Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces

Mourad Kerboua,Amar Debbouche,Dumitru Baleanu

doi:10.14232/ejqtde.2014.1.58

Abstract

We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlin- ear differential equations of Sobolev type in Hilbert spaces. We use Holder's inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted di- rectly from deterministic control problems for the main results. A new set of sufficient conditions is formulated and proved for the fractional stochastic control system to be approximately controllable. An example is given to illustrate the abstract results.

Highlights

We are concerned with the following fractional stochastic nonlocal system of Sobolev type
In the present literature there is only a limited number of papers that deal with the approximate controllability of fractional stochastic systems [27], as well as with the existence and controllability results of fractional evolution equations of Sobolev type [26]
In [24], the authors proved the approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces

Summary

Introduction

We are concerned with the following fractional stochastic nonlocal system of Sobolev type. In the present literature there is only a limited number of papers that deal with the approximate controllability of fractional stochastic systems [27], as well as with the existence and controllability results of fractional evolution equations of Sobolev type [26]. R. Sakthivel et al [37] studied the approximate controllability of a class of dynamic control systems described by nonlinear fractional stochastic differential equations in Hilbert spaces. In [24], the authors proved the approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces. We present a new concept in stochastic analysis that we present a nonlocal condition given in stochastic term together with Riemann–Liouville fractional derivative, we use this tool to establish the approximate controllability of Sobolev type fractional deterministic nonlocal stochastic control systems in Hilbert spaces.

Preliminaries

Approximate controllability

Example