Impulsive Exponential Stabilization of Functional Differential Systems with Infinite Delay

Abstract

By using the Razumikhin technique and Lyapunov functions, we investigated the impulsive exponential stabilization of functional differential systems with infinite delay. A new result on the exponential stabilization by impulses is gained. Our result shows that impulses can make unstable systems stable. A numerical example is given to illustrate the feasibility of the result.

1. Introduction

Recently, there are many results of impulsive stability for delay systems as impulses can make unstable systems stable and stable systems unstable after impulse effects; see [1–21] and references therein. The problem of stabilizing the solutions by imposing proper impulsive control for delayed system now attracts more and more authors' attentions; see [22–26]. For example, in [22, 26], the authors have investigated impulsive stabilization of second-order differential equations with finite delay. The main tools used are Lyapunov functionals, stability theory, and control by impulses. In [23], by employing the Razumikhin technique and Lyapunov functions, several global exponential stability criteria are established for general impulsive differential equations with finite delay. However, not much has been developed in the direction of the stabilization theory of impulsive functional differential systems, especially for infinite delays of impulsive functional differential systems. This is due to some theoretical and technical difficulties; see [14, 16–21, 24]. In [24], Luo and Shen studied impulsive stabilization of functional differential equations with infinite delay. By using Lyapunov functions and Razumikhin techniques, some Razumikhin type theorems on uniform asymptotical stability are obtained. However, to the best of the authors' knowledge, there is little work on the impulsive exponential stabilization of functional differential systems with infinite delay.

The aim of this work is to establish a criterion on the impulsive exponential stabilization of functional differential systems with infinite delay by using Lyapunov functions and the Razumikhin technique. Our result shows that functional differential equations with infinite delay may be exponentially stabilized by impulses. Moreover, to some degree, the result we obtained is less conservative and more feasible than that given in [23].

This paper is organized as follows. In Section 2, we introduce some notations and definitions. Section 3 is devoted to the main results, and a numerical example is given to demonstrate the effectiveness of our result. In the last section, concluding remarks are given in Section 4.

2. Preliminaries

Let ℝ denote the set of real numbers,ℝ₊ the set of nonnegative real numbers, and ℝ^𝑛 the 𝑛-dimensional real space equipped with the Euclidean norm ‖⋅‖. For any 𝑡≥𝑡₀≥0>−𝛼≥−∞, let 𝑓(𝑡,𝑥(𝑠)) where 𝑠∈[𝑡−𝛼,𝑡] or 𝑓(𝑡,𝑥(⋅)) be a Volterra type functional. In the case when 𝛼=+∞, the interval [𝑡−𝛼,𝑡] is understood to be replaced by (−∞,𝑡].

Consider the following impulsive functional differential systems: 𝑥^′(𝑡)=𝑓(𝑡,𝑥(⋅)),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,Δ𝑥|_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑡_𝑘,𝑥_𝑡⁻_𝑘),𝑘=1,2…, (2.1) where the impulse times 𝑡_𝑘 satisfy 0≤𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯,lim_𝑘→+∞𝑡_𝑘=+∞, and 𝑥^′ denotes the right-hand derivative of 𝑥. 𝑓∈𝐶([𝑡_𝑘−1,𝑡_𝑘)×ℂ,ℝ^𝑛),𝑓(𝑡,0)=0. ℂ is an open set in 𝑃𝐶([−𝛼,0],ℝ^𝑛), where 𝑃𝐶([−𝛼,0],ℝ^𝑛)={𝜓∶[−𝛼,0]→ℝ^𝑛 is continuous everywhere except at finite number of points 𝑡_𝑘, at wh{ich 𝜓(𝑡⁺_𝑘) and 𝜓(𝑡⁻_𝑘) exist and 𝜓(𝑡⁺_𝑘)=𝜓(𝑡_𝑘)}. Define 𝑃𝐶𝐵(𝑡)={𝑥∈ℂ∶𝑥isbounded}. For 𝜓∈𝑃𝐶𝐵(𝑡), the norm of 𝜓 is defined by ‖𝜓‖=sup_{−𝛼≤𝜃≤0}|𝜓(𝜃)|. For any 𝜎≥0, let 𝑃𝐶𝐵_𝛿(𝜎)={𝜓∈𝑃𝐶𝐵(𝜎)∶‖𝜓‖<𝛿}. Let 𝐾₁={𝑎∈𝐶(ℝ₊,ℝ₊)∣𝑎(0)=0 and 𝑎(𝑡)>0 for 𝑡>0 and 𝑎 is strictly increasing in 𝑡}.

