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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 376464, 13 pages
http://dx.doi.org/10.1155/2012/376464
Research Article

New Results on Global Exponential Stability of Impulsive Functional Differential Systems with Delayed Impulses

School of Mathematical Sciences, Anhui University, Hefei 230601, China

Received 20 June 2012; Accepted 7 August 2012

Academic Editor: Elena Braverman

Copyright © 2012 Pei Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and the reproduction in any medium, provided the original work is properly cited.

Abstract

By using the Lyapunov functions and the Razumikhin techniques, the exponential stability of impulsive functional differential systems with delayed impulses is investigated. The obtained results have shown that the system will stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous flows, and they improve and complement ones from some recent works. An example is provided to illustrate the effectiveness and the advantages of the results obtained.

1. Introduction

There has been a growing interest in the theory of impulsive dynamical systems in the past decades because of their applications to various problems arising in communications, control technology, impact mechanics, electrical engineering, medicine, biology and so forth; see the monographs [1, 2] and the papers [38] and the references therein. In particular, special attention has been focused on stability and impulsive stabilization of impulsive functional differential systems (IFDSs) (see, e.g., [926]).

However, in these previous works on stability of IFDSs, the authors always suppose that the state variables on the impulses are only related to the present state variables. But in most cases, it is more applicable that the state variables on the impulses that we add are also related to the past ones. For example, it is more realistic in practice if the impulsive control depends on a past state due to a time lag between the time when the observation of the state is made and the time when the feedback control reaches the system.

In fact, there have been several attempts in the literature to study the stability and control problems of a particular class of IFDSs with delayed impulses (see, e.g., [2736]). Lian et al. [27] investigated the optimal control problem of linear continuous-time systems possessing delayed discrete-time controllers in networked control systems. For nonlinear impulsive systems, Khadra et al. studied the impulsive synchronization problem coupled by linear delayed impulses in [28]. In addition, in [2934], the authors investigate the uniform asymptotic stability and global exponential stability of general IFDSs: ̇𝑥(𝑡)=𝑓𝑡,𝑥𝑡,𝑡𝑡𝑘,𝑡𝑡0,𝑡Δ𝑥𝑘=𝐼𝑘𝑡𝑘,𝑥𝑡𝑘,𝑘+.(1.1) But in these stability analyses, the effects of time delay on the impulses have been ignored. For example in [3134], the Lyapunov function was assumed to be satisfied 𝑉(𝑡𝑘,𝜑(0)+𝐼𝑘(𝑡𝑘,𝜑))(1+𝑑𝑘)𝑉(𝑡𝑘,𝜑(0)).

Very recently, in [35], Zhang and Sun established some sufficient conditions for uniform stability, uniform asymptotical stability, and practical stability of a particular class of IFDSs with delayed impulses: ̇𝑥(𝑡)=𝑓𝑡,𝑥𝑡,𝑡𝑡𝑘,𝑡𝑡0,𝑡Δ𝑥𝑘=𝐼𝑘𝑥𝑡𝑘+𝐽𝑘𝑥𝑡𝑘𝜏,𝑘+.(1.2) However, their results are only valid for some specific systems due to the restrictive requirements on the continuous flows and impulsive gain. Lin et al. [36] investigated the exponential stability and uniform stability of the following more generalized IFDSs with delayed impulses: ̇𝑥(𝑡)=𝑓𝑡,𝑥𝑡,𝑡𝑡𝑘,𝑡𝑡0,𝑡Δ𝑥𝑘=𝐼𝑘𝑥𝑡𝑘+𝐽𝑘𝑥𝑡𝑘,𝑘+.(1.3) But those results can only been applied to the systems with stable discrete dynamics since their results need the strong condition of impulsive gain 𝑑𝑘+𝑒𝑘<1.

Motivated by the above discussions, in this paper, we further study the exponential stability of IFDSs with delayed impulses. Different from the previous works on exponential stability of IFDSs with/without delayed impulses [18, 31, 34, 36], we will divide the systems into two classes: the system with stable continuous dynamics and unstable discrete dynamics, the systems with unstable continuous dynamics and stable discrete dynamics. The first class of impulsive systems corresponds to the case when the continuous dynamics are subjected to impulsive perturbations, while the second class of impulsive systems corresponds to the case when impulses are employed to stabilize the unstable continuous dynamics. This idea is enlightened in part by the works Chen and Zheng [37] about the uncertain impulsive systems. By using the Lyapunov functions and the Razumikhin techniques, some global exponential stability criteria are derived. The results obtained improve and complement some recent works. It is worth mentioning that our results shown that the system will be stable if the impulses' frequency and amplitude are suitably related to the increase or decrease of the continuous flows. Moreover, some results obtained can be applied to IFDSs with any time delay. In the end, an example is provided to illustrate the effectiveness and the advantages of the results obtained.

