Optimization of non-cylindrical domains for the exact null controllability of the 1D wave equation

Université Clermont Auvergne, Laboratoire de Mathématiques Blaise Pascal CNRS-UMR 6620, Campus des Cézeaux, 63178 Aubière cedex, France.

^* Corresponding author: Arnaud.Munch@uca.fr

Received: 16 July 2020
Accepted: 19 January 2021

Abstract

This work is concerned with the null controllability of the one-dimensional wave equation over non-cylindrical distributed domains. The controllability in that case has been obtained by Castro et al. [SIAM J. Control Optim. 52 (2014)] for domains satisfying the usual geometric optic condition. We analyze the problem of optimizing the non-cylindrical support q of the control of minimal L²(q)-norm. In this respect, we prove a uniform observability inequality for a class of domains q satisfying the geometric optic condition. The proof based on the d’Alembert formula relies on arguments from graph theory. Numerical experiments are discussed and highlight the influence of the initial condition on the optimal domains.