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

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.

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 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.

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).

Let 𝐻,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, E⟨𝜔(𝑡),𝑥⟩_𝐾𝜔(𝑠),𝑦_𝐾=(𝑡∧𝑠)𝑄𝑥,𝑦_𝐾,𝑥,𝑦∈𝐾, (1) 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 the 𝑄-Wiener process 𝜔(𝑡), we introduce the subspace 𝐾₀=𝑄^1/2(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 {𝑒_𝑖}_𝑖≥1 in 𝐾, a bounded sequence of nonnegative real numbers 𝜆_𝑖 such that 𝑄𝑒_𝑖=𝜆_𝑖𝑒_𝑖,𝑖=1,2,…, and a sequence {𝛽_𝑖(𝑡)}_𝑖≥1 of independent Brownian motions such that ⟨𝜔(𝑡),𝑒⟩=^∞_𝑖=1𝜆_𝑖𝑒_𝑖,𝑒𝛽_𝑖(𝑡),𝑒∈𝐾, (2) and 𝐵_𝑡=𝐵^𝜔_𝑡, where 𝐵^𝜔_𝑡 is the sigma algebra generated by {𝜔(𝑠)∶0≤𝑠≤𝑡}. Let ℒ⁰₂=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 𝜓²_ℒ20=tr𝜓𝑄^1/2𝜓𝑄^1/2∗forany𝜓∈ℒ⁰₂. (3) Clearly, for any bounded operators 𝜓∈ℒ(𝐾,𝐻), this norm reduces to ‖𝜓‖²_ℒ02=tr(𝜓𝑄𝜓^∗).

Let ℝ 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).

Let PC^𝑏_ℱ0([−𝜏,0],𝐻)(PC^𝑏_ℱ𝑡([−𝜏,0],𝐻)) denote the family of all bounded ℱ₀(ℱ_𝑡)-measurable, PC([−𝜏,0],𝐻)-valued random variables 𝜙, satisfying ‖𝜙‖²_ℒ2=sup_{−𝜏≤𝜃≤0}E‖𝜙(𝜃)‖²_𝐻.

In this paper, we consider the following neutral stochastic partial functional differential equation with impulses: 𝑑𝑥(𝑡)((=(𝑡)(((((𝑡)+𝑔𝑡,𝑥𝑡−𝜏))𝐴𝑥+𝑓𝑡,𝑥𝑡−𝜏))𝑑𝑡+𝜎𝑡,𝑥𝑡−𝜏))𝑑𝜔,𝑡≥0,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡_𝑘,𝑡=𝑡_𝑘𝑥,𝑘∈ℤ,₀(𝑠)=𝜙(𝑠)∈PC^𝑏_ℱ0([−𝜏,0],𝐻),𝑠∈[−𝜏,0],a.s, (4) 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→ℒ⁰₂ are jointly continuous functions; the fixed moment of time 𝑡_𝑘 satisfies 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯, and lim_𝑘→+∞=+∞; 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘) represent the right and left limits of 𝑥(𝑡) at 𝑡=𝑡_𝑘,𝑘=1,2,…, respectively; Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘) represents the jump in the state 𝑥 at time 𝑡_𝑘 with 𝐼_𝑘 determining the size of the jump.

We also assume that 0∈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 ‖𝑥‖_𝛼=(−𝐴)^𝛼𝑥_𝐻,𝑥∈𝐷(−𝐴)^𝛼, (5) defines a norm on 𝐷((−𝐴)^𝛼).