2. Preliminaries

Let denote the set of real numbers, + the set of nonnegative real numbers, + the set of positive integers, and 𝑛 the 𝑛-dimensional real space equipped with the Euclidean norm ||. Let 𝜏>0 and PC([𝜏,0];𝑛)={𝜑[𝜏,0]𝑛|𝜑(𝑡+)=𝜑(𝑡) for all 𝑡[𝜏,0), 𝜑(𝑡) exist and 𝜑(𝑡)=𝜑(𝑡) for all, but at most a finite number of points 𝑡(𝜏,0]} be with the norm 𝜑=sup𝜏𝜃0|𝜑(𝜃)|, where 𝜑(𝑡+) and 𝜑(𝑡) denote the right-hand and left-hand limits of function 𝜑(𝑡) at 𝑡, respectively. Denote PC([𝑡0𝜏,𝑏];+)={𝜑[𝑡0𝜏,𝑏]+𝜑ispiecewisecontinuous} for 𝑏>𝑡0, and PC([𝑡0𝜏,);+)={𝜑|𝜑|[𝑡0𝜏,𝑏]PC([𝑡0𝜏,𝑏];+) for all 𝑏>𝑡0𝜏}.

Consider the IFDS in which the state variables on the impulses are related to the time delay: ̇𝑥(𝑡)=𝑓𝑡,𝑥𝑡,𝑡𝑡𝑘,𝑡𝑡0,𝑡Δ𝑥𝑘=𝐼𝑘𝑡𝑘𝑡,𝑥𝑘+𝐽𝑘𝑡𝑘,𝑥𝑡𝑘,𝑘+𝑥𝑡0[],=𝜙(𝑠),𝑠𝜏,0(2.1) where 𝑥𝑛, 𝑓+×𝑛, 𝐼𝑘+×𝑛𝑛, 𝐽𝑘+×𝑛, 𝜙PC([𝜏,0];𝑛), is a open set in PC([𝜏,0];𝑛). and The fixed moments of impulse times {𝑡𝑘,𝑘+} satisfy 0𝑡0<𝑡1<<𝑡𝑘<,𝑡𝑘 (as 𝑘), Δ𝑥(𝑡𝑘)=𝑥(𝑡𝑘)𝑥(𝑡𝑘); 𝑥𝑡, 𝑥𝑡PC([𝜏,0];𝑛) are defined by 𝑥𝑡=𝑥(𝑡+𝜃), 𝑥𝑡=𝑥(𝑡+𝜃) for 𝜃[𝜏,0], respectively.

Throughout this paper, we assume that 𝑓,𝐼𝑘, and 𝐽𝑘, 𝑘+, satisfy the necessary conditions for the global existence and uniqueness of solutions for all 𝑡𝑡0, see [6, 3033]. Then for any 𝜙PC([𝜏,0];𝑛), there exists a unique function satisfying system (2.1) denoted by 𝑥(𝑡;𝑡0,𝜙), which is continuous on the right-hand side and limitable on the left-hand side. Moreover, we assume that 𝑓(𝑡,0)0, 𝐼𝑘(𝑡𝑘,0)0 and 𝐽𝑘(𝑡𝑘,0)0, 𝑘+, which imply that 𝑥(𝑡)0 is a solution of (2.1), which is called the trivial solution.

At the end of this section, let us introduce the following definitions.

Definition 2.1. A function 𝑉[𝑡0𝜏,)×𝑛+ belongs to class 𝑣0 if (i)𝑉is continuous on each of the sets [𝑡𝑘1,𝑡𝑘)×𝑛, and for each 𝑥𝑛, 𝑡[𝑡𝑘1,𝑡𝑘), 𝑘+, lim(𝑡,𝑦)(𝑡𝑘,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑡𝑘,𝑥) exists;(ii)𝑉(𝑡,𝑥) is locally Lipschitz in 𝑥𝑛, and 𝑉(𝑡,0)0 for all 𝑡𝑡0.

Definition 2.2. Given a function 𝑉𝑣0, the upper right-hand Dini derivative of 𝑉 with respect to system (2.1) is defined by 𝐷+𝑉(𝑡,𝜓(0))=limsup0+1[],𝑉(𝑡+,𝜓(0)+𝑓(𝑡,𝜓))𝑉(𝑡,𝜓(0))(2.2) for (𝑡,𝜓)[𝑡0,)×PC([𝜏,0];𝑛).

