A Study on Controllability of a Class of Impulsive Fractional Nonlinear Evolution Equations with Delay in Banach Spaces

Daliang Zhao

doi:10.3390/fractalfract5040279

Abstract

Under a new generalized definition of exact controllability we introduced and with a appropriately constructed time delay term in a special complete space to overcome the delay-induced-difficulty, we establish the sufficient conditions of the exact controllability for a class of impulsive fractional nonlinear evolution equations with delay by using the resolvent operator theory and the theory of nonlinear functional analysis. Nonlinearity in the system is only supposed to be continuous rather than Lipschitz continuous by contrast. The results obtained in the present work are generalizations and continuations of the recent results on this issue. Further, an example is presented to show the effectiveness of the new results.

Highlights

This paper’s primary objective is to investigate the exact controllability of the following impulsive fractional nonlinear evolution equations with delay in Banach spaces: γD x (t) = Ax (t) + f (t, x (t), xt ) + Bu(t), a.e. t ∈ I := [0, a], (1)∆x = x − x = Ii ( x), i = 1, 2, · · ·, m,x (t) = φ(t), t ∈ [−b, 0], where D γ represents the Caputo derivative of order γ ∈ (0, 1)
Inspired by the abovementioned papers and the ideas adopted in [13], in this work, we present a new depiction of the exact controllability of the system (1) by using the theory of resolvent operator and the theory of nonlinear functional analysis
C0 -semigroup based on probability density function [24] is replaced by resolvent operators without compact conditions, which is different from most of the existing literatures such as [7,10,11,12,17,21,22,25,26]. (iii) With the properly defined delay item in a corresponding complete space we introduced, we have solved the delay-induced-difficulty during the investigation of exact controllability by measures of noncompactness

Summary

Introduction

This paper’s primary objective is to investigate the exact controllability of the following impulsive fractional nonlinear evolution equations with delay in Banach spaces: γ. Some excellent results of exact controllability for various fractional differential equations have been established recently [7,10,11,12,13,14,15,16,17,18,19,20,21,22,23], but the limitation is that the functions in the systems are either Lipschitz continuous, compact or satisfy some special growth suppositions. We point out that nonlinearities and impulsive items in these papers satisfy special growth assumptions [21], Lipschitz condition [12,22], and semigroups together with the resolvent operators of some systems possess compactness, which still show the limitation to a certain extent in practical problems It seems interesting whether the exact controllability of the impulsive fractional evolution equations with delay can be established via noncompact resolvent operators together with the nonlinearity satisfying continuity rather than Lipschitz continuity. An example is worked out in the last section to illustrate our theory of the main results

Preliminaries

Main Results

Examples

Conclusions