NUMERICAL SHOOTING METHODS FOR OPTIMAL BOUNDARY CONTROL AND EXACT BOUNDARY CONTROL OF 1-D WAVE EQUATIONS

Abstract. Numerical solutions of optimal Dirichlet boundary control problems for linear and semilinear wave equations are studied. The optimal control problem is reformulated as a system of equations (an optimality system) that consists of an initial value problem for the underlying (linear or semilinear) wave equation and a terminal value problem for the adjoint wave equation. The discretized optimality system is solved by a shooting method. The convergence properties of the numerical shooting method in the context of exact controllability are illustrated through computational experiments. In particular, in the case of the linear wave equation, convergent approximations are obtained for both smooth minimum L2-norm Dirichlet control and generic, non-smooth minimum L2-norm Dirichlet controls. Key words. Controllability, optimal control, wave equation, shooting method, finite difference method