1_𝜏𝑑𝑢. (20) Since the right hand side of the inequality is increasing function, we have 𝑒^{𝛼(𝑡−𝑡𝑘)}𝑥_{𝑖𝑘+1(𝑡−𝑡𝑘)}−𝑥^𝑖𝑘+1_𝜏≤𝑀𝛿+𝐷+𝑀𝐿^{𝑡−𝑡𝑘}₀𝑒^𝛼𝑢𝑥_{𝑖𝑘+1𝑢}−𝑥^𝑖𝑘+1_𝜏𝑑𝑢. (21) Applying the Bellman-Gronwall inequality to (21) yields 𝑥_{𝑖𝑘+1(𝑡−𝑡𝑘)}−𝑥^𝑖𝑘+1_𝜏=𝑥_𝑡−𝑥^𝑖𝑘+1_𝜏≤𝑀𝛿+𝐷𝑒^{−(𝛼−𝑀𝐿)(𝑡−𝑡𝑘)}. (22) Let 𝑡=𝑡_𝑘+1, and then we get 𝑥_𝑡𝑘+1−𝑥^𝑖𝑘+1_𝜏≤𝑀𝛿+𝐷𝑒^{−(𝛼−𝑀𝐿)(𝑡𝑘+1−𝑡𝑘)}≤𝛿, (23) and (18) is proved. The proof is completed.

Remark 4. Lemma 3 shows that if the initial function 𝜑 is in the 𝛿 neighborhood of 𝑥^𝑖0, then the solution 𝑥(𝑡,0,𝜑,𝜎) with the switching law 𝜎 whose dwell time satisfies 𝜇≥max{𝜏,(ln(𝑀(𝛿+𝐷)/𝛿))/(𝛼−𝑀𝐿)} has the following property: at the 𝑘th switching point, the function 𝑥(𝑡_𝑘+𝜃,0,𝜑,𝜎)(𝜃∈[−𝜏,0]) is in the 𝛿 neighborhood of 𝑥^𝑖𝑘.

Theorem 5. Let 𝜇 be the dwell time of the switching law 𝜎=((𝑡₁,𝑖₁),…,(𝑡_𝑘,𝑖_𝑘),…) and 𝛿>0 is a constant. If 𝜇≥max{𝜏,ln(𝑀(𝛿+𝐷)/𝛿)/(𝛼−𝑀𝐿)}, then 𝜑∈𝐵_𝛿(𝑥^𝑖0) implies that the solution 𝑥(𝑡,0,𝜑,𝜎) of (1) satisfies 𝑥(𝑡)∈𝐵_𝑀𝛿(Γ).

