Approximate controllability of semi-linear heat equation with Non-instantaneous impulses, memory and delay

TL;DR

This paper establishes the approximate controllability of a semilinear heat equation incorporating non-instantaneous impulses, memory, and delay, using a novel approach that bypasses fixed point theorems and demonstrates robustness of control.

Contribution

It introduces a new technique for controllability analysis that avoids fixed point theorems and applies to complex systems with impulses, memory, and delays.

Findings

Controllability is robust under impulses, memory, and delays.

A new method for control without fixed point theorems.

Open problems and a framework for future research.

Abstract

The semilinear heat equation with non-instantaneous impulses \textbf{(NII)}, memory, and delay is considered and its approximate controllability is obtained. This is done by employing a technique that avoids fixed point theorems and pulls back the control solution to a fixed curve in a short time interval. We demonstrate, once again, that the controllability of the system is robust under the influence of non-instantaneous impulses, memory, and delays. Finally, we present some open problems and a possible general framework to study the controllability of non-instantaneous impulses semilinear systems.