The Asymptotic Behavior for a Class of Impulsive Delay Differential Equations

Abstract

This paper is concerned with asymptotical behavior for a class of impulsive delay differential equations. The new criteria for determining attracting sets and attracting basin of the impulsive system are obtained by developing the properties of quasi-invariant sets. Examples and numerical simulations are given to illustrate the effectiveness of our results. In addition, we show that the impulsive effects may play a key role to these asymptotical properties even though the solutions of corresponding nonimpulsive systems are unbounded.

1. Introduction

Impulsive delay differential equations have attracted increasing interests since time delays and impulsive effects commonly exist in many fields such as population dynamics, automatic control, drug administration, and communication networks [1–4]. In past two decades, its asymptotical behaviors such as stability and attractivity of the equilibrium point or periodical solutions have been deeply studied for impulsive functional differential equations (see, [5–18]).

However, under impulsive perturbation, the solutions may not be attracted to an equilibrium point or periodical trajectory but to some bounded region. In this case, it is interesting to investigate the attracting set and attracting basin, that is, the region attracting the solutions and the range in which initial values vary when remaining the attractivity for impulsive delay differential equations. In [19], Xu and Yang first give the method to estimate global attracting set and invariant set for impulsive delayed systems by developing delayed differential inequalities. The techniques are further developed to study global attractivity for some complex impulsive systems such as impulsive neutral differential equations [20, 21] and impulsive stochastic systems [22]. But the techniques and methods given in the existing publications are invalid for determining locally attracting set and attracting basin for impulsive delay differential equations.

In this paper, our objective is to mainly discuss the asymptotical behavior on (locally) attracting set and its attracting basin for a class of impulsive delay differential equations. Based on the quasi-invariant properties, we estimate the existence range of attracting set and attracting basin of the impulsive delay systems by solving algebraic equations and employing differential inequality technique. Examples are given to illustrate the effectiveness of our method and show that the asymptotic behavior of the impulsive systems may be different from one of the corresponding continuous systems.

2. Preliminaries

Let 𝑁 be the set of all positive integers, 𝑅^𝑛 the space of 𝑛-dimensional real column vectors, and 𝑅^𝑚×𝑛 the set of 𝑚×𝑛 real matrices. For 𝐴,𝐵∈𝑅^𝑚×𝑛 or 𝐴,𝐵∈𝑅^𝑛, 𝐴≥𝐵(𝐴≤𝐵,𝐴>𝐵,𝐴<𝐵) means that each pair of corresponding elements of 𝐴 and 𝐵 satisfies the inequality “≥(≤,>,<).” 𝑅^𝑛₊={𝑥∈𝑅^𝑛∣𝑥≥0}, 𝐸=(1,1,…,1)^𝑇∈𝑅^𝑛, and 𝐼 denotes an 𝑛×𝑛 unit matrix.

Let 𝜏>0 and 𝑡₀<𝑡₁<𝑡₂<⋯ be the fixed points with lim_𝑘→∞𝑡_𝑘=∞ (called impulsive moments).

𝐶[𝑋,𝑌] denotes the space of continuous mappings from the topological space 𝑋 to the topological space 𝑌. Let 𝐶=^Δ𝐶[[−𝜏,0],𝑅^𝑛] especially.

Morever, PC=^Δ{𝜙∶[−𝜏,0]→𝑅^𝑛∣𝜙(𝑡⁺)=𝜙(𝑡) for 𝑡∈[−𝜏,0), 𝜙(𝑡⁻) exists for 𝑡∈(−𝜏,0], 𝜙(𝑡⁻)=𝜙(𝑡) for all but at most a finite number of points 𝑡∈(−𝜏,0]}. PC is a space of piecewise right-hand continuous functions which is a nature extension of the phrase space 𝐶.

We define PC[[𝑡₀,∞),𝑅^𝑛]=^Δ{𝜓∶[𝑡₀,∞)→𝑅^𝑛∣𝜓(𝑡) is continuous at 𝑡≠𝑡_𝑘, 𝜓(𝑡⁺_𝑘) and 𝜓(𝑡⁻_𝑘) exist, 𝜓(𝑡_𝑘)=𝜓(𝑡⁺_𝑘), for 𝑘∈𝑁}.

