# American Institute of Mathematical Sciences

June  2014, 3(2): 287-304. doi: 10.3934/eect.2014.3.287

## Observability of rectangular membranes and plates on small sets

 1 Département de mathématique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex 2 Sapienza Università di Roma, Sapienza Università di Roma, Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sezione Matematica, Via A. Scarpa n.16 00161 Roma

Received  November 2013 Revised  April 2014 Published  May 2014

Since the works of Haraux and Jaffard we know that rectangular plates may be observed by subregions not satisfying the geometrical control condition. We improve these results by observing only on an arbitrarily short segment inside the domain. The estimates may be strengthened by observing on several well-chosen segments.
In the second part of the paper we establish various observability theorems for rectangular membranes by applying Mehrenberger's recent generalization of Ingham's theorem.
Citation: Vilmos Komornik, Paola Loreti. Observability of rectangular membranes and plates on small sets. Evolution Equations & Control Theory, 2014, 3 (2) : 287-304. doi: 10.3934/eect.2014.3.287
##### References:
 [1] C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory, Bol. Un. Mat. Ital. B (8), 2 (1999), 33-63.  Google Scholar [2] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95. doi: 10.1023/A:1020806811956.  Google Scholar [3] 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), 1024-1065. doi: 10.1137/0330055.  Google Scholar [4] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957.  Google Scholar [5] S. Gasmi and A. Haraux, $N$-cyclic functions and multiple subharmonic solutions of Duffing's equation, J. Math. Pures Appl., 97 (2012), 411-423. doi: 10.1016/j.matpur.2009.08.005.  Google Scholar [6] A. Haraux, On a completion problem in the theory of distributed control of wave equations, in Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, Vol. X (Paris, 1987-1988), Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991, 241-271.  Google Scholar [7] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465.  Google Scholar [8] L. F. Ho, Observabilité frontière de l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443-446.  Google Scholar [9] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426.  Google Scholar [10] S. Jaffard, Contrôle interne exact des vibrations d'une plaque carrée, C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 759-762.  Google Scholar [11] S. Jaffard, Contrôle interne exact des vibrations d'une plaque rectangulaire, Portugalia Math., 47 (1990), 423-429.  Google Scholar [12] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.  Google Scholar [13] V. Komornik and P. Loreti, Multiple-point internal observability of membranes and plates, Appl. Anal., 90 (2011), 1545-1555. doi: 10.1080/00036811.2011.569497.  Google Scholar [14] V. Komornik and B. Miara, Cross-like internal observability of rectangular membranes, Evol. Equations and Control Theory, 3 (2014), 135-146. doi: 10.3934/eect.2014.3.135.  Google Scholar [15] J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar [16] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Masson, Paris, 1988.  Google Scholar [17] P. Loreti, On some gap theorems, in European Women in Mathematics-Marseille 2003, CWI Tract, 135, Centrum Wisk. Inform., Amsterdam, 2005, 39-45.  Google Scholar [18] P. Loreti and M. Mehrenberger, An Ingham type proof for a two-grid observability theorem, ESAIM Control Optim. Calc. Var., 14 (2008), 604-631. doi: 10.1051/cocv:2007062.  Google Scholar [19] P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim., 35 (1997), 641-653. doi: 10.1137/S036301299526962X.  Google Scholar [20] M. Mehrenberger, An Ingham type proof for the boundary observability of a N-d wave equation, C. R. Math. Acad. Sci. Paris, 347 (2009), 63-68. doi: 10.1016/j.crma.2008.11.002.  Google Scholar [21] Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation, J. Fourier Anal. Appl., 19 (2013), 514-544. doi: 10.1007/s00041-013-9267-4.  Google Scholar [22] G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation, Trans. Amer. Math. Soc., 361 (2009), 951-977. doi: 10.1090/S0002-9947-08-04584-4.  Google Scholar

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##### References:
 [1] C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory, Bol. Un. Mat. Ital. B (8), 2 (1999), 33-63.  Google Scholar [2] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95. doi: 10.1023/A:1020806811956.  Google Scholar [3] 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), 1024-1065. doi: 10.1137/0330055.  Google Scholar [4] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957.  Google Scholar [5] S. Gasmi and A. Haraux, $N$-cyclic functions and multiple subharmonic solutions of Duffing's equation, J. Math. Pures Appl., 97 (2012), 411-423. doi: 10.1016/j.matpur.2009.08.005.  Google Scholar [6] A. Haraux, On a completion problem in the theory of distributed control of wave equations, in Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, Vol. X (Paris, 1987-1988), Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991, 241-271.  Google Scholar [7] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465.  Google Scholar [8] L. F. Ho, Observabilité frontière de l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443-446.  Google Scholar [9] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426.  Google Scholar [10] S. Jaffard, Contrôle interne exact des vibrations d'une plaque carrée, C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 759-762.  Google Scholar [11] S. Jaffard, Contrôle interne exact des vibrations d'une plaque rectangulaire, Portugalia Math., 47 (1990), 423-429.  Google Scholar [12] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.  Google Scholar [13] V. Komornik and P. Loreti, Multiple-point internal observability of membranes and plates, Appl. Anal., 90 (2011), 1545-1555. doi: 10.1080/00036811.2011.569497.  Google Scholar [14] V. Komornik and B. Miara, Cross-like internal observability of rectangular membranes, Evol. Equations and Control Theory, 3 (2014), 135-146. doi: 10.3934/eect.2014.3.135.  Google Scholar [15] J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar [16] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Masson, Paris, 1988.  Google Scholar [17] P. Loreti, On some gap theorems, in European Women in Mathematics-Marseille 2003, CWI Tract, 135, Centrum Wisk. Inform., Amsterdam, 2005, 39-45.  Google Scholar [18] P. Loreti and M. Mehrenberger, An Ingham type proof for a two-grid observability theorem, ESAIM Control Optim. Calc. Var., 14 (2008), 604-631. doi: 10.1051/cocv:2007062.  Google Scholar [19] P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim., 35 (1997), 641-653. doi: 10.1137/S036301299526962X.  Google Scholar [20] M. Mehrenberger, An Ingham type proof for the boundary observability of a N-d wave equation, C. R. Math. Acad. Sci. Paris, 347 (2009), 63-68. doi: 10.1016/j.crma.2008.11.002.  Google Scholar [21] Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation, J. Fourier Anal. Appl., 19 (2013), 514-544. doi: 10.1007/s00041-013-9267-4.  Google Scholar [22] G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation, Trans. Amer. Math. Soc., 361 (2009), 951-977. doi: 10.1090/S0002-9947-08-04584-4.  Google Scholar
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