Lemma 1 (see Pazy [16]). Suppose that 0∈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 each 𝑥∈𝐷((−𝐴)^𝛼), 𝑆(𝑡)(−𝐴)^𝛼𝑥=(−𝐴)^𝛼𝑆(𝑡)𝑥; (6)(ii) there exists 𝑀_𝛼>0 such that (−𝐴)^𝛼𝑆(𝑡)≤𝑀_𝛼𝑡^−𝛼𝑒^−𝛾𝑡. (7)

Definition 2. A stochastic process 𝑥(𝑡),𝑡∈ℝ⁺, is called a mild solution of the system (4), if(i)𝑥(𝑡) is an ℱ_𝑡(𝑡≥0) adapted process,(ii)𝑥(𝑡)∈𝐻 has a càdlàg path on 𝑡∈ℝ⁺ almost surely,(iii) for arbitrary 𝑡∈ℝ⁺, we have 𝑥(𝑡)(𝑡𝜙(0)((−=𝑆)+𝑔0,𝜙−𝑔𝑡,𝑥𝑡−𝜏))_𝑡0()𝑔((+𝐴𝑆𝑡−𝑠𝑠,𝑥𝑠−𝜏))𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠−𝜏))𝑑𝑠+^𝑡₀𝑆(𝑡−𝑠)𝜎(𝑠,𝑥(𝑠−𝜏))𝑑𝜔(𝑠)+_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘, (8)where 𝑥₀(⋅)∈PC^𝑏_ℱ0([−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∈PC^𝑏_ℱ0([−𝜏,0],𝐻), E‖𝑥(𝑡)‖²≤𝑀𝜙²_ℒ2𝑒^−𝜆𝑡,𝑡≥0. (9)

3. Exponential Stability

For system (4), 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, 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦≤𝐿_𝑓𝑥−𝑦,𝑓(𝑡,0)=0,𝜎(𝑡,𝑥)−𝜎(𝑡,𝑦)_ℒ02≤𝐿_𝜎𝑥−𝑦,𝜎(𝑡,0)=0. (10)(A2) There exist 𝛼∈(0,1],𝐿_𝑔>0 such that for any 𝑥,𝑦∈𝐻 and 𝑡≥0, 𝑔(𝑡,𝑥)∈𝐷(−𝐴)^𝛼,(−𝐴)^𝛼𝑔(𝑡,𝑥)−(−𝐴)^𝛼𝑔𝑡,𝑦≤𝐿_𝑔,𝑔()𝑥−𝑦𝑡,0=0. (11)(A3) There exist some positive numbers 𝑞_𝑘(𝑘=1,2,…) such that 𝐼_𝑘(𝑥)−𝐼_𝑘𝑦≤𝑞_𝑘𝑥−𝑦,𝐼_𝑘(0)=0,𝑘=1,2,…, (12)

for any 𝑥,𝑦∈𝐻 and ∑^∞_𝑘=1𝑞_𝑘<∞.

