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

Authors: Nan Ding
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

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.


---

## Body

## 1. 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 pth moment of mild solution to second-order 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 [6–15].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 impulsive-integral 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 impulsive-integral inequality to investigate the global attracting set and exponential p-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 Section2, 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.


## 2. Preliminaries

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 ℱ0 contains all ℙ-null sets).LetH,K be real separable Hilbert spaces, let and ℒ(K,H) be the space of bounded linear operators mapping K into H. For convenience, we will use the same notations ∥·∥ to denote the norms in H,K, and ℒ(K,H) without any confusion. Let {ω(t):t≥0} denotes a K-valued {ℱ}t≥0-Wiener process defined on (Ω,ℱ,{ℱ}t≥0,ℙ) with covariance operator Q; that is,
(1)E〈ω(t),x〉K〈ω(s),y〉K=(t∧s)〈Qx,y〉K,x,y∈K,
where Q is a positive self-adjoint, trace class operator on K, 〈·,·〉K denotes the inner product of K, and 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 theQ-Wiener process ω(t), we introduce the subspace K0=Q1/2(K) of K which, endowed with the inner product 〈u,v〉K0=〈Q-1/2u,Q-1/2v〉K, is a Hilbert space. We assume that there exists a complete orthonormal system {ei}i≥1 in K, a bounded sequence of nonnegative real numbers λi such that Qei=λiei,i=1,2,…, and a sequence {βi(t)}i≥1 of independent Brownian motions such that
(2)〈ω(t),e〉=∑i=1∞‍λi〈ei,e〉βi(t),e∈K,
and Bt=Btω, where Btω is the sigma algebra generated by {ω(s):0≤s≤t}. Let ℒ20=ℒ2(K0,H) denote the space of all Hilbert-Schmidt operators from K0 into H. It turns out to be a separable Hilbert space, equipped with the norm
(3)∥ψ∥ℒ022=tr((ψQ1/2)(ψQ1/2)*)foranyψ∈ℒ20.
Clearly, for any bounded operators ψ∈ℒ(K,H), this norm reduces to ∥ψ∥ℒ202=tr(ψQψ*).Letℝ and ℤ be the sets of real and integer numbers, respectively; let ℝ+=[0,+∞) and C(X,Y) denote the space of continuous mapping from the topological space X to the topological space Y. Particularly, 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.PC
(
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-) exist, φ(t+)=φ(t)}, where J⊂ℝ is a bounded interval and φ(t+) and φ(t-) denote the right-hand and left-hand limits of the function φ(t), respectively. Particularly, let PC≜PC([-τ,0],H).LetPCℱ0b([-τ,0],H)(PCℱtb([-τ,0],H)) denote the family of all bounded ℱ0(ℱt)-measurable, PC([-τ,0],H)-valued random variables ϕ, satisfying ∥ϕ∥ℒ22=sup-τ≤θ≤0E∥ϕ(θ)∥H2.In this paper, we consider the following neutral stochastic partial functional differential equation with impulses:(4)d[x(t)+g(t,x(t-τ))]=[Ax(t)+f(t,x(t-τ))]dt+σ(t,x(t-τ))dω(t),t≥0,t≠tk,Δx(tk)=x(tk+)-x(tk-)=Ik(x(tk)),t=tk,k∈ℤ,x0(s)=ϕ(s)∈PCℱ0b([-τ,0],H),s∈[-τ,0],a.s,
where A:D(A)⊂H→H is the infinitesimal generator of an analytic semigroup of linear operator S(t)t≥0 on a Hilbert space H; f,g:ℝ+×H→H and σ:ℝ+×H→ℒ20 are jointly continuous functions; the fixed moment of time tk satisfies 0<t1<t2<⋯<tk<⋯, and limk→+∞=+∞; x(tk+) and x(tk-) represent the right and left limits of x(t) at t=tk,k=1,2,…, respectively; Δx(tk)=x(tk+)-x(tk-) represents the jump in the state x at time tk with Ik determining the size of the jump.We also assume that0∈ρ(-A), the resolvent set of -A. Then, it is possible to define the fractional power (-A)α for some 0<α≤1 as a closed linear operator with its domain D((-A)α); furthermore, the subspace D((-A)α) is dense in H, and the expression
(5)∥x∥α=∥(-A)αx∥H,x∈D((-A)α),
defines a norm on D((-A)α).Lemma 1 (see Pazy [16]).
Suppose that0∈ρ(-A), then, we know that there exist constants M≥1,γ>0 such that ∥S(t)∥≤Me-γt for t≥0, and for every 0<α≤1(i)
we have for eachx∈D((-A)α),
(6)S(t)(-A)αx=(-A)αS(t)x;(ii)
there existsMα>0 such that
(7)∥(-A)αS(t)∥≤Mαt-αe-γt.Definition 2.
A stochastic processx(t),t∈ℝ+, is called a mild solution of the system (4), if(i)
x(t) is an ℱt(t≥0) adapted process,(ii)
x(t)∈H has a càdlàg path on t∈ℝ+ almost surely,(iii)
for arbitraryt∈ℝ+, we have
(8)x(t)=S(t)[ϕ(0)+g(0,ϕ)]-g(t,x(t-τ))-∫0t‍AS(t-s)g(s,x(s-τ))ds+∫0t‍S(t-s)f(s,x(s-τ))ds+∫0t‍S(t-s)σ(s,x(s-τ))dω(s)+∑0<tk<t‍S(t-tk)Ik(x(tk)),where x0(·)∈PCℱ0b([-τ,0],H), a.s.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 λ>0 and M≥1 such that for any solution x(t) with the initial condition ϕ∈PCℱ0b([-τ,0],H),
(9)E∥x(t)∥2≤M∥ϕ∥ℒ22e-λt,t≥0.


