New Controllability Results of Fractional Nonlocal Semilinear Evolution Systems with Finite Delay

Daliang Zhao,Juan Mao

doi:10.1155/2020/7652648

Abstract

In the present paper, sufficient conditions ensuring the complete controllability for a class of semilinear fractional nonlocal evolution systems with finite delay in Banach spaces are derived. The new results are obtained under a weaker definition of complete controllability we introduced, and then the Lipschitz continuity and other growth conditions for the nonlinearity and nonlocal item are not required in comparison with the existing literatures. In addition, an appropriate complete space and a corresponding time delay item are introduced to conquer the difficulties caused by time delay. Our main tools are properties of resolvent operators, theory of measure of noncompactness, and Mönch fixed point theorem.

Highlights

In recent decades, various research studies on fractional differential systems are growing vigorously, which are due to the enormous scope and extensive applications of fractional calculus theory in mathematics, biology, physics, economics, and engineering science [1,2,3,4,5,6,7,8,9].As we all know, time delay effects exist widely in various fields such as communication security, weather predicting, and population dynamics. e difficulties in the study of these fields lie in the time lag effect for the systems caused by the delay
E proposed fractional evolution system (1) here, which generalizes the case of integral order differential equations studied in [41] about the complete controllability, has more extensive and valid applications in contrast to the abovementioned literatures [11, 21,22,23,24,25] as follows
In order to conquer the inconveniences caused by time delay in the study of complete controllability, for each u ∈ C(I, X), t ∈ I, and the function φ(t) in (1), we draw into the function ut defined by u(t + θ), t + θ ≥ 0, ut(θ) 􏼨 φ(t + θ), t + θ ≤ 0, (2)

Summary

Introduction

Various research studies on fractional differential systems are growing vigorously, which are due to the enormous scope and extensive applications of fractional calculus theory in mathematics, biology, physics, economics, and engineering science [1,2,3,4,5,6,7,8,9]. There is scarcely any results on complete controllability of fractional nonlocal evolution equations with delay in Banach spaces, except [21, 23, 24]. E current paper, inspired by the aforementioned analyses, is to address the complete controllability of the following fractional nonlocal semilinear evolution equations with finite delay in Banach spaces: Dqu(t) Au(t) + f 􏼨. E proposed fractional evolution system (1) here, which generalizes the case of integral (first) order differential equations studied in [41] about the complete controllability, has more extensive and valid applications in contrast to the abovementioned literatures [11, 21,22,23,24,25] as follows.

Preliminaries

Main Results

An Example

Conclusions