Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation

¹ Laboratory of Mathematics Informatics and Systems (LAMIS), University of Larbi Tebessi, 12002 Tebessa, Algeria.
² Departamento de Matemática e Informática, ETSI Caminos, Canales y Puertos, Polytecnical University of Madrid, 28040 Madrid, Spain.

^* Corresponding author: carlos.castro@upm.es

Received: 14 October 2020
Accepted: 31 May 2021

Abstract

We propose a numerical method to approximate the exact averaged boundary control of a family of wave equations depending on an unknown parameter σ. More precisely the control, independent of σ, that drives an initial data to a family of final states at time t = T, whose average in σ is given. The idea is to project the control problem in the finite dimensional space generated by the first N eigenfunctions of the Laplace operator. When applied to a single (nonparametric) wave equation, the resulting discrete control problem turns out to be equivalent to the Galerkin approximation proposed by F. Bourquin et al. [C.R. Acad. Sci. Paris 313 I (1991) 757–760]. We give a convergence result of the discrete controls to the continuous one. The method is illustrated with several examples in 1-d and 2-d in a square domain and allows us to give some conjectures on the averaged controllability for the continuous problem.