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)inf_𝑘=1,2,…(𝑡_𝑘−𝑡_𝑘−1)=𝜃>0,(A5)0<((18𝑀²_1−𝛼𝐿²_𝑔𝛾^1−2𝛼Γ(2𝛼−1)+3𝑀²𝐿²_𝑓𝛾⁻¹+3𝑀²𝐿²_𝜎)𝑒^𝛾𝜏+(1/𝜃)ln(6𝑀²(∑^∞_𝑘=1𝑞_𝑘)²))/(1−18‖(−𝐴)^−𝛼‖²𝐿²_𝑔𝑒^𝛾𝜏)<𝛾hold for 𝛼∈(1/2,1], where Γ(⋅) is the Gamma function and 𝑀,𝑀_1−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 E‖𝑥(𝑡)‖²𝑆(𝑡𝜙(0)((−=E)+𝑔0,𝜙−𝑔𝑡,𝑥𝑡−𝜏))_𝑡0()𝑔((+𝐴𝑆𝑡−𝑠𝑠,𝑥𝑠−𝜏))𝑑𝑠_𝑡0𝑆()𝑓((+𝑡−𝑠𝑠,𝑥𝑠−𝜏))𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝜎(𝑠,𝑥(𝑠−𝜏))𝑑𝜔(𝑠)+_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)𝐼_𝑘(𝑥(𝑡_𝑘))²𝑆(𝑡𝜙(0)((−≤3E)+𝑔0,𝜙−𝑔𝑡,𝑥𝑡−𝜏))_𝑡0𝐴𝑆(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠−𝜏))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)𝐼_𝑘(𝑥(𝑡_𝑘))²+3E^𝑡₀𝑆(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠−𝜏))𝑑𝑠²+3E^𝑡₀𝑆(𝑡−𝑠)𝜎(𝑠,𝑥(𝑠−𝜏))𝑑𝜔(𝑠)²≤18E𝑆(𝑡)[𝜙(0)+𝑔(0,𝜙)]²+18E𝑔(𝑡,𝑥(𝑡−𝜏))²+18E^𝑡₀−𝐴𝑆(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠−𝜏))𝑑𝑠²+6E_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)𝐼_𝑘(𝑥(𝑡_𝑘))²+3E^𝑡₀𝑆(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠−𝜏))𝑑𝑠²+3E^𝑡₀𝑆(𝑡−𝑠)𝜎(𝑠,𝑥(𝑠−𝜏))𝑑𝜔(𝑠)²=⁶_𝑖=1𝐽_𝑖. (13) It follows from (A2) that 𝐽₁=18E𝑆(𝑡)[𝜙(0)+𝑔(0,𝜙)]²≤18‖𝑆(𝑡)‖²×E𝜙(0)+(−𝐴)^−𝛼(−𝐴)^𝛼𝑔0,𝜙²≤18𝑀²𝑒^−2𝛾𝑡E𝜙+(−𝐴)^−𝛼𝐿_𝑔𝜙²≤18𝑀²𝑒^−𝛾𝑡1+(−𝐴)^−𝛼𝐿_𝑔²𝜙²_ℒ2. (14) By (A2), we can get 𝐽₂=18E𝑔(𝑡,𝑥(𝑡−𝜏))²=18E(−𝐴)^−𝛼(−𝐴)^𝛼𝑔(𝑡,𝑥(𝑡−𝜏))²≤18(−𝐴)^−𝛼2𝐿²_𝑔E‖𝑥(𝑡−𝜏)‖². (15) Employing Lemma 1, (A2), and Hölder inequality, we obtain 