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Issue		ESAIM: COCV Volume 5, 2000
Page(s)		591 - 622
DOI		10.1051/cocv:2000123

DOI: 10.1051/cocv:2000123

ESAIM: COCV, November 2000, Vol. 5, 591-622

Stabilisation frontière de problèmes de Ventcel

Amar Heminna
Institut de Mathématiques, USTHB, BP. 32, EL-Alia, 16111 Alger, Algérie ; (hemina@usa.net)

Reçu le 29 novembre 1999. Révisé le 9 juin, le 17 juillet, le 29 août et le 25 septembre 2000

Abstract: The problem of boundary stabilization for the isotropic linear elastodynamic system and the wave equation with Ventcel's conditions are considered (see [12]). The boundary observability and the exact controllability were etablished in [11]. We prove here the enegy decay to zero for the elastodynamic system with stationary Ventcel's conditions by introducing a nonlinear boundary feedback. We also give a boundary feedback leading to arbitrarily large energy decay rates for the elastodynamic system with evolutive Ventcel's conditions. A spectral study proves, finally, that the natural feedback is not sufficient to assure the exponential decay in the case of the wave equation with Ventcel's conditions.

Résumé: On considère le problème de la stabilisation frontière de problèmes de Ventcel pour le système linéaire isotrope de l'élasticité et pour l'équation des ondes (cf. [12]). L'observabilité et la contrôlabilité ont été établies dans [11]. On montre ici la décroissance vers zéro de l'énergie pour le système de l'élasticité avec conditions de Ventcel stationnaires par un feedback non linéaire ; on montre aussi la décroissance exponentielle arbitrairement grande de l'énergie de la solution du système de l'élasticité par des feedbacks frontières. On montre enfin, par une étude spectrale, que le feedback naturel est insuffisant pour assurer la décroissance exponentielle de l'énergie dans le cas de l'équation des ondes.

Keywords and phrases: Élasticité, ondes, problème de Ventcel, contrôlabilité, stabilisation.

AMS Subject Classification: 93B03, 93B05, 93D15.

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