Proof. We use mathematical induction to prove the result. When 𝑡∈[0,𝑡₁], 𝜌𝑥≤𝑥𝑡,0,𝜑,𝜎,Γ𝑡,0,𝜑,𝜎−𝑥𝑡,0,𝑥^𝑖0=𝑥,𝜎_𝑖1𝑡,0,𝜑,𝜎−𝑥_𝑖1𝑡,0,𝑥^𝑖0=𝑥_𝑖1(𝑡)−𝑥_𝑖1(𝑡=𝑒_{𝐴𝑖1𝑡})𝜑(0)+^𝑡₀𝑒^{𝐴𝑖1(𝑡−𝑢)}𝑓_𝑖1𝑥_𝑖1(𝑢−𝜏)𝑑𝑢−𝑒^{𝐴𝑖1𝑡}𝑥^𝑖0−^𝑡₀𝑒^{𝐴𝑖1(𝑡−𝑢)}𝑓_𝑖1𝑥_𝑖1()≤𝑒𝑢−𝜏𝑑𝑢_{𝐴𝑖1𝑡}𝜑(0)−𝑥^𝑖0+^𝑡₀𝑒^{𝐴𝑖1(𝑡−𝑢)}𝑓_𝑖1𝑥_𝑖1(𝑢−𝜏)−𝑓_𝑖1𝑥_𝑖1(𝑢−𝜏)𝑑𝑢≤𝑀𝑒^−𝛼𝑡𝜑(0)−𝑥^𝑖0+𝐿𝑀^𝑡₀𝑒^{−𝛼(𝑡−𝑢)}𝑥_𝑖1(𝑢−𝜏)−𝑥_𝑖1(𝑢−𝜏)𝑑𝑢≤𝑒^−𝛼𝑡𝑀𝛿+𝐿𝑀^𝑡₀𝑒^𝛼𝑢𝑥_𝑖1𝑢−𝑥_{𝑖1𝑢𝜏}𝑑𝑢, (24) which implies that 𝑒^𝛼𝑡𝑥_𝑖1𝑡−𝑥_{𝑖1𝑡𝜏}≤𝑀𝛿+𝐿𝑀^𝑡₀𝑒^𝛼𝑢𝑥_𝑖1𝑢−𝑥_{𝑖1𝑢𝜏}𝑥𝑑𝑢,_𝑖1𝑡−𝑥_{𝑖1𝑡𝜏}≤𝑀𝛿𝑒^{−(𝛼−𝐿𝑀)𝑡}≤𝑀𝛿. (25) Thus, for 𝑡∈[0,𝑡₁] and 𝜑∈𝐵_𝛿, 𝜌𝑥𝑡,0,𝜑,𝜎,Γ≤𝑀𝛿. (26) Suppose that, when 𝑡∈[𝑡_𝑘−1,𝑡_𝑘], (26) holds. Now we prove that, when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1], 𝜌𝑥𝑡,0,𝜑,𝜎,Γ≤𝑀𝛿. (27) In fact, by Lemma 3, we know that 𝑥_𝑡𝑘∈𝐵_𝛿(𝑥^𝑖𝑘) and 𝜌𝑥≤𝑥−𝑥𝑡,0,𝜑,𝜎,Γ𝑡,0,𝜑,𝜎^𝑖𝑘_𝑖𝑘+1𝑡−𝑡_𝑘,0,𝑥^𝑖𝑘=𝑥_𝑖𝑘+1𝑡−𝑡_𝑘,0,𝑥_𝑡𝑘,𝜎−𝑥^𝑖𝑘_𝑖𝑘+1𝑡−𝑡_𝑘,0,𝑥^𝑖𝑘=𝑥_𝑖𝑘+1𝑡−𝑡_𝑘−𝑥^𝑖𝑘_𝑖𝑘+1𝑡−𝑡_𝑘=𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘)}𝑥_𝑡𝑘(0)+^{𝑡−𝑡𝑘}₀𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘−𝑢)}𝑓_𝑖𝑘+1𝑥_𝑖𝑘+1(𝑢−𝜏)𝑑𝑢−𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘)}𝑥^𝑖𝑘−^{𝑡−𝑡𝑘}₀𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘−𝑢)}𝑓_𝑖𝑘+1𝑥^𝑖𝑘_𝑖𝑘+1(𝑢−𝜏)𝑑𝑢≤𝑀𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑥_𝑡𝑘(0)−𝑥^𝑖𝑘+𝐿𝑀^{𝑡−𝑡𝑘}₀𝑒^{−𝛼(𝑡−𝑡𝑘−𝑢)}𝑥_𝑖𝑘+1(𝑢−𝜏)−𝑥^𝑖𝑘_𝑖𝑘+1(𝑢−𝜏)𝑑𝑢≤𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑀𝛿+𝐿𝑀^{𝑡−𝑡𝑘}₀𝑒^𝛼𝑢𝑥_{𝑖𝑘+1𝑢}−𝑥^𝑖𝑘_{𝑖𝑘+1𝑢𝜏}𝑑𝑢, (28) which implies that 𝑥_𝑖𝑘+1(𝑡−𝑡_𝑘)−𝑥^𝑖𝑘_𝑖𝑘+1(𝑡−𝑡_𝑘)_𝜏≤𝑀𝛿𝑒^{−(𝛼−𝐿𝑀)(𝑡−𝑡𝑘)}≤𝑀𝛿 (29) and thus 𝜌𝑥𝑡,0,𝜑,𝜎,Γ≤𝑥_𝑖𝑘+1(𝑡−𝑡_𝑘)−𝑥^𝑖𝑘_𝑖𝑘+1(𝑡−𝑡_𝑘)_𝜏≤𝑀𝛿. (30) Thus, for all 𝑡≥0, we have 𝜌𝑥𝑡,0,𝜑,𝜎,Γ≤𝑀𝛿. (31) The proof is completed.

