Comptes Rendus
Partial Differential Equations/Optimal Control
Uniform observability estimates for the 1-D discretized wave equation and the random choice method
Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 505-510.

In this Note, we consider the 1-dimensional wave equation, discretized by means of Glimm's random choice method. We prove that for almost every choice of the random parameter, the observability estimate is true asymptotically, uniformly with respect to the discretization parameters.

Dans cette Note, nous considérons l'équation des ondes unidimensionnelle discrétisée selon la méthode du choix aléatoire due à J. Glimm. Nous établissons que pour presque tout choix de la variable aléatoire, l'estimée d'observabilité pour cette équation est vraie asymptotiquement, uniformément en les paramètres de discrétisation.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.03.012

Jean-Michel Coron 1, 2, 3; Sylvain Ervedoza 4; Olivier Glass 1, 2

1 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005 Paris, France
2 CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
3 Institut Universitaire de France
4 Laboratoire de mathématiques de Versailles, Université de Versailles Saint Quentin en Yvelines, 45, avenue des États-Unis, 78035 Versailles, France
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Jean-Michel Coron; Sylvain Ervedoza; Olivier Glass. Uniform observability estimates for the 1-D discretized wave equation and the random choice method. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 505-510. doi : 10.1016/j.crma.2009.03.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.012/

[1] J. Glimm Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., Volume 18 (1965), pp. 697-715

[2] R. Glowinski; J.-L. Lions; J. He Exact and approximate controllability for distributed parameter systems. A numerical approach, Encyclopedia of Mathematics and its Applications, vol. 117, Cambridge University Press, Cambridge, 2008

[3] G.R. Grimmett; D.R. Stirzaker Probability and Random Processes, Oxford University Press, New York, 1992

[4] J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués, Tomes 1 & 2, Masson, RMA 8 & 9, Paris 1988

[5] E. Zuazua Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., Volume 47 (2005) no. 2, pp. 197-243

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