Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces

TL;DR

This paper establishes the existence and approximate controllability of non-autonomous impulsive functional evolution inclusions in Banach spaces, addressing gaps in previous literature and providing concrete examples.

Contribution

It introduces new sufficient conditions for approximate controllability of impulsive evolution systems in Banach spaces, correcting prior misunderstandings and extending existing theories.

Findings

Proved existence of mild solutions using evolution families and fixed point theorem.

Derived optimal control in feedback form for linear-quadratic regulator problem.

Established approximate controllability under new sufficient conditions.

Abstract

In this paper, we are concerned with the approximate controllability results for a class of impulsive functional differential control systems involving time dependent operators in Banach spaces. First, we show the existence of a mild solution for non-autonomous functional impulsive evolution inclusions in separable reflexive Banach spaces with the help of the evolution family and a generalization of the Leray-Schauder fixed point theorem for multi-valued maps. In order to establish sufficient conditions for the approximate controllability of our problem, we first consider a linear-quadratic regulator problem and obtain the optimal control in the feedback form, which contains the resolvent operator consisting of duality mapping. With the help of this optimal control, we prove the approximate controllability of the linear system and hence derive sufficient conditions for the approximate controllability of our problem. Moreover, in this paper, we rectify several shortcomings of the related works available in the literature, namely, proper identification of resolvent operator in Banach spaces, characterization of phase space in the presence of impulsive effects and lack of compactness of the operator $h(\cdot)\mapsto \int_{0}^{\cdot}\mathrm{U}(\cdot,s)h(s)\mathrm{d} s : \mathrm{L}^{1}([0,T];\mathbb{Y}) \rightarrow \mathrm{C}([0,T];\mathbb{Y}),$ where $\mathbb{Y}$ is a Banach space and $\mathrm{U}(\cdot,\cdot)$ is the evolution family, etc. Finally, we provide a concrete example to illustrate the efficiency of our results.