For each 𝑘=1,2…,𝐼_𝑘(𝑡,𝑥)∈𝐶([𝑡₀,∞)×ℝ^𝑛,ℝ^𝑛),𝐼_𝑘(𝑡,0)=0, and for any 𝜌>0, there exists a 𝜌₁>0(0<𝜌₁<𝜌) such that 𝑥∈𝑆(𝜌₁) implies that 𝑥+𝐼_𝑘(𝑡_𝑘,𝑥)∈𝑆(𝜌), where 𝑆(𝜌)={𝑥∶‖𝑥‖<𝜌,𝑥∈ℝ^𝑛}.

For any given 𝜎≥𝑡₀, system (2.1) is supplemented with initial conditions of the form 𝑥_𝜎=𝜙, (2.2) where 𝜙∈𝑃𝐶([−𝛼,0],ℝ^𝑛).

In this paper, we assume that the solution for the initial problem (2.1)-(2.2) does exist and is unique which will be written in the form 𝑥(𝑡,𝜎,𝜙); see [1, 4, 13]. Since 𝑓(𝑡,0)=0,𝐼(𝑡,0)=0,𝑘=1,2,…, then 𝑥(𝑡)=0 is a solution of (2.1)-(2.2), which is called the trivial solution. In this paper, we always assume that the solution 𝑥(𝑡,𝜎,𝜙) of (2.1)-(2.2) can be continued to ∞ from the right of 𝜎.

We introduce some definitions as follows.

Definition 2.1 (see [1]). The function 𝑉∶[−𝛼,∞)×ℂ→ℝ₊ belongs to class 𝑣₀ if
(i)𝑉 is continuous on each of the sets [𝑡_𝑘−1,𝑡_𝑘)×ℂ and lim_{(𝑡,𝜑)→(𝑡}⁻_𝑘,𝜓)𝑉(𝑡,𝜑)=𝑉(𝑡⁻_𝑘,𝜓) exists;(ii)𝑉(𝑡,𝑥) is locally Lipschitzian in 𝑥 and 𝑉(𝑡,0)≡0.

Definition 2.2 (see [1]). Let 𝑉∈𝑣₀, for any (𝑡,𝜓)∈[𝑡_𝑘−1,𝑡_𝑘)×ℂ, the upper right-hand Dini derivative of 𝑉(𝑡,𝑥) along the solution of (2.1)-(2.2) is defined by 𝐷⁺𝑉(𝑡,𝜓(0))=limsup_ℎ→0+{𝑉(𝑡+ℎ,𝜓(0)+ℎ𝑓(𝑡,𝜓))−𝑉(𝑡,𝜓(0))}ℎ. (2.3)

Definition 2.3 (see [1]). Assume that 𝑥(𝑡)=𝑥(𝑡,𝜎,𝜙) is the solution of (2.1)-(2.2) through (𝜎,𝜙). Then the trivial solution of (2.1)-(2.2) is said to be exponentially stable if for any 𝜀>0,𝜎≥𝑡₀, there exist constants 𝜆>0 and 𝛿=𝛿(𝜀)>0 such that 𝜙∈𝑃𝐶𝐵_𝛿(𝜎) implies ‖𝑥(𝑡)‖<𝜀⋅𝑒^{−𝜆(𝑡−𝜎)},𝑡≥𝜎.

3. Main Results

In this section, we shall develop Lyapunov-Razumikhin methods and establish some theorems which provide sufficient conditions for exponential stability of the trivial solution of (2.1)-(2.2).

