Controllability for Retarded Semilinear Neutral Control Systems of Fractional Order in Hilbert Spaces

Daewook Kim,Jin-Mun Jeong

doi:10.3390/math9060671

Abstract

In this paper, we discuss the approximate controllability for a class of retarded semilinear neutral control systems of fractional order by investigating the relations between the reachable set of the semilinear retarded neutral system of fractional order and that of its corresponding linear system. The research direction used here is to find the conditions for nonlinear terms so that controllability is maintained even in perturbations. Finally, we will show a simple example to which the main result can be applied.

Highlights

Let H and V be two complex Hilbert spaces so that V is a dense subspace of H
We study the approximate controllability of the following semilinear retarded neutral functional differential control system of fractional order: (
By Theorems 1 and 2, we get that the approximate controllability of the general retarded linear differential system corresponding to (44) with g ≡ 0 and F ≡ 0 is equivalent to the condition for the H-approximate controllability of the semilinear system (44) for any α > 12

Summary

Introduction

Let H and V be two complex Hilbert spaces so that V is a dense subspace of H. We study the approximate controllability of the following semilinear retarded neutral functional differential control system of fractional order:. The controller B is a bounded linear operator from U to H, where U is a Banach space of control variables. The result assert the equivalence condition between the controllability for the retarded neutral control system of fractional order and one for the associated the linear system excluded the nonlinear term. The paper is organized as follows—in Section 2, we deal with the regularity and structure for solutions of semilinear fractional order retarded neutral functional differential equations and introduce basic properties. We will show a simple example to which the main result can be applied

Retarded Linear Equations

Semilinear Fractional Order Differential Equations

Approximate Reachable Sets

Example

Conclusions