Optimal control of the undamped linear wave equation with measure valued controls

Measure valued optimal control problems governed by the linear wave equation are analyzed. The space of vector measures M(Ω_c, L²(I)) is chosen as control space and the corresponding total variation norm as the control cost functional. The support of the controls (sparsity pattern) is time-independent, which is desired in many applications, e.g., inverse problems or optimal actuator placement. New regularity results for the linear wave equation are proven and used to show the well-posedness of the control problem in all three space dimensions. Furthermore first order optimality conditions are derived and structural properties of the optimal control are investigated. Higher regularity of optimal controls in time is shown on the basis of the regularity results for the state. Finally the optimal control problem is used to solve an inverse source problem.