Exponential Stability in Mean Square for Neutral Stochastic Partial Functional Differential Equations with Impulses

We discuss the exponential stability in mean square of mild solution for neutral stochastic partial functional differential equations with impulses. By applying impulsive Gronwall-Bellman inequality, the stochastic analytic techniques, the fractional power of operator, and semigroup theory, we obtain some completely new sufficient conditions ensuring the exponential stability in mean square of mild solution for neutral stochastic partial functional differential equations with impulses. Finally, an example is provided to illustrate the obtained theory.


Introduction
Stochastic partial differential equations have attracted the attention of many authors, and many valuable results on the existence, uniqueness, and stability of mild solution have been established.For example, Ren and Sakthivel [1] have established the existence and uniqueness of mild solution for a class of second-order neutral stochastic evolution equations with infinite delay and Poisson jumps by means of the successive approximation and the continuous dependence of solutions on the initial data by means of a corollary of the Bihari inequality; Sakthivel et al. [2,3] have discussed the existence and uniqueness of square-mean pseudo almost automorphic mild solutions for stochastic fractional differential equations by using the stochastic analysis theory, fixed point strategy, and the existence of mild solution to nonlinear stochastic fractional differential by using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directed from deterministic fractional equations, respectively; Sakthivel et al. [4,5] also have derived the exponential stability of mild solutions to the second-order stochastic evolution equations with Poisson jumps by applying stochastic analysis theory, and the existence and asymptotic stability in th moment of mild solution to secondorder nonlinear neural stochastic differential equations with the help of fixed point theory, stochastic analysis technique, and semigroup theory, respectively.Besides stochastic effects, impulsive effects likewise exist in real-world models.It is to be noted that there has been increasing interest in the study of the existence, uniqueness, and stability of mild solution for stochastic partial functional differential equations with impulses due to its wide applications in various sciences, and many significant results have been obtained [6][7][8][9][10][11][12][13][14][15].
To the best of the author's knowledge, there are only few works about the exponential stability of mild solution to neutral stochastic partial functional differential equations with impulses.One of the reasons is that the mild solutions do not have stochastic differentials, so Itô formula fails to deal with this problem.Another reason is that when we consider the exponential stability of mild solution for stochastic partial functional differential equations with impulses, impulsive effects on the system brings about many difficulties, since the corresponding theory for such problem has not yet been fully developed.For example, Sakthivel and Luo [8,9] have discussed the asymptotic stability for mild solution of impulsive stochastic partial differential equations by using the fixed point theorem which can be regarded as an excellent tool to derive the exponential stability for mild solution to stochastic partial differential with delays in Luo [10,11]; this very useful method may be difficult and even ineffective for the exponential stability of such system with impulses; some other methods used in Caraballo and Liu [12], Wan and Duan [13], and so forth are also ineffective for this problem, since mild solutions do not have stochastic differentials; in addition, Chen [14] established an impulsiveintegral inequality to investigate the exponential stability of stochastic partial differential equation with delays and impulses, but it is not effective for neutral type.For the previous reasons, Long et al. [15] established a new impulsiveintegral inequality to investigate the global attracting set and exponential -stability of stochastic neutral partial functional differential equations with impulses.However, the methods of studying the exponential stability of mild solution for neutral stochastic partial functional differential equations with impulses is still not abundant.
Motivated by the previous discussion, in this paper, we discuss exponential stability in mean square of mild solution for neutral stochastic partial functional differential equations with impulses.By applying impulsive Gronwall-Bellman inequality, the stochastic analytic techniques, inequality technique, the fractional power of operator, and semigroup theory, we obtain some completely new sufficient conditions to ensure the exponential stability in mean square of mild solution for stochastic partial functional differential equations with impulses.
The rest of this paper is organized as follows.In Section 2, we present some basic notations, definitions, and auxiliary results.In Section 3, sufficient conditions are derived to ensure the exponential stability in mean square for mild solution.Finally, an example is given to demonstrate the obtained results.

Preliminaries
Let (Ω, F, {F} ≥0 , P) be a complete probability space with a filtration {F} ≥0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all P-null sets).
Let ,  be real separable Hilbert spaces, let and L(, ) be the space of bounded linear operators mapping  into .For convenience, we will use the same notations ‖⋅‖ to denote the norms in , , and L(, ) without any confusion.Let {() :  ≥ 0} denotes a -valued {F} ≥0 -Wiener process defined on (Ω, F, {F} ≥0 , P) with covariance operator ; that is, where  is a positive self-adjoint, trace class operator on , ⟨⋅, ⋅⟩  denotes the inner product of , and E denotes the mathematical expectation.In particular, we call such () :  ≥ 0, a -valued -Wiener process with respect to {F} ≥0 .In order to define stochastic integrals with respect to the -Wiener process (), we introduce the subspace  0 =  1/2 () of  which, endowed with the inner product ⟨, V⟩  0 = ⟨ −1/2 ,  −1/2 V⟩  , is a Hilbert space.We assume that there exists a complete orthonormal system {  } ≥1 in , a bounded sequence of nonnegative real numbers   such that   =     ,  = 1, 2, . .., and a sequence {  ()} ≥1 of independent Brownian motions such that and   =    , where    is the sigma algebra generated by {() : 0 ≤  ≤ }.Let L 0 2 = L 2 ( 0 , ) denote the space of all Hilbert-Schmidt operators from  0 into .It turns out to be a separable Hilbert space, equipped with the norm Clearly, for any bounded operators  ∈ L(, ), this norm reduces to ‖‖ 2 Let R and Z be the sets of real and integer numbers, respectively; let R + = [0, +∞) and (, ) denote the space of continuous mapping from the topological space  to the topological space .Particularly,  ≜ ([−, 0], R) denotes the family of all continuous R-valued functions  defined on [−, 0] with the norm ‖‖  = sup −≤≤0 ‖()‖, where  is a positive constant.
In this paper, we consider the following neutral stochastic partial functional differential equation with impulses: where  : () ⊂  →  is the infinitesimal generator of an analytic semigroup of linear operator () ≥0 on a Hilbert space ; ,  : R + ×  →  and  : R + ×  → L 0 2 are jointly continuous functions; the fixed moment of time We also assume that 0 ∈ (−), the resolvent set of −.Then, it is possible to define the fractional power (−)  for some 0 <  ≤ 1 as a closed linear operator with its domain ((−)  ); furthermore, the subspace ((−)  ) is dense in , and the expression defines a norm on ((−)  ).
Definition 3. The mild solution of system ( 4) is said to be exponentially stable in mean square if there exists a pair of positive constants  > 0 and  ≥ 1 such that for any solution () with the initial condition

Exponential Stability
For system (4), we impose the following assumptions.
Theorem 5. Suppose that all the conditions of Theorem 4 hold.
Then the mild solution of system (4) is exponential stable almost surely.
Proof.The proof is quite similar to the proof of Theorem 5.1 in [18], and we omit it here.

Conclusion
In this paper, neutral stochastic partial functional differential equation with impulses has been investigated.By utilizing impulsive inequality, the stochastic analytic technique, inequality technique, the fractional power of operator, and semigroup theory, some completely new sufficient conditions are obtained to guarantee the exponential stability of mild solution for neutral stochastic partial functional differential equation with impulses.Finally, it should be pointed out that it is possible to extend the results obtained in this paper to other neutral stochastic impulsive differential equations, such as neutral stochastic impulsive partial differential equations with Poisson jumps and so forth.