For 𝑥∈𝑅^𝑛, 𝐴∈𝑅^𝑛×𝑛, 𝜙∈𝐶 or 𝜙∈PC, we define [𝑥]⁺=𝑥₁,𝑥₂,…,𝑥_𝑛^𝑇,[𝐴]⁺=𝑎_{𝑖𝑗𝑛×𝑛},𝜙⁺_𝜏=𝜙_1𝜏,𝜙_2𝜏,…,𝜙_𝑛𝜏, (1) where ‖𝜙_𝑖‖_𝜏=sup_{𝑠∈[−𝜏,0]}‖𝜙_𝑖(𝑠)‖ and ‖⋅‖ is an norm in 𝑅^𝑛.

In this paper, we will consider a impulsive delay differential equations: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑓𝑡,𝑥_𝑡,𝑡≠𝑡_𝑘,𝑡≥𝑡₀,Δ𝑥=𝐵𝑥𝑡⁻_𝑘+𝐼_𝑘𝑡⁻_𝑘,𝑥𝑡⁻_𝑘,𝑘∈𝑁, (2) where 𝑥̇(𝑡) denotes the right-hand derivative of 𝑥(𝑡), Δ𝑥=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘), 𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘), 𝐴 = diag{𝑎₁,𝑎₂,…,𝑎_𝑛}, 𝐵 = diag{𝑏₁,𝑏₂,…,𝑏_𝑛}, 𝑓∈𝐶[[𝑡_𝑘−1,𝑡_𝑘)×PC,𝑅^𝑛], and the limit lim_{(𝑡,𝜙)→(𝑡}⁻_𝑘,𝜑)𝑓(𝑡,𝜙)=𝑓(𝑡⁻_𝑘,𝜑) exists, 𝐼_𝑘∈𝐶[[𝑡₀,∞)×𝑅^𝑛,𝑅^𝑛], and 𝑥_𝑡∈PC is defined by 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠),𝑠∈[−𝜏,0].

A function 𝑥(𝑡)∶[𝑡₀−𝜏,∞)→𝑅^𝑛 is called to be a solution of (2) through (𝑡₀,𝜙), if 𝑥(𝑡)∈PC[[𝑡₀,∞),𝑅^𝑛] as 𝑡≥𝑡₀, and satisfies (2) with the initial condition 𝑥𝑡₀+𝑠=𝜙(𝑠),𝑠∈[−𝜏,0],𝜙∈PC. (3) Throughout the paper, we always assume that for any 𝜙∈PC, system (2) has at least one solution through (𝑡₀,𝜙), denoted by 𝑥(𝑡,𝑡₀,𝜙) or 𝑥_𝑡(𝑡₀,𝜙) (simply 𝑥(𝑡) and 𝑥_𝑡 if no confusion should occur), where 𝑥_𝑡(𝑡₀,𝜙)=𝑥(𝑡+𝑠,𝑡₀,𝜙)∈PC, 𝑠∈[−𝜏,0].

In this paper, we need the following definitions involving attracting set, attracting basin, the quasi-invariant set of impulsive systems, and monotonous vector functions.

Definition 1. The set 𝑆⊂PC is called to be an attracting set of (2), and 𝐷⊂PC is called an attraction basin of 𝑆, if for any initial value 𝜙∈𝐷, the solution 𝑥_𝑡(𝑡₀,𝜙) converges to 𝑆 as 𝑡→+∞. That is, dist𝑥_𝑡𝑡₀,𝜙,𝑆⟶0,a𝑠𝑡⟶+∞, (4) where dist(𝜑,𝑆)=inf_𝜓∈𝑆 dist(𝜑,𝜓), dist(𝜑,𝜓)=sup_{𝑠∈[−𝜏,0]}‖𝜑(𝑠)−𝜓(𝑠)‖, for 𝜑∈PC.

Definition 2. The set 𝐷⊂PC is called to be a positive quasi-invariant set of (2), if there is a positive diagonal matrix 𝐿 = diag{𝑙_𝑖} such that for any initial value 𝜙∈𝐷, the solutions 𝑥_𝑡(𝑡₀,𝜙) satisfy 𝐿𝑥_𝑡(𝑡₀,𝜙)∈𝐷, for 𝑡≥𝑡₀. When 𝐿=𝐼(identity matrix) especially, the set 𝐷 is called positively invariant.