## 3. Exponential Stability

For system (4), we impose the following assumptions.(A1)
There exist constantsLf>0,Lσ>0 such that for any x,y∈H and t≥0,
(10)∥f(t,x)-f(t,y)∥≤Lf∥x-y∥,f(t,0)=0,∥σ(t,x)-σ(t,y)∥ℒ20≤Lσ∥x-y∥,σ(t,0)=0.(A2)
There existα∈(0,1],Lg>0 such that for any x,y∈H and t≥0,
(11)g(t,x)∈D((-A)α),∥(-A)αg(t,x)-(-A)αg(t,y)∥≤Lg∥x-y∥,g(t,0)=0.(A3)
There exist some positive numbersqk(k=1,2,…) such that
(12)∥Ik(x)-Ik(y)∥≤qk∥x-y∥,Ik(0)=0,k=1,2,…,for anyx,y∈H and ∑k=1∞‍qk<∞.Under the assumptions, (A1)–(A3), the existence and uniqueness of mild solution to the system (4) are easily shown by using Picard iterative method.Theorem 4.
Suppose that the assumptions (A1)–(A3) hold; furthermore, and the following assumptions(A4)
infk=1,2,…(tk-tk-1)=θ>0,(A5)
0<((18M1-α2Lg2γ1-2αΓ(2α-1)+3M2Lf2γ-1+3M2Lσ2)eγτ+(1/θ)ln(6M2(∑k=1∞‍qk)2))/(1-18∥(-A)-α∥2Lg2eγτ)<γhold for α∈(1/2,1], where Γ(·) is the Gamma function and M,M1-α, and γ are corresponding constants in Lemma 1. Then the mild solution of system (4) is exponentially stable in mean square.Proof.
From (8), for any t≥0, we can get
(13)E∥x(t)∥2=E∥∑0<tk<t‍S(t)[ϕ(0)+g(0,ϕ)]-g(t,x(t-τ))-∫0t‍AS(t-s)g(s,x(s-τ))ds+∫0t‍S(t-s)f(s,x(s-τ))ds+∫0t‍S(t-s)σ(s,x(s-τ))dω(s)+∑0<tk<t‍S(t-tk)Ik(x(tk))∥2≤3E∥∑0<tk<t‍S(t)[ϕ(0)+g(0,ϕ)]-g(t,x(t-τ))-∫0t‍AS(t-s)g(s,x(s-τ))ds+∑0<tk<t‍S(t-tk)Ik(x(tk))∥2+3E∥∫0t‍S(t-s)f(s,x(s-τ))ds∥2+3E∥∫0t‍S(t-s)σ(s,x(s-τ))dω(s)∥2≤18E∥S(t)[ϕ(0)+g(0,ϕ)]∥2+18E∥g(t,x(t-τ))∥2+18E∥∫0t‍-AS(t-s)g(s,x(s-τ))ds∥2+6E∥∑0<tk<t‍S(t-tk)Ik(x(tk))∥2+3E∥∫0t‍S(t-s)f(s,x(s-τ))ds∥2+3E∥∫0t‍S(t-s)σ(s,x(s-τ))dω(s)∥2=∑i=16‍Ji.
It follows from (A2) that
(14)J1=18E∥S(t)[ϕ(0)+g(0,ϕ)]∥2≤18∥S(t)∥2×E(∥ϕ(0)∥+∥(-A)-α∥∥(-A)αg(0,ϕ)∥)2≤18M2e-2γtE(∥ϕ∥+∥(-A)-α∥Lg∥ϕ∥)2≤18M2e-γt(1+∥(-A)-α∥Lg)2∥ϕ∥ℒ22.
By (A2), we can get
(15)J2=18E∥g(t,x(t-τ))∥2=18E∥(-A)-α(-A)αg(t,x(t-τ))∥2≤18∥(-A)-α∥2Lg2E∥x(t-τ)∥2.
Employing Lemma 1, (A2), and Hölder inequality, we obtain
(16)J3=18E∥∫0t‍-AS(t-s)g(s,x(s-τ))ds∥2≤18E(∫0t‍∥(-A)1-αS(t-s)∥×∥(-A)αg(s,x(s-τ))∥ds∫0t‍)2≤18M1-α2Lg2E×(∫0t‍e-γ(t-s)(t-s)α-1∥x(s-τ)∥ds)2≤18M1-α2Lg2(∫0t‍e-γ(t-s)(t-s)2(α-1)ds)×(∫0t‍e-γ(t-s)E∥x(s-τ)∥2ds)≤18M1-α2Lg2γ1-2αΓ(2α-1)×∫-τt-τ‍eγτe-γ(t-s)E∥x(s)∥2ds≤18M1-α2Lg2γ1-2αΓ(2α-1)×[∫0t‍γ-1e-γt(eγτ-1)∥ϕ∥ℒ22+eγτ∫0t-τ‍e-γ(t-s)E∥x(s)∥2ds].