Definition 2.3. The trivial solution of system (2.1) or, simply, system (2.1) is said to be globally exponentially stable if there exist positive constants 𝛼 and 𝐶 such that for any initial data 𝑥𝑡0=𝜙PC([𝜏,0];𝑛), the solution 𝑥(𝑡;𝑡0,𝜙) satisfies ||𝑥𝑡;𝑡0||,𝜙𝐶𝜙𝑒𝛼(𝑡𝑡0),𝑡𝑡0.(2.3)

3. Main Results

In this section, we shall analyze the global exponential stability of system (2.1) by employing the Razumikhin techniques and the Lyapunov functions.

Theorem 3.1. Assume that there exist functions 𝑉𝑣0, 𝑐PC([𝑡0𝜏,);+), several positive constants 𝑐1, 𝑐2, ̃𝑐, 𝑝, 𝑞, and nonnegative constants 𝜌1, 𝜌2, 𝜌1+𝜌21 such that(i)𝑐1|𝑥|𝑝𝑉(𝑡,𝑥)𝑐2|𝑥|𝑝, for all (𝑡,𝑥)[𝑡0𝜏,)×𝑛;(ii)𝑉(𝑡𝑘,𝜑(0))𝜌1(1+𝜇𝑘)𝑉(𝑡𝑘,𝜑(0))+𝜌2(1+𝜇𝑘)sup𝜃[𝜏,0]𝑉(𝑡𝑘+𝜃,𝜑(𝜃)),  for each 𝑘+ and 𝜑𝑃𝐶([𝜏,0];𝑛), where 𝜇𝑘, 𝑘+, are nonnegative constants with Σ𝑘=1𝜇𝑘<;(iii)𝐷+𝑉(𝑡,𝜑(0))𝑐(𝑡)𝑉(𝑡,𝜑(0)), for all 𝑡𝑡0, 𝑡𝑡𝑘, 𝑘+, 𝜑PC([𝜏,0];𝑛), whenever 𝑉(𝑡+𝜃,𝜑)<𝑞𝑉(𝑡,𝜑(0)), 𝜃[𝜏,0];(iv)𝜌1+𝜌2𝑒̃𝑐𝜏<𝑞<𝑒̃𝑐𝜚, inf𝑡0𝑥00085𝑡0𝑐(𝑡)̃𝑐, where 𝜚=inf𝑘+{𝑡𝑘𝑡𝑘1}.
Then the trivial solution of system (2.1) is globally exponentially stable and the convergence rate should not be greater than (1/𝑝)(̃𝑐(ln𝑞/𝜚)).

