Finite-approximate controllability of fractional stochastic evolution equations with nonlocal conditions

Yongxiang Li,Yonghong Ding

doi:10.1186/s13660-020-02354-4

Abstract

This paper deals with the finite-approximate controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. We establish sufficient conditions for the finite-approximate controllability of the control system when the compactness conditions or Lipschitz conditions for the nonlocal term and uniform boundedness conditions for the nonlinear term are not required. The discussion is based on the fixed point theorem, approximation techniques and diagonal argument. In the end, an example is presented to illustrate the abstract theory. Our result improves and extends some relevant results in this area.

Highlights

We shall be concerned with the finite-approximate controllability for following fractional evolution equations with nonlocal initial conditions of the form
There are few works that have reported about the study of controllability problems for stochastic evolution equations with nonlocal conditions, see [30, 31], and the authors suppose that the nonlocal item g is a completely continuous map in these papers
As a special case, the methods used in the present paper can be applied to study the finite-approximate controllability of deterministic systems with nonlocal conditions by suitably introducing the abstract space and norm

Summary

Introduction

We shall be concerned with the finite-approximate controllability (see Definition 2.8 in Sect. 2) for following fractional evolution equations with nonlocal initial conditions of the form. In order to establish the main results, the assumption of compactness or Lipschitz condition on the nonlocal term plays an important role in these articles This restriction is too strong and is not usually satisfied in practical applications. There are few works that have reported about the study of controllability problems for stochastic evolution equations with nonlocal conditions, see [30, 31], and the authors suppose that the nonlocal item g is a completely continuous map in these papers. As a special case, the methods used in the present paper can be applied to study the finite-approximate controllability of deterministic systems with nonlocal conditions by suitably introducing the abstract space and norm. 3, we state and prove finite-approximate controllability results for fractional stochastic evolution equation with nonlocal conditions.

Preliminaries

L02 ds

Application