Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions

Yanchao Zang,Junping Li

doi:10.1186/1687-2770-2013-193

Abstract

In this paper, the approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions and infinite delay in Hilbert spaces is studied. By using the Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory, some sufficient conditions are given for the approximate controllability of the system. At the end, an example is given to illustrate the application of our result. MSC:65C30, 93B05, 34K40, 34K45.

Highlights

The purpose of this paper is to prove the existence and approximate controllability of mild solutions for a class of fractional impulsive neutral stochastic differential equations with nonlocal conditions described in the form h(t, xt
To the best of our knowledge, it seems that little is known about approximate controllability of fractional impulsive neutral stochastic differential equations with infinite delay and nonlocal conditions
In order to study the approximate controllability for the fractional control system ( ), we introduce the following linear fractional differential system

Summary

Introduction

Takes values in the real separable Hilbert space H; A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup of a bounded linear operators T(t), t ≥ , in the Hilbert space H. Benchohra et al in [ ] considered the VIP for a particular class of fractional neutral functional differential equations with infinite delay. Mahmudov [ ] investigated the controllability of infinite dimensional linear stochastic systems, and in [ ] Dauer and Mahmudov extended the results to semilinear stochastic evolution equations with finite delay. To the best of our knowledge, it seems that little is known about approximate controllability of fractional impulsive neutral stochastic differential equations with infinite delay and nonlocal conditions. Let A be the infinitesimal generator of an analytic semigroup {T(t)}t≥ of uniformly bounded linear operators on H, and in this paper, we always assume that T(t) is compact.

Cv is defined by

Br H

MT α

Cv ds

We define the operator