Proof. Set 𝐿=𝑘=1(1+𝜇𝑘); from the condition Σ𝑘=1𝜇𝑘<, we known that 1𝐿<. Fix any initial data 𝜙PC([𝜏,0];𝑛) and write 𝑥(𝑡;𝑡0,𝜙)=𝑥(𝑡), 𝑉(𝑡,𝑥(𝑡))=𝑉(𝑡) simply. From condition (iv), we can choose a small enough constant 𝛾>0 such that 𝑒𝛾𝜏𝜌1+𝜌2𝑒̃𝑐𝜏<𝑞<𝑒(̃𝑐𝛾)𝜚,𝛾<̃𝑐.(3.1)
Set ̃𝑞=𝑞𝑒𝛾𝜏>1, choose 𝑀>0 such that ̃𝑞𝑐2<𝑀. Define 𝑊(𝑡)=𝑒𝛾(𝑡𝑡0)𝑉(𝑡). In the following, we shall show that 𝑊(𝑡)𝐿𝑀𝜙𝑝,𝑡𝑡0.(3.2) In order to do so, we first prove that 𝑊(𝑡)<𝑀𝜙𝑝𝑡,𝑡0𝜏,𝑡1.(3.3) It is noted that 𝑊𝑡0+𝜃𝑐2𝜙𝑝<1𝑀̃𝑞𝜙𝑝<𝑀𝜙𝑝[].,𝜃𝜏,0(3.4) So it only needs to prove 𝑊(𝑡)<𝑀𝜙𝑝𝑡,𝑡0,𝑡1.(3.5) We assume, on the contrary, there exist some 𝑡(𝑡0,𝑡1) such that 𝑊(𝑡)𝑀𝜙𝑝. Set 𝑡𝑡=inf𝑡0,𝑡1𝑊(𝑡)𝑀𝜙𝑝.(3.6) Note that 𝑊(𝑡) is continuous on 𝑡[𝑡0,𝑡1), then 𝑡(𝑡0,𝑡1) and 𝑊𝑡=𝑀𝜙𝑝,𝑊(𝑡)<𝑀𝜙𝑝𝑡,𝑡0𝜏,𝑡.(3.7) Define 𝑡𝑡=sup𝑡0,𝑡1𝑊(𝑡)𝑀̃𝑞𝜙𝑝,(3.8) then 𝑡(𝑡0,𝑡) and 𝑊𝑡=1𝑀̃𝑞𝜙𝑝1,𝑊(𝑡)>𝑀̃𝑞𝜙𝑝𝑡,𝑡,𝑡.(3.9) Consequently, for all 𝑡[𝑡,𝑡], 𝑊(𝑡+𝜃)𝑀𝜙𝑝[],̃𝑞𝑊(𝑡),𝜃𝜏,0(3.10) which implies that 𝑉(𝑡+𝜃)=𝑒𝛾(𝑡+𝜃𝑡0)𝑊(𝑡+𝜃)̃𝑞𝑒𝛾(𝑡+𝜃𝑡0)𝑊(𝑡)̃𝑞𝑒𝛾𝜏[].𝑉(𝑡)=𝑞𝑉(𝑡),𝜃𝜏,0(3.11) Then it follows from condition (iii) that one has that 𝐷+𝑊(𝑡)=𝑒𝛾(𝑡𝑡0)𝛾𝑉(𝑡)+𝐷+𝑡𝑉(𝑡)(𝛾𝑐(𝑡))𝑊(𝑡),𝑡,𝑡,(3.12) which leads to 𝑊𝑡𝑡𝑊𝑒𝑡𝑡(𝛾𝑐(𝑠))d𝑠𝑡𝑊𝑒(𝛾̃𝑐)(𝑡𝑡)1̃𝑞𝑀𝜙𝑝<𝑀𝜙𝑝,(3.13) this is a contradiction. Thus (3.5) holds.
Now we assume that for some 𝑚+,𝑚1, 𝑊(𝑡)<𝑀𝑚𝜙𝑝𝑡,𝑡0𝜏,𝑡𝑚,(3.14) where 𝑀1=𝑀, 𝑀𝑚=𝑀1𝑖𝑚1(1+𝜇𝑖) for 𝑚2. We will prove that 𝑊(𝑡)<𝑀𝑚+1𝜙𝑝𝑡,𝑡𝑚,𝑡𝑚+1.(3.15) To do this, we first claim 𝑊𝑡𝑚𝑒+𝜃(̃𝑐𝛾)𝜏𝑀̃𝑞𝑚𝜙𝑝[,𝜃𝜏,0).(3.16) Suppose not, then there exists ̃𝜃[𝜏,0) such that 𝑊(𝑡𝑚+̃𝜃)>(𝑒(̃𝑐𝛾)𝜏/̃𝑞)𝑀𝑚𝜙𝑝. Without lose generality, we assume 𝑡𝑚+̃𝜃(𝑡𝑙1,𝑡𝑙], 𝑙+,𝑙𝑚.
There are two cases to be considered.