Definition 3. Let Ω⊂𝑅^𝑛. The vector function 𝐹(𝑥)∶Ω→𝑅^𝑛 is called to be monotonically nondecreasing in Ω, if for any 𝑥^′,𝑥^′′∈Ω, 𝑥^′≤𝑥^′′ implies 𝐹(𝑥^′)≤𝐹(𝑥^′′).

3. Main Results

In this paper, we always make the following assumptions. (𝐻₁) There exist nonnegative constants 𝜃,𝜚 such that 0<𝜃≤𝑡_𝑘−𝑡_𝑘−1≤𝜚, for 𝑘∈𝑁.(𝐻₂)[𝑓(𝑡,𝜑)]⁺≤𝑝([𝜑]⁺_𝜏) for 𝑡≥𝑡₀ and 𝜑∈PC, where the vector function 𝑝(⋅)∶𝑅^𝑛₊→𝑅^𝑛₊ is continuous and monotonically nondecreasing in 𝑅^𝑛₊.(𝐻₃) [𝐼_𝑘(𝑡,𝑥)]⁺≤𝑞([𝑥]⁺), for 𝑡≥𝑡₀, 𝑘∈𝑁 and 𝑥∈𝑅^𝑛, where the vector function 𝑞(⋅)∶𝑅^𝑛₊→𝑅^𝑛₊ is continuous and monotonically nondecreasing in 𝑅^𝑛₊.

To obtain attractivity, we first give the quasi-invariant properties of (2).

Theorem 4. Assume that in addition to (𝐻₁)–(𝐻₃), there is a vector 𝑧^∗≥0 such that 𝑝𝑀𝑧^∗+𝑊𝐼−𝑒^{−𝑊𝜃−1}𝑞𝑀𝑧^∗−𝑊𝑧^∗<0, (5) where 𝑊 = diag{𝑤₁,…,𝑤_𝑛}, 𝑀 = diag{𝑚₁,…,𝑚_𝑛}, 𝑤_𝑖>0,𝑚_𝑖≥1,𝑖=1,2,…,𝑛, are defined by 𝑤_𝑖=−𝑎_𝑖−ln1+𝑏_𝑖𝜚,if0<1+𝑏_𝑖<1,−𝑎_𝑖−ln1+𝑏_𝑖𝜃,if1+𝑏_𝑖𝑚≥1,_𝑖=11+𝑏_𝑖,if0<1+𝑏_𝑖<1,1,if1+𝑏_𝑖≥1. (6) Then, the set 𝐷={𝜙∈PC∣[𝜙]⁺_𝜏≤𝑧^∗} is a positive quasi-invariant set of (2). When 𝑀=𝐼 especially, 𝐷 is a positive invariant set of (2).