Remark 6. (i) The result in Theorem 5 shows that if the initial function 𝜑 is in the 𝛿 neighborhood of 𝑥^𝑖0, then the solution 𝑥(𝑡,0,𝜑,𝜎) with the switching law whose dwell time satisfies 𝜇≥max{𝜏,(ln(𝑀(𝛿+𝐷)/𝛿))/(𝛼−𝑀𝐿)} is in the 𝑀𝛿 neighborhood of the set Γ.
(ii) Theorem 5 not only shows the boundedness of the switched system but also gives the region in which the solution will remain.

By the same procedure of Lemma 3 and Theorem 5, we can study the boundedness of the solution of the following nonlinear switched multidelays system: 𝑥̇=𝐴_𝛾𝑥+^𝑃_𝑗=1𝑓_𝛾𝑗𝑥,𝑥𝑡−𝜏_𝛾𝑗,𝛾∈𝕀≜{1,2,…,𝑁}, (32) where 𝑥∈𝑅^𝑛, 𝑓_𝛾𝑗∈𝐶(𝑅^𝑛×𝑅^𝑛,𝑅^𝑛), 𝐴_𝛾∈𝑅^𝑛×𝑛, and 0≤𝜏_𝛾𝑗≤𝜏.

Assume that (H1)’ there exist 𝛼>0,𝑀≥1 such that ‖𝑒^𝐴𝛾𝑡‖≤𝑀𝑒^−𝛼𝑡;(H2)’ there exists 𝐿, 0<𝐿<𝛼/2𝑃𝑀, such that ‖𝑓_𝛾𝑗(𝑥₁,𝑥₂)−𝑓_𝛾𝑗(𝑦₁,𝑦₂)‖≤𝐿[‖𝑥₁−𝑦₁‖+‖𝑥₂−𝑦₂‖],𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝑅^𝑛, 𝛾∈𝕀, and 𝑗=1,2,…,𝑃;(H3)’𝐴_𝛾𝑥+∑^𝑃_𝑗=1𝑓_𝛾𝑗(𝑥,𝑥)=0 has only one solution 𝑥^𝛾∈𝑅^𝑛 and 𝑥^𝑗≠𝑥^𝑙 for 𝑗≠𝑙, 𝑗,𝑙∈𝕀. Let Γ^′={𝑥^𝑗_𝑖(𝑡;0,𝑥^𝑗),𝑖,𝑗∈𝕀,𝑡∈𝑅⁺}. Similarly, we can obtain the following.

Lemma 7. Let 𝜇 be the dwell time of the switching law 𝜎=((𝑡₁,𝑖₁),…,(𝑡_𝑘,𝑖_𝑘),…), and 𝛿>0 is a constant. If 𝜇≥max{𝜏,(ln(𝑀(𝛿+𝐷)/𝛿))/(𝛼−2𝑃𝑀𝐿)}, then 𝜑∈𝐵_𝛿(𝑥^𝑖0) implies that the solution 𝑥(𝑡,0,𝜑,𝜎) of (32) satisfies 𝑥_𝑡𝑘∈𝐵_𝛿(𝑥^𝑖𝑘).

Theorem 8. Let 𝜇 be the dwell time of the switching law 𝜎=((𝑡₁,𝑖₁),…,(𝑡_𝑘,𝑖_𝑘),…), and 𝛿>0 is a constant. If 𝜇≥max{𝜏,(ln(𝑀(𝛿+𝐷)/𝛿))/(𝛼−2𝑃𝑀𝐿)}, then 𝜑∈𝐵_𝛿(𝑥^𝑖0) implies that the solution 𝑥(𝑡,0,𝜑,𝜎) of (32) satisfies 𝑥(𝑡)∈𝐵_𝑀𝛿(Γ^′) for all 𝑡∈𝑅⁺.

