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

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 pth 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.

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][17][18][19][20][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][26][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.

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.
Then, for  ∈ (0, 1], (ii) there exist constants   > 0 and  > 0 such that, for Now we will state and prove the main results on stability.
Theorem 4. Suppose that ( 1 )-( 5 ) hold.Then system (1) has a unique mild solution and is mean square exponentially stable, if the initial data Ψ is mean square exponentially stable and Here l = E(∑  =1 |  |) and  1− and  are defined by (5).
According to [5], we similarly have the following.
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.
Corollary 7. Assume that the conditions in Theorem 4 hold, but ( 2 ) 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 10.Besides, it should be pointed out that the proposed method in the paper can be employed to consider the th moment ( ≥ 2) exponential stability to system (1).

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.