Proof. Let 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝜙) be a solution of (2) through (𝑡₀,𝜙). It is easily verified that the following formula for the variation of parameters is valid: 𝑥(𝑡)=𝐾𝑡,𝑡₀𝜙(0)+^𝑡_𝑡0𝐾(𝑡,𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠+_{𝑡0<𝑡𝑘≤𝑡}𝐾𝑡,𝑡_𝑘𝐼_𝑘𝑡⁻_𝑘,𝑥𝑡⁻_𝑘,𝑡≥𝑡₀, (7) where 𝐾(𝑡,𝑠) is the Cauchy matrix of linear impulsive system 𝑦̇(𝑡)=𝐴𝑦(𝑡),𝑡≠𝑡_𝑘,Δ𝑦𝑡⁺_𝑘=𝐵𝑦𝑡⁻_𝑘,𝑘∈𝑁. (8) According to the representation of the Cauchy matrix (see page 74 [2]), 𝐾(𝑡,𝑠)=𝑒^{𝐴(𝑡−𝑠)}_{𝑠<𝑡𝑘≤𝑡}(𝐼+𝐵),𝑡≥𝑠≥𝑡₀. (9) Since 0<𝜃≤𝑡_𝑘−𝑡_𝑘−1≤𝜚, for 𝑘∈𝑁, we obtain the following estimate: _{𝑠<𝑡𝑘≤𝑡}1+𝑏_𝑖≤1+𝑏_𝑖^{((𝑡−𝑠)/𝜚)−1}=11+𝑏_𝑖𝑒^{(ln∣1+𝑏𝑖∣/𝜚)(𝑡−𝑠)},if0<1+𝑏_𝑖<1,1+𝑏_𝑖^{(𝑡−𝑠)/𝜃}=𝑒^{(ln∣1+𝑏𝑖∣/𝜃)(𝑡−𝑠)},if1+𝑏_𝑖≥1. (10) In terms of the definition of 𝑀 and 𝑊, [𝐾(𝑡,𝑠)]⁺≤𝑀𝑒^{−𝑊(𝑡−𝑠)},𝑡≥𝑠≥𝑡₀. (11) By (7) and (11) and the assumptions (𝐻₂) and (𝐻₃), then [𝑥(𝑡)]⁺≤𝑀𝑒^{−𝑊(𝑡−𝑡0)}𝜙⁺_𝜏+𝑀^𝑡_𝑡0𝑒^{−𝑊(𝑡−𝑠)}𝑝𝑥_𝑠⁺_𝜏𝑑𝑠+𝑀_{𝑡0<𝑡𝑘≤𝑡}𝑒^{−𝑊(𝑡−𝑡𝑘)}𝑞𝑥𝑡⁻_𝑘⁺,𝑡≥𝑡₀. (12) Since 𝑡_𝑘−𝑡_𝑘−1≥𝜃>0 and 𝑊 = diag{𝑤₁,…,𝑤_𝑛}>0, we have _{𝑡0<𝑡𝑘≤𝑡}𝑒^{−𝑊(𝑡−𝑡𝑘)}=_{𝑡0<𝑡𝑘≤𝑡}𝑒^{−𝑊(𝑡−𝑡𝑘)}𝐼−𝑒^−𝑊𝜃𝐼−𝑒^{−𝑊𝜃−1}≤_{𝑡0<𝑡𝑘≤𝑡}𝑒^{−𝑊(𝑡−𝑡𝑘)}𝐼−𝑒^{−𝑊(𝑡𝑘−𝑡𝑘−1)}𝐼−𝑒^{−𝑊𝜃−1}=_{𝑡0<𝑡𝑘≤𝑡}𝑒^{−𝑊(𝑡−𝑡𝑘)}−𝑒^{−𝑊(𝑡−𝑡𝑘−1)}𝐼−𝑒^{−𝑊𝜃−1}≤𝐼−𝑒^{−𝑊(𝑡−𝑡0)}𝐼−𝑒^{−𝑊𝜃−1}. (13) From