Case 1. 𝑊(𝑡)>(𝑒(̃𝑐𝛾)𝜏/̃𝑞)𝑀𝑚𝜙𝑝 over 𝑡[𝑡𝑙1,𝑡𝑚+̃𝜃).
By assumption (3.14), for all 𝑡[𝑡𝑙1,𝑡𝑚+̃𝜃), we get 𝑊(𝑡+𝜃)<𝑀𝑚𝜙𝑝<𝑒(̃𝑐𝛾)𝜏𝑀𝑚𝜙𝑝[].<̃𝑞𝑊(𝑡),𝜃𝜏,0(3.17) Thus, by conditions (iii)-(iv) and inequalities (3.10)–(3.13), we have 𝑊𝑡𝑚+̃𝜃𝑡𝑊𝑙1𝑒(𝛾̃𝑐)(𝑡𝑚+̃𝜃𝑡𝑙1)<𝑀𝑚𝜙𝑝𝑒(̃𝑐𝛾)𝜏𝑒(𝛾̃𝑐)(𝑡𝑚𝑡𝑙1)𝑒(̃𝑐𝛾)𝜏𝑞𝑚𝑙+1𝑀𝑚𝜙𝑝<𝑒(̃𝑐𝛾)𝜏𝑀̃𝑞𝑚𝜙𝑝.(3.18) This is a contradiction.
Case 2. There are some 𝑡[𝑡𝑙1,𝑡𝑚+̃𝜃) such that 𝑊(𝑡)>(𝑒(̃𝑐𝛾)𝜏/̃𝑞)𝑀𝑚𝜙𝑝.
In this case, define 𝑡𝑡=sup𝑡𝑙1,𝑡𝑚+̃𝜃𝑒𝑊(𝑡)(̃𝑐𝛾)𝜏𝑀̃𝑞𝑚𝜙𝑝.(3.19) Then 𝑡[𝑡𝑙1,𝑡𝑚+̃𝜃) and 𝑊𝑡=𝑒(̃𝑐𝛾)𝜏𝑀̃𝑞𝑚𝜙𝑝𝑒,𝑊(𝑡)>(̃𝑐𝛾)𝜏𝑀̃𝑞𝑚𝜙𝑝,𝑡𝑡,𝑡𝑚+̃𝜃.(3.20) So from assumption (3.14), for any 𝑡[𝑡,𝑡𝑚+̃𝜃), we have 𝑊(𝑡+𝜃)<𝑀𝑚𝜙𝑝<𝑒(̃𝑐𝛾)𝜏𝑀𝑚𝜙𝑝[].̃𝑞𝑊(𝑡),𝜃𝜏,0(3.21) It follows from condition (iii) that 𝑊𝑡𝑚+̃𝜃𝑊𝑡=𝑒(̃𝑐𝛾)𝜏𝑀̃𝑞𝑚𝜙𝑝.(3.22) This is also a contradiction. Hence, inequality (3.16) holds.
Similarly, we can prove 𝑊𝑡𝑚1𝑀̃𝑞𝑚𝜙𝑝.(3.23) Then it follows from (3.16), (3.23), and condition (ii) that we obtain 𝑊𝑡𝑚𝜌11+𝜇𝑚𝑊𝑡𝑚+𝜌21+𝜇𝑚𝑒𝛾𝜏sup[]𝜃𝜏,0𝑊𝑡𝑚𝜌+𝜃1+𝜌2𝑒̃𝑐𝜏𝑀̃𝑞𝑚+1𝜙𝑝<𝑀𝑚+1𝜙𝑝.(3.24) Now we suppose that (3.15) is not true, let 𝑡𝑡=inf𝑡𝑚,𝑡𝑚+1𝑊(𝑡)𝑀𝑚+1𝜙𝑝.(3.25) Then 𝑡(𝑡𝑚,𝑡𝑚+1) and 𝑊𝑡=𝑀𝑚+1𝜙𝑝,𝑊(𝑡)<𝑀𝑚+1𝜙𝑝𝑡,𝑡𝑚,𝑡.(3.26)
If 𝑊(𝑡)>(1/̃𝑞)𝑀𝑚+1𝜙𝑝 for all 𝑡[𝑡𝑚,𝑡], set 𝑡=𝑡𝑚; otherwise, let 𝑡𝑡=sup𝑡𝑚,𝑡1𝑊(𝑡)𝑀̃𝑞𝑚+1𝜙𝑝.(3.27) Thus for all 𝑡[𝑡,𝑡], we have 𝑊(𝑡+𝜃)𝑀𝑚+1𝜙𝑝[].̃𝑞𝑊(𝑡),𝜃𝜏,0(3.28) It follows from condition (iii) that 𝐷+𝑊(𝑡)=𝑒𝛾(𝑡𝑡0)𝛾𝑉(𝑡)+𝐷+𝑡𝑉(𝑡)(𝛾𝑐(𝑡))𝑊(𝑡),𝑡,𝑡,(3.29) which implies 𝑊𝑡𝑡𝑊𝑒(𝛾̃𝑐)(𝑡𝑡)𝑡𝑊<𝑀𝑚+1𝜙𝑝.(3.30) This is a contradiction. Therefore, (3.15) holds.
By mathematical induction, (3.15) holds for any 𝑚+. That is, (3.2) holds, which implies that ||||𝑥(𝑡)𝐶𝜙𝑒(𝛾/𝑝)(𝑡𝑡0),𝑡𝑡0,(3.31) where 𝐶=(LM/𝑐1)1/𝑝. This completes the proof.

