On Approximate Controllability of Second-Order Neutral Partial Stochastic Functional Integrodifferential Inclusions with Infinite Delay and Impulsive Effects

Zuomao Yan

doi:10.1155/2015/925209

Abstract

We discuss the approximate controllability of second-order impulsive neutral partial stochastic functional integrodifferential inclusions with infinite delay under the assumptions that the corresponding linear system is approximately controllable. Using the fixed point strategy, stochastic analysis, and properties of the cosine family of bounded linear operators combined with approximation techniques, a new set of sufficient conditions for approximate controllability of the second-order impulsive partial stochastic integrodifferential systems are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result.

Highlights

Impulsive effects exist widely in many evolution processes in which states are changed abruptly at certain moments of time, involving fields such as physics, chemical technology, population dynamics, biotechnology, and economics; see [1,2,3,4] and the references therein
In addition to impulsive effects, stochastic effects likewise exist in real systems
A lot of dynamical systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, and other areas of science

Summary

Introduction

Impulsive effects exist widely in many evolution processes in which states are changed abruptly at certain moments of time, involving fields such as physics, chemical technology, population dynamics, biotechnology, and economics; see [1,2,3,4] and the references therein. The authors in [5,6,7] studied the existence of mild solutions for a class of abstract impulsive neutral stochastic functional differential and integrodifferential equations with infinite delay in Hilbert spaces. We consider the approximate controllability of the following second-order impulsive neutral partial stochastic functional integrodifferential inclusions with infinite delay in Hilbert spaces of the form d [x󸀠 (t) − g (t, xt, x󸀠 (t))] ∈ [Ax (t) + Bu (t)] dt t. To the best of the author’s knowledge, there are no results about the existence and approximate controllability of mild solutions for second-order impulsive second-order neutral partial stochastic functional integrodifferential inclusions with infinite delay, which is expressed in the form of (1).

Preliminaries

Existence of Solutions for Impulsive Stochastic Control System

Approximate Controllability of Impulsive Stochastic Control System

Example