Abstract
The problem of stability for nonlinear impulsive stochastic functional differential equations with delayed impulses is addressed in this paper. Based on the comparison principle and an impulsive delay differential inequality, some exponential stability and asymptotical stability criteria are derived, which show that the system will be stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous stochastic flows. The obtained results complement ones from some recent works. Two examples are discussed to illustrate the effectiveness and advantages of our results.
1. Introduction
Impulsive dynamical equations have received considerable attention during the recent decades since they provide a natural framework for mathematical modeling of many real world evolutionary processes where the states undergo abrupt changes at certain instants (see [1–7]). In particular, more researchers have given special interests to the stability and stabilization analysis of impulsive functional differential equations (IFDEs) and there are extensive literatures in this field (see [8–14] and reference therein).
In the current literature concerning IFDEs, the impulses are assumed to take the form , which indicates that the state “jump” at the impulse times is 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, in the transmission of the impulse information, input delays are often encountered (see, e.g., [15, 16]). So, it is more meaningful if the above impulses are modified as . Recently, there have been several attempts in the literature to study the stability and control problems of IFDEs with delayed impulse (IFDEs-DI). For example, by using Lyapunov functions couples with Razumikhin techniques, some Razumikhin-type asymptotic stability and exponential stability criteria for IFDEs-DI were established in [17–19], and some Lyapunov-based sufficient conditions for the exponential stability of the equations were derived in [20].
On the other hand, stochastic perturbations are unavoidable in real equations (see [21, 22] and reference therein). In recent years, the stability analysis of impulsive stochastic functional equations which include delay equations is interesting to many investigators, and many results of stability criteria of these equations have been reported (see, e.g., [23–29]). Very recently, [30, 31] took environment noise into account and generalized delayed impulses to stochastic equations. In particular, applying the Lyapunov functions couples with Razumikhin techniques, [30] investigates both moment and almost sure exponential stability of impulsive stochastic functional differential equations with delayed impulses (ISFDEs-DI), and several Razumikhin-type criteria on the exponential stability and uniform stability in terms of two measures for the equations were established in [31]. But it is worth noting that the stability analysis in [30] and the effects of time delay on the impulses have been ignored. And in [30, 31], the authors only consider the case that the impulsive stabilization. Moreover, it is well known that the Razumikhin techniques are very effective in the study of stability problems for ordinary and functional differential equations. However, when we use the Razumikhin techniques, we need to choose an appropriate minimal class of functionals relative to which the derivative of the Lyapunov function or Lyapunov functional is estimated, which is not entirely convenient.
Motivated by the above discussion, in this paper, we will further investigate the stability of ISFDEs-DI. By using the comparison principle and an impulsive delay differential inequality, some exponential and asymptotical stability criteria are derived, which are more convenient to be applied than those Razumikhin-type conditions. Our results complement ones from some recent works and show that the ISFDE-ID will be stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the corresponding continuous stochastic flows. The rest of the paper is organized as follows. In Section 2, some relevant notations and definitions are presented. In Section 3, the comparison principle, an impulsive delay differential inequality, and several criteria on the exponential stability and asymptotical stability are established. Section 4 provides two illustrative examples to demonstrate the applications of the obtained results. Finally, conclusions are drawn in Section 5.
2. Preliminaries
Throughout this paper, unless otherwise specified, we let be a complete probability space with a filtration satisfying the usual conditions; that is, it is right continuous and contains all -null sets. Let be a -dimensional Brownian motion defined on the probability space. Let denote the set of positive integers, the -dimensional real Euclidean space, and the space of real matrices. stands for the identity matrix of appropriate dimensions. For , denotes the Euclidean norm. For , denotes spectral norm of the matrix . Denote by the minimum eigenvalue of a matrix. If is a vector or matrix, its transpose is denoted by .
Let and for all , exist and let for all but at most a finite number of points be with the norm , where and denote the right-hand and left-hand limits of function at , respectively. Denote = ∈ for all .
For and , let denote the family of all -measurable -valued random variables such that , where stands for the mathematical expectation operator with respect to the given probability measure . And denote the family of all measurable -valued random variables , such that . Let be the family of all bounded -valued functions, and let be the family of all measurable -valued functions.
Consider the following ISFDE-DI: where the initial value , , . Both and are Borel measurable. represents the impulsive perturbation of at time . The fixed moments of impulse times satisfy (as ). are the impulse input delays satisfying .