Theorem 3.1. Assume that there exist functions 𝑉∈𝑣₀,𝑤∈𝐾₁ and positive constants 𝑝,𝑐₁,𝜂,𝜆,𝛽_𝑘≥0,𝑘∈ℤ₊ and 𝛾∈(1,𝑀^⋆],𝑀^⋆>1 such that the following conditions hold:
(i)𝑐₁‖𝑥‖^𝜂≤𝑉(𝑡,𝑥)≤𝑤(‖𝑥‖),(𝑡,𝑥)∈[−𝛼,∞)×𝑆(𝜌),(ii) for any 𝜎≥𝑡₀ and 𝜓∈𝑃𝐶([−𝛼,0],𝑆(𝜌)), if 𝛾𝑒^{𝜆(𝑡−𝜎)}𝑉(𝑡,𝜓(0))≥𝑉(𝑡+𝜃,𝜓(𝜃)), −𝛼≤𝜃≤0,𝑡≠𝑡_𝑘,𝑘∈ℤ₊, then 𝐷⁺𝑉(𝑡,𝜓(0))≤𝑝𝑉(𝑡,𝜓(0)), (3.1)(iii) for all (𝑡_𝑘,𝜓)∈𝑅₊×𝑃𝐶([−𝛼,0],𝑆(𝜌₁)),𝑉(𝑡_𝑘,𝜓(0)+𝐼_𝑘(𝑡_𝑘,𝜓))≤𝛽_𝑘𝑉(𝑡⁻_𝑘,𝜓(0)),𝑘∈ℤ₊, where 𝛽_𝑘 satisfies ∏^∞_𝑘=1max{𝛽_𝑘𝑀^⋆,1}<∞,(iv)𝑡_𝑘−𝑡_𝑘−1<ln𝛾/(𝑝+𝜆),𝑘∈ℤ₊,then the trivial solution of (2.1)-(2.2) is exponentially stable with the approximate exponential convergence rate 𝜆/𝜂.

