Geodesic
Observability of square membranes by Fourier series methods
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 8 (2015) no. 3, pp. 127-140
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{VYURU_2015_8_3_a7,
     author = {V. Komornik and P. Loreti},
     title = {Observability of square membranes by {Fourier} series methods},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {127--140},
     year = {2015},
     volume = {8},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2015_8_3_a7/}
}
TY  - JOUR
AU  - V. Komornik
AU  - P. Loreti
TI  - Observability of square membranes by Fourier series methods
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
PY  - 2015
SP  - 127
EP  - 140
VL  - 8
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VYURU_2015_8_3_a7/
LA  - en
ID  - VYURU_2015_8_3_a7
ER  - 
%0 Journal Article
%A V. Komornik
%A P. Loreti
%T Observability of square membranes by Fourier series methods
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
%D 2015
%P 127-140
%V 8
%N 3
%U http://geodesic.mathdoc.fr/item/VYURU_2015_8_3_a7/
%G en
%F VYURU_2015_8_3_a7
V. Komornik; P. Loreti. Observability of square membranes by Fourier series methods. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 8 (2015) no. 3, pp. 127-140. http://geodesic.mathdoc.fr/item/VYURU_2015_8_3_a7/

[1] Lasiecka I., Triggiani R., “Regularity of Hyperbolic Equations under $L_2(0,T; L_2(\Gamma ))$ Boundary Terms”, Appl. Math. and Optimiz., 10 (1983), 275–286 | DOI | MR | Zbl

[2] Lions J.-L., Contrôle des systèmes distribués singuliers, Gauthier-Villars, Paris, 1983 | MR

[3] Ho L. F., “Observabilité frontière de l'équation des ondes”, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443–446 | MR | Zbl

[4] Lions J.-L., “Exact Controllability, Stabilizability, and Perturbations for Distributed Systems”, Siam Rev., 30 (1988), 1–68 | DOI | MR | Zbl

[5] Lions J.-L., Contrôlabilité exacte et stabilisation de systèmes distribués, v. I, II, Masson, Paris, 1988 | MR

[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., 30 (1992), 1024–1065 | DOI | MR | Zbl

[7] Komornik V., “Controlabilité Exacte en un Temps Minimal”, C. R. Acad. Sci. Paris Sér. I Math., 304 (1987), 223–225 | MR | Zbl

[8] Mehrenberger M., “An Ingham Type Proof for the Boundary Observability of a N-d Wave Equation”, C. R. Math. Acad. Sci. Paris, 347:1–2 (2009), 63–68 | DOI | MR | Zbl

[9] Ingham A. E., “Some Trigonometrical Inequalities with Applications in the Theory of Series”, Math. Z., 41 (1936), 367–379 | DOI | MR

[10] Komornik V., Miara B., “Cross-Like Internal Observability of Rectangular Membranes”, Evol. Equations and Control Theory, 3:1 (2014), 135–146 | DOI | MR | Zbl

[11] Komornik V., Loreti P., “Observability of Rectangular Membranes and Plates on Small Sets”, Evol. Equations and Control Theory, 3:2 (2014), 287–304 | DOI | MR | Zbl

[12] Komornik V., Loreti P., Fourier Series in Control Theory, Springer-Verlag, New York, 2005 | MR | Zbl

[13] Gasmi S., Haraux A., “N-Cyclic Functions and Multiple Subharmonic Solutions of Duffing's Equation”, J. Math. Pures Appl., 97 (2012), 411–423 | DOI | MR | Zbl

[14] Baiocchi C., Komornik V., Loreti P., “Ingham Type Theorems and Applications to Control Theory”, Bol. Un. Mat. Ital. B. Series 8, 2:1 (1999), 33–63 | MR | Zbl

[15] Baiocchi C., Komornik V., Loreti P., “Ingham–Beurling Type Theorems with Weakened Gap Conditions”, Acta Math. Hungar., 97:1–2 (2002), 55–95 | DOI | MR | Zbl

[16] Fadeev D. K., Sominsky D. K., Problems in Higher Algebra, Mir, M., 1972

[17] Haraux A., “Séries lacunaires et controle semi-interne des vibrations d'une plaque rectangulaire”, J. Math. Pures Appl., 68 (1989), 457–465 | MR | Zbl

[18] Komornik V., Loreti P., “Ingham Type Theorems for Vector-Valued Functions and Observability of Coupled Linear Systems”, SIAM J. Control Optim., 37 (1998), 461–485 | DOI | MR

[19] Komornik V., Loreti P., “Multiple-Point Internal Observability of Membranes and Plates”, Appl. Anal., 90:10 (2011), 1545–1555 | DOI | MR | Zbl

[20] Loreti P., “On Some Gap Theorems”, Proceedings of the 11th Meeting of EWM, CWI Tract, 2005 | MR

[21] Loreti P., Mehrenberger M., “An Ingham Type Proof for a Two-Grid Observability Theorem”, ESAIM Control Optim. Calc. Var., 14:3 (2008), 604–631 | DOI | MR | Zbl

[22] Loreti P., Valente V., “Partial Exact Controllability for Spherical Membranes”, SIAM J. Control Optim., 35 (1997), 641–653 | DOI | MR | Zbl