the strict inequality (5), there is an enough small number 𝜀>0 such that 𝑝(𝑀𝑧)+𝑊𝐼−𝑒^{−𝑊𝜃−1}𝑞()𝑧𝑀𝑧−𝑊𝑧<0,_Δ=𝑧^∗+𝜀𝐸>0. (14) In the following, we will prove that [𝜙]⁺_𝜏<𝑧 implies [𝑥(𝑡)]⁺=𝑥𝑡,𝑡₀,𝜙⁺<𝑀𝑧,𝑡≥𝑡₀. (15) Otherwise, from the piecewise continuity of 𝑥(𝑡), there must be an integer 𝑖 and 𝑡^∗>𝑡₀ such that 𝑥_𝑖𝑡^∗≥𝑚_𝑖𝑧_𝑖,[𝑥(𝑡)]⁺≤𝑀𝑧,𝑡₀≤𝑡<𝑡^∗. (16)(17) By using (12), (13), (14), (17), 𝑊>0, and the monotonicity of 𝑝(⋅),𝑞(⋅), we can get 𝑥𝑡^∗+≤𝑒^{−𝑊(𝑡∗−𝑡0)}𝑀𝜙⁺_𝜏+𝑀^𝑡∗_𝑡0𝑒^{−𝑊(𝑡∗−𝑠)}𝑝(𝑀𝑧)𝑑𝑠+𝑀_{𝑡0<𝑡𝑘≤𝑡∗}𝑒^{−𝑊(𝑡∗−𝑡𝑘)}𝑞(𝑀𝑧)<𝑒^{−𝑊(𝑡∗−𝑡0)}𝑀𝑧+𝑀𝐼−𝑒^{−𝑊(𝑡∗−𝑡0)}𝑊⁻¹𝑝(𝑀𝑧)+𝑀𝐼−𝑒^{−𝑊(𝑡∗−𝑡0)}𝐼−𝑒^{−𝑊𝜃−1}𝑞(𝑀𝑧)=𝑒^{−𝑊(𝑡∗−𝑡0)}𝑀𝑊⁻¹×𝑊𝑧−𝑝(𝑀𝑧)−𝑊𝐼−𝑒^{−𝑊𝜃−1}𝑞(𝑀𝑧)+𝑊⁻¹𝑀𝑝(𝑀𝑧)+𝑀𝐼−𝑒^{−𝑊𝜃−1}𝑞(𝑀𝑧)<𝑀𝑊⁻¹𝑊𝑧−𝑝(𝑀𝑧)−𝑊𝐼−𝑒^{−𝑊𝜃−1}𝑞(𝑀𝑧)+𝑊⁻¹𝑀𝑝(𝑀𝑧)+𝑀𝐼−𝑒^{−𝑊𝜃−1}𝑞(𝑀𝑧)=𝑀𝑧. (18) This contradicts (16), and so (15) holds. Letting 𝜀→0, from (15), we have for any 𝜙∈𝐷 (i.e., [𝜙]⁺_𝜏≤𝑧^∗), 𝑥𝑡,𝑡₀,𝜙⁺≤𝑀𝑧^∗,thatis,𝑀⁻¹𝑥𝑡,𝑡₀,𝜙⁺≤𝑧^∗,𝑡≥𝑡₀. (19) Therefore, the set 𝐷={𝜙∈PC∣[𝜙]⁺_𝜏≤𝑧^∗} is a positive quasi-invariant set of (2). When 𝑀=𝐼 especially, 𝐷 is a positive invariant set of (2). The proof is complete.

