[Un stabilisateur localisé non standard pour la poutre de Timoshenko]
Dans cette note est examinée la stabilisation de la poutre de Timoshenko avec un amortissement localisé. L'amortissement est lié à la somme des vitesses de tassement et de cisaillement angulaire ; ce travail généralise au système de Timoshenko un résultat antérieur de Haraux, établi pour un système d'équations d'ondes ordinaires. D'abord, nous montrons que la stabilité forte a lieu si et seulement si le support du contrôle rencontre une extrémité de l'intervalle considéré. Puis nous utilisons la combinaison de la méthode des multiplicateurs avec la méthode du domaine des fréquences pour démontrer la stabilité exponentielle du semi-groupe associé quand le support du contrôle rencontre une extrémité de l'intervalle considéré. Quand la vitesse de propagation de l'onde générée par le tassement et celle de l'onde générée par l'angle de cisaillement sont distinctes, la preuve est semblable à celle connue pour deux ondes amorties de la même manière. Cependant, quand les deux vitesses sont égales, une identité importante perd sa validité, et la preuve se poursuit par l'introduction d'une équation auxiliaire dont la solution joue un rôle prépondérant dans les estimations ultérieures.
The stabilization of the Timoshenko beam system with localized damping is examined. The damping involves the sum of the bending and shear angle velocities; this work generalizes an earlier result of Haraux, established for a system of ordinary wave equations, to the Timoshenko system. First, we show that strong stability holds if and only if the boundary of the support of the feedback control intersects that of the interval under consideration. Next, we use the frequency domain method combined with the multipliers technique to prove the exponential stability of the associated semigroup when the damping support is a neighborhood of one endpoint of the interval under consideration. When the speed of propagation of the wave generated by the bending and that of the wave generated by the shear angle are distinct, the proof is similar to what is known for two ordinary waves similarly damped. However, when the two speeds are equal, an important identity breaks down, and the proof is carried out by the introduction of an appropriate auxiliary equation whose solution plays a critical role in subsequent estimates.
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@article{CRMATH_2015__353_3_247_0,
author = {Tebou, Louis},
title = {A localized nonstandard stabilizer for the {Timoshenko} beam},
journal = {Comptes Rendus. Math\'ematique},
pages = {247--253},
publisher = {Elsevier},
volume = {353},
number = {3},
year = {2015},
doi = {10.1016/j.crma.2015.01.004},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.01.004/}
}
TY - JOUR AU - Tebou, Louis TI - A localized nonstandard stabilizer for the Timoshenko beam JO - Comptes Rendus. Mathématique PY - 2015 SP - 247 EP - 253 VL - 353 IS - 3 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2015.01.004/ DO - 10.1016/j.crma.2015.01.004 LA - en ID - CRMATH_2015__353_3_247_0 ER -
%0 Journal Article %A Tebou, Louis %T A localized nonstandard stabilizer for the Timoshenko beam %J Comptes Rendus. Mathématique %D 2015 %P 247-253 %V 353 %N 3 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2015.01.004/ %R 10.1016/j.crma.2015.01.004 %G en %F CRMATH_2015__353_3_247_0
Tebou, Louis. A localized nonstandard stabilizer for the Timoshenko beam. Comptes Rendus. Mathématique, Tome 353 (2015) no. 3, pp. 247-253. doi : 10.1016/j.crma.2015.01.004. http://archive.numdam.org/articles/10.1016/j.crma.2015.01.004/
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