Alternating and variable controls for the wave equation^*

¹ Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma, P.le A. Moro 5, 00185 Roma, Italy.
² Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, via A. Scarpa 16, 00161 Roma, Italy.

^** Corresponding author: agresti@mat.uniroma1.it

Received: 15 January 2018
Accepted: 23 March 2019

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.