Mean Square Exponential Stability for Stochastic Functional Differential Equations with Impulses

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, stochastic functional differential equations with impulses are considered. By employing Gronwall-Bellman inequality, the stochastic analytic technique and the properties of operator semigroup, the sufficient conditions ensuring the exponential stability in mean square for mild solution of such system are obtained. Our results can generalize and improve the existing works.


Introduction
Stochastic partial differential equations have attracted the attention of many authors and many valuable results on the stability of the solution have been established, see [4,5,6,9,11,13] and references therein.For example, Taniguchi [13] has considered the exponential stability for stochastic partial differential equations by the energy inequality; Caraballo and Liu [9] have investigated the exponential stability for mild solution to stochastic partial differential equations with delays by utilizing the Gronwall inequality; Liu and Shi [6] have considered the exponential stability for stochastic partial functional differential equations by means of the Razuminkhin-type theorem; Taniguchi [11] has proved the almost sure exponential stability of mild solution for stochastic partial functional differential equation by using the analytic technique; Luo [4][5] has discussed asymptotic stability of stochastic partial differential equations with infinite delays and exponential stability for mild solutions of stochastic partial differential equation with delays by fixed point theorem, respectively.
On the other hand, impulsive effects likewise exist in a wide variety of evolutionary processes, for example, medicine and biology, economics, mechanics, electronics and telecommunications, etc., in which many sudden and abrupt changes occur instantaneously, in the form of impulses.Many interesting results on impulsive effects have been obtained, see [1,3] and references therein.When we consider the exponential stability for mild solutions of stochastic partial functional differential equations with impulses, the main difficulty mainly comes from impulsive effects on the system since the corresponding theory for such problem has not yet been fully developed.Many excellent tools to derive the exponential stability for mild solution of stochastic partial functional differential equations may be difficult and even ineffective for the exponential stability of such system with impulses.Therefore, there is few results on the stability for stochastic partial functional differential equations with impulses, see [7][8] and references therein.
Motivated by the above discussion, in this paper, by using Gronwall-Bellman inequality, the stochastic analytic techniques and the properties of operator semigroup, we obtain some new sufficient conditions to ensure the exponential stability in mean square for mild solution of stochastic partial functional differential equations with impulses.Our results can generalize and improve the existing works.
The rest of this paper is organized as follows.In section 2, we present some basic notations and definitions.In section 3, sufficient conditions are derived to ensure the exponential stability in mean square for mild solution.

Preliminary Notes
Let H, K be two real separable Hilbert spaces and L(K, H) be the space of bounded linear operators mapping K into H.For convenience, we shall use the same notations • to denote the norms in H, K and L(K, H) without any confusion.Let (Ω, F, {F} t≥0 , P) be a complete probability space with a filtration {F} t≥0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all P-null sets).Let {ω(t) : t ≥ 0} denote a K-valued {F} t≥0 -Wiener process defined on (Ω, F, {F} t≥0 , P) with covariance operator Q, i.e., where Q is a positive self-adjoint, trace class operator on K, < •, • > K denotes the inner product of K, E denotes the mathematical expectation.In particular, we call such ω(t) : t ≥ 0, a K-valued Q-Wiener process with respect to {F} t≥0 .
In order to define stochastic integrals with respect to the Q-Wiener process ω(t), we introduce the subspace > K is a Hilbert space.We assume that there exists a complete orthonormal system {e i } i≥1 in K, a bounded sequence of nonnegative real numbers λ i such that Qe i = λ i e i , i = 1, 2 Let R and Z be the sets of real and integer numbers, respectively; R + = [0, +∞) and C(X, Y ) denotes the space of continuous mapping from the topological space X to the topological space Y .Especially, C = C([−τ, 0], R) denotes the family of all continuous R-valued functions φ defined on [−τ, 0] with the norm φ τ = sup −τ ≤θ≤0 φ(θ) , where τ is a positive constant.
P C(J, R n ) = {ϕ : J → R n is continuous for all but at most a finite number of points t ∈ J and at these points t ∈ J, ϕ(t + ) and ϕ(t + ) exists, ϕ(t + ) = ϕ(t)}, where J ⊂ R is a bounded interval, ϕ(t + ) and ϕ(t + ) denote the right-hand and left-hand limits of the function ϕ(t), respectively.Especially, let H .In this paper, we consider the following stochastic partial functional differential equation with impulses: where We also assume 0 ∈ ρ(−A), the resolvent set of −A.Then we know that there exist constant M ≥ 0, γ > 0 such that

Definition 2.2
The mild solution of system ( 1) is said to be exponentially stable in mean square if there exists a pair of positive constants λ > 0 and M ≥ 1 such that for any solution x(t) with the initial condition

Main Results
For system (1), we impose the following assumptions: (A1) There exist constants L f > 0, L σ > 0 such that for any x, y ∈ H and t ≥ 0, Under the assumptions:(A1)-(A2), the existence and uniqueness of mild solution to the system (1) is easily shown by using Picard iterative method.

Theorem 3.1 Suppose the assumptions (A1)-(A2) hold, and we further assume that the following conditions
hold.Then the mild solution of system ( 1) is exponentially stable in mean square.
Proof.From (3), for any t ≥ 0, we can get It follows from (2) that Combining (A2) with Hölder inequality, we can get From (A1) and Hölder inequality, we obtain Using (A1) and Burkholder-type inequality, we obtain Substituting ( 4)-( 9) into (3), we have where On the other hand, by (A4), one has Thereby, (10) can be rewritten as From the assumption (A5), it implies that the mild solution of system ( 1) is exponentially stable in mean square.This completes the proof.
Theorem 3.2 Suppose that all the conditions of Theorem 3.1 hold.Then the mild solution of system ( 1) is exponential stable almost surely.
Proof.The proof is quite similar to the proof of Theorem 5.1 in [4], we omit it here.
If I k (•) ≡ 0, then system (1) becomes stochastic partial functional differential equations: Corollary 3.3 Assume (A1) holds and the following condition holds.Then the mild solution of system (11) is exponentially stable in mean square.

ACKNOWLEDGEMENTS.
The author would like to thank the referee and the editor for their careful comments and valuable suggestions on this work.This work is supported by Youth Foundation of Chongqing Three Gorges University (No.12QN24), Foundation for Professor and Doctor of Chongqing Three Gorges University (No.12ZZ36), Scientific Research Foundation of Chongqing Three Gorges University (No.12RC12).
is the infinitesimal generator of an analytic semigroup of linear operator S(t) t≥0 on a Hilbert space H; f : R − k ) represents the jump in the state x at time t k with I k determining the size of the jump.