Optimal design of the damping set for the stabilization of the wave equation. (English) Zbl 1105.49005
Summary: We consider the problem of optimizing the shape and position of the damping set for the internal stabilization of the linear wave equation in , . In a first theoretical part, we reformulate the problem into an equivalent non-convex vector variational one using a characterization of divergence-free vector fields. Then, by means of gradient Young measures, we obtain a relaxed formulation of the problem in which the original cost density is replaced by its constrained quasi-convexification. This implies that the new relaxed problem is well-posed in the sense that there exists a minimizer and, in addition, the infimum of the original problem coincides with the minimum of the relaxed one. In a second numerical part, we address the resolution of the relaxed problem using a first-order gradient descent method. We present some numerical experiments which highlight the influence of the over-damping phenomena and show that for large values of the damping potential the original problem has no minimizer. We then propose a penalization technique to recover the minimizing sequences of the original problem from the optimal solution of the relaxed one.
|49J20||Optimal control problems with PDE (existence)|
|35B37||PDE in connection with control problems (MSC2000)|
|35L05||Wave equation (hyperbolic PDE)|
|93C20||Control systems governed by PDE|
|49J45||Optimal control problems involving semicontinuity and convergence; relaxation|
|49M30||Other numerical methods in calculus of variations|