A remark on observability of the wave equation with moving boundary

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.