Proof. From condition (iii), there exists constant 𝑀>0 such that ^∞_𝑘=1max{𝛽_𝑘𝑀^⋆,1}≤𝑀. (3.2) For any 𝜀∈(0,𝜌₁), we choose 𝛿=𝛿(𝜀)>0 such that 𝑤(𝛿)≤(𝑀^⋆𝑀)⁻¹min{𝑐₁𝜀^𝜂,𝜀}. Then, for any 𝜎≥𝑡₀, let 𝑥(𝑡)=𝑥(𝑡,𝜎,𝜙) be a solution of (2.1)-(2.2) through (𝜎,𝜙). We shall show for any 𝜙∈𝑃𝐶𝐵_𝛿(𝜎)‖𝑥(𝑡)‖<𝜀⋅𝑒^{−(𝜆/𝜂)(𝑡−𝜎)},𝑡≥𝜎. (3.3) For convenience, suppose 𝑉(𝑡)=𝑉(𝑡,𝑥(𝑡)) and 𝑉(𝜎)=max{sup_{−𝛼≤𝜃≤0}𝑉(𝜎+𝜃,𝜓(𝜃)),𝑀^⋆𝑉(𝜎)}. (3.4) Since 𝜙∈𝑃𝐶𝐵_𝛿(𝜎), we have 𝑉(𝜎)≤𝑀^⋆𝑤(𝛿).
Suppose that 𝜎∈[𝑡_𝑙−1,𝑡_𝑙),𝑙∈ℤ₊. Next we prove for 𝑡∈[𝜎,𝑡_𝑙)∪[𝑡_𝑘,𝑡_𝑘+1),𝑘≥𝑙, 𝑉(𝑡)≤𝑉(𝜎)(∏^𝑘_𝑠=𝑙max{𝛽_𝑠𝑀^⋆,1})𝑒^{−𝜆(𝑡−𝜎)},𝑡≥𝜎. (3.5) First, it is clear that for 𝑡∈[𝜎−𝛼,𝜎]𝑐₁‖𝑥(𝑡)‖^𝜂≤𝑉(𝑡)≤𝑉(𝜎)≤𝑀^⋆𝑤(𝛿)≤𝑀⁻¹min{𝑐₁𝜀^𝜂,𝜀}<𝑐₁𝜀^𝜂, (3.6) which implies ‖𝑥(𝑡)‖<𝜀<𝜌₁,𝑡∈[𝜎−𝛼,𝜎].
We next claim that (3.5) holds for all 𝑡∈[𝜎,𝑡_𝑙), that is, 𝑉(𝑡)≤𝑉(𝜎)𝑒^{−𝜆(𝑡−𝜎)},𝑡∈[𝜎,𝑡_𝑙). (3.7) In order to do this, let Γ(𝑡)=𝑉(𝑡)𝑒^{𝜆(𝑡−𝜎)},𝑡≥𝜎,𝑉(𝑡),𝜎−𝛼≤𝑡≤𝜎, (3.8) then it is obvious that Γ(𝑡)≥𝑉(𝑡) for all 𝑡≥𝜎−𝛼.
Next we prove Γ(𝑡)≤𝑉(𝜎) for 𝑡∈[𝜎,𝑡_𝑙), which implies that (3.5) holds for 𝑡∈[𝜎,𝑡_𝑙). Suppose that this assertion is false, then there exists some 𝑡∈[𝜎,𝑡_𝑙) such that Γ(𝑡)>𝑉(𝜎).
Let 𝑡^⋆=inf{𝑡∈[𝜎,𝑡_𝑙)∣Γ(𝑡)≥𝑉(𝜎)}, (3.9) then in view of Γ(𝜎)=𝑉(𝜎)≤𝑀^⋆−1𝑉(𝜎)<𝑉(𝜎), we get 𝑡^⋆∈(𝜎,𝑡_𝑙),Γ(𝑡^⋆)=𝑉(𝜎),Γ(𝑡)<𝑉(𝜎),𝑡∈[𝜎,𝑡^⋆). (3.10) Considering (3.6), we also obtain 𝑉(𝑡)≤Γ(𝑡)<𝑉(𝜎)∀𝑡∈[𝜎−𝛼,𝑡^⋆). (3.11) On the other hand, considering 𝛾∈(1,𝑀^⋆], we get Γ(𝑡^⋆1)=𝑉(𝜎)>𝛾1𝑉(𝜎),Γ(𝜎)=𝑉(𝜎)<𝑀^⋆1𝑉(𝜎)≤𝛾𝑉(𝜎). (3.12) So we can define 𝑡^⋆⋆=sup{𝑡∈[𝜎,𝑡^⋆1]∣Γ(𝑡)≤𝛾𝑉(𝜎)}, (3.13) which implies that 𝑡^⋆⋆∈[𝜎,𝑡^⋆1),𝛾𝑉(𝜎)=Γ(𝑡^⋆⋆1),𝛾𝑉(𝜎)<Γ(𝑡),𝑡∈(𝑡^⋆⋆,𝑡^⋆]. (3.14) It follows from (3.11) that 𝛾𝑉(𝑡)𝑒^{𝜆(𝑡−𝜎)}=𝛾Γ(𝑡)≥𝑉(𝜎)≥𝑉(𝑠),𝜎−𝛼≤𝑠≤𝑡,𝑡∈[𝑡^⋆⋆,𝑡^⋆]. (3.15) Using condition (ii), the inequality 𝐷⁺𝑉(𝑡)≤𝑝𝑉(𝑡) holds for all 𝑡∈[𝑡^⋆⋆,𝑡^⋆]. Hence, we obtain for 𝑡∈[𝑡^⋆⋆,𝑡^⋆]𝐷⁺Γ(𝑡)=𝐷⁺𝑉(𝑡)𝑒^{𝜆(𝑡−𝜎)}+𝜆𝑉(𝑡)𝑒^{𝜆(𝑡−𝜎)}=𝑒^{𝜆(𝑡−𝜎)}(𝐷⁺𝑉(𝑡)+𝜆𝑉(𝑡))≤𝑒^{𝜆(𝑡−𝜎)}(𝑝𝑉(𝑡)+𝜆𝑉(𝑡))=𝑉(𝑡)𝑒^{𝜆(𝑡−𝜎)}(𝑝+𝜆)≤(𝑝+𝜆)Γ(𝑡). (3.16) From (iv), we define 𝜏=̇max_𝑘∈ℤ+{𝑡_𝑘−𝑡_𝑘−1}, then 𝜏(𝑝+𝜆)<ln𝛾.
Thus, we have ^𝑡⋆_𝑡⋆⋆𝑑Γ(𝑠)=Γ(𝑠)^Γ(𝑡⋆)_Γ(𝑡^⋆⋆₎𝑑𝑠𝑠=₍𝑉_{𝜎)(1/𝛾)}𝑉_(𝜎)𝑑𝑠𝑠≥ln𝛾>𝜏𝑝+𝜆𝜏. (3.17) However, ^Γ(𝑡⋆)_{Γ(𝑡⋆⋆)}𝑑𝑠𝑠≤^𝑡⋆_𝑡⋆⋆(𝑝+𝜆)𝑑𝑠≤^𝑡⋆⋆_𝑡^+𝜏_⋆⋆(𝑝+𝜆)𝑑𝑠=𝜏𝑝+𝜆𝜏. (3.18) This is a contradiction. Therefore, we obtain Γ(𝑡)≤𝑉(𝜎),𝑡∈[𝜎,𝑡_𝑙), which implies that (3.5) holds for all 𝑡∈[𝜎,𝑡_𝑙).
Meanwhile, we get for 𝑡∈[𝜎,𝑡_𝑙)𝑐₁‖𝑥‖^𝜂≤𝑉(𝑡)≤𝑉(𝜎)𝑒^{−𝜆(𝑡−𝜎)}≤𝑉(𝜎)<𝑀^⋆𝑤(𝛿)≤𝑀⁻¹min{𝑐₁𝜀^𝜂,𝜀}<𝑐₁𝜀^𝜂, (3.19) which implies that 𝑥(𝑡⁻_𝑙)∈𝑆(𝜌₁),𝑥(𝑡_𝑙)∈𝑆(𝜌).
