Observability of Square Membranes by Fourier Series Methods

Fourier series methods have been successfully applied in control theory for a long time. Some theorems, however, resisted this approach. Some years ago, Mehrenberger succeeded in establishing the boundary observability of vibrating rectangular membranes (and of analogous higher dimensional problems) by developing an ingenious generalization of Ingham's classical theorem on nonharmonic Fourier series. His method turn out to be useful for other applications as well. We improve Mehrenberger's approach by a shorter proof, and we improve and generalize some earlier applications.

Keywords

observability; nonharmonic Fourier series; Ingham's theorem; wave equation.

References

1. Lasiecka I., Triggiani R. Regularity of Hyperbolic Equations under L_2(0,T; L_2(Gamma )) Boundary Terms. Appl. Math. and Optimiz., 1983, vol. 10, pp. 275-286. DOI: 10.1007/BF01448390
2. Lions J.-L. Controle des systemes distribues singuliers. Paris, Gauthier-Villars, 1983.
3. Ho L.F. Observabilite frontiere de l'equation des ondes. C. R. Acad. Sci. Paris Ser. I Math., 1986, vol. 302, pp. 443-446.
4. Lions J.-L. Exact Controllability, Stabilizability, and Perturbations for Distributed Systems. Siam Rev., 1988, vol. 30, pp. 1-68. DOI: 10.1137/1030001
5. Lions J.-L. Controlabilite exacte et stabilisation de systemes distribues I-II. Masson, Paris, 1988.
6. Bardos C., Lebeau G., Rauch J. Sharp Sufficient Conditions for the Observation, Control and Stabilization of Waves from the Boundary. SIAM J. Control Optim., 1992, vol. 30, pp. 1024-1065. DOI: 10.1137/0330055
7. Komornik V. Controlabilite Exacte en un Temps Minimal. C. R. Acad. Sci. Paris Ser. I Math., 1987, vol. 304, pp. 223-225.
8. Mehrenberger M. An Ingham Type Proof for the Boundary Observability of a N-d Wave Equation. C. R. Math. Acad. Sci. Paris, 2009, vol. 347, no. 1-2, pp. 63-68. DOI: 10.1016/j.crma.2008.11.002
9. Ingham A.E. Some Trigonometrical Inequalities with Applications in the Theory of Series. Math. Z., 1936, vol. 41, pp. 367-379. DOI: 10.1007/BF01180426
10. Komornik V., Miara B. Cross-Like Internal Observability of Rectangular Membranes. Evol. Equations and Control Theory, 2014, vol. 3, no. 1, pp. 135-146. DOI: 10.3934/eect.2014.3.135
11. Komornik V., Loreti P. Observability of Rectangular Membranes and Plates on Small Sets. Evol. Equations and Control Theory, 2014, vol. 3, no. 2, pp. 287-304. DOI: 10.3934/eect.2014.3.287
12. Komornik V., Loreti P. Fourier Series in Control Theory. New York, Springer-Verlag, 2005.
13. Gasmi S., Haraux A. N-Cyclic Functions and Multiple Subharmonic Solutions of Duffing's Equation. J. Math. Pures Appl., 2012, vol. 97, pp. 411-423. DOI: 10.1016/j.matpur.2009.08.005
14. Baiocchi C., Komornik V., Loreti P. Ingham Type Theorems and Applications to Control Theory. Bol. Un. Mat. Ital. B. Series 8, 1999, vol. 2, no. 1, pp. 33-63.
15. Baiocchi C., Komornik V., Loreti P. Ingham-Beurling Type Theorems with Weakened Gap Conditions. Acta Math. Hungar., 2002, vol. 97, no. 1-2, pp. 55-95. DOI: 10.1023/A:1020806811956
16. Fadeev D.K., Sominsky D.K. Problems in Higher Algebra. Moscow, Mir, 1972.
17. Haraux A. Series lacunaires et controle semi-interne des vibrations d'une plaque rectangulaire. J. Math. Pures Appl., 1989, vol. 68, pp. 457-465.
18. Komornik V., Loreti P. Ingham Type Theorems for Vector-Valued Functions and Observability of Coupled Linear Systems. SIAM J. Control Optim., 1998, vol. 37, pp. 461-485. DOI: 10.1137/S0363012997317505
19. Komornik V., Loreti P. Multiple-Point Internal Observability of Membranes and Plates. Appl. Anal., 2011, vol. 90, no. 10, pp. 1545-1555. DOI: 10.1080/00036811.2011.569497
20. Loreti P. On Some Gap Theorems. Proceedings of the 11th Meeting of EWM, CWI Tract, 2005.
21. Loreti P., Mehrenberger M. An Ingham Type Proof for a Two-Grid Observability Theorem ESAIM Control Optim. Calc. Var., 2008, vol. 14, no. 3, pp. 604-631.
22. Loreti P., Valente V. Partial Exact Controllability for Spherical Membranes. SIAM J. Control Optim., 1997, vol. 35, pp. 641-653. DOI: 10.1137/S036301299526962X