Based on the obtained quasi-invariant set, we have the following

Theorem 5. Let Δ(𝑧)=𝑝(𝑧)+𝑊𝐼−𝑒^{−𝑊𝜃−1}𝑞(𝑧)−𝑀⁻¹𝑊𝑧,𝑧∈𝑅^𝑛₊. (20) Assume that all conditions in Theorem 4 hold. Define Ω₁=𝑧∈𝑅^𝑛₊(),Ω∣Δ𝑀𝑧<0₂=𝑧∈𝑅^𝑛₊(𝑧),Ω∣Δ<0₃=𝑧∈𝑅^𝑛₊(𝑧),Ω∣Δ≥0_∗1=_{𝑧∗∈Ω1}𝑧∈𝑅^𝑛₊∣𝑧≤𝑧^∗,Ω^∗₂=_{𝑧∗∈Ω2}𝑧∈𝑅^𝑛₊∣𝑧≤𝑧^∗. (21) Then, 𝑆={𝜙∈PC∣[𝜙]⁺_𝜏∈Ω^∗₂∩Ω₃} is an attracting set of (2) and 𝐷={𝜙∈PC∣[𝜙]⁺_𝜏∈Ω^∗₁} is the attracting basin of 𝑆.

Proof. From (5) and the definitions of the above sets, then 𝑧^∗∈Ω₁, 𝑀𝑧^∗∈Ω₂, 0∈Ω^∗₁, 0∈Ω^∗₂, 0∈Ω₃. Obviously, Ω₁,Ω₂,Ω^∗₁,Ω^∗₂,Ω₃ and Ω^∗₂∩Ω₃ are nonempty, and so the definitions of the sets of 𝑆 and 𝐷 are valid. For any 𝜙∈𝐷, there is a 𝑧^∗∈Ω₁ satisfying [𝜙]⁺_𝜏≤𝑧^∗. According to Theorem 4, we obtain [𝑥(𝑡)]⁺=𝑥𝑡,𝑡₀,𝜙⁺≤𝑀𝑧^∗∈Ω₂,∀𝑡≥𝑡₀. (22) That is, 𝜎=^Δlimsup_𝑡→∞[𝑥(𝑡)]⁺∈Ω^∗₂. (23) Then, for any given 𝜀>0, there is a 𝑇₁>𝑡₀ such that [𝑥(𝑡)]⁺≤𝜀𝐸+𝜎,𝑡≥𝑇₁. (24) In light of 𝑊 = diag{𝑤_𝑖}>0, for the above 𝜀>0 and 𝑇₁, we can find an enough large 𝑇₂>0 such that ^∞_𝑇2𝑒^−𝑊𝑠𝑑𝑠≤𝜀𝐼,_{𝑡0<𝑡𝑘≤𝑇1}𝑒^{−𝑊(𝑡−𝑡𝑘)}≤𝜀𝐼,𝑡>𝑇₂. (25) Using (12), (13), (22), (24), and (25), we have for 𝑡≥𝜏+𝑇₁+𝑇₂, [𝑥(𝑡)]⁺≤𝑒^{−𝑊(𝑡−𝑡0)}𝑀𝜙⁺_𝜏+^𝑡_𝑡0𝑒^{−𝑊(𝑡−𝑠)}𝑀𝑝𝑥_𝑠⁺_𝜏+𝑑𝑠_{𝑡0<𝑡𝑘≤𝑡}𝑀𝑒^{−𝑊(𝑡−𝑡𝑘)}𝑞𝑥𝑡⁻_𝑘⁺≤𝑒^{−𝑊(𝑡−𝑡0)}𝑀𝜙⁺_𝜏+^𝑡−𝑇2_𝑡0+^𝑡_𝑡−𝑇2𝑒^{−𝑊(𝑡−𝑠)}𝑀𝑝𝑥_𝑠⁺_𝜏+𝑑𝑠_{𝑡0<𝑡𝑘≤𝑇1}+_{𝑇1<𝑡𝑘≤𝑡}𝑀𝑒^{−𝑊(𝑡−𝑡𝑘)}𝑞𝑥𝑡⁻_𝑘⁺≤𝑒^{−𝑊(𝑡−𝑡0)}𝑀𝑧^∗+^∞_𝑇2𝑒^−𝑊𝑠𝑀𝑝𝑀𝑧^∗+𝑑𝑠_{𝑡𝑡−𝑇2}𝑒^{−𝑊(𝑡−𝑠)}𝑀𝑝(𝜀𝐸+𝜎)𝑑𝑠+𝑀_{𝑡0<𝑡𝑘≤𝑇1}𝑒^{−𝑊(𝑡−𝑡𝑘)}𝑞𝑀𝑧^∗+𝑀_{𝑇1<𝑡𝑘≤𝑡}𝑒^{−𝑊(𝑡−𝑡𝑘)}𝑞(𝜀𝐸+𝜎)≤𝑒^{−𝑊(𝑡−𝑡0)}𝑀𝑧^∗+𝜀𝑀𝑝𝑀𝑧^∗+𝐼−𝑒^−𝑊𝑇2𝑊⁻¹𝑀𝑝(𝜀𝐸+𝜎)+𝜀𝑀𝑞𝑀𝑧^∗+𝑀𝐼−𝑒^{−𝑊(𝑡−𝑇1)}𝐼−𝑒^{−𝑊𝜃−1}𝑞(𝜀𝐸+𝜎)≤𝑒^{−𝑊(𝑡−𝑡0)}𝑀𝑧^∗+𝜀𝑀𝑝𝑀𝑧^∗+𝑞𝑀𝑧^∗+𝑊⁻¹𝑀𝑝(𝜀𝐸+𝜎)+𝑀𝐼−𝑒^{−𝑊𝜃−1}𝑞(𝜀𝐸+𝜎). (26) This implies that 𝜎=limsup_𝑡→+∞[𝑥(𝑡)]⁺≤𝜀𝑀𝑝𝑀𝑧^∗+𝑞𝑀𝑧^∗+𝑊⁻¹𝑀𝑝(𝜀𝐸+𝜎)+𝑀𝐼−𝑒^{−𝑊𝜃−1}𝑞(𝜀𝐸+𝜎). (27) Letting 𝜖→0⁺, then 𝜎≤𝑊⁻¹𝑀𝑝(𝜎)+𝑀𝐼−𝑒^{−𝑊𝜃−1}𝑞(𝜎). (28) That is, Δ(𝜎)≥0 and 𝜎∈Ω₃. Thus, 𝜎∈Ω^∗₂∩Ω₃. (29) From the definition of 𝜎 and 𝑆, dist (𝑥_𝑡(𝑡₀,𝜙),𝑆)→0 as 𝑡→+∞. The proof is complete.

From the above theorems, we can obtain sufficient conditions ensuring global attractivity and stability in the following corollaries.