Furthermore, note that 𝑉(𝑡_𝑙)≤𝛽_𝑙𝑉(𝑡⁻_𝑙)≤𝛽_𝑙𝑉(𝜎)𝑒^{−𝜆(𝑡𝑙−𝜎)}, (3.20) we next prove 𝑉(𝑡)≤max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎)𝑒^{−𝜆(𝑡−𝜎)},𝑡∈[𝑡_𝑙,𝑡_𝑙+1), (3.21) which is equal to prove Γ(𝑡)≤max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎),𝑡∈[𝑡_𝑙,𝑡_𝑙+1). (3.22) Suppose that this assertion is not true, then there exists some 𝑡∈[𝑡_𝑙,𝑡_𝑙+1) such that Γ(𝑡)>max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎).
Let 𝑡^∗=inf{𝑡∈[𝑡_𝑙,𝑡_𝑙+1)∣Γ(𝑡)≥max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎)}. (3.23) Then we know 𝑡^∗∈(𝑡_𝑙,𝑡_𝑙+1),Γ(𝑡^∗)=max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎),Γ(𝑡)<max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎),𝑡∈[𝑡_𝑙,𝑡^∗). (3.24) Also, in view of the fact Γ(𝑡)<𝑉(𝜎) for 𝑡∈[𝜎−𝛼,𝑡_𝑙), we obtain Γ(𝑡)<max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎)∀𝑡∈[𝜎−𝛼,𝑡^∗). (3.25) Note that 𝛾∈(1,𝑀^⋆], then we have Γ(𝑡^∗)=max{𝛽_𝑙𝑀^⋆1,1}𝑉(𝜎)>𝛾max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎),Γ(𝑡_𝑙)≤𝛽_𝑙𝑉(𝜎)𝑒^{−𝜆(𝑡𝑙−𝜎)}<𝛽_𝑙1𝑉(𝜎)≤𝛾𝛽_𝑙𝑀^⋆1𝑉(𝜎)≤𝛾max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎). (3.26)(3.27) So we can define 𝑡^∗∗=sup{𝑡∈[𝑡_𝑙,𝑡^∗1]∣Γ(𝑡)≤𝛾max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎)}. (3.28) Then, we obtain 𝑡^∗∗∈[𝑡_𝑙,𝑡^∗),Γ(𝑡^∗∗1)=𝛾max{𝛽_𝑙𝑀^⋆1,1}𝑉(𝜎),Γ(𝑡)>𝛾max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎),𝑡∈(𝑡^∗∗,𝑡^∗]. (3.29) Thus, combining (3.25), we get 𝛾𝑉(𝑡)𝑒^{𝜆(𝑡−𝜎)}=𝛾Γ(𝑡)≥max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎)≥Γ(𝑠)≥𝑉(𝑠),𝜎−𝛼≤𝑠≤𝑡,𝑡∈[𝑡^∗∗,𝑡^∗]. (3.30) Similarly, by assumptions (ii), (iv), as the proof of (3.16), we can obtain 𝐷⁺Γ(𝑡)≤(𝑝+𝜆)Γ(𝑡),𝑡∈[𝑡^∗∗,𝑡^∗]. (3.31) Consequently, we have ^𝑡∗_𝑡∗∗𝑑Γ(𝑠)=Γ(𝑠)^Γ(𝑡∗)_Γ(𝑡^∗∗₎𝑑𝑠𝑠=^{max{𝛽𝑙𝑀⋆,1}(𝜎)}_(1/𝛾)max{𝑉_{𝛽𝑙𝑀⋆,1}}𝑉_(𝜎)𝑑𝑠𝑠≥ln𝛾>𝑝𝜏+𝜆𝜏, (3.32) where 𝜏=̇max_𝑘∈ℤ+{𝑡_𝑘−𝑡_𝑘−1}.
However, we note ^𝑡∗_𝑡∗∗𝑑Γ(𝑠)=Γ(𝑠)^Γ(𝑡∗)_Γ(𝑡^∗∗₎𝑑𝑠𝑠≤^𝑡∗∗_𝑡^+𝜏_∗∗(𝑝+𝜆)𝑑𝑠=𝑝𝜏+𝜆𝜏, (3.33) which is a contradiction. So (3.21) holds.
Note that 𝑉(𝑡_𝑙+1)≤𝛽_𝑙+1𝑉(𝑡⁻_𝑙+1)≤𝛽_𝑙+1max{𝛽_𝑙𝑀^⋆,1}𝑉(𝜎)𝑒^{−𝜆(𝑡𝑙+1−𝜎)}. (3.34) Similarly, we can prove 𝑉(𝑡)≤max{𝛽_𝑙+1𝑀^⋆⋅max{𝛽_𝑙𝑀^⋆,1},max{𝛽_𝑙𝑀^⋆,1}}𝑉(𝜎)𝑒^{−𝜆(𝑡−𝜎)},𝑡∈[𝑡_𝑙+1,𝑡_𝑙+2), (3.35) that is, 𝑉(𝑡)≤max{𝛽_𝑙𝑀^⋆,1}max{𝛽_𝑙+1𝑀^⋆,1}𝑉(𝜎)𝑒^{−𝜆(𝑡−𝜎)},𝑡∈[𝑡_𝑙+1,𝑡_𝑙+2). (3.36) By simple induction hypothesis, we may prove that for 𝑡∈[𝜎,𝑡_𝑙)∪[𝑡_𝑘,𝑡_𝑘+1),𝑘≥𝑙, 𝑉(𝑡)≤𝑉(𝜎)(∏^𝑘_𝑠=𝑙max{𝛽_𝑠𝑀^⋆,1})𝑒^{−𝜆(𝑡−𝜎)},𝑡≥𝜎. (3.37) Hence, (3.5) holds.
Under the help of conditions (i), (iii), and the definition of 𝑉(𝜎), we arrive at 𝑐₁‖𝑥‖^𝜂≤≤𝑉(𝑡)𝑉(𝜎)(^𝑘_𝑠=𝑙max{𝛽_𝑠𝑀^⋆,1})𝑒^{−𝜆(𝑡−𝜎)}≤𝑤(𝛿)𝑀^⋆(^𝑘_𝑠=𝑙max{𝛽_𝑠𝑀^⋆,1})𝑒^{−𝜆(𝑡−𝜎)}≤𝑤(𝛿)𝑀^⋆𝑀𝑒^{−𝜆(𝑡−𝜎)}≤min{𝑐₁𝜀^𝜂,𝜀}𝑒^{−𝜆(𝑡−𝜎)}≤𝑐₁𝜀^𝜂𝑒^{−𝜆(𝑡−𝜎)},𝑡≥𝜎, (3.38) which implies that ‖𝑥(𝑡)‖<𝜀⋅𝑒^{−(𝜆/𝜂)(𝑡−𝜎)},𝑡≥𝜎. (3.39) Therefore, (3.3) holds. The proof of Theorem 3.1 is therefore complete.

