Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation
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. 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. in reference [2]. 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.