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

TL;DR

This paper introduces a numerical approach to approximate averaged boundary controls for wave equations with unknown velocities, ensuring convergence from discrete to continuous solutions in finite-dimensional spaces.

Contribution

It develops a finite-dimensional projection method for controlling wave equations with unknown parameters, providing convergence guarantees and practical illustrations.

Findings

The discrete control solutions converge to the continuous control.

The method successfully controls wave equations in 1D and 2D domains.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

We propose a numerical method to approximate the exact averaged boundary control of a family of wave equations depending on an unknown parameter sigma. More precisely the control, independent of sigma, that drives an initial data to a family of final states at time t = T, whose average in sigma 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. The resulting discrete control problem has solution whenever the continuous one has it, and 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.