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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 $\Bbb {R}^N$, $N=1,2$. 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:
49J20Optimal control problems with PDE (existence)
35B37PDE in connection with control problems (MSC2000)
35L05Wave equation (hyperbolic PDE)
93C20Control systems governed by PDE
49J45Optimal control problems involving semicontinuity and convergence; relaxation
49M30Other numerical methods in calculus of variations
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References:
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