4. Attractiveness

In this section, we first give the concept of an attractor of the switched system with switching law, and then we study the existence of the attractor of the nonlinear switched delay system (1). We will establish sufficient conditions which guarantee the existence of the attractor and furthermore we can find out the switching law and the attractor of the switching delay system (1).

Definition 9. A set Ω⊂𝑅^𝑛 is called an attractor of the switched system (1) with a switching law 𝜎, if there exists a 𝛿>0 such that, for any 𝜀>0, there exists a 𝑇>0, such that, for any 𝜑∈𝐵_𝛿(Ω)∩𝐶_𝜏, 𝑡≥𝑇 implies 𝜌(𝑥(𝑡,0,𝜑,𝜎),Ω)≤𝜀.

Remark 10. According to Definition 9, Ω is an attractor of the switched system (1) with a switching law 𝜎 if and only if there exists 𝛿>0 such that 𝜑∈𝐵_𝛿(Ω) implies lim_𝑡→∞𝑥𝑡,0,𝜑,𝜎=0. (33) In this section, we will prove that Γ is the attractor of system (1). For convenience, we first show that 𝜑∈𝐵_𝛿(Γ)⟹lim_𝑡→∞𝑥𝑡,0,𝜑,𝜎=0 (34) is equivalent to 𝜑∈𝐵_𝛿𝑥^𝑖⟹lim_𝑡→∞𝑥𝑡,0,𝜑,𝜎=0. (35)

Lemma 11. For any given 𝜀>0, 𝜆>0, there exists 𝑇(𝜀,𝜆)=ln(𝑀Δ/𝜆𝜀)/(𝛼−𝑀𝐿)>0 such that if 𝜑∈𝐵_𝜀(Γ), 𝜑∈𝐶_𝜏, then the solution 𝑥_𝑖(𝑡,0,𝜑) of the 𝑖𝑡ℎ subsystem of (1) satisfies 𝑥_𝑖𝑡∈𝐵_𝜆𝜀(𝑥^𝑖),𝑡≥𝑇, where Δ≜sup{‖𝑥−𝑦‖,𝑥,𝑦∈𝐵_𝜀(Γ)}., If 𝜑∈𝐵_𝜀(𝑥^𝑗) and 𝜑∈𝐶_𝜏 especially, then one can take 𝑇(𝜀,𝜆)=ln(𝑀(𝜀+𝐷)/𝜆𝜀)/(𝛼−𝑀𝐿)>0 such that when 𝑡≥𝑇 implies the solution of the 𝑖𝑡ℎ subsystem of (1) 𝑥_𝑖(𝑡,0,𝜑) satisfies 𝑥_𝑖𝑡∈𝐵_𝜆𝜀(𝑥^𝑖).