𝐽₃=18E^𝑡₀−𝐴𝑆(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠−𝜏))𝑑𝑠²≤18E^𝑡₀(−𝐴)^1−𝛼𝑆(×)𝑡−𝑠)(−𝐴^𝛼𝑔(𝑠,𝑥(𝑠−𝜏))𝑑𝑠²≤18𝑀²_1−𝛼𝐿²_𝑔E×^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}(𝑡−𝑠)^𝛼−1‖𝑥(𝑠−𝜏)‖𝑑𝑠²≤18𝑀²_1−𝛼𝐿²_𝑔^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}(𝑡−𝑠)^2(𝛼−1)×𝑑𝑠_𝑡0𝑒^{−𝛾(𝑡−𝑠)}E‖𝑥(𝑠−𝜏)‖²𝑑𝑠≤18𝑀²_1−𝛼𝐿²_𝑔𝛾^1−2𝛼Γ()×2𝛼−1_{𝑡−𝜏−𝜏}𝑒^𝛾𝜏𝑒^{−𝛾(𝑡−𝑠)}E‖𝑥(𝑠)‖²𝑑𝑠≤18𝑀²_1−𝛼𝐿²_𝑔𝛾^1−2𝛼Γ()×𝛾2𝛼−1₋₁𝑒^−𝛾𝑡𝑒^𝛾𝜏−1𝜙²_ℒ2+𝑒^𝛾𝜏₀^𝑡−𝜏𝑒^{−𝛾(𝑡−𝑠)}E‖𝑥(𝑠)‖²𝑑𝑠. (16) Combining Lemma 1 and (A3), with Hölder inequality, we can get 𝐽₄=6E_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)𝐼_𝑘(𝑥(𝑡_𝑘))²≤6E_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘²≤6𝑀²E_{0<𝑡𝑘<𝑡}𝑒^{−𝛾(𝑡−𝑡𝑘)}𝑞_𝑘𝑥𝑡_𝑘²≤6𝑀^2∞_𝑘=1𝑞_{𝑘0<𝑡𝑘<𝑡}𝑒^{−𝛾(𝑡−𝑡𝑘)}𝑞_𝑘E𝑥𝑡_𝑘²≤6𝑀^2∞_𝑘=1𝑞_𝑘²_{0<𝑡𝑘<𝑡}𝑒^{−𝛾(𝑡−𝑡𝑘)}E𝑥𝑡_𝑘². (17) From Lemma 1, (A1) and Hölder inequality, we obtain 𝐽₅=3E^𝑡₀𝑆(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠−𝜏))𝑑𝑠²≤3E^𝑡₀‖(𝑆(𝑡−𝑠)‖𝑓(𝑠,𝑥(𝑠−𝜏))𝑑𝑠²≤3𝑀²𝐿²_𝑓E^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}‖𝑥(𝑠−𝜏)‖𝑑𝑠²≤3𝑀²𝐿²_𝑓^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}×𝑑𝑠_𝑡0𝑒^{−𝛾(𝑡−𝑠)}E‖𝑥(𝑠−𝜏)‖²𝑑𝑠≤3𝑀²𝐿²_𝑓𝛾⁻¹𝛾⁻¹𝑒^−𝛾𝑡𝑒^𝛾𝜏−1𝜙²_ℒ2+𝑒^𝛾𝜏₀^𝑡−𝜏𝑒^{−𝛾(𝑡−𝑠)}E‖𝑥(𝑠)‖²𝑑𝑠. (18) Using (A1) and Burkholder-type inequality, we obtain 𝐽₆=3E^𝑡₀𝑆(𝑡−𝑠)𝜎(𝑠,𝑥(𝑠−𝜏))𝑑𝜔(𝑠)²≤3E^𝑡₀‖𝑆(𝑡−𝑠)‖²‖𝜎(𝑠,𝑥(𝑠−𝜏))‖²_ℒ02𝑑𝑠≤3𝑀²𝐿²_𝜎^𝑡₀𝑒^{−2𝛾(𝑡−𝑠)}E‖𝑥(𝑠−𝜏)‖²𝑑𝑠≤3𝑀²𝐿²_𝜎𝛾⁻¹𝑒^−𝛾𝑡𝑒^𝛾𝜏−1𝜙²_ℒ2+𝑒^𝛾𝜏₀^𝑡−𝜏𝑒^{−𝛾(𝑡−𝑠)}E‖𝑥(𝑠)‖²𝑑𝑠. (19)
