Approximate controllability of non-instantaneous impulsive fractional evolution equations of order $1<\alpha<2$ with state-dependent delay in Banach spaces

TL;DR

This paper investigates the approximate controllability of non-instantaneous impulsive fractional evolution equations with state-dependent delay in Banach spaces, providing conditions for both linear and nonlinear cases and validating with an example.

Contribution

It introduces new sufficient conditions for approximate controllability of fractional evolution equations with impulses and delays, including both linear and nonlinear systems.

Findings

Established controllability conditions for linear fractional systems.

Extended results to nonlinear fractional systems.

Validated results with a concrete example.

Abstract

The current article examines the approximate controllability problem for non-instantaneous impulsive fractional evolution equations of order $1<\alpha<2$ with state-dependent delay in separable reflexive Banach spaces. In order to establish sufficient conditions for the approximate controllability of our problem, we first formulate the linear-regulator problem and obtain the optimal control in feedback form. By using this optimal control, we deduce the approximate controllability of the linear fractional control system of order $1<\alpha<2$. Further, we derive sufficient conditions for the approximate controllability of the nonlinear problem. Finally, we provide a concrete example to validate the efficiency of the derived results.