Combining Lemma 1 and (A3), with Hölder inequality, we can get
(17)J4=6E∥∑0<tk<t‍S(t-tk)Ik(x(tk))∥2≤6E(∑0<tk<t‍∥S(t-tk)∥∥Ik(x(tk))∥)2≤6M2E(∑0<tk<t‍e-γ(t-tk)qk∥x(tk)∥)2≤6M2(∑k=1∞‍qk)(∑0<tk<t‍e-γ(t-tk)qkE∥x(tk)∥2)≤6M2(∑k=1∞‍qk)2(∑0<tk<t‍e-γ(t-tk)E∥x(tk)∥2).
From Lemma 1, (A1) and Hölder inequality, we obtain
(18)J5=3E∥∫0t‍S(t-s)f(s,x(s-τ))ds∥2≤3E(∫0t‍∥(S(t-s)∥∥f(s,x(s-τ))∥ds)2≤3M2Lf2E(∫0t‍e-γ(t-s)∥x(s-τ)∥ds)2≤3M2Lf2(∫0t‍e-γ(t-s)ds)×(∫0t‍e-γ(t-s)E∥x(s-τ)∥2ds)≤3M2Lf2γ-1[∫0t-τ‍γ-1e-γt(eγτ-1)∥ϕ∥ℒ22+eγτ∫0t-τ‍e-γ(t-s)E∥x(s)∥2ds].
Using (A1) and Burkholder-type inequality, we obtain
(19)J6=3E∥∫0t‍S(t-s)σ(s,x(s-τ))dω(s)∥2≤3E∫0t‍∥S(t-s)∥2∥σ(s,x(s-τ))∥ℒ202ds≤3M2Lσ2∫0t‍e-2γ(t-s)E∥x(s-τ)∥2ds≤3M2Lσ2[∫0t-τ‍γ-1e-γt(eγτ-1)∥ϕ∥ℒ22+eγτ∫0t-τ‍e-γ(t-s)E∥x(s)∥2ds].
Substituting (14)–(19) into (13), we have(20)E∥x(t)∥2eγt≤18M2(1+∥(-A)-α∥Lg)2∥ϕ∥ℒ22+18∥(-A)-α∥2Lg2E{t-τ<0}∥x(t-τ)∥2eγt+18∥(-A)-α∥2Lg2×E{t-τ≥0}∥x(t-τ)∥2eγt+18M1-α2Lg2γ1-2αΓ(2α-1)×[γ-1(eγτ-1)∥ϕ∥ℒ22+eγτ∫0t‍E∥x(s)∥2eγsds]+6M2(∑k=1∞‍qk)2(∑0<tk<t‍E∥x(tk)∥2eγtk)+3M2Lf2γ-1×[γ-1(eγτ-1)∥ϕ∥ℒ22+eγτ∫0t‍E∥x(s)∥2eγsds]+3M2Lσ2×[γ-1(eγτ-1)∥ϕ∥ℒ22+eγτ∫0t‍E∥x(s)∥2eγsds]≤18M2(1+∥(-A)-α∥Lg)2∥ϕ∥ℒ22+18∥(-A)-α∥2Lg2eγτ∥ϕ∥ℒ22+18∥(-A)-α∥2Lg2eγτE∥x(t)∥2eγt+18M1-α2Lg2γ-2αΓ(2α-1)(eγτ-1)∥ϕ∥ℒ22+18M1-α2Lg2γ1-2αΓ(2α-1)eγτ×∫0t‍E∥x(s)∥2eγsds+6M2(∑k=1∞‍qk)2(∑0<tk<t‍E∥x(tk)∥2eγtk)+3M2Lf2γ-2(eγτ-1)∥ϕ∥ℒ22+3M2Lf2γ-1eγτ∫0t‍E∥x(s)∥2eγsds+3M2Lσ2γ-1(eγτ-1)∥ϕ∥ℒ22+3M2Lσ2eγτ∫0t‍E∥x(s)∥2eγsds.From assumption (A5), we can get
(21)E∥x(t)∥2eλt≤C∥ϕ∥ℒ22+p∫0t‍E∥x(s)∥2eγsds+β∑0<tk<t‍E∥x(tk)∥2eγtk,
where C≜(1-18∥(-A)-α∥2Lg2eγτ)-1[18M2(1+∥(-A)-α∥Lg)2+18∥(-A)-α∥2Lg2eγτ+18M1-α2Lg2γ-2αΓ(2α-1)(eγτ-1)+3M2Lf2γ-2(eγτ-1)+3M2Lσ2γ-1(eγτ-1)], p≜(1-18∥(-A)-α∥2Lg2eγτ)-1[18M1-α2Lg2γ1-2αΓ(2α-1)eγτ+3M2Lf2γ-1eγτ+3M2Lσ2eγτ],andβ≜(1-18∥(-A)-α∥2Lg2eγτ)-16M2(∑k=1∞‍qk)2.
According to Gronwall-Bellmen’s inequality [17], we have
(22)E∥x(t)∥2≤C∥ϕ∥ℒ22∏0<tk<t‍(1+β)e-(γ-p)t.
On the other hand, by (A4), one has
(23)∏0<tk<t‍(1+β)≤(1+β)t/θ=e(t/θ)ln(1+β).
