Le problème d'optimisation de la forme et de la position de la zone de dissipation pour l'équation des ondes peut être mal posé. En utilisant une reformulation variationnelle et la théorie de la mesure de Young, on présente dans cette note une méthode générale pour relaxer ce type de problème. A partir de la mesure de Young optimal associée au problème relaxé bien posé, on obtient des informations concernant les suites minimisantes pour le problème original ainsi que des propriétés de continuité sur la fonction coût relaxée.
The problem of determining the optimal damping set for the stabilization of the wave equation may be not well-posed. By means of a vector variational reformulation and use of gradient Young measures, we present a general methodology to relax this kind of problems. From the optimal Young measure associated with the relaxed problem, we obtain information concerning minimizing sequences for the original problem as well as continuity properties of the relaxed cost function.
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@article{CRMATH_2006__343_5_371_0, author = {M\"unch, Arnaud and Pedregal, Pablo and Periago, Francisco}, title = {A variational approach to a shape design problem for the wave equation}, journal = {Comptes Rendus. Math\'ematique}, pages = {371--376}, publisher = {Elsevier}, volume = {343}, number = {5}, year = {2006}, doi = {10.1016/j.crma.2006.07.013}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2006.07.013/} }
TY - JOUR AU - Münch, Arnaud AU - Pedregal, Pablo AU - Periago, Francisco TI - A variational approach to a shape design problem for the wave equation JO - Comptes Rendus. Mathématique PY - 2006 SP - 371 EP - 376 VL - 343 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2006.07.013/ DO - 10.1016/j.crma.2006.07.013 LA - en ID - CRMATH_2006__343_5_371_0 ER -
%0 Journal Article %A Münch, Arnaud %A Pedregal, Pablo %A Periago, Francisco %T A variational approach to a shape design problem for the wave equation %J Comptes Rendus. Mathématique %D 2006 %P 371-376 %V 343 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2006.07.013/ %R 10.1016/j.crma.2006.07.013 %G en %F CRMATH_2006__343_5_371_0
Münch, Arnaud; Pedregal, Pablo; Periago, Francisco. A variational approach to a shape design problem for the wave equation. Comptes Rendus. Mathématique, Tome 343 (2006) no. 5, pp. 371-376. doi : 10.1016/j.crma.2006.07.013. http://www.numdam.org/articles/10.1016/j.crma.2006.07.013/
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