Exponential Stability of Jump-Diffusion Systems with Neutral Term and Impulses

Yang, Hua; Liu, Jianguo; Jiang, Feng

doi:https://doi.org/10.1155/2015/192083

Mathematical Problems in Engineering

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Recent Developments on the Stability and Control of Stochastic Systems

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Volume 2015 | Article ID 192083 | https://doi.org/10.1155/2015/192083

Exponential Stability of Jump-Diffusion Systems with Neutral Term and Impulses

Hua Yang,^1,2Jianguo Liu,¹and Feng Jiang³

Academic Editor: Leonid Shaikhet

Received13 Jan 2015

Accepted11 Feb 2015

Published10 Dec 2015

Abstract

We study jump-diffusion systems with neutral term and impulses. Under some conditions, we prove that the jump-diffusion systems with neutral term and impulses are mean square and almost surely exponentially stable. Finally, we give an example to describe the theoretical results.

1. Introduction

Recently, stochastic partial differential systems (SPDS) are often used to describe some evolution phenomena in studying pattern recognition and engineering [1, 2]. Dynamic behavior of solutions for SPDS has been discussed by many researchers [3–8].

In the practical application, there exists often impulsive disturbance under specific circumstances [9, 10]. For example, in [11, 12], Zhu et al. discussed stability behavior of stochastic impulsive systems. Sakthivel and Luo [13] discussed asymptotics of stochastic impulsive systems. Further, in [14], Jiang and Shen studied asymptotic behavior for stochastic impulsive infinite delays systems. Chen et al. [15] discussed stability of stochastic impulsive systems by inequality technique.

In addition, many models such as population models and circuits models often include the derivative terms of the current state and past state, which are often described as neutral systems [16–21]. Meanwhile, there are also a few works on jump diffusions, which are discussed extensively. For example, Zhu [22] discussed the long-time behavior of the solution including the th moment asymptotic stability and almost sure stability for stochastic jump systems. In [23, 24], the authors established dynamical behavior of stochastic jump systems and stochastic jump biological model. Cui et al. [25–27] studied the existence, uniqueness, and some stability of stochastic jump systems. Luo and Taniguchi [28] discussed the existence of solutions of neutral stochastic jump systems under non-Lipschitz condition. Ren and Sakthivel [29, 30] discussed dynamic behavior of second-order jump-diffusion systems.

The rest of the paper is organized as follows. In Section 2, we give some preliminaries on mild solution. Then we give some conditions to guarantee stability of mild solution by the fixed point theory in Section 3. In Section 4, an example is presented to show our conclusions.

2. Preliminaries

Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions [31]. Let and Moreover, let and be real separable Hilbert spaces with norms , and let be the space of all bounded linear operators from into . In this work, is the norms of operators. The notation denotes the family of all -measurable functions from into with the norm .

Let be a -valued Wiener process on the probability space with a trace class operator on . being the set of all -Hilbert-Schmidt operators from to . For the construction, the reader is referred to [19, 25, 31, 32]. Assume that , , is a stationary -Poisson point process with characteristic measure . defined by for . Let , which is independent of . For the Poisson measure, see [21].

Suppose that , , is an analytic semigroup with its infinitesimal generator [14]. For the analytic semigroup, see Pazy [32, Page 60–75]. In the paper, assume that . According to Pazy [32], a linear closed operator () can be defined on .

Consider a jump-diffusion system with neutral term and impulses: with the initial data Here , , and are continuous. Consider , , where and , .

Definition 1. A process , , , is said to be the mild solution to system (1) if (i) is a -adapted, càdlàg process and is almost surely square-integrable on ;(ii)for satisfies and

To establish exponential stability [7, 10, 20] of system (1), we need the following hypotheses.(), where is a positive constant.()There exists such that, for , , ()There exist positive constants such that, for , , ()There exist constants such that, for , ()One has for .

Remark 2. We should point out that it is clear that system (1) has a trivial solution when by ()–().

