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2017 | Vol. 23, nr 1 | 43--51
Tytuł artykułu

A remark on observability of the wave equation with moving boundary

Warianty tytułu
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].
Wydawca

Rocznik
Strony
43--51
Opis fizyczny
Bibliogr. 14 poz.
Twórcy
autor
  • UR Analysis and Control of PDEs, UR13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia, kais.ammari@fsm.rnu.tn
autor
  • UR Analyse Non-Linéaire et Géométrie, UR13ES32, Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia, ahmed.bchatnia@fst.utm.tn
autor
Bibliografia
  • [1] J. W. S. Cassals, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge, 1966.
  • [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).
Typ dokumentu
Bibliografia
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