Remark 3.2. In Theorem 3.1, the impulsive condition is not straightforward to be verified. To simplify the result, we introduce some more testable conditions.

Corollary 3.3. Assume that there exist functions 𝑉∈𝑣₀,𝑤∈𝐾₁, and positive constants 𝑀^⋆,𝑝,𝑐₁,𝜂,𝜆,𝛽_𝑘≥0,𝑘∈ℤ₊ and 𝛾∈(1,𝑒^𝑀⋆] such that conditions (i), (ii), (iv) in Theorem 3.1 hold, moreover, suppose that
(iii^′) for all (𝑡_𝑘,𝜓)∈𝑅₊×𝑃𝐶([−𝛼,0],𝑆(𝜌₁)),𝑉(𝑡_𝑘,𝜓(0)+𝐼_𝑘(𝑡_𝑘,𝜓))≤𝛽_𝑘𝑉(𝑡⁻_𝑘,𝜓(0)),𝑘∈ℤ₊, where 𝛽_𝑘≥𝑒^−𝑀⋆ and there exists a constant 𝑀>0 such that ^𝑛∑_𝑘=𝑙ln𝛽_𝑘+(𝑛−𝑙+1)𝑀^⋆<𝑀∀𝑛≥𝑙,𝑙∈ℤ₊, (3.40) then the trivial solution of (2.1)-(2.2) is exponentially stable with the approximate exponential convergence rate 𝜆/𝜂.

