Stability of Stochastic Differential Delay Systems with Delayed Impulses

and Applied Analysis 3 3. Main Results Before establishing the main results, we derive the following lemma, which is useful to present the main results. Lemma 2. Let assumptions (A1) and (A2) hold. Suppose that infk∈N{tk − tk−1} = β1 and (l1 − 1)β1 < d ≤ l1β1 for some positive integer l1. Then E|X (t)| p ≤ K1E 󵄩󵄩󵄩󵄩ξ 󵄩󵄩󵄩󵄩 p τ , t ∈ [t0 − τ, t0 + d] , (9) whereK1 = 31(1 + h) l e 3 p−1 L(d p +d p/2 . Proof. Since (l1 − 1)β1 < d ≤ l1β1, the maximum number of impulsive times on the interval (t0, t0 + d] is l1. Suppose that the impulsive instants on (t0, t0 +d] are ti, 1 ≤ i ≤ m ≤ l1. For t ∈ (t0, t1), using (A1), we have E|X (t)| p = E 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ξ (0) + ∫ t t 0 f (Xs, s) ds + ∫ t t 0 g (Xs, s) dB (s) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 p ≤ 3 p−1 [E 󵄩󵄩󵄩󵄩ξ 󵄩󵄩󵄩󵄩 p τ + (t − t0) p−1 E∫ t t 0 󵄨󵄨󵄨󵄨f (Xs, s) 󵄨󵄨󵄨󵄨 p ds +(t − t0) (p−2)/2 E∫ t t 0 󵄨󵄨󵄨󵄨g (Xs, s) 󵄨󵄨󵄨󵄨 p ds] ≤ 3 p−1 [E 󵄩󵄩󵄩󵄩ξ 󵄩󵄩󵄩󵄩 p τ + L(t − t0) p−1 ∫ t t 0 E sup t 0 −r≤u≤s |X (u)| p ds +L(t − t0) (p−2)/2 ∫ t t 0 E sup t 0 −r≤u≤s |X (u)| p ds] ≤ 3 p−1 E 󵄩󵄩󵄩󵄩ξ 󵄩󵄩󵄩󵄩 p τ + 3 p−1 L × [(t − t0) p−1 + (t − t0) (p−2)/2 ]


Introduction
Impulsive dynamical systems have attracted considerable interest in science and engineering in recent years because they provide a natural framework for mathematical modeling of many real world problems where the reactions undergo abrupt changes [1][2][3]. These systems have found important applications in various fields, such as control systems with communication constraints [4], sampled-data systems [5,6], and mechanical systems [7]. On the other hand, impulsive control based on impulsive systems can provide an efficient way to deal with plants that cannot endure continuous control inputs [3]. In recent years, the impulsive control theory has been generalized from deterministic systems to stochastic systems and has been shown to have wide applications [8].
In most of recent research results, the impulses are usually assumed to take the following form: Δ ( ) = ( + ) − ( − ) = ( ( − ), ), which indicates the state jump at the impulse time. However, time delays inevitably occurred in the transmission of the impulsive information. Hence, input delays should be considered (see e.g., [5,16]). In the context of stability of deterministic differential equations with delayed impulses, there have appeared several results in the literature (see e.g., [17][18][19]). For example, in [17], the asymptotic stability is investigated for a class of delay-free autonomous systems with the impulses of Δ ( + ) = 1 (( − ) − ), and a sufficient asymptotic stability condition is proposed involving the sizes of impulse input delays. In [19], Chen and Zheng considered more general impulses taking the form Δ ( + ) = ( ( − ), (( − ) − )) and obtained some criteria of exponential stability for nonlinear time-delay systems with delayed impulse effects.
However, most of the existing results of the stability for systems with delayed impulses were considered for the deterministic differential systems. It is noticed that many real world systems are disturbed by stochastic factors. Therefore, it seems interesting to study the stability of stochastic delay differential systems with delayed impulses. Recently, the exponential stability is investigated for impulsive stochastic functional differential system in [20], and exponential stability and uniform stability in terms of two measures were obtained for stochastic differential systems with delayed impulses. Motivated by the above works, the aim of this paper is to study th moment and almost sure exponential stability of a stochastic delay differential system with delayed impulses. It is shown that an unstable stochastic delay system can be successfully stabilized by delayed impulses. Moreover, it is also shown that if a continuous dynamic system is stable, then, under some conditions, the delayed impulses do not destroy the stability of the systems. Our results can generalize some existing results in [20,21].
The paper is organized as follows. In Section 2, we introduce the notations and definitions. We establish several stability criteria for stochastic differential delay systems with delayed impulses in Section 3. In Section 4, two examples are given to illustrate the effectiveness of our results.