Thereby, (22) can be rewritten as
(24)E∥x(t)∥2≤C∥ϕ∥ℒ22∏0<tk<t‍(1+β)e-(γ-p-ln(1+β)/θ)t,
which implies the mild solution of system (4) is exponentially stable in mean square.
This completes the proof.Theorem 5.
Suppose that all the conditions of Theorem4 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.If the impulsive effects,Ik(·)≡0(k=1,2,…), system (4) is turned into the following neutral stochastic partial functional differential equations:
(25)d[x(t)+g(t,x(t-τ))]=[Ax(t)+f(t,x(t-τ))]dt+σ(t,x(t-τ))dω(t),t≥0,x0(s)=ϕ(s)∈Cℱ0b([-τ,0],H),s∈[-τ,0],a.s.Corollary 6.
Assume that (A1)-(A2) hold, and the following condition(26)0<(9M1-α2Lg2γ1-2αΓ(2α-1)+3M2Lf2γ-1+3M2Lσ2)eγτ1-9∥(-A)-α∥2Lg2eγτ<γ
holds for α∈(1/2,1]. Then the mild solution of system (25) is exponentially stable in mean square.Ifg(t,·)≡0, then system (4) becomes stochastic partial functional differential equations with impulses as follows:
(27)dx(t)=[Ax(t)+f(t,x(t-τ))]dt+σ(t,x(t-τ))dω(t),t≥0,t≠tk,Δx(tk)=x(tk+)-x(tk-)=Ik(x(tk)),t=tk,k∈ℤ,x0(s)=ϕ(s)∈PCℱ0b([-τ,0],H),s∈[-τ,0],a.s.Corollary 7.
Assume that (A1) and (A3) hold, and the following conditions(A4)
infk=1,2,…(tk-tk-1)=θ>0,(A5)
(3M2Lf2γ-1+3M2Lσ2)eγτ+(1/θ)ln(6M2(∑k=1∞‍qk)2)<γhold. Then the mild solution of system (27) is exponentially stable in mean square.Ifg(t,·)≡0 and Ik(·)≡0, then system (4) becomes stochastic partial functional differential equations as follows:
(28)dx(t)=[Ax(t)+f(t,x(t-τ))]dt+σ(t,x(t-τ))dω(t),t≥0,x0(s)=ϕ(s)∈Cℱ0b([-τ,0],H),s∈[-τ,0],a.s.Corollary 8.
Assume that (A1) holds, and the following condition(29)(3M2Lf2γ-1+3M2Lσ2)eγτ<γ
holds. Then the mild solution of system (28) is exponentially stable in mean square.Remark 9.
In [12], Caraballo and Liu have studied the exponential stability in p(p≥2)-moment of mild solution to (28) by utilizing Gronwall inequality, and the monotone decreasing behaviors of the delays are imposed. Particularly, when τ(t)≡τ,δ(t)≡τ, the condition ensuring the exponential stability in mean square is (3M2Lf2γ-1+3M2Lσ2)eγτ<γ. From Corollary 8, it is obvious that our results improve the results in [12].


