Cross-like internal observability of rectangular membranes

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March 2014, 3(1): 135-146. doi: 10.3934/eect.2014.3.135

Cross-like internal observability of rectangular membranes

Vilmos Komornik ^1, and Bernadette Miara ^2,

1.	Département de mathématique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France
2.	Université Paris-Est, Cité Descartes-Champs-sur-Marne, 5, boulevard Descartes, 77454 Marne la Vallée, France

Received October 2013 Revised December 2013 Published February 2014

Abstract
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We present a new way to establish internal observability results for the wave equation. Our method is based on some variants of Ingham's theorem on nonharmonic Fourier series, due to Loreti, Valente and Mehrenberger.

Keywords: Observability, membrane, Fourier series, Ingham type theorem..

Mathematics Subject Classification: Primary: 35L05; Secondary: 93B0.

Citation: Vilmos Komornik, Bernadette Miara. Cross-like internal observability of rectangular membranes. Evolution Equations & Control Theory, 2014, 3 (1) : 135-146. doi: 10.3934/eect.2014.3.135

References:

[1]	C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. doi: 10.1137/0330055.
[2]	A. Haraux, Contrôlabilité exacte d'une membrane rectangulaire au moyen d'une fonctionnelle analytique localisée,, C. R. Acad. Sci. Paris Sér. I Math., 306* (1988), 125.*
[3]	A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,, J. Math. Pures Appl., 68 (1989), 457.
[4]	A. Haraux, On a completion problem in the theory of distributed control of wave equations,, Nonlinear partial differential equations and their applications. Collège de France Seminar, 220 (1991), 1987.
[5]	A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series,, Math. Z., 41 (1936), 367. doi: 10.1007/BF01180426.
[6]	V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer-Verlag, (2005).
[7]	J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001.
[8]	J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Contrôlabilité exacte,, Masson, (1988).
[9]	P. Loreti and M. Mehrenberger, An Ingham type proof for a two-grid observability theorem,, ESAIM Control Optim. Calc. Var., 14 (2008), 604. doi: 10.1051/cocv:2007062.
[10]	P. Loreti and V. Valente, Partial exact controllability for spherical membranes,, SIAM J. Control Optim., 35 (1997), 641. doi: 10.1137/S036301299526962X.
[11]	M. Mehrenberger, An Ingham type proof for the boundary observability of a N-d wave equation,, C. R. Math. Acad. Sci. Paris, 347* (2009), 63. doi: 10.1016/j.crma.2008.11.002.*
[12]	Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation,, J. Fourier Anal. Appl., 19 (2013), 514. doi: 10.1007/s00041-013-9267-4.

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References:

[1]	C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. doi: 10.1137/0330055.
[2]	A. Haraux, Contrôlabilité exacte d'une membrane rectangulaire au moyen d'une fonctionnelle analytique localisée,, C. R. Acad. Sci. Paris Sér. I Math., 306* (1988), 125.*
[3]	A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,, J. Math. Pures Appl., 68 (1989), 457.
[4]	A. Haraux, On a completion problem in the theory of distributed control of wave equations,, Nonlinear partial differential equations and their applications. Collège de France Seminar, 220 (1991), 1987.
[5]	A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series,, Math. Z., 41 (1936), 367. doi: 10.1007/BF01180426.
[6]	V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer-Verlag, (2005).
[7]	J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001.
[8]	J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Contrôlabilité exacte,, Masson, (1988).
[9]	P. Loreti and M. Mehrenberger, An Ingham type proof for a two-grid observability theorem,, ESAIM Control Optim. Calc. Var., 14 (2008), 604. doi: 10.1051/cocv:2007062.
[10]	P. Loreti and V. Valente, Partial exact controllability for spherical membranes,, SIAM J. Control Optim., 35 (1997), 641. doi: 10.1137/S036301299526962X.
[11]	M. Mehrenberger, An Ingham type proof for the boundary observability of a N-d wave equation,, C. R. Math. Acad. Sci. Paris, 347* (2009), 63. doi: 10.1016/j.crma.2008.11.002.*
[12]	Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation,, J. Fourier Anal. Appl., 19 (2013), 514. doi: 10.1007/s00041-013-9267-4.

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