Remark 3.2. The parameters 𝜌1 and 𝜌2 in condition (ii) describe the influence of impulses on the stability of the underlying continuous systems. When 𝜌1+𝜌21, the Lyapunov function 𝑉 may jump up along the state trajectories of system (2.1) at impulsive time instant 𝑡𝑘. Thus the impulses may be viewed as disturbances, that is, they potentially destroy the stability of continuous system. In this case, it is required that the impulses do not occur too frequently. Theorem 3.1 tells us to what extent we can relax the restriction on the impulses to keep the exponential stability property of the original continuous system.

Theorem 3.3. Assume that there exist functions 𝑉𝑣0, 𝑐PC([𝑡0𝜏,);+), several positive constants 𝑐1, 𝑐2, ̃𝑐, 𝑝, 𝑞, and nonnegative constants 𝜌1, 𝜌2, 𝜌1+𝜌2<1 such that (i)𝑐1|𝑥|𝑝𝑉(𝑡,𝑥)𝑐2|𝑥|𝑝, for all (𝑥,𝑡)𝑛×[𝑡0𝜏,);(ii)𝑉(𝑡𝑘,𝜑(0))𝜌1(1+𝜇𝑘)𝑉(𝑡𝑘,𝜑(0))+𝜌2(1+𝜇𝑘)sup𝜃[𝜏,0]𝑉(𝑡𝑘+𝜃,𝜑(𝜃)), for each 𝑘+ and 𝜑𝑃([𝜏,0];𝑛), where 𝜇𝑘, 𝑘+, are nonnegative constants with 𝑘=1𝜇𝑘<;(iii)𝐷+𝑉(𝑡,𝜑(0))𝑐(𝑡)𝑉(𝑡,𝜑(0)), for all 𝑡𝑡0, 𝑡𝑡𝑘, 𝑘+, 𝜑PC([𝜏,0];𝑛), whenever 𝑉(𝑡+𝜃,𝜑)<𝑞𝑉(𝑡,𝜑(0)), 𝜃[𝜏,0];(iv)𝑞>1/(𝜌1+𝜌2)>𝑒̃𝑐𝜚, ̃𝑐𝜚sup𝑡𝑡0𝑡𝑡+𝜚𝑐(𝑠)𝑑𝑠, where 𝜚=sup𝑘+{𝑡𝑘𝑡𝑘1}.
Then the trivial solution of system (2.1) is globally exponentially stable for any time delay 𝜏(0,) and the convergence rate should not be greater than (1/𝑝)((ln𝑞/𝜚)̃𝑐).

Proof . From condition (iv), we can choose a small enough constant 𝛾>0 such that 𝑒𝑞>𝛾𝜏𝜌1+𝜌2𝑒𝛾𝜏>1𝜌1+𝜌2𝑒𝛾𝜏>𝑒(̃𝑐+𝛾)𝜚,𝑞𝑒𝛾𝜏>1.(3.32) Set ̃𝑞=𝑞𝑒𝛾𝜏. The following proof can be completed by using the similar arguments as in the proof of Theorem 3.1, so it is omitted.

Remark 3.4. When 𝜌1+𝜌2<1, the Lyapunov function 𝑉 may jump down along the state trajectories of system (2.1) at impulsive time instant 𝑡𝑘. Thus the impulses may be viewed impulsive stabilizing, that is, they may be used to stabilize the continuous system if the original continuous system is not stable. In this case, the impulses must be frequent and their amplitude must be suitably related the growth rate of 𝑉.

Remark 3.5. If 𝑐(𝑡)𝑐, then Theorem 3.3 becomes Theorem 3.1 in [36] with 𝑑𝑘=𝜌1(1+𝜇𝑘), 𝑒𝑘=𝜌2(1+𝜇𝑘), 𝑑𝑘+𝑒𝑘<1. Obviously, Theorem 3.3 in this paper has a wider adaptive range than those in [36].
Let 𝐽𝑘0 in system (2.1), then we have the following IFDS (see [923, 26]): 𝑥̇𝑥(𝑡)=𝑓𝑡,𝑡,𝑡𝑡𝑘,𝑡𝑡0,𝑡Δ𝑥𝑘=𝐼𝑘𝑥𝑡𝑘,𝑡𝑘,𝑘+,𝑥𝑡0[].=𝜙(𝑠),𝑠𝜏,0(3.33) For system (3.33), we have the following results by Theorems  3.1 and 3.2, respectively.

Corollary 3.6. Assume that there exist functions 𝑉𝑣0, 𝑐PC([𝑡0𝜏,);+), and several positive constants 𝑐1, 𝑐2, ̃𝑐, 𝑝, 𝑞, and a constant 𝜌1 such that (i)𝑐1|𝑥|𝑝𝑉(𝑡,𝑥)𝑐2|𝑥|𝑝, for all (𝑥,𝑡)𝑛×[𝑡0𝜏,);(ii)𝑉(𝑡𝑘,𝜑(0))𝜌(1+𝜇𝑘)𝑉(𝑡𝑘,𝜑(0)), for each 𝑘+ and 𝜑𝑃([𝜏,0];𝑛), where 𝜇𝑘, 𝑘+, are nonnegative constants with 𝑘=1𝜇𝑘<;(iii)𝐷+𝑉(𝑡,𝜑(0))𝑐(𝑡)𝑉(𝑡,𝜑(0)), for all 𝑡𝑡0, 𝑡𝑡𝑘,𝑘+,𝜑PC([𝜏,0];𝑛), whenever 𝑉(𝑡+𝜃,𝜑)<𝑞𝑉(𝑡,𝜑(0)), 𝜃[𝜏,0];(iv)𝜌<𝑞<𝑒̃𝑐𝜚, inf𝑡𝑡0𝑐(𝑡)̃𝑐, where 𝜚=inf𝑘+{𝑡𝑘𝑡𝑘1}.
Then the trivial solution of system (3.33) is globally exponentially stable for any time delay 𝜏(0,) and the convergence rate should not be greater than (1/𝑝)(̃𝑐(ln𝑞/𝜚)).