Substituting (14)–(19) into (13), we haveE‖𝑥(𝑡)‖²𝑒^𝛾𝑡≤18𝑀²1+(−𝐴)^−𝛼𝐿_𝑔²𝜙²_ℒ2+18(−𝐴)^−𝛼2𝐿²_𝑔E_{{𝑡−𝜏<0}}‖𝑥(𝑡−𝜏)‖²𝑒^𝛾𝑡+18(−𝐴)^−𝛼2𝐿²_𝑔×E_{{𝑡−𝜏≥0}}‖𝑥(𝑡−𝜏)‖²𝑒^𝛾𝑡+18𝑀²_1−𝛼𝐿²_𝑔𝛾^1−2𝛼Γ()×𝛾2𝛼−1₋₁𝑒^𝛾𝜏−1𝜙²_ℒ2+𝑒^𝛾𝜏𝑡₀E‖𝑥(𝑠)‖²𝑒^𝛾𝑠𝑑𝑠+6𝑀^2∞_𝑘=1𝑞_𝑘²_{0<𝑡𝑘<𝑡}E𝑥𝑡_𝑘²𝑒^𝛾𝑡𝑘+3𝑀²𝐿²_𝑓𝛾⁻¹×𝛾⁻¹𝑒^𝛾𝜏−1𝜙²_ℒ2+𝑒^𝛾𝜏𝑡₀E‖𝑥(𝑠)‖²𝑒^𝛾𝑠𝑑𝑠+3𝑀²𝐿²_𝜎×𝛾⁻¹𝑒^𝛾𝜏−1𝜙²_ℒ2+𝑒^𝛾𝜏𝑡₀E‖𝑥(𝑠)‖²𝑒^𝛾𝑠𝑑𝑠≤18𝑀²1+(−𝐴)^−𝛼𝐿_𝑔²𝜙²_ℒ2+18(−𝐴)^−𝛼2𝐿²_𝑔𝑒^𝛾𝜏𝜙²_ℒ2+18(−𝐴)^−𝛼2𝐿²_𝑔𝑒^𝛾𝜏E‖𝑥(𝑡)‖²𝑒^𝛾𝑡+18𝑀²_1−𝛼𝐿²_𝑔𝛾^−2𝛼Γ(2𝛼−1)𝑒^𝛾𝜏−1𝜙²_ℒ2+18𝑀²_1−𝛼𝐿²_𝑔𝛾^1−2𝛼Γ(2𝛼−1)𝑒^𝛾𝜏×^𝑡₀E‖𝑥(𝑠)‖²𝑒^𝛾𝑠𝑑𝑠+6𝑀^2∞_𝑘=1𝑞_𝑘²_{0<𝑡𝑘<𝑡}E𝑥(𝑡_𝑘)²𝑒^𝛾𝑡𝑘+3𝑀²𝐿²_𝑓𝛾⁻²𝑒^𝛾𝜏−1𝜙²_ℒ2+3𝑀²𝐿²_𝑓𝛾⁻¹𝑒^𝛾𝜏𝑡₀E‖𝑥(𝑠)‖²𝑒^𝛾𝑠𝑑𝑠+3𝑀²𝐿²_𝜎𝛾⁻¹𝑒^𝛾𝜏−1𝜙²_ℒ2+3𝑀²𝐿²_𝜎𝑒^𝛾𝜏𝑡₀E‖𝑥(𝑠)‖²𝑒^𝛾𝑠𝑑𝑠. (20)From assumption (A5), we can get E‖𝑥(𝑡)‖²𝑒^𝜆𝑡≤𝐶𝜙²_ℒ2+𝑝^𝑡₀E‖𝑥(𝑠)‖²𝑒^𝛾𝑠𝑑𝑠+𝛽_{0<𝑡𝑘<𝑡}E𝑥(𝑡_𝑘)²𝑒^𝛾𝑡𝑘, (21) where 𝐶≜(1−18‖(−𝐴)^−𝛼‖²𝐿²_𝑔𝑒^𝛾𝜏)⁻¹[18𝑀²(1+‖(−𝐴)^−𝛼‖𝐿_𝑔)²+18‖(−𝐴)^−𝛼‖²𝐿²_𝑔𝑒^𝛾𝜏+18𝑀²_1−𝛼𝐿²_𝑔𝛾^−2𝛼Γ(2𝛼−1)(𝑒^𝛾𝜏−1)+3𝑀²𝐿²_𝑓𝛾⁻²(𝑒^𝛾𝜏−1)+3𝑀²𝐿²_𝜎𝛾⁻¹(𝑒^𝛾𝜏−1)], 𝑝≜(1−18‖(−𝐴)^−𝛼‖²𝐿²_𝑔𝑒^𝛾𝜏)⁻¹[18𝑀²_1−𝛼𝐿²_𝑔𝛾^1−2𝛼Γ(2𝛼−1)𝑒^𝛾𝜏+3𝑀²𝐿²_𝑓𝛾⁻¹𝑒^𝛾𝜏+3𝑀²𝐿²_𝜎𝑒^𝛾𝜏],and𝛽≜(1−18‖(−𝐴)^−𝛼‖²𝐿²_𝑔𝑒^𝛾𝜏)⁻¹6𝑀²(∑^∞_𝑘=1𝑞_𝑘)².
According to Gronwall-Bellmen’s inequality [17], we have E‖𝑥(𝑡)‖²≤𝐶𝜙²_{ℒ20<𝑡𝑘<𝑡}1+𝛽𝑒^{−(𝛾−𝑝)𝑡}. (22) On the other hand, by (A4), one has 1+𝛽≤_{0<𝑡𝑘<𝑡}1+𝛽^𝑡/𝜃=𝑒^{(𝑡/𝜃)ln(1+𝛽)}. (23) Thereby, (22) can be rewritten as E‖𝑥(𝑡)‖²≤𝐶𝜙²_{ℒ20<𝑡𝑘<𝑡}1+𝛽𝑒^{−(𝛾−𝑝−ln(1+𝛽)/𝜃)𝑡}, (24) 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 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.

