Uniform observability of the one-dimensional wave equation for non-cylindrical domains. Application to the control's support optimization

TL;DR

This paper establishes uniform observability for the 1D wave equation in non-cylindrical domains, enabling control support optimization, with theoretical proofs and numerical experiments demonstrating the influence of initial conditions.

Contribution

It proves uniform observability in a class of non-cylindrical domains and addresses the optimization of control support, extending previous controllability results.

Findings

Uniform observability holds in a class of non-cylindrical domains.

Optimal control support depends on initial conditions.

Numerical experiments illustrate the theoretical results.

Abstract

This work is concerned with the distributed controllability of the one-dimensional wave equation over non-cylindrical domains. The controllability in that case has been obtained in [Castro-Cindea-Munch, Controllability of the linear one-dimensional wave equation with inner moving forces, SIAM J. Control Optim 2014] for domains satisfying the usual geometric optics condition. In the present work, we first show that the corresponding observability property holds true uniformly in a precise class of non-cylindrical domains. Within this class, we then consider, for a given initial datum, the problem of the optimization of the control support and prove its well-posedness. Numerical experiments are then discussed and highlight the influence of the initial condition on the optimal domain.