A minimum effort optimal control problem for the wave equation

A minimum effort optimal control problem for the undamped wave equation is considered which involves L ^∞‐control costs. Since the problem is non‐differentiable a regularized problem is introduced. Uniqueness of the solution of the regularized problem is proven and the convergence of the regularized solutions is analyzed. Further, a semi‐smooth Newton method is formulated to solve the regularized problems and its superlinear convergence is shown. Thereby special attention has to be paid to the well‐posedness of the Newton iteration. Numerical examples confirm the theoretical results.