Corollary 3.7. Assume that there exist functions 𝑉𝑣0, 𝑐PC([𝑡0𝜏,);+) and several positive constants 𝑐1, 𝑐2, ̃𝑐, 𝑝, 𝑞, and a constant 𝜌<1 such that (i)𝑐1|𝑥|𝑝𝑉(𝑡,𝑥)𝑐2|𝑥|𝑝, for all (𝑥,𝑡)𝑛×[𝑡0𝜏,);(ii)𝑉(𝑡𝑘,𝜑(0))𝜌(1+𝜇𝑘)𝑉(𝑡𝑘,𝜑(0)), for each 𝑘+, 𝜑𝑃([𝜏,0];𝑛), where 𝜇𝑘, 𝑘+, are nonnegative constants with 𝑘=1𝜇𝑘<;(iii)𝐷+𝑉(𝑡,𝜑(0))𝑐(𝑡)𝑉(𝑡,𝜑(0)), for all 𝑡𝑡0, 𝑡𝑡𝑘, 𝑘+, 𝜑PC([𝜏,0];𝑛), whenever 𝑉(𝑡+𝜃,𝜑)<𝑞𝑉(𝑡,𝜑(0)), 𝜃[𝜏,0];(iv)𝑞>1/𝜌>𝑒̃𝑐𝜚, ̃𝑐𝜚sup𝑡0𝑥00085𝑡0𝑡𝑡+𝜚𝑐(𝑠)𝑑𝑠, where 𝜚=sup𝑘+{𝑡𝑘𝑡𝑘1}. Then the trivial solution of system (3.33) is globally exponentially stable for any time delay 𝜏(0,) and the convergence rate should not be greater than (1/𝑝)((ln𝑞/𝜚)̃𝑐).

Remark 3.8. If 𝑐(𝑡)𝑐>0, 𝜇𝑘0, 𝑘+, then Theorems  3.1 and 3.2 in [25] follow from Corollaries 3.6 and 3.7, respectively.

4. Example

In this section, an example is given to show the effectiveness and advantages of our results.

Example 4.1. Consider the following IFDS (see [35, 36]): ̇𝑥(𝑡)=𝑎𝑥(𝑡)+𝑏𝑥(𝑡𝜏),𝑡𝑡𝑘𝑥𝑡,𝑡>0,𝑘𝑡=𝑐𝑥𝑘𝑡+𝑑𝑥𝑘𝜏,𝑘+,(4.1) where 𝑥, 𝜏>0.
In the following, we will divide the system (4.1) into two classes to consider.
Case 1. 𝑎0 and 0<|𝑐|+|𝑑|<1.

Property 1. The trivial solution of system (4.1) is globally exponentially stable with impulse time sequences that satisfy sup𝑘+𝑡𝑘𝑡𝑘1||𝑑||||𝑑||<|𝑐|+ln|𝑐|+𝑎|||𝑑||+||𝑏||.𝑐|+(4.2)

Proof . From equality (4.2), one can choose a small enough constant >0 such that ||𝑑|||𝑐|+>0,sup𝑘+𝑡𝑘𝑡𝑘1||𝑑||||𝑑||<|𝑐|+ln|𝑐|+𝑎||𝑑||+||𝑏||.|𝑐|+(4.3)
Let 𝑉(𝑡,𝑥)=|𝑥|. By calculation, we have 𝐷+||||+||𝑏||||||||𝑏||𝑉(𝑡,𝜑(0))𝑎𝜑(0)𝜑(𝜏)=𝑎𝑉(𝑡,𝜑(0))+𝑉(𝑡,𝜑),(4.4) for all 𝑡𝑡𝑘, 𝑘+ and 𝜑PC([𝜏,0];). By taking 𝑝=1, 𝑐1=𝑐2=1, 𝜌1=|𝑐|, 𝜌2=|𝑑|, 𝑞=1/(|𝑐|+|𝑑|), ̃𝑐=𝑐(𝑡)𝑎+(|𝑏|/(|𝑐|+|𝑑|)), and 𝜇𝑘0, 𝑘+ in Theorem 3.3, it is easy to obtain Property 1.

