Optimal location of the support of the control for the 1-D wave equation: numerical investigations

We consider in this paper the homogeneous 1-D wave equation defined on Ω ⊂ R. Using the Hilbert Uniqueness Method, one may define, for each subset ω ⊂ Ω, the exact control vω of minimal L (ω × (0, T ))-norm which drives to rest the system at a large enough time T > 0. We address the question of the optimal position of ω which minimizes the functional J : ω → ||vω||L2(ω×(0,T )). We express the shape derivative of J as an integral on ∂ω×(0, T ) independently of any adjoint solution. This expression leads to a descent direction for J 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 character of the problem by considering its convexification.