Corollary 6. Assume that (𝐻₁)–(𝐻₃) hold with 𝑝𝜑⁺_𝜏=𝑃𝜑⁺_𝜏𝑝+𝜇,𝑃=_{𝑖𝑗𝑛×𝑛}≥0,𝜇=𝜇₁,…,𝜇_𝑛^𝑇𝑞[𝑥]≥0,₊=𝑄[𝑥]⁺𝑞+𝜈,𝑄=_{𝑖𝑗𝑛×𝑛}≥0,𝜈=𝜈₁,…,𝜈_𝑛^𝑇≥0. (30) If the spectral radius 𝜌(Λ)<1,whereΛ=𝑊⁻¹𝑀𝑃+𝑀𝐼−𝑒^{−𝜃𝑊−1}𝑄, (31) then 𝐷={𝜙∈PC∣[𝜙]⁺_𝜏≤𝑍=^Δ(𝐼−Λ)⁻¹𝑊⁻¹(𝜇+𝑊[𝐼−𝑒^−𝜃𝑊]⁻¹𝜈)} is a positive quasi-invariant set of (2), and 𝑆={𝜙∈PC∣[𝜙]⁺_𝜏≤(𝐼−Λ)⁻¹𝑊⁻¹𝑀(𝜇+𝑊[𝐼−𝑒^−𝜃𝑊]⁻¹𝜈)} is a global attracting set of (2).

Proof. Since 𝑝(𝑧)=𝑃𝑧+𝜇 and 𝑞(𝑧)=𝑄𝑧+𝜈, we directly calculate Δ(𝑧)=𝑀⁻¹𝑊(Λ−𝐼)𝑧+𝜇+𝑊𝐼−𝑒^{−𝜃𝑊−1}Δ()()𝜈,𝑀𝑧=𝑊Λ−𝐼𝑧+𝜇+𝑊𝐼−𝑒^{−𝜃𝑊−1}𝜈. (32) Without loss of generality, we assume that 𝜇,𝜈>0. Since 𝜌(Λ)<1, (𝐼−Λ)⁻¹ exists and (𝐼−Λ)⁻¹≥0 (see [23]), and so 𝑍>0. For any 𝜅>0, we take 𝑧^∗=(1+𝜅)𝑍>0 in Theorem 4 and verify the condition (5): Δ𝑀𝑧^∗=−𝜅𝜇+𝑊𝐼−𝑒^{−𝜃𝑊−1}𝜈<0. (33) According to Theorem 4, when 𝜅→0, we deduce that 𝐷 is a positive quasi-invariant set of (2). Furthermore, by (33), (1+𝜅)𝑍∈Ω₁,(1+𝜅)𝑀𝑍∈Ω₂. (34) From the arbitrariness of 𝜅, we obtain Ω^∗₁=Ω^∗₂=𝑅^𝑛₊. Moreover, Ω₃=𝑧∈𝑅^𝑛₊(𝑧)=∣Δ≥0𝑧∈𝑅^𝑛₊∣𝑀⁻¹𝑊(Λ−𝐼)𝑧+𝜇+𝑊𝐼−𝑒^{−𝜃𝑊−1}=𝜈≥0𝑧∈𝑅^𝑛₊∣(𝐼−Λ)𝑧≤𝑊⁻¹𝑀𝜇+𝑊𝐼−𝑒^{−𝜃𝑊−1}𝜈⊂𝑧∈𝑅^𝑛₊∣𝑧≤(𝐼−Λ)⁻¹𝑊⁻¹𝑀×𝜇+𝑊𝐼−𝑒^{−𝜃𝑊−1}𝜈. (35) It follows from Theorem 5 that 𝑆^′={𝜙∈PC∣[𝜙]⁺_𝜏∈Ω₃} is a global attracting set of (2) and 𝑆 is also a global attracting set due to 𝑆^′⊂𝑆. The proof is complete.

Corollary 7. Assume that all conditions in Corollary 6 hold with 𝜇=𝜈=0. Then, the zero solution 𝑥(𝑡)=0 of (2) is globally asymptotically stable.

4. Illustrative Examples

The following illustrative examples will demonstrate the effectiveness of our results and also show the different asymptotical behaviors between the impulsive system and the corresponding continuous system.

