# 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 and 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. [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. [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. [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. [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. [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. [7] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465. [8] L. F. Ho, Observabilité frontière de l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443-446. [9] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426. [10] S. Jaffard, Contrôle interne exact des vibrations d'une plaque carrée, C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 759-762. [11] S. Jaffard, Contrôle interne exact des vibrations d'une plaque rectangulaire, Portugalia Math., 47 (1990), 423-429. [12] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005. [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. [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. [15] J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001. [16] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Masson, Paris, 1988. [17] P. Loreti, On some gap theorems, in European Women in Mathematics-Marseille 2003, CWI Tract, 135, Centrum Wisk. Inform., Amsterdam, 2005, 39-45. [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. [19] P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim., 35 (1997), 641-653. doi: 10.1137/S036301299526962X. [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. [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. [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.

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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. [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. [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. [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. [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. [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. [7] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465. [8] L. F. Ho, Observabilité frontière de l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443-446. [9] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426. [10] S. Jaffard, Contrôle interne exact des vibrations d'une plaque carrée, C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 759-762. [11] S. Jaffard, Contrôle interne exact des vibrations d'une plaque rectangulaire, Portugalia Math., 47 (1990), 423-429. [12] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005. [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. [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. [15] J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001. [16] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Masson, Paris, 1988. [17] P. Loreti, On some gap theorems, in European Women in Mathematics-Marseille 2003, CWI Tract, 135, Centrum Wisk. Inform., Amsterdam, 2005, 39-45. [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. [19] P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim., 35 (1997), 641-653. doi: 10.1137/S036301299526962X. [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. [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. [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.
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