As a standing hypothesis, we assume that for any there exists a unique stochastic process satisfying (1) denoted by , which is continuous on the right-hand side and limitable on the left-hand side (see [32]). Moreover, we assume that , , and for all , ; then (1) admits a trivial solution .
We introduce the following scalar IFDE-DI as the comparison system: where the initial value ; is defined as , . is continuous, Lebesgue measurable, and nondecreasing with respect to the last argument; , are continuous and nondecreasing. Assume that , , and ; then system (2) admits a trivial solution . We further assume that for any , there exists a unique solution to system (2) on denoted by (see [5, 6]) which is continuous on the right-handside and limitable on the left-hand side.
For convenience, we introduce the following function classes:= , continuous and strictly increasing, . . . .
At the end of this section, let us introduce the following definitions.
Definition 1 (see [23, 26]). The trivial solution of (1) is said to be as follows.(i)th moment stable if, for any , there exists such that whenever .(ii)th moment asymptotically stable if it is th moment stable and there exists such that whenever .(iii)th moment globally exponentially stable if there is a pair of positive constants , such that for all . When , it is usually said to be globally exponentially stable in mean square.
Definition 2 (see [26]). A function belongs to class if (i) is continuous on each of the sets and for each , , and , exists;(ii) is continuously once differentiable in and twice in in each of the sets , .
If , define an operator from to by where
3. Main Results
In this section, we will develop an impulsive delay differential inequality and comparison principles and establish some criteria on th moment exponential stability and asymptotical stability for (1).
Lemma 3 (impulsive delay differential inequality). Assume that , , , , , , , and(i) for each ;(ii), where .
Then any solution of the scalar impulsive delay differential inequality problem
satisfies
where is the unique positive solution of .
Proof . Set , . For each , by the second inequality of (8), we have
where .
On the other hand, for any , ,
For , integrating inequality (11) from to , we obtain
this implies that
For , by the same method, together with (10), (11), and (13), we have
By induction, we have, for , ,
Thus, for , we get
Let be impulse points in , . In view of condition (i), we get
where is the first impulsive point before and satisfies . Submitting this into inequality (16), then, for ,
Let . Then condition (ii) implies . Moreover, and . Hence has a unique positive solution . Next, we claim that
Since
So we only need to prove (19) for . Suppose not, then there exists a such that
Thus from (18), (22), and , we see that
which is a contradiction. Therefore, (19) holds. This completes the proof.
Lemma 4 (comparison principle). Assume that there exists a function such that(i) for any , ;(ii) for all , .Then, provided , , where is the solution process to (1).
Proof. For any and sufficiently small satisfying , by the Itô formula together with condition (i), we have
this implies that
Write simply. Now supposing that for each , , we claim that
Consider the system
where is a constant. We claim that for .
In fact, if this is not true, then from the continuity of and in , we know that there exist a and a sufficiently small constant such that and
Thus . On the other hand, by condition (i), we obtain that
This is a contradiction. So holds for all . Let ; then , and hence inequality (27) holds.
Noting that and are nondecreasing, by (27) and condition (ii), we get
Thus, it follows from (27) and (31) that
Similar to the previous process, we have when . By induction, it follows that , . The proof is complete.
Theorem 5. Assume that there exist functions , , and such that (i) for any ;(ii) for any , ;(iii) for all , .Then the stability properties of the trivial solution of IFDE-DI (2) imply the corresponding stability properties of the trivial solution of ISFDE-DI (1). Moreover, if condition (i) is replaced by(i*)there exist positive constants , , and such that for all then the global exponential stability of the trivial solution of IFDE-DI (2) implies that th moment global exponential stability of ISFDE-DI (1).
Proof. Firstly, assume that the trivial solution of IFDE-DI (2) is stable. Let ; then for given , there exists such that and
Let , . From conditions (ii) and (iii) and Lemma 4, we get that
Let and ; then by condition (i) and , we have . Hence, by (34) and (35), we have
If , then by conditions (i) and (36), we have
that is, the trivial solution of ISFDE-DI (1) is stable.
Next, let us suppose that the trivial solution of IFDE-DI (2) is asymptotically stable. This implies that the trivial solution of ISFDE-DI (1) is stable. Let , . Since is attractive, for any , there exist and such that
Choose . Note the fact that implies . Then by (35) and (37), we get
which implies that the trivial solution of ISFDE-DI (1) is asymptotically stable.
Thirdly, let us suppose that the trivial solution of IFDE-DI (2) is globally exponentially stable and condition holds. Then, there exists a couple of positive constants and such that
Let , . Then by (35) and (40), we get for all . Thus, by condition (i*), it yields that
Hence, the trivial solution of ISFDE-DI (1) is th moment globally exponentially stable. The proof is complete.
