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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 ${ℝ}^{N}$, $N=1,2$. 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.
##### MSC:
 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