Nous considérons la stabilité des équations des ondes avec un amortissement visco-élastique distribué autour de la frontière du domaine. Nous montrons que l'énergie du système tend vers zéro uniformément et exponentiellement pour toute donnée initiale d'énergie finie.
We consider the stability of wave equations with local viscoelastic damping distributed around the boundary of domain. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy.
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@article{CRMATH_2004__339_11_769_0,
author = {Kangsheng Liu and Bopeng Rao},
title = {Stabilit\'e exponentielle des \'equations des ondes avec amortissement local de {Kelvin{\textendash}Voigt}},
journal = {Comptes Rendus. Math\'ematique},
pages = {769--774},
publisher = {Elsevier},
volume = {339},
number = {11},
year = {2004},
doi = {10.1016/j.crma.2004.09.029},
language = {fr},
}
TY - JOUR AU - Kangsheng Liu AU - Bopeng Rao TI - Stabilité exponentielle des équations des ondes avec amortissement local de Kelvin–Voigt JO - Comptes Rendus. Mathématique PY - 2004 SP - 769 EP - 774 VL - 339 IS - 11 PB - Elsevier DO - 10.1016/j.crma.2004.09.029 LA - fr ID - CRMATH_2004__339_11_769_0 ER -
Kangsheng Liu; Bopeng Rao. Stabilité exponentielle des équations des ondes avec amortissement local de Kelvin–Voigt. Comptes Rendus. Mathématique, Volume 339 (2004) no. 11, pp. 769-774. doi : 10.1016/j.crma.2004.09.029. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.029/
[1] Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992), pp. 1024-1065
[2] Spectrum and stability for elastic systems with global or local Kelvin–Voigt damping, SIAM J. Appl. Math., Volume 59 (1999), pp. 651-668
[3] Unique continuation for elliptic operators: a geometric-variational approach, Commun. Pure Appl. Math., Volume 40 (1987), pp. 347-366
[4] The Analysis of Linear Partial Differential Operators, Springer-Verlag, Berlin, 1985
[5] Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, Volume 1 (1985), pp. 43-56
[6] Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Commun. Pure Appl. Math., Volume 54 (2001), pp. 339-360
[7] Control of wave process with distributed controls supported on a subregion, SIAM J. Control Optim., Volume 21 (1983), pp. 68-85
[8] Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping, SIAM J. Control Optim., Volume 36 (1998) no. 3, pp. 1086-1098
[9] Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., Volume 53 (2002), pp. 265-280
[10] K. Liu, B. Rao, Exponential stability for the wave equations with local Kelvin–Voigt damping, en préparation
[11] Compensation spectrale et taux de décroissance optimal de l'énergie de systèmes partiellement amortis, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003), pp. 531-536
[12] On the spectrum of -semigroups, Trans. Am. Math. Soc., Volume 284 (1984), pp. 847-857
[13] Control, observation and polynomial decay for a coupled heat-wave system, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003), pp. 823-828
[14] Polynomial decay and control of a 1-d model for fluid–structure interaction, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003), pp. 745-750
[15] Exponential decay for the semilinear wave equation with localized damping, Commun. Partial Differential Equations, Volume 15 (1990), pp. 205-235
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