Preliminaries
Throughout this paper, let (Ω, F, ) be a complete probability space with some filtration {F } ⩾0 satisfying the usual conditions (i.e., the filtration is increasing and right continuous while F 0 contains all null sets). Let = ( ( ), ≥ 0) be an -dimensional F -adapted Brownian motion. For ∈ R , | | denotes the Euclidean norm of . For −∞ < < < ∞, we say that a function from [ , ] to R is piecewise continuous, if the function has at most a finite number of jumps discontinuous on In this paper, we consider the following stochastic delay differential systems with delayed impulses: where { , ∈ N} is a strictly increasing sequence such that → ∞ as → ∞; { ≥ 0, ∈ N} are the impulsive input delays satisfying = max and = max{ , }. As a standing hypothesis, we assume that , , and are assumed to satisfy necessary assumptions so that, for any ∈ F ([− , 0]; R ), system (1) has a unique global solution, denoted by ( ; ), and, moreover, ( ; ) ∈ F ([− , 0]; R ).
The purpose of this paper is to discuss the stability of system (1). Let us begin with the following definition. Definition 1. The trivial solution of system (1) is said to be as follows.
(1) th moment exponentially stable, if, for any initial data or, equivalently, where and are positive constants independent of 0 .
Abstract and Applied Analysis 3

Main Results
Before establishing the main results, we derive the following lemma, which is useful to present the main results.

Lemma 2. Let assumptions
which implies Using the Gronwall inequality, it follows that According to (A 2 ), we get It follows that Hence, Repeating the above argument gives that, for ∈ [ 0 − , ], Since there are no impulses on ( , 0 + ], we obtain This completes the proof. When the continuous dynamics in system (1) is unstable, the following theorem shows that the system (1) can be stabilized by the delayed impulses. Theorem 3. Let the assumptions in Lemma 2 hold. Assume that there exist positive constants 1 , 2 , 1 , and and ≥ 1 such that where ∏ ∞ =1 < ∞; Then the trivial solution of system (1) is th moment exponentially stable.
Abstract and Applied Analysis 5 Remark 4. In Theorem 3, the positive constant is introduced in (H 3 ), where > 1 and ≤ 1 are allowed. As mentioned in [13], the constant is introduced in (H 3 ), which makes it possible to tolerate certain perturbations in the overall impulsive stabilization process; that is, it is not strictly required by Theorem 3 that each impulse contributes to stabilize the system; there can exist some destabilized impulses. Moreover, when 3 −2 = 1/2, 3 −1 = 1/2, 3 = 4, for ∈ N, we have Π ∞ =1 < 5 and ∑ ∞ =1 ( − 1) = +∞. Then, Theorem 3 can be used, but the results in [20,21] cannot be applicable to this case.
In the following theorem, we will show that if the continuous dynamics is stable, then, under some condition, the system is still stable with the delayed impulsive effects.

Theorem 5. Assume that the assumptions in Lemma 2 hold.
Suppose that there exist positive constants 1 , 2 , 2 , and and ≥ 1 such that Then the trivial solution of system (1) is th moment exponentially stable.
Remark 6. When the continuous system in system (1) is stable, the system (1) can always be stable with stabilized impulses. Thus, 1 + 2 < 1 is permissible in Theorem 5, and only one constraint − > 1 is assumed for constant . However, 1 + 2 ≥ 1 and 1 + 2̃> are necessary in Theorem 3.2 of [20]. Thus, in this aspect, Theorem 5 is more general than the results existing in [20].
The following theorem shows that the trivial solution of system (1) is almost sure exponentially stable, under some additional conditions. Theorem 7. Suppose that ≥ 1 and the conditions in Theorem 3 or Theorem 5 hold. Then, the trivial solution of system (1) is almost sure exponentially stable.
Proof. Using Theorem 3 or Theorem 5, we derive that the trivial solution of system (1) is th moment exponentially stable. Therefore, there exists a positive constant 1 such that It is obvious that Combining the Hölder inequality with (A 1 ) and (56) implies that By virtue of Burkholder-Davis-Gundy inequality, (A 1 ), and (56), we have where ( ) is a positive constant depending on only. Thanks to (A 2 ) and (56), we see that Substituting (58)-(60) into (57) gives that where 2 is a positive constant. Then for all ∈ (0, ) and ∈ N, we have Abstract and Applied Analysis 7 Using the Borel-Cantelli Lemma, we see that there exists an 0 ( ) such that, for almost all ∈ Ω, ≥ 0 ( ), where ≤ ≤ ( + 1) . It follows that Consequently, Let → 0; then the result follows.

Numerical Examples
In this section, two numerical examples are given to show the effectiveness of the main results derived in the preceding section.
Example 8. Consider a stochastic delay differential system with delayed impulses as follows:    It can be seen in Figures 1 and 2 that unstable continuous dynamics of system (66) can be successfully stabilized by delayed impulses.
It can be seen from Figures 3 and 4 that the delayed impulses can robust the stability of the system (68).

Conclusion
The th moment and almost sure exponential stability are investigated in this paper. Using Razumikhin methods, several sufficient conditions are established for stability of stochastic delay differential systems with delayed impulses. Finally, two numerical simulation examples are offered to verify the effectiveness of the main results.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.