3. Main Results

In the section, we will state and prove our main results on mean square and almost surely exponential stability to system (1) by the fixed point theory. To prove our main results, we firstly give a useful lemma.

Lemma 3 (see [18, 32]). Under (), assume that . Then, for , (i)for , ;(ii)there exist constants and such that, for ,

Now we will state and prove the main results on stability.

Theorem 4. Suppose that ()–() hold. Then system (1) has a unique mild solution and is mean square exponentially stable, if the initial data is mean square exponentially stable and Here and and are defined by (5).

Proof. Let be the Banach space of with the norm and there exist and such that, for , Define an operator by for and for , Now we will prove that the operator has a fixed point in . Without loss of generality, we suppose that . Let . We firstly claim that . Let and we then have from (8)Note that the initial data is mean square exponentially stable; that is, there exist, for , such that , . By () and (), we have Then (5) together with () and () yieldsBy (), we have By the properties of the martingales, we have By () and (), we have From (9) to (14), we can see obviously that there exist and such that Next we claim that is càdlàg on . Let , , and ; we have from (8) that We can easily see that as , , and . Moreover, by the properties of the martingales, we have the fact that when , Consequently, we obtain that .
We finally claim is contractive. From (8), , Similar to (10)–(14), we have Here .
Consequently, we have Then if (6) holds, is contractive. Therefore, system (1) has a unique and is mean square exponentially stable if (6) holds. This proof is complete.

According to [5], we similarly have the following.

Theorem 5. Under the conditions in Theorem 4, system (1) is almost surely exponentially stable.

If , system (1) becomes with the initial data .

From Theorems 4 and 5, we have the following.

Corollary 6. Assume that the conditions in Theorem 4 hold, but (6) is replaced with the following condition: Then system (22) admits a unique mild solution and is mean square and almost surely exponentially stable.

If and , system (1) becomes with the initial data .

Corollary 7. Assume that the conditions in Theorem 4 hold, but () and (6) are replaced with the following condition: Then system (24) has a unique mild solution and is mean square and almost surely exponentially stable.

Remark 8. We think that the results of the paper can be generalized to infinite delay systems. Systems (22) and (24) have been discussed in [14] and [13], respectively, which focus on asymptotic stability of mild solution. Also by Theorem 4 system (1) without impulses is also mean square and almost surely exponential stability under some conditions, which has been studied in [25]. However, it is well known that there are great differences on the method between the time-delay cases, in particular when considering a problem involved in perturbation. In the paper, we mainly focus on exponential stability. In the sense, [13, 14, 25] are generalized to more extensive systems.

Remark 9. In particular, when , , system (1) without jumps, impulses, and neutral term reduces to SPDS, which is mean square and almost surely exponential stability if . When , Luo [5] showed that system (1) without jumps, impulses, and neutral term is mean square exponentially stable to this system. In the sense, the result of the paper improves the result of [5].

Remark 10. Besides, it should be pointed out that the proposed method in the paper can be employed to consider the th moment () exponential stability to system (1).

4. Illustrative Example

Example 1. Consider a jump-diffusion system with neutral term and impulses: with , , , where , and .
Let and . The operator is defined by with and then is given by and the domain Since, for , , from Pazy [32, Page 70], we have Obviously, ()–() are satisfied with Thus, by Theorems 4 and 5, system (26) is mean square and almost surely exponentially stable if where is defined by (5).

5. Concluding Remarks

In this paper, we have discussed jump-diffusion systems with neutral term and impulses. Some conditions on mean square and almost surely exponential stability of the mild solutions to the jump-diffusion systems with neutral term and impulses are derived by the fixed point theory. The obtained results extend some earlier results to the case of SPDS with neutral term and jump and impulses. Finally, the results of this paper are demonstrated well with an example.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (61304067 and 61071136), the Natural Science Foundation of Hubei Province of China (2013CFB443), and Research Fund for the Doctoral Program of Higher Education of China (20110142110069).

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Copyright © 2015 Hua Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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