Existence Solution and Controllability of Sobolev Type Delay Nonlinear Fractional Integro-Differential System

Hamdy Ahmed,Ahmed Ghanem,Hassan El-Owaidy,Mahmoud El-Borai

doi:10.3390/math7010079

Abstract

Fractional integro-differential equations arise in the mathematical modeling of various physical phenomena like heat conduction in materials with memory, diffusion processes, etc. In this manuscript, we prove the existence of mild solution for Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. We establish the sufficient conditions for the approximate controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. In addition, we prove the exact null controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. Finally, an example is given to illustrate the obtained results.

Highlights

Fractional differential equations have been considered greatly by research community in various aspects due to its salient features for real world problems
We study the existence of the mild solution for Sobolev-type impulsive fractional differential equations and we discuss the sufficient conditions for approximate controllability and null controllability of the same problem
By using fractional calculus and fixed point theorems with the resolvent operator, we proved the existence of a mild solution for a Sobolev type nonlinear fractional delay integro-differential system with an impulsive condition

Summary

Introduction

Fractional differential equations have been considered greatly by research community in various aspects due to its salient features for real world problems (see [1,2,3,4,5,6,7]). Controllability problems for different kinds of dynamical systems have been studied by several authors (see [8,9,10,11,12,13,14,15]). Up to now, no work has been reported yet regarding the null controllability of Sobolev type nonlinear fractional delay integro-differential system with the impulsive condition of order 1 < q < 2. Motivated by these facts, we study the existence of the mild solution for Sobolev-type impulsive fractional differential equations and we discuss the sufficient conditions for approximate controllability and null controllability of the same problem

Preliminaries

Existence Solution

Approximate Controllability

Exact Null Controllability

Application

Conclusions