Optimal Control of Wave Energy Dissipation via a Mobile Damping Actuator

This work studies an optimal control problem for a wave equation with a moving localized damping, modeled by the characteristic function of a ball whose center evolves according to a controlled second-order dynamical system. The objective is to minimize the H1-norm of the time derivative of the wave at a final time, while penalizing the control effort through a quadratic cost on the acceleration. We first establish the local well-posedness of the coupled system, consisting of a damped wave equation and an ordinary differential equation for the damping center. Under suitable compactness and regularity assumptions on the admissible controls, we prove the existence of at least one optimal control via the direct method of the calculus of variations. We then derive first-order necessary optimality conditions through an extended Lagrangian formalism, yielding a coupled system involving the primal state, two adjoint states, and a pointwise relation linking the optimal control to the adjoint variable. The gradient of the cost functional is computed explicitly, showing that the sensitivity is concentrated on the boundary of the moving damping zone. Finally, a numerical implementation based on a gradient descent algorithm is proposed, using a Newmark scheme for the wave equation and a Verlet scheme for the trajectory, with regularization of the nonsmooth indicator function. The resulting algorithm provides a systematic approach for computing optimal damping trajectories in applications such as vibration suppression and noise control.