Alternating and variable controls for the wave equation

TL;DR

This paper investigates the exact observability of the wave equation with boundary controls that vary over time, providing conditions in one and multiple dimensions, and exploring specific control exchange scenarios.

Contribution

It introduces new conditions for exact observability with variable boundary controls, using Fourier series in 1D and multiplier methods in higher dimensions.

Findings

Exact observability conditions depend on control timing and location.

Single exchange and constant rate exchange scenarios are analyzed.

Connections between 1D and multi-dimensional cases are discussed.

Abstract

The present article discusses the exact observability of the wave equation when the observation subset of the boundary is variable in time. In the one-dimensional case, we prove an equivalent condition for the exact observability, which takes into account only the location in time of the observation. To this end we use Fourier series. Then we investigate the two specific cases of single exchange of the control position, and of exchange at a constant rate. In the multi-dimensional case, we analyse sufficient conditions for the exact observability relying on the multiplier method. In the last section, the multi-dimensional results are applied to specific settings and some connections between the one and multi-dimensional case are discussed; furthermore some open problems are presented.