Proof. Let 𝑥_𝑖(𝑡,0,𝜑)≜𝑥_𝑖(𝑡) be the solution of the 𝑖th subsystem of (1) and 𝜑∈𝐵_𝜀(Γ), and then 𝑥_𝑖(𝑡)−𝑥^𝑖=𝑒^𝐴𝑖𝑡𝜑(0)+^𝑡₀𝑒^{𝐴𝑖(𝑡−𝑢)}𝑓_𝑖𝑥_𝑖(𝑢−𝜏)𝑑𝑢−𝑒^𝐴𝑖𝑡𝑥^𝑖−^𝑡₀𝑒^{𝐴𝑖(𝑡−𝑢)}𝑓_𝑖𝑥^𝑖≤𝑒𝑑𝑢_𝐴𝑖𝑡𝜑(0)−𝑥^𝑖+^𝑡₀𝑒^{𝐴𝑖(𝑡−𝑢)}𝑓_𝑖𝑥_𝑖(𝑢−𝜏)−𝑓_𝑖𝑥^𝑖𝑑𝑢≤𝑀𝑒^−𝛼𝑡𝜑(0)−𝑥^𝑖+^𝑡₀𝑀𝐿𝑒^{−𝛼(𝑡−𝑢)}𝑥_𝑖(𝑢−𝜏)−𝑥^𝑖𝑑𝑢, (36) which implies that 𝑒^𝛼𝑡𝑥_𝑖𝑡−𝑥^𝑖_𝜏≤𝑀𝜑−𝑥^𝑖_𝜏+𝑀𝐿^𝑡₀𝑒^𝛼𝑢𝑥_𝑖𝑢−𝑥^𝑖_𝜏𝑑𝑢 (37) and thus 𝑥_𝑖𝑡−𝑥^𝑖_𝜏≤𝑀Δ𝑒^{−(𝛼−𝑀𝐿)𝑡}. (38) So, when 𝑡>𝑇=ln(𝑀Δ/𝜆𝜀)/(𝛼−𝑀𝐿), we have 𝑥_𝑖𝑡−𝑥^𝑖_𝜏≤𝜆𝜀andthus𝑥_𝑖𝑡∈𝐵_𝜆𝜀𝑥^𝑖. (39)
If 𝜑∈𝐵_𝜀(𝑥^𝑗) specially, it follows from (37), that 𝑒^𝛼𝑡𝑥_𝑖𝑡−𝑥^𝑖_𝜏≤𝑀𝜑−𝑥^𝑖_𝜏+𝑀𝐿^𝑡₀𝑒^𝛼𝑢𝑥_𝑖𝑢−𝑥^𝑖_𝜏𝑑𝑢≤𝑀𝜑−𝑥^𝑗_𝜏+𝑥^𝑗−𝑥^𝑖+𝑀𝐿^𝑡₀𝑒^𝛼𝑢𝑥_𝑖𝑢−𝑥^𝑖_𝜏𝐷𝑑𝑢≤𝑀𝜀++𝑀𝐿^𝑡₀𝑒^𝛼𝑢𝑥_𝑖𝑢−𝑥^𝑖_𝜏𝑑𝑢, (40) which implies that 𝑥_𝑖𝑡−𝑥^𝑖_𝜏≤𝑀𝜀+𝐷𝑒^{−(𝛼−𝑀𝐿)𝑡} (41) and thus, when 𝑡≥𝑇=ln(𝑀(𝜀+𝐷)/𝜆𝜀)/(𝛼−𝑀𝐿), we have ‖𝑥_𝑖𝑡−𝑥^𝑖‖_𝜏≤𝜆𝜀; that is, 𝑥_𝑖𝑡∈𝐵_𝜆𝜀(𝑥^𝑖). This completes the proof.

Theorem 12. Suppose that the dwell time 𝜇 of the switching law 𝜎={(𝑡₁,𝑖₁),(𝑡₂,𝑖₂),…} satisfies 𝑡₁≥ln(𝑀Δ/𝜆𝜀)/(𝛼−𝑀𝐿). If 𝜇≥max{𝜏,ln(𝑀Δ/𝜆𝜀)/(𝛼−𝑀𝐿)}, then the set Γ is an attractor of the switched delay system (1).

