Approximate controllability of fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in Banach spaces

TL;DR

This paper establishes sufficient conditions for the approximate controllability of fractional order impulsive evolution equations with delays in Banach spaces, using resolvent operators and fixed point theorems.

Contribution

It provides new criteria for approximate controllability of fractional impulsive systems with delays, addressing gaps in existing literature and including detailed analysis of linear systems.

Findings

Derived sufficient conditions for approximate controllability

Analyzed the controllability of linear fractional systems

Provided an illustrative example demonstrating the results

Abstract

The present paper deals with the control problems governed by fractional non-instantaneous impulsive functional evolution equations with state-dependent delay involving Caputo fractional derivatives in Banach spaces. The main objective of this work is to formulate sufficient conditions for the approximate controllability of the considered system in separable reflexive Banach spaces. We have exploited the resolvent operator technique and Schauder's fixed point theorem in the proofs to achieve this goal. The approximate controllability of linear system is discussed in detail, which lacks in the existing literature. We also provide an example to illustrate the efficiency of the developed results. Moreover, we point out some shortcomings of the existing works in the context of characterization of mild solution and phase space, and approximate controllability of fractional order impulsive systems in Banach spaces.