Locally distributed desensitizing controls for the wave equation

Louis Tebou

doi:10.1016/j.crma.2008.02.019

Optimal Control

Locally distributed desensitizing controls for the wave equation
[Contrôles insensibilisant localement distribués pour l'équation des ondes]

Présenté par : Gilles Lebeau

Louis Tebou ¹

¹ Department of Mathematics, Florida International University, Miami, FL 33199, USA

Comptes Rendus. Mathématique, Volume 346 (2008) no. 7-8, pp. 407-412.

Résumé
Abstract

Nous considérons dans un domaine borné une équation des ondes avec des données initiales incomplètes. Pour ce système, nous construisons des contrôles localement distribués qui insensibilisent une certaine norme de la solution du système. Ce résultat est nouveau pour les dimensions d'espace supérieures ou égales à deux. La méthode de démonstration allie une application judicieuse de l'inégalité de Carleman, et une technique de localisation.

We consider the wave equation with partially known initial data in a bounded domain. For this system, we construct locally distributed controls that desensitize a certain norm of the state. This result is new in space dimensions greater than one. The method of proof combines a judicious application of the Carleman estimate, and a localization technique.

Reçu le : 2007-11-28
Accepté le : 2008-02-18
Publié le : 2008-04-01

DOI : 10.1016/j.crma.2008.02.019

Affiliations des auteurs :

Louis Tebou ¹

¹ Department of Mathematics, Florida International University, Miami, FL 33199, USA

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Louis Tebou. Locally distributed desensitizing controls for the wave equation. Comptes Rendus. Mathématique, Volume 346 (2008) no. 7-8, pp. 407-412. doi : 10.1016/j.crma.2008.02.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.02.019/

Bibliographie
Cité par

[1] C. Bardos; G. Lebeau; J. Rauch Sharp sufficient conditions for the observation, control and stabilization from the boundary, SIAM J. Control Optim., Volume 30 (1992), pp. 1024-1065

[2] O. Bodart; C. Fabre Controls insensitizing the norm of the solution of a semilinear heat equation, J. Math. Anal. Appl., Volume 195 (1995) no. 3, pp. 658-683

[3] O. Bodart; M. Gonzàlez-Burgos; R. Pérez-García Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. Partial Differential Equations, Volume 29 (2004) no. 7–8, pp. 1017-1050

[4] O. Bodart; M. Gonzàlez-Burgos; R. Pérez-García A local result on insensitizing controls for a semilinear heat equation with nonlinear boundary Fourier conditions, SIAM J. Control Optim., Volume 43 (2004) no. 3, pp. 955-969

[5] R. Dáger Insensitizing controls for the 1-D wave equation, SIAM J. Control Optim., Volume 45 (2006) no. 5, pp. 1758-1768

[6] T. Duyckaerts; X. Zhang; E. Zuazua On the optimality of the observability for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 25 (2008), pp. 1-41

[7] E. Fernández-Cara; G.C. Garcia; A. Osses Controls insensitizing the observation of a quasi-geostrophic ocean model, SIAM J. Control Optim., Volume 43 (2005) no. 5, pp. 1616-1639

[8] A. Haraux On a completion problem in the theory of distributed control of wave equations, Paris, 1987–1988 (Pitman Res. Notes Math. Ser.), Volume vol. 220, Longman Sci. Tech., Harlow (1991), pp. 241-271

[9] V. Komornik Exact Controllability and Stabilization. The Multiplier Method, RAM, Masson & John Wiley, Paris, 1994

[10] J.-L. Lions Quelques notions dans l'analyse et le contrôle de systèmes à données incomplètes, Málaga, 1989, Univ. Málaga, Málaga (1990), pp. 43-54 (in Spanish)

[11] J.L. Lions Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués, vol. 1, RMA, vol. 8, Masson, Paris, 1988

[12] S. Micu; J.H. Ortega; L. de Teresa An example of ϵ-insensitizing controls for the heat equation with no intersecting observation and control regions, Appl. Math. Lett., Volume 17 (2004) no. 8, pp. 927-932

[13] L. Tebou, Desensitizing controls for some semilinear hyperbolic equations, in preparation

[14] L. de Teresa Controls insensitizing the norm of the solution of a semilinear heat equation in unbounded domains, ESAIM Control Optim. Calc. Var., Volume 2 (1997), pp. 125-149

[15] L. de Teresa Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations, Volume 25 (2000) no. 1–2, pp. 39-72

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