Proof. From Lemma 11, we can see that we only need to prove that, for any 𝜑∈𝐶_𝜏, 𝜑∈𝐵₁(∪^𝑁_𝑖=1𝑥^𝑖), the solution 𝑥(𝑡,0,𝜑,𝜎) satisfies lim_𝑡→∞𝜌(𝑥(𝑡,0,𝜑,𝜎,),Γ)=0. In fact, we take 𝜀=1 and 𝜆=1/2 in Lemma 11, and then 𝜑∈𝐶_𝜏 and 𝜑∈𝐵₁(𝑥^𝑖0)(𝑖₀∈𝕀) implies that 𝑥_𝑡𝑘∈𝐵_1/2^𝑘(𝑥^𝑖𝑘) if 𝜇≥ln[2𝑀(1+𝐷)]/(𝛼−𝑀𝐿), 𝑘=1,2,… . On the other hand, for 𝑡∈[0,𝑡₁], we have 𝜌𝑥𝑥𝑡,0,𝜑,𝜎,Γ=𝜌_𝑖1𝑡,0,𝜑,Γ≜𝜌𝑥_𝑖1(𝑡)𝑥,Γ=inf_𝑖1(𝑡)≤𝑥−𝑦,𝑦∈Γ_𝑖1(𝑡)−𝑥_𝑖1𝑡,0,𝑥^𝑖0=𝑥_𝑖1(𝑡)−𝑥^𝑖0_𝑖1(𝑡=𝑒_{𝐴𝑖1𝑡})𝜑(0)+^𝑡₀𝑒^{𝐴𝑖1(𝑡−𝑢)}𝑓_𝑖1𝑥_𝑖1(𝑢−𝜏)𝑑𝑢−𝑒^{𝐴𝑖1𝑡}𝑥^𝑖0−^𝑡₀𝑒^{𝐴𝑖1(𝑡−𝑢)}𝑓_𝑖1𝑥^𝑖0_𝑖1()≤𝑒𝑢−𝜏𝑑𝑢_{𝐴𝑖1𝑡}𝜑(0)−𝑥^𝑖0+^𝑡₀𝑒^{𝐴𝑖1(𝑡−𝑢)}𝑓_𝑖1𝑥_𝑖1(𝑢−𝜏)−𝑓_𝑖1𝑥^𝑖0_𝑖1(𝑢−𝜏)𝑑𝑢≤𝑀𝑒^−𝛼𝑡𝜑(0)−𝑥^𝑖0+^𝑡₀𝑀𝐿𝑒^{−𝛼(𝑡−𝑢)}𝑥_𝑖1(𝑢−𝜏)−𝑥^𝑖0_𝑖1(𝑢−𝜏)𝑑𝑢, (42) which implies that 𝑒^𝛼𝑡𝑥_𝑖1(𝑡)−𝑥^𝑖0_𝑖1(𝑡)≤𝑀𝜑−𝑥^𝑖0_𝜏+^𝑡₀𝑀𝐿𝑒^𝛼𝑢𝑥_𝑖1𝑢−𝑥^𝑖0_{𝑖1𝑢𝜏}𝑑𝑢. (43) Since the right hand side of the above inequality is an increasing function, we have 𝑒^𝛼𝑡𝑥_𝑖1𝑡−𝑥^𝑖0_{𝑖1𝑡𝜏}≤𝑀𝜑−𝑥^𝑖0_𝜏+^𝑡₀𝑀𝐿𝑒^𝛼𝑢𝑥_𝑖1𝑢−𝑥^𝑖0_{𝑖1𝑢𝜏}𝑥𝑑𝑢,_𝑖1𝑡−𝑥^𝑖0_{𝑖1𝑡𝜏}≤𝑀𝜑−𝑥^𝑖0_𝜏𝑒^{−(𝛼−𝑀𝐿)𝑡}, (44) and thus 𝜌𝑥𝑡,0,𝜑,𝜎,Γ≤𝑥_𝑖1(𝑡)−𝑥_𝑖1(𝑡)≤𝑀, (45) since 𝜑∈𝐵₁(𝑥^𝑖0). When 𝑡∈[𝑡₁,𝑡₂], since 𝑥_𝑡1∈𝐵_1/2(𝑥^𝑖1), we have 𝜌𝑥𝑥𝑡,0,𝜑,𝜎,Γ=𝜌_𝑖2𝑡−𝑡₁,0,𝑥_𝑡1,Γ≜𝜌𝑥_𝑖2𝑡−𝑡₁𝑥,Γ=inf_𝑖2𝑡−𝑡₁≤𝑥−𝑦,𝑦∈Γ_𝑖2𝑡−𝑡₁−𝑥_𝑖2𝑡−𝑡₁,0,𝑥^𝑖1=𝑥_𝑖2𝑡−𝑡₁−𝑥^𝑖1_𝑖2𝑡−𝑡₁=𝑒^{𝐴𝑖2(𝑡−𝑡1)}𝑥_𝑡1(0)+^𝑡−𝑡1₀𝑒^{𝐴𝑖2(𝑡−𝑡1−𝑢)}𝑓_𝑖2𝑥_𝑖2(𝑢−𝜏)𝑑𝑢−𝑒^{𝐴𝑖2𝑡}𝑥^𝑖1−^𝑡−𝑡1₀𝑒^{𝐴𝑖2(𝑡−𝑡1−𝑢)}𝑓_𝑖2𝑥^𝑖1_𝑖2()≤𝑒𝑢−𝜏𝑑𝑢_{𝐴𝑖2(𝑡−𝑡1)}𝑥_𝑡1(0)−𝑥^𝑖1+^𝑡−𝑡1₀𝑒^{𝐴𝑖2(𝑡−𝑡1−𝑢)}×𝑓_𝑖2𝑥_𝑖2(𝑢−𝜏)−𝑓_𝑖2𝑥^𝑖1_𝑖2(𝑢−𝜏)𝑑𝑢≤𝑀𝑒^{−𝛼(𝑡−𝑡1)}𝑥_𝑡1(0)−𝑥^𝑖1+^𝑡−𝑡1₀𝑀𝐿𝑒^{−𝛼(𝑡−𝑡1−𝑢)}𝑥_𝑖2(𝑢−𝜏)−𝑥^𝑖1_𝑖2(𝑢−𝜏)𝑑𝑢, (46) which