Optimal Design of the Support for the Control for the 2-D Wave Equation: A Numerical Method

Abstract

We consider in this paper the homogeneous 2-D wave equation defined on $\Omega \subset \mathbb{R}^2$. Using the Hilbert Uniqueness Method, one may associate to a suitable fixed subset $\omega \subset \Omega$, the control $v_{\omega}$ of minimal $L^2 (\omega \times (0, T))$-norm which drives to rest the system at a time $T>0$ large enough. We address the question of the optimal position of $\omega$ which minimize the functional $J : \omega \rightarrow ||v_{\omega}||_{L^2(\omega \times (0,T))}$. Assuming $\omega \in C^1(\Omega)$, we express the shape derivative of $J$ as a curvilinear integral on $∂\omega \times (0,T)$ independently of any adjoint solution. This expression leads to a descent direction and permits to define a gradient algorithm efficiently initialized by the topological derivative associated to $J$. The numerical approximation of the problem is discussed and numerical experiments are presented in the framework of the level set approach. We also investigate the well-posedness of the problem by considering its relaxation.