If the impulsive effects, 𝐼_𝑘(⋅)≡0(𝑘=1,2,…), system (4) is turned into the following neutral stochastic partial functional differential equations: 𝑑𝑥(𝑡)((=(𝑡)(((((𝑡)𝑥+𝑔𝑡,𝑥𝑡−𝜏))𝐴𝑥+𝑓𝑡,𝑥𝑡−𝜏))𝑑𝑡+𝜎𝑡,𝑥𝑡−𝜏))𝑑𝜔,𝑡≥0,₀(𝑠)=𝜙(𝑠)∈𝐶^𝑏_ℱ0([−𝜏,0],𝐻),𝑠∈[−𝜏,0],a.s. (25)

Corollary 6. Assume that (A1)-(A2) hold, and the following condition 0<9𝑀²_1−𝛼𝐿²_𝑔𝛾^1−2𝛼Γ(2𝛼−1)+3𝑀²𝐿²_𝑓𝛾⁻¹+3𝑀²𝐿²_𝜎𝑒^𝛾𝜏1−9(−𝐴)^−𝛼2𝐿²_𝑔𝑒^𝛾𝜏<𝛾 (26) holds for 𝛼∈(1/2,1]. Then the mild solution of system (25) is exponentially stable in mean square.

If 𝑔(𝑡,⋅)≡0, then system (4) becomes stochastic partial functional differential equations with impulses as follows: (𝑡)=(𝑡)(((((𝑡)𝑑𝑥𝐴𝑥+𝑓𝑡,𝑥𝑡−𝜏))𝑑𝑡+𝜎𝑡,𝑥𝑡−𝜏))𝑑𝜔,𝑡≥0,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡_𝑘,𝑡=𝑡_𝑘𝑥,𝑘∈ℤ,₀(𝑠)=𝜙(𝑠)∈PC^𝑏_ℱ0([−𝜏,0],𝐻),𝑠∈[−𝜏,0],a.s. (27)

Corollary 7. Assume that (A1) and (A3) hold, and the following conditions(A4)inf_𝑘=1,2,…(𝑡_𝑘−𝑡_𝑘−1)=𝜃>0,(A5)(3𝑀²𝐿²_𝑓𝛾⁻¹+3𝑀²𝐿²_𝜎)𝑒^𝛾𝜏+(1/𝜃)ln(6𝑀²(∑^∞_𝑘=1𝑞_𝑘)²)<𝛾hold. Then the mild solution of system (27) is exponentially stable in mean square.

If 𝑔(𝑡,⋅)≡0 and 𝐼_𝑘(⋅)≡0, then system (4) becomes stochastic partial functional differential equations as follows: (𝑡)=(𝑡)(((((𝑡)𝑥𝑑𝑥𝐴𝑥+𝑓𝑡,𝑥𝑡−𝜏))𝑑𝑡+𝜎𝑡,𝑥𝑡−𝜏))𝑑𝜔,𝑡≥0,₀(𝑠)=𝜙(𝑠)∈𝐶^𝑏_ℱ0([−𝜏,0],𝐻),𝑠∈[−𝜏,0],a.s. (28)