Example 8. Consider a scalar nonlinear impulsive delay system 𝑥̇(𝑡)=0.2𝑥(𝑡)+0.2𝑥²(𝑡−1)+0.1,𝑡≠𝑡_𝑘,𝑘∈𝑁,𝑡≥𝑡₀𝑡=0,Δ𝑥=−0.6𝑥⁻_𝑘+0.1𝑥²𝑡⁻_𝑘+0.1sin𝑒^𝑡𝑘,𝑡_𝑘=𝑡_𝑘−1+0.15. (36) According to Theorems 4 and 5, we have 𝜃=𝜚=0.15, 𝐴=0.2, 𝐵=−0.6, 𝑀=2.5, 𝑊≐5.9086, 𝑝(𝑧)=0.2𝑧²+0.1, 𝑞(𝑧)=0.1𝑧²+0.1, Δ(𝑧)≐1.2052𝑧²−2.3634𝑧+1.1052, and so Ω₁=𝑧∈𝑅₊()=(),Ω∣Δ𝑀𝑧<00.3079,0.4765_∗1=[],Ω0,0.4765₂=𝑧∈𝑅₊(𝑧)=(),Ω∣Δ<00.7698,1.1913_∗2=[],Ω0,1.1913₃=𝑧∈𝑅₊(𝑧)=[]∪[),Ω∣Δ≥00,0.76981.1913,+∞_∗2∩Ω₃=[0,0.7698]. (37) Thus, 𝑆={𝜙∈PC∣[𝜙]⁺_𝜏≤0.7698} is an attracting set of (36), and 𝐷={𝜙∈PC∣[𝜙]⁺_𝜏≤0.4765} is an attracting basin of 𝑆. However, solutions of the corresponding continuous system (i.e., Δ𝑥=0 in (36)) may be unbounded. Taking the initial condition 𝜙(𝑠)=0.2, 𝑠∈[−1,0], Figure 1 shows the different asymptotic behavior between the solution of (36) with no impulse and one with impulses.

(a)

(b)

Figure 1 

The trajectory of (36) with: (a) no impulse (i.e., Δ𝑥=0) and (b) impulses.

Example 9. Consider a 2-dimensional impulsive delay system 𝑥̇₁(𝑡)=𝑥₁(𝑡)+0.5sin𝑥₁(𝑡−1)−0.4𝑥₂()𝑥̇𝑡−1−0.5,𝑡≥0,₂(𝑡)=−4𝑥₂(𝑡)−0.5𝑥₁(𝑡−1)+0.4cos𝑥₂(𝑡−1)+0.5,𝑡≠𝑡_𝑘,Δ𝑥₁=−0.5𝑥₁𝑡⁻_𝑘+0.1cos𝑥₁𝑡⁻_𝑘+0.5sin𝑒^𝑡𝑘,𝑡_𝑘=0.1𝑘,Δ𝑥₂=0.1𝑥₂𝑡⁻_𝑘+0.2sin𝑥₂𝑡⁻_𝑘−0.5cos𝑒^𝑡𝑘,𝑘∈𝑁. (38) According to Corollary 6, we have 𝜃=𝜚=0.1, 𝐴 = diag{1,−4}, 𝐵 = diag{−0.5,0.1}, 𝑀 = diag{2,1}, 𝑊 = diag{5.9315,3.0469}, 𝑝(𝑧)=𝑃𝑧+𝜇, 𝑞(𝑧)=𝑄𝑧+𝜈, Λ=𝑊⁻¹𝑀𝑃+𝑀[𝐼−𝑒^−𝜃𝑊]⁻¹𝑄, where ,,𝑃=0.50.40.50.4,𝑄=0.1000.2𝜇=𝜈=0.50.5,Λ=0.61560.13490.16410.8928 (39) and so 𝜌(Λ)=0.9575<1. Therefore, 𝐷={𝜙∈PC∣[𝜙]⁺_𝜏≤(1.7105,2.5228)^𝑇} is a positive quasi-invariant set of (38), and 𝑆={𝜙∈PC∣[𝜙]⁺_𝜏≤(1.8637,2.8720)^𝑇} is a global attracting set of (38). Figure 2 shows the asymptotic properties of solutions of (38) under the different initial conditions.

Figure 2 

Global attracting set of (38).

Acknowledgments

This work is supported partially by the National Natural Science Foundation of China under Grant nos. 10971240, 61263020, and 61004042, the Key Project of Chinese Education Ministry under Grant no. 212138, the Natural Science Foundation of Chongqing under Grant CQ CSTC 2011BB0117, and the Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ120630.