Theorem 6. Assume that there exist a function , positive constants , , , and , constants and , and such that (i) for any ;(ii) for any , ;(iii) for all , ;(iv) for each ;(v) where . Then the trivial solution of ISFDE-DI (1) is th moment globally exponentially stable.
Proof. Let , , , and . We obtain the comparison system (2). It is easy to verify that all conditions of Theorem 5 are satisfied and so the global exponential stability of the trivial solution of IFDE-DI (2) implies that th moment global exponential stability of ISFDE-DI (1).
Furthermore, let be the unique positive solution of . Using conditions (ii) and (iii), we find
Thus from conditions (iv) and (v) and Lemma 3, we obtain that
which implies that the trivial solution of IFDE-DI (2) is globally exponentially stable. The proof of Theorem 6 is complete.
Remark 7. An impulsive stochastic dynamical system can be viewed as a hybrid one comprised of two components: a continuous stochastic dynamic and a discrete dynamic. Theorem 6 can be used to deal will all three cases: the system with stable continuous stochastic dynamic and unstable discrete dynamic, the system with unstable continuous stochastic dynamic and stable discrete dynamic, and the system with stable continuous stochastic dynamic and stable discrete dynamic. When , the continuous stochastic dynamic of (1) may be stable. In this case, in order to ensure the stability of the entire system, the delayed impulses’ frequency and amplitude , should be suitably related to the decrease of continuous flows; that is, conditions (iv) and (v) hold. In this sense, Theorem 6 can be used to deal with the robust stabling of continuous stochastic dynamic subject to delayed impulsive perturbations. When , the continuous stochastic dynamic of (1) may be unstable and the stability of the entire system is determined by the delayed impulse effects. In this case, we need to require that the delayed impulses’ frequency and amplitude should be suitablly related to the decrease of of continuous flows.
Remark 8. It is noted that the exponential stability analysis in [30, 31] only considers the case of impulsive stabilization. In this sense, Theorem 6 has a wider adaptive range.
4. Examples
In this section, the effectiveness and advantages of the results derived in the preceding section will be illustrated by two examples.
Example 1. Consider the two-dimensional nonlinear impulsive stochastic delay equation in the form
where , , . If there exists a positive constant such that
then (44) is globally exponentially stable for any bounded impulsive input delays .
Denote . Choose the Lyapunov function ; then for any , we have
for .
Take , , , , , , , . It is easy to check that all conditions of Theorem 6 are satisfied under conditions (45), which means that (44) is globally mean square exponentially stable for any bounded impulsive input delays .
Remark. It is noted that (44) without impulses is globally mean square exponentially stable and the impulses are destabilizing since . Hence, the existing stability theorems in [30, 31] fail to work. This shows that our results have a wider adaptive range.
Example 2. Consider the following impulsive stochastic delayed neural network:
where
with .
It is noted that (47) without impulse is not stable, and its simulation with delay and initial data and are shown in Figures 1 and 2.
In the following, applying Theorem 5, we will show that under impulsive control law, (47) is mean square exponentially stable if .
Denote . Choose . Then condition (i) of Theorem 5 holds with , for .
Thus, the comparison system is which according to case (iii) of Corollary 1 in [19] is globally exponentially stable for any bounded impulsive input delays if . Hence, we conclude by Theorem 6 that system (47) is mean square exponentially stable if . With the same initial value, the simulations of the impulsive stochastic delay neural network (47) under the delayed impulsive control law , , are shown in Figures 3 and 4.
5. Conclusions
This paper has investigated the exponential stability of ISFDEs-DI based on the comparison approach and an impulsive delay differential inequality. Some criteria on the th moment global exponential stability are established. The obtained results complement some recent works. Two examples have been given to illustrate the effectiveness and the advantages of the results obtained. One of the drawbacks of the proposed method is perhaps that our results require the condition and thus cannot deal with the time delay system with . There will be future work to establish a criterion for the above system.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11226247, 11301004), the 211 Project of Anhui University (32030018/33010205), the Anhui Provincial Nature Science Foundation (1308085QA15/1308085MA01), the Natural Science Foundation of Jiangsu Province (BK20130239), the Research Fund for the Doctoral Program of Higher Education of China (20130094120015), the Key Foundation of Anhui Education Bureau (KJ2012A019), and the Research Fund for the Doctoral Program of Higher Education (20103401120002).