Especially, let 𝛾=𝑀^⋆,𝛽_𝑘=1/𝛾>0 in Theorem 3.1, then we can obtain the following results whose proof is similar and thus omitted.

Corollary 3.4. Assume that there exist functions 𝑉∈𝑣₀,𝑤∈𝐾₁ and constants 𝑝,𝑐₁,𝜂,𝛾>1,𝜆>0 such that the following conditions hold:
(i)𝑐₁‖𝑥‖^𝜂≤𝑉(𝑡,𝑥)≤𝑤(‖𝑥‖),(𝑡,𝑥)∈[−𝛼,∞)×𝑆(𝜌);(ii)for any 𝜎≥𝑡₀ and 𝜓∈𝑃𝐶([−𝛼,0],𝑆(𝜌)), if 𝛾𝑒^{𝜆(𝑡−𝜎)}𝑉(𝑡,𝜓(0))≥𝑉(𝑡+𝜃,𝜓(𝜃)), −𝛼≤𝜃≤0,𝑡≠𝑡_𝑘,𝑘∈ℤ₊, then 𝐷⁺𝑉(𝑡,𝜓(0))≤𝑝𝑉(𝑡,𝜓(0));(iii)for all (𝑡_𝑘,𝜓)∈𝑅₊×𝑃𝐶([−𝛼,0],𝑆(𝜌₁)),𝑉(𝑡_𝑘,𝜓(0)+𝐼_𝑘(𝑡_𝑘,𝜓))≤(1/𝛾)𝑉(𝑡⁻_𝑘,𝜓(0)),𝑘∈ℤ₊;(iv)𝑡_𝑘−𝑡_𝑘−1<ln𝛾/(𝑝+𝜆),𝑘∈ℤ₊,then the trivial solution of (2.1)-(2.2) is exponentially stable with the approximate exponential convergence rate 𝜆/𝜂.

Remark 3.5. If 𝛼<+∞, then the exponential stability of system (2.1)-(2.2) has been investigated extensively in [23] under the following assumptions (where the definition of 𝑝,𝛼^⋆,𝑐₁,𝑐,𝜆,𝜏,𝑑_𝑘, see [23]):
(i)𝑐₁‖𝑥‖^𝑝≤𝑉(𝑡,𝑥)≤𝑐₂‖𝑥‖^𝑝, for any 𝑡∈ℝ₊ and 𝑥∈ℝ^𝑛;(ii)𝐷⁺𝑉(𝑡,𝜑(0))≤𝑐𝑉(𝑡,𝜑(0)), for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘),𝑘∈ℤ₊, whenever 𝑞𝑉(𝑡,𝜑(0))≤𝑉(𝑡+𝑠,𝜑(𝑠)) for 𝑠∈[−𝜏,0], where 𝑞≥max{𝑒^𝛼⋆𝑐,𝑒^2𝜆𝛼⋆} is a constant;(iii)𝑉(𝑡_𝑘,𝜑(0)+𝐼_𝑘(𝑡_𝑘,𝜑))≤𝑑_𝑘𝑉(𝑡_𝑘,𝜑(0)), where 𝑑_𝑘>0,∀𝑘∈ℤ₊, are constants;(iv)𝜏≤𝑡_𝑘−𝑡_𝑘−1≤𝛼^⋆ and ln(𝑑_𝑘)+(𝜆+𝑐)𝛼^⋆<−𝜆(𝑡_𝑘+1−𝑡_𝑘),𝑘∈ℤ₊.It is easily seen that these conditions are more restrictive than ones given in Theorem 3.1. For example, let 𝑑_𝑘=𝑑(constant), 𝑡_𝑘+1−𝑡_𝑘=1, then it is necessary that condition ln(𝑑)<−(𝜆+𝑐)𝛼^⋆−𝜆 holds (see [23]). Note in our Corollary 3.4, we only require that ln(𝑑)<−𝜆−𝑐, where here 𝛾=1/𝑑,𝑝=𝑐 in Theorem 3.1. Moreover, we see that condition 𝑡_𝑘+1−𝑡_𝑘>𝜏 is not necessary in our results, which are milder than the restrictions in [23].

