Approximate Controllability of Linear Fractional Impulsive Evolution Equations in Hilbert Spaces

TL;DR

This paper studies the approximate controllability of linear fractional impulsive evolution equations in Hilbert spaces, providing a resolvent-based criterion and illustrating its application with an example.

Contribution

It introduces a necessary and sufficient condition for approximate controllability of fractional impulsive systems using impulsive resolvent operators, extending classical controllability criteria.

Findings

Derived an explicit mild solution representation combining fractional and impulsive operators

Established a resolvent convergence criterion for approximate controllability

Provided a concrete example demonstrating the theoretical results

Abstract

This paper investigates the approximate controllability of linear fractional impulsive evolution equations in Hilbert spaces. The system under consideration involves the Caputo fractional derivative of order $0<\alpha\leq 1$, a closed linear operator generating a strongly continuous semigroup, and instantaneous state jumps governed by bounded linear impulse operators. We first derive an explicit representation of the mild solution by combining fractional solution operators with impulsive operators. Using this representation, we characterize the approximate controllability of the system through a necessary and sufficient condition expressed in terms of the convergence of an associated family of impulsive resolvent operators. This resolvent condition extends the classical criterion for approximate controllability to the fractional impulsive setting. To illustrate the applicability of our theoretical results, a concrete example is provided. The analysis presented here bridges the gap between the well-established theory for integer-order impulsive systems and the more complex fractional case, highlighting the distinct challenges and solutions arising from the interplay of fractional dynamics and impulsive effects.