## 4. Example

Example 1. We consider the following neutral stochastic partial functional differential equation with impulses:
(30)d[x(t)+a1x(t-1)]=[∂2∂z2x(t)+a2x(t-1)]dt+a3x(t-1)dw(t),0≤z≤π,t≥0,t≠tk,Δx(tk)=Ik(x(tk-))=b1k2x(tk-),t=tk,k=1,2,…,x0(s)=ϕ(s)∈PCℱtb([-1,0],L2[0,π]),x(t,0)=x(t,π)=0,-1≤s≤0,
where ai>0,i=1,2,3,b1≤0 are constants and w(t) denote the standard cylindrical Wiener process.LetH=L2[0,π],H1=W2,2(0,π)∩W01,2(0,π). Define bounded linear operator A:H1→H by
(31)Ax=∂2x∂z2∈H,∀x∈H1.Then we get(32)Ax=∑n=1∞‍n2〈x,en〉Hen,x∈H1,
where en(z)=2/πsinnz,n=1,2,… is the set of eigenvector of -A.The bounded linear operator(-A)3/4 is given by
(33)(-A)3/4x=∑n=1∞‍n3/2〈x,en〉Hen,
with domain
(34)D((-A)3/4)=H3/4={x∈H,∑n=1∞‍n3/2〈x,en〉Hen∈H}.We can easily know thatA is the infinitesimal generator of an analytic semigroup S(t),t≥0, in H and
(35)S(t)x=∑n=1∞‍e-n2t〈x,en〉Hen,x∈H.
Furthermore, we know that ∥S(t)∥≤e-π2t,t≥0.Letg(t,x(t-τ))=a1x(t-1),f(t,x(t-τ))=a2x(t-1),σ(t,x(t-τ))=a3x(t-1),andtk+1=tk+1.Then we can get(36)M=1,γ=π2,τ=1,Lf=a2,Lg=a1∥(-A)3/4∥,Lσ=a3,θ=1,qk=b1k2,∥(-A)3/4∥=1,∥(-A)-3/4∥≤1π3/2.
From Theorem 4, we know that the mild of system (30) is exponentially stable in mean square provided that
(37)[18M1/42a12π1/2+3a22π2+3a32]eπ2+lnπ46<π2-18a12eπ2π.


## 5. 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.



---

*Source: 729159-2013-05-30.xml*