Remark 3.6. To author's knowledge, there is little work on exponential stability of impulsive differential systems with infinite delay with 𝐷⁺𝑉≥0. Our result allows for significant increases in 𝑉 between impulses as long as the decreases of 𝑉 at impulse times balance it properly, which shows that differential equations with infinite delay may be exponentially stabilized by impulses.

In the following, an example is given to demonstrate the effectiveness of our result.

Example 3.7. Consider the following equations: 𝑥^′1(𝑡)=(5+1𝑒18^−0.1𝑡1)𝑥(𝑡)−15⁰_−∞𝑒^{𝑠−0.1𝑡}1𝑥(𝑡+𝑠)𝑑𝑠,𝑡≥0,𝑡≠𝑘,𝑥(𝑘)=√𝑒𝑥(𝑘⁻𝑥),𝑘=1,2…,_𝜎=𝜙. (3.41) Choose 𝑉(𝑡)=𝑥², we easily observe 𝜂=2,𝑐₁=1,𝑤(𝑠)=𝑠²,𝑡_𝑘−𝑡_𝑘−1=̇𝜏=1. Let 𝛾=𝑒,𝜆=0.2,𝑝=2/5+2/18+(2√𝑒)/15. It is easy to check that 𝜏(𝑝+𝜆)≈0.37<1=ln𝛾.
Suppose that 𝜎=𝑡₀=0, then in view of Corollary 3.3, 𝛾𝑒^{𝜆(𝑡−𝜎)}𝑉(𝑡,𝜓(0))≥𝑉(𝑡+𝜃,𝜓(𝜃)),𝜃≤0, implies that √𝑒𝑒^0.1𝑡|𝑥(𝑡)|>|𝑥(𝑡+𝜃)|,𝜃≤0.
Hence, 𝐷⁺1𝑉(𝑡,𝑥(⋅))=2𝑥(𝑡){(5+1𝑒18^−0.1𝑡1)𝑥(𝑡)−15⁰_−∞𝑒^{𝑠−0.1𝑡}𝑥(𝑡+𝑠)𝑑𝑠}≤2𝑥²1(𝑡){(5+1𝑒18^−0.1𝑡1)+15⁰_−∞𝑒^{𝑠−0.1𝑡}√𝑒𝑒^0.1𝑡𝑑𝑠}≤2𝑥²1(𝑡){(5+1𝑒18^−0.1𝑡√)+𝑒15⁰_−∞𝑒^𝑠𝑑𝑠}≤2𝑥²1(𝑡){5+1+√18𝑒}15≤𝑝𝑉(𝑡). (3.42) Besides, 𝑉(𝑘)=𝑥²(𝑘)=(1/𝑒)𝑥²(𝑘⁻)=(1/𝛾)𝑉(𝑘⁻).
By Corollary 3.3, the trivial solution of (3.41) is exponentially stable with the approximate exponential convergence rate 0.1. Taking initial values: 𝜙=10𝑒^0.2𝑡,𝑡≤0. The numerical simulations are illustrated in Figures 1 and 2.

Figure 1 

Figure 2 

Remark 3.8. From above example, we see that the trivial solution of system (3.41) without impulses is unstable. However, after impulsive control, the trivial solution becomes exponentially stable. This implies that differential systems with infinite delay may be exponentially stabilized by impulses and impulses can make unstable systems stable.

4. Conclusions

In this paper, we study exponential stability of impulsive differential systems with infinite delay. A new sufficient criterion ensuring exponential stability is gained by using the Razumikhintechnique and Lyapunov functions. Our result shows that differential equations with infinite delay may be exponentially stabilized by impulses. Also, the result here (with 𝛼<+∞) is discussed from the point of view of its comparison with the earlier result. An example is given to illustrate the feasibility of the result.

Acknowledgments

The authors are deeply grateful to the Associate Editor and two reviewers for their careful reading of this paper and helpful comments, which have been very useful for improving the quality of this paper.