Remark 4.2. In this case, the impulses are used to stabilize the unstable original continuous system. In [35], under assumption that 𝑎, 𝑏, 𝑐, 𝑑>0, and 𝑐+𝑑<1, Zhang and Sun obtained that system (4.1) is uniformly stable if the impulses’ instances satisfy sup𝑘+𝑡𝑘𝑡𝑘1<2(𝑐+𝑑)2ln(𝑐+𝑑)(2𝑎+𝑏)(𝑐+𝑑)2;+𝑏(4.5) Lin et al. [36] derived that system (4.1) is exponentially stable if sup𝑘+𝑡𝑘𝑡𝑘11<2(𝑐+𝑑)ln(𝑐+𝑑).𝑎(𝑐+𝑑)+𝑏(4.6) Obviously, under condition 𝑎, 𝑏, 𝑐, 𝑑>0, and 𝑐+𝑑<1, we get ||𝑑||||𝑑|||𝑐|+ln|𝑐|+𝑎|||𝑑||+||𝑏||𝑐|+=(𝑐+𝑑)ln(𝑐+𝑑)1𝑎(𝑐+𝑑)+𝑏>2(𝑐+𝑑)ln(𝑐+𝑑)𝑎(𝑐+𝑑)+𝑏,(4.7) and one can also verify that (𝑐+𝑑)ln(𝑐+𝑑)𝑎>(𝑐+𝑑)+𝑏2(𝑐+𝑑)2ln(𝑐+𝑑)(2𝑎+𝑏)(𝑐+𝑑)2+𝑏.(4.8) So our results are less conservative than those in [35, 36].
Case 2. 𝑎<0 and |𝑐|+|𝑑|1.

Property 2. Suppose that system’s parameters 𝑎, 𝑏, 𝑐, 𝑑 and time delay 𝜏 satisfy ||𝑑|||𝑐|+2𝑒2𝑎𝜏||𝑏||<2𝑎+||𝑏||.(4.9) Then the trivial solution of system (4.1) is globally exponentially stable with impulse time sequences that satisfy inf𝑘+𝑡𝑘𝑡𝑘1>||𝑑||2𝑎𝜏2ln|𝑐|+||𝑏||+||𝑏||||𝑑||2𝑎+|𝑐|+2𝑒2𝑎𝜏.(4.10)

Proof. From equalities (4.9) and (4.10), we can choose a small enough constant >0 such that ||𝑑|||𝑐|+2𝑒2𝑎𝜏<||𝑑|||𝑐|+2𝑒2𝑎𝜏||𝑏||+<2𝑎+||𝑏||,inf𝑘+𝑡𝑘𝑡𝑘1||𝑑||>ln|𝑐|+2𝑒2𝑎𝜏+2||𝑏||+||𝑏||||𝑑||2𝑎+|𝑐|+2𝑒2𝑎𝜏.+(4.11) Set 𝑞=(|𝑐|+|𝑑|)2𝑒2𝑎𝜏+, then one can conclude that ||𝑏||||𝑏||||𝑑||2𝑎++𝑞<0,|𝑐|+2𝑒(2𝑎+|𝑏|+𝑞|𝑏|)𝜏||𝑏||<𝑞<2𝑎+||𝑏||.(4.12)
Let 𝑉(𝑡,𝑥)=(1/2)𝑥2. By calculation, we have 𝐷+1𝑉(𝑡,𝜑(0))𝑎+2||𝑏||𝜑21(0)+2||𝑏||𝜑2||𝑏||||𝑏||(𝜏)=2𝑎+𝑉(𝑡,𝜑(0))+𝑉(𝑡,𝜑),(4.13) for all 𝑡𝑡𝑘, 𝑘+, and 𝜑PC([𝜏,0];). By taking 𝑝=2, 𝑐1=𝑐2=2, 𝜌1=|𝑐|(|𝑐|+|𝑑|), 𝜌2=|𝑑|(|𝑐|+|𝑑|), ̃𝑐=𝑐(𝑡)(2𝑎+|𝑏|+𝑞|𝑏|), and 𝜇𝑘0, 𝑘+ in Theorem 3.3, we can obtain Property 2.

Remark 4.3. In this case, the underlying continuous system is stable, the impulses are disturbances, which potentially destroy the stability of continuous system. So the existing results in [35, 36] are invalid for this case.

5. Conclusions

This paper has studied the exponential stability of IFDSs in which the state variables on the impulses are related to the time delay. By using the Razumikhin techniques and the Lyapunov functions, some criteria on the global exponential stability are established. The obtained results improve and complement some recent works. An example has been given to illustrate the effectiveness and the advantages of the results obtained.

Acknowledgments

This work was supported by the 211 Project of the Anhui University (32030018/33010205/KJTD002B), the Research Fund for Doctor Station of the Ministry of Education of China (20113401110001), the Key Natural Science Foundation (KJ2009A49), the Foundation of Anhui Education Bureau (KJ2012A019), and the National Natural Science Foundation of China (11126179).

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