Approximate controllability of impulsive fractional stochastic differential equations with state-dependent delay

Xiaozhi Zhang,Chuanxi Zhu,Chenggui Yuan

doi:10.1186/s13662-015-0412-z

Xiaozhi Zhang, Chuanxi Zhu + Show 1 more

Open Access

https://doi.org/10.1186/s13662-015-0412-z

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Abstract

This work is concerned with the approximate controllability of nonlinear fractional impulsive stochastic differential system under the assumption that the corresponding linear system is approximately controllable. Using fractional calculus, stochastic analysis, and the technique of stochastic control theory, a new set of sufficient conditions for the approximate controllability of a fractional impulsive stochastic differential system is obtained. The results in this paper are generalizations and continuations of the recent results on this issue. An example is given to illustrate the efficiency of the main results.

Highlights

1 Introduction In the last few decades, fractional differential systems have provided an excellent tool in electrochemistry, physics, porous media, control theory, engineering etc., due to the descriptions of memory and hereditary properties of various materials and processes
Stochastic control theory is a stochastic generalization of classical control theory [, ]
The biggest difficulty is the analysis of a stochastic control system and stochastic calculations induced by the stochastic process

Summary

Introduction

In the last few decades, fractional differential systems have provided an excellent tool in electrochemistry, physics, porous media, control theory, engineering etc., due to the descriptions of memory and hereditary properties of various materials and processes. By using the stochastic analysis technique and the methods directly from deterministic control problems, Sakthivel et al [ ] considered the approximate controllability of fractional stochastic evolution equations. Ahmed [ ] considered the approximate controllability of impulsive neutral stochastic differential equations with finite delay and fractional Brownian motion in a Hilbert space, and a new set of sufficient conditions for approximate controllability was formulated and proved. The stochastic functional differential equations with state-dependent delay have many important applications in mathematical models of real phenomena, and the study of this type of equations has received much attention in recent years. Sakthivel and Ren [ ] studied the approximate controllability of fractional differential equations with state-dependent delay. An example is given to illustrate the effectiveness of the main results

Preliminaries

Main results