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. 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.