Study on Control Problem of a Impulsive Neutral Integro-Differential Equations with Fading Memory

TL;DR

This paper investigates the approximate controllability of impulsive neutral integro-differential equations with fading memory in Banach spaces, establishing theoretical results and illustrating them with an example.

Contribution

It introduces a method to analyze controllability of complex integro-differential equations with impulses and memory, extending existing theories to semilinear systems.

Findings

Established approximate controllability for linear systems.

Proved existence of mild solutions for semilinear systems.

Provided a detailed example illustrating the theoretical results.

Abstract

This article addresses control problems for semilinear impulsive neutral integro-differential equations with memory in a Banach space. It investigates the approximate controllability of linear and semilinear systems and proves the establishment of mild solutions in the semilinear setting. The approach involves constructing a resolvent family for the corresponding integro-differential equation of linear type without memory. The results for the linear system are established first, then extended to the semilinear scenario, followed by a detailed example to illustrate the theoretical findings.