implies that 𝑒^{𝛼(𝑡−𝑡1)}𝑥_𝑖2𝑡−𝑥^𝑖1_{𝑖2𝑡𝜏}≤𝑀𝑥_𝑡1−𝑥^𝑖1_𝜏+^𝑡−𝑡1₀𝑀𝐿𝑒^𝛼𝑢𝑥_𝑖2𝑢−𝑥^𝑖1_{𝑖2𝑢𝜏}𝑥𝑑𝑢,_𝑖2𝑡−𝑥^𝑖1_{𝑖2𝑡𝜏}≤𝑀𝑥_𝑡1−𝑥^𝑖1_𝜏𝑒^{−(𝛼−𝑀𝐿)(𝑡−𝑡1)}, (47) and thus 𝜌𝑥𝑡,0,𝜑,𝜎,Γ≤𝑥_𝑖2𝑡−𝑥^𝑖1_{𝑖2𝑡𝜏}≤12𝑀, (48) since 𝑥_𝑡1∈𝐵_1/2(𝑥^𝑖1). Suppose that, for 𝑡∈[𝑡_𝑘−1,𝑡_𝑘], 𝜌𝑥≤1𝑡,0,𝜑,𝜎,Γ2^𝑘−1𝑀. (49) When 𝑡∈[𝑡_𝑘,𝑡_𝑘+1], since 𝑥_𝑡𝑘∈𝐵_1/2^𝑘(𝑥^𝑖𝑘), (take 𝜀=1/2^𝑘 in Lemma 11), we have 𝜌𝑥𝑥𝑡,0,𝜑,𝜎,Γ=𝜌_𝑖𝑘+1𝑡−𝑡_𝑘,0,𝑥_𝑡𝑘𝑥,Γ=𝜌_𝑖𝑘+1𝑡−𝑡_𝑘𝑥,Γ=inf_𝑖𝑘+1𝑡−𝑡_𝑘≤𝑥−𝑦,𝑦∈Γ_𝑖𝑘+1𝑡−𝑡_𝑘−𝑥_𝑖𝑘+1𝑡−𝑡_𝑘,0,𝑥^𝑖𝑘≜𝑥_𝑖𝑘+1𝑡−𝑡_𝑘−𝑥^𝑖𝑘_𝑖𝑘+1𝑡−𝑡_𝑘=𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘)}𝑥_𝑡𝑘(0)+^{𝑡−𝑡𝑘}₀𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘−𝑢)}𝑓_𝑖𝑘+1𝑥_𝑖𝑘+1(𝑢−𝜏)𝑑𝑢−𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘)}𝑥^𝑖𝑘−^{𝑡−𝑡𝑘}₀𝑒^{𝐴𝑖𝑘+1(𝑡−𝑡𝑘−𝑢)}𝑓_𝑖𝑘+1𝑥^𝑖𝑘_𝑖𝑘+1(𝑢−𝜏)𝑑𝑢≤𝑀𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑥_𝑡𝑘(0)−𝑥^𝑖𝑘+𝑀𝐿^{𝑡−𝑡𝑘}₀𝑒^{−𝛼(𝑡−𝑡𝑘−𝑢)}𝑥_𝑖𝑘+1(𝑢−𝜏)−𝑥^𝑖𝑘_𝑖𝑘+1(𝑢−𝜏)𝑑𝑢 (50) which implies that 𝑥_{𝑖𝑘+1𝑡}−𝑥^𝑖𝑘_{𝑖𝑘+1𝑡𝜏}≤𝑀𝑥_𝑡𝑘−𝑥^𝑖𝑘_𝜏𝑒^{−(𝛼−𝑀𝐿)(𝑡−𝑡1)}≤12^𝑘𝑀 (51) and thus 𝜌𝑥≤1𝑡,0,𝜑,𝜎,Γ2^𝑘𝑀,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (52) Therefore, we have lim_𝑡→∞𝜌𝑥𝑡,0,𝜑,𝜎,Γ=0, (53) which means that Γ is the attractor of system (1) with a switching law 𝜎.
Similarly, we can study the existence of the attractor of the switched delay system (32) with the assumptions (H1)’–(H3)’ and obtain the following results.

