A remark on observability of the wave equation with moving boundary

• Autorzy: Ammari, K. , Bchatnia, A. , El Mufti, K.
• Języki publikacji: EN

Abstrakty

EN

We deal with thewave equation with assigned moving boundary (0 < x < a(t)) uponwhich Dirichlet or mixed boundary conditions are specified. Here a(t) is assumed to move slower than light and periodically. Moreover, a is continuous, piecewise linear with two independent parameters. Our major concern will be an observation problem which is based measuring, at each t > 0, of the transverse velocity at a(t). The key to the results is the use of a reduction theorem by Yoccoz [14].

Słowa kluczowe

PL

obserwowalność; liczba obrotu

EN

strings with moving ends; observability; rotation number

• Czasopismo: Journal of Applied Analysis
• Rocznik: 2017
• Tom: Vol. 23, nr 1
• Strony: 43--51
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Ammari, K.

• UR Analysis and Control of PDEs, UR13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia,

autor

Bchatnia, A.

• UR Analyse Non-Linéaire et Géométrie, UR13ES32, Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia,

autor

El Mufti, K.

• UR Analysis and Control of PDEs, UR13ES64, ISCAE, University of Manouba, Manouba, Tunisia,,

Bibliografia

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• [2] C. Castro, Exact controllability of the 1-d wave equation from a moving interior point, ESAIM Control Optim. Calc. Var. 19 (2010), 301-316.
• [3] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-Dflexible Multi-Structures, Math. Appl. (Berlin) 50, Springer, Berlin, 2006.
• [4] R. de la Llave and N. P. Petrov, Theory of circle maps and the problem of one-dimensional optical resonator with a periodically moving wall, Phys. Rev. E (3) 59 (1999), 6637-6651.
• [5] J. Dittrich, P. Duclos and N. Gonzalez, Stability and instability of the wave equation solutions in a pulsating domain, Rev. Math. Phys. 10 (1998), 925-962.
• [6] N. Gonzalez, L’équation des ondes dans un domaine dépendant du temps, Ph.D. thesis, University of Toulon and Czech Technical University, 1997.
• [7] N. Gonzalez, An example of pure stability for the wave equation with moving boundary, J. Math. Anal. Appl. 228 (1998), 51-59.
• [8] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5-234.
• [9] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monogr. Math., Springer, New York, 2005.
• [10] S. Lang, Introduction to Diophantine Approximations, 2nd ed., Springer, New York, 1995.
• [11] M. Yamaguchi, Periodic solutions of nonlinear equations of string with periodically oscillating boundaries, Funkcial. Ekvac. 45 (2002), 397-416.
• [12] M. Yamaguchi, One dimensional wave equations in domain with quasiperiodically moving boundaries and quasiperiodic dynamical systems, J. Math. Kyoto Univ. 45 (2005), 57-97.
• [13] M. Yamaguchi and H. Yoshida, Nonhomogeneous string problem with periodically moving boundaries, Fields Inst. Commun. 25 (2000), 565-574.
• [14] J. C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une conditio diophantienne, Ann. Sci. Éc. Norm. Supér. (4) 17 (1984), 333-359.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).

Identyfikatory

DOI

10.1515/jaa-2017-0007