Optimal design of the damping set for the stabilization of the wave equation |
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Authors: | Arnaud Münch Francisco Periago |
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Institution: | a Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, UMR CNRS 6623, 16 route de Gray, 25030 Besançon, France b Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain c Departamento de Matemática Aplicada y Estadística, ETSI Industriales, Universidad Politécnica de Cartagena, 30203 Cartagena, Spain |
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Abstract: | We consider the problem of optimizing the shape and position of the damping set for the internal stabilization of the linear wave equation in RN, 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. |
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