Lemma 13. For any given 𝜀>0, 𝜆>0, there exists 𝑇()(𝑀())𝜀,𝜆=max𝜏,ln𝛿+Δ/𝛿𝛼−2𝑃𝑀𝐿, (54) such that if 𝜑∈𝐵_𝜀(Γ^′) and 𝜑∈𝐶_𝜏, then the solution 𝑥_𝑖(𝑡,0,𝜑) of the 𝑖𝑡ℎ subsystem of (32) satisfies 𝑥_𝑖𝑡∈𝐵_𝜆𝜀(𝑥^𝑖),𝑡≥𝑇. If 𝜑∈𝐵_𝜀(𝑥^𝑗) especially, then one can take 𝑇(𝜀,𝜆)=ln(𝑀(𝜀+𝐷)/𝜆𝜀)/(𝛼−2𝑃𝑀𝐿)>0 such that when 𝑡≥𝑇 implies the solution 𝑥_𝑖(𝑡,0,𝜑) of the 𝑖𝑡ℎ subsystem of (32) satisfies 𝑥_𝑖𝑡∈𝐵_𝜆𝜀(𝑥^𝑖),𝑡≥𝑇.

Theorem 14. Suppose that the dwell time 𝜇 of the switching law 𝜎={(𝑡₁,𝑖₁),(𝑡₂,𝑖₂),…} satisfies 𝑡₁≥ln(𝑀Δ/𝜆𝜀)/(𝛼−2𝑃𝑀𝐿); if 𝜇≥max{𝜏,ln(𝑀Δ/𝜆𝜀)/(𝛼−2𝑃𝑀𝐿)}, then the set Γ^′ is an attractor of the switched delay system (32).

Acknowledgments

This work was partially supported by the NSFC (nos. 11071257, 61290325, and 11171019), the ARC Discovery Projects, HUST Startup Research Fund, HUST Independent Innovation Research Fund (GF and Natural Science 2013), the Natural Science Foundation of Beijing (no. 1122009), and the Grant of Beijing Education Committee Key Project (no. KZ201310028031). General Program of Science and Technology Development Project of Beijing Municipal Education Commission (no. KM200810028001), and Beijing Municipal Commission of Education (KM2014).