Approximate controllability of non-autonomous second order impulsive functional evolution equations in Banach spaces

TL;DR

This paper studies the approximate controllability of complex second order non-autonomous impulsive evolution equations in Banach spaces, providing new theoretical conditions and applying them to wave equations with delays.

Contribution

It introduces novel sufficient conditions for approximate controllability of non-autonomous impulsive systems in Banach spaces, including the use of resolvent operators and fixed point theorems.

Findings

Established controllability conditions for second order impulsive systems.

Applied theoretical results to wave equations with delays.

Extended existing literature on non-autonomous evolution equations.

Abstract

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder's fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we verify our results to examine the approximate controllability of the non-autonomous wave equation with non-instantaneous impulses and finite delay in the application section.