Corollary 8. Assume that (A1) holds, and the following condition 3𝑀²𝐿²_𝑓𝛾⁻¹+3𝑀²𝐿²_𝜎𝑒^𝛾𝜏<𝛾 (29) 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 (3𝑀²𝐿²_𝑓𝛾⁻¹+3𝑀²𝐿²_𝜎)𝑒^𝛾𝜏<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: 𝑑𝑥(𝑡)+𝑎₁𝑥(=𝜕𝑡−1)₂𝜕𝑧²𝑥(𝑡)+𝑎₂𝑥(𝑡−1)𝑑𝑡+𝑎₃𝑥(𝑡−1)𝑑𝑤(𝑡),0≤𝑧≤𝜋,𝑡≥0,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘=𝑏₁𝑘²𝑥𝑡⁻_𝑘,𝑡=𝑡_𝑘𝑥,𝑘=1,2,…,₀(𝑠)=𝜙(𝑠)∈PC^𝑏_ℱ𝑡[−1,0],𝐿²[],𝑥()()0,𝜋𝑡,0=𝑥𝑡,𝜋=0,−1≤𝑠≤0, (30) where 𝑎_𝑖>0,𝑖=1,2,3,𝑏₁≤0 are constants and 𝑤(𝑡) denote the standard cylindrical Wiener process.

Let 𝐻=𝐿²[0,𝜋],𝐻₁=𝑊^2,2(0,𝜋)∩𝑊₀^1,2(0,𝜋). Define bounded linear operator 𝐴∶𝐻₁→𝐻 by 𝜕𝐴𝑥=²𝑥𝜕𝑧²∈𝐻,∀𝑥∈𝐻₁. (31)

Then we get 𝐴𝑥=^∞_𝑛=1𝑛²𝑥,𝑒_𝑛𝐻𝑒_𝑛,𝑥∈𝐻₁, (32) where 𝑒_𝑛(𝑧)=2/𝜋sin𝑛𝑧,𝑛=1,2,… is the set of eigenvector of −𝐴.

The bounded linear operator (−𝐴)^3/4 is given by (−𝐴)^3/4𝑥=^∞_𝑛=1𝑛^3/2𝑥,𝑒_𝑛𝐻𝑒_𝑛, (33) with domain 𝐷(−𝐴)^3/4=𝐻_3/4=𝑥∈𝐻,^∞_𝑛=1𝑛^3/2𝑥,𝑒_𝑛𝐻𝑒_𝑛∈𝐻. (34)

We can easily know that 𝐴 is the infinitesimal generator of an analytic semigroup 𝑆(𝑡),𝑡≥0, in 𝐻 and 𝑆(𝑡)𝑥=^∞_𝑛=1𝑒^−𝑛2𝑡𝑥,𝑒_𝑛𝐻𝑒_𝑛,𝑥∈𝐻. (35) Furthermore, we know that ‖𝑆(𝑡)‖≤𝑒^−𝜋2𝑡,𝑡≥0.

Let 𝑔(𝑡,𝑥(𝑡−𝜏))=𝑎₁𝑥(𝑡−1),𝑓(𝑡,𝑥(𝑡−𝜏))=𝑎₂𝑥(𝑡−1),𝜎(𝑡,𝑥(𝑡−𝜏))=𝑎₃𝑥(𝑡−1),and𝑡_𝑘+1=𝑡_𝑘+1.

Then we can get 𝑀=1,𝛾=𝜋²,𝜏=1,𝐿_𝑓=𝑎₂,𝐿_𝑔=𝑎₁(−𝐴)^3/4,𝐿_𝜎=𝑎₃𝑞,𝜃=1,_𝑘=𝑏₁𝑘²,(−𝐴)^3/4=1,(−𝐴)^−3/4≤1𝜋^3/2. (36) From Theorem 4, we know that the mild of system (30) is exponentially stable in mean square provided that 18𝑀²_1/4𝑎²₁𝜋^1/2+3𝑎²₂𝜋²+3𝑎²₃𝑒^𝜋2𝜋+ln⁴6<𝜋²−18𝑎²₁𝑒^𝜋2𝜋. (37)

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.

Acknowledgments

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 the Key Foundation of Chongqing Three Gorges University (no. 12ZD19).