A remark on observability of the wave equation with moving boundary

TL;DR

This paper investigates the observability of the wave equation with a periodically moving boundary, focusing on boundary velocity measurements and employing a reduction theorem to establish results.

Contribution

It introduces a new analysis of wave equations with moving boundaries, utilizing a reduction theorem to address observability with boundary velocity measurements.

Findings

Established conditions for observability with moving boundary

Demonstrated effectiveness of reduction theorem in this context

Provided insights into wave behavior with periodic boundary motion

Abstract

We deal with the wave equation with assigned moving boundary ($0<x<a(t)$) upon which Dirichlet or mixed boundary conditions are specified, here $a(t)$ is assumed to move slower than the light and periodically. Moreover $a$ is continuous, piecewise linear with two independent parameters. Our major concern will be an observation problem which is based measuring, at each $t>0$ of the transverse velocity at $a(t)$. The key to the results is the use of a reduction theorem.