Bilinear optimal control for a wave equation

We consider the problem of optimal control of a wave equation. A bilinear control is used to bring the state solutions close to a desired profile under a quadratic cost of control. We establish the existence of solutions of the underlying initial boundary-value problem and of an optimal control that minimizes the cost functional. We derive an optimality system by formally differentiating the cost functional with respect to the control and evaluating the result at an optimal control. We establish existence and uniqueness of the solution of the optimality system and thus determine the unique optimal control in terms of the solution of the optimality system.