Comptes Rendus
Calculus of Variations
A variational approach to a shape design problem for the wave equation
Comptes Rendus. Mathématique, Volume 343 (2006) no. 5, pp. 371-376.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.07.013
Arnaud Münch 1; Pablo Pedregal 2; Francisco Periago 3

1 Laboratoire de mathématiques, Université de Franche-Comté, UMR CNRS 6623, 25030 Besançon, France
2 Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain
3 Departamento de Matemática Aplicada y Estadística, ETSI Industriales, Universidad Politécnica, 30203 Cartagena, Spain
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Arnaud Münch; Pablo Pedregal; Francisco Periago. A variational approach to a shape design problem for the wave equation. Comptes Rendus. Mathématique, Volume 343 (2006) no. 5, pp. 371-376. doi : 10.1016/j.crma.2006.07.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.07.013/

[1] S.J. Cox Designing for optimal energy absorption: the damped wave equation, Int. Ser. Numer. Math., vol. 126, Birkhäuser, 1998, pp. 103-109

[2] F. Fahroo; K. Ito Variational formulation of optimal damping designs, Contemp. Math., Volume 209 (1997), pp. 95-114

[3] P. Hebrard; A. Henrot Optimal shape and position of the actuators for the stabilization of a string, Systems Control Lett., Volume 48 (2003), pp. 199-209

[4] A. Münch, Optimal internal stabilization of a damped wave equation by a level set approach, Prépublication du laboratoire de mathématiques de Besançon 01/05, 2005

[5] A. Münch, P. Pedregal, F. Periago, Optimal design of the damping set for the stabilization of the wave equation, J. Differential Equations, in press

[6] P. Pedregal Parametrized Measures and Variational Principles, Birkhäuser, 1997

[7] P. Pedregal Vector variational problems and applications to optimal design, ESAIM:COCV, Volume 11 (2005), pp. 357-381

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