A numerical method for exact boundary controllability problems for the wave equation

The computational approximation of exact boundary controllability problems for the wave equation in two dimensions is studied. A numerical method is defined that is based on the direct solution of optimization problems that are introduced in order to determine unique solutions of the controllability problem. The uniqueness of the discrete finite-difference solutions obtained in this manner is demonstrated. The convergence properties of the method are illustrated through computational experiments. Efficient implementation strategies for the method are also discussed. It is shown that for smooth, minimum L2-norm Dirichlet controls, the method results in convergent approximations without the need to introduce regularization. Furthermore, for the generic case of nonsmooth Dirichlet controls, convergence with respect to L2 norms is also numerically demonstrated. One of the strengths of the method is the flexibility it allows for treating other controls and other minimization criteria; such generalizations are discussed. In particular, the minimum H1-norm Dirichlet controllability problem is approximated and solved, as are minimum regularized L2-norm Dirichlet controllability problems with small penalty constants. Finally, a discussion is provided about the differences between our method and existing methods; these differences may explain why our methods provide convergent approximations for problems for which existing methods produce divergent approximations unless they are regularized in some manner.