Robust Exponential Stability of Mild Solutions to Impulsive Stochastic Neutral Integro-Differential Equations with Delay

In this paper, we study the robust properties of exponential stability of mild solutions for some classes of impulsive stochastic integro-differential perturbed system with function of finite delay time and supported this stability by some important result.


Introduction
The Impulsive systems arise naturally in various fields, such as mechanical systems and biological systems, economics, etc. see [8].Impulsive dynamical systems exhibit the continuous evolutions of the states typically described by ordinary differential equations coupled with instantaneous state jumps or impulses.And the presence of impulses implies that the trajectories of the system do not necessarily preserve the basic properties of the non-impulsive dynamical systems.To this end the theory of impulsive differential systems has emerged as an important area of investigation in applied sciences and impulsive stochastic integro-differential equations studied in [1].In the last few years many papers have Sameer Qasim Hasan and Amna Hassan Ibrahim been published about the stability of impulsive integro-differential systems [3].In particular, the exponential stability of mild solutions of various stochastic delay differential equations has been established [7].The conditions ensuring the exponential stability given by [6] and [5].The aim of this paper is to study the existence and exponential stability of a class of impulsive control stochastic integro-differential equations of mild solutions by using a new integral inequality. .Suppose that T(t) is an analytic semigroup with its infinitesimal generator A; for literature relating to semigroup theory, we suggest Pazy [2] .We suppose 0 p (A),the resolvent set of -A.For any α[0,1], it is possible to define the fractional power (-A) α which is a closed linear operator with its domain D(((-A) α )) .In this paper we consider the following impulsive stochastic neutral integro-differential equations with finite delay:

Preliminaries
In a real separable Hilbert space X, where u :

 
are continuous ; A is the infinitesimal generator of semigroup of bounded linear operators T(t), t  0 in X; B : UX is a linear bounded operator u() U Hilbert space of admissible control functions and; Ik: XX,.Furthermore, the fixed moments of time tk satisfy 0< t1 < ... < tm < limk→∞, tk =∞, x (t + k ) and x(t - k ) represent the right and left limits of x(t) at t = tk, respectively.Also, ∆x (tk) = x (t + k )x(t - k ), represents the jump in the state x at time tk with Ik determining the size of the jump.Let τ > 0 and C = C([-τ, 0];X) denote the family of all right continuous functions with lefthand limits η from [-τ, 0] to X .The space C is assumed to be equipped with the norm|| η || C = sup θ∈[-τ 0] |η(θ)|.Here C b F0 ([-τ, 0], X) is the family of all almost surely bounded, F0-measurable, continuous random variables from [-τ, 0] to X.

t T t z x p z t x t p t A Bw T t s z t x t p t ds t t t T t s F t x ds T t s G t y dw s T t s x s s dw s
Lemma2.1:If (H3) holds, then for any β ∈ (0, 1]:

Stability of mild solutions [5] [6] [7]
In this section, to establish sufficient conditions ensuring the exponential stability in p moment (p ≥ 2) for a mild solution to Eq. (1), we firstly establish a new integral inequality to overcome the difficulty when the neutral term and impulsive effects are present.

(YF
the usual conditions.Moreover, let X,Y be two real separable Hilbert space and let L(Y.X) denote the space of all bounded linear operators from Y to X and let L(Y.X) denoted the space of all bounded operators from Y to X .For simplicity, we use the notation |.| to denote the norm in X,Y and |||| to denote the operator norm in L(X.X) and L(Y.X).Let of X,Y, respectively .Let {ŵ (t): t≥0} denote a Y-valued Wiener process defined on the probability space 0 for all x, y  Y, where Q is positive , self -adjoin a, trace class operator on Y.In particular, we dente by w(t) Y-valued Q-Wiener process with respect to 0 {} t F t  .We assume that there exists a complete orthonormal system {ei } in Y, a bounded sequence of nonnegative real numbers i is the ϭ-algebra generated by{w(t):0≤s≤t}.Let 0 1\2 L = L (Q Y;X) 22 be the space of all Hilbert-Schmidt operators from 1\2