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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

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.  Google Scholar

[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.   Google Scholar

[3]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,, J. Math. Pures Appl., 68 (1989), 457.   Google Scholar

[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.   Google Scholar

[5]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series,, Math. Z., 41 (1936), 367.  doi: 10.1007/BF01180426.  Google Scholar

[6]

V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer-Verlag, (2005).   Google Scholar

[7]

J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1.  doi: 10.1137/1030001.  Google Scholar

[8]

J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Contrôlabilité exacte,, Masson, (1988).   Google Scholar

[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.  Google Scholar

[10]

P. Loreti and V. Valente, Partial exact controllability for spherical membranes,, SIAM J. Control Optim., 35 (1997), 641.  doi: 10.1137/S036301299526962X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.   Google Scholar

[3]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,, J. Math. Pures Appl., 68 (1989), 457.   Google Scholar

[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.   Google Scholar

[5]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series,, Math. Z., 41 (1936), 367.  doi: 10.1007/BF01180426.  Google Scholar

[6]

V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer-Verlag, (2005).   Google Scholar

[7]

J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1.  doi: 10.1137/1030001.  Google Scholar

[8]

J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Contrôlabilité exacte,, Masson, (1988).   Google Scholar

[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.  Google Scholar

[10]

P. Loreti and V. Valente, Partial exact controllability for spherical membranes,, SIAM J. Control Optim., 35 (1997), 641.  doi: 10.1137/S036301299526962X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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