Qualitative properties for the 1-D Impulsive Wave Equation: Controllability and Observability

TL;DR

This paper investigates the impulsive wave equation in one dimension, establishing existence of solutions and analyzing controllability and observability properties when controls act impulsively on subdomains at specific times.

Contribution

It provides new results on the existence, impulse approximate controllability, and observability of the 1-D impulsive wave equation, advancing understanding of control properties for impulsive PDEs.

Findings

Existence of solutions for the impulsive wave equation.

Impulse approximate controllability when controls act on subdomains.

Impulse observability results for the wave equation.

Abstract

In this paper, we establish some important results for the impulsive wave equation. We begin by proving the existence of a solution. Then, we study the impulse approximate controllability where the control function acts on a subdomain $\omega$ and at one instant of time $\tau \in (0, T)$. Afterward, we study impulse observability