[Une méthode générale pour obtenir des taux de décroissance de l'énergie des équations d'évolution avec dissipation-mémoire]
On étudie le problème de la stabilisation des équations de type hyperbolique par un feedback-mémoire distribué. L'objet de cette Note est de montrer qu'il existe une méthode constructive générale qui permet d'obtenir un taux de décroissance de l'énergie en fonction du comportement au voisinage de l'infini du noyau. Cette méthode permet de retrouver de manière naturelle les résultats connus (cas exponentiel, polynômial, …) mais aussi de définir une classe très générale et quasi-optimale de noyaux à laquelle elle s'applique. Elle permet de montrer sous une condition, aussi très générale, que l'énergie des solutions décroit au moins aussi vite que le noyau à l'infini.
This Note is concerned with stabilization of hyperbolic systems by a distributed memory feedback. We present here a general method which gives energy decay rates in terms of the asymptotic behavior of the kernel at infinity. This method, which allows us to recover in a natural way the known cases (exponential, polynomial, …), applies to a large quasi-optimal class of kernels. It also provides sharp energy decay rates compared to the ones that are available in the literature. We give a general condition under which the energy of solutions is shown to decay at least as fast as the kernel at infinity.
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@article{CRMATH_2009__347_15-16_867_0,
author = {Alabau-Boussouira, Fatiha and Cannarsa, Piermarco},
title = {A general method for proving sharp energy decay rates for memory-dissipative evolution equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {867--872},
publisher = {Elsevier},
volume = {347},
number = {15-16},
year = {2009},
doi = {10.1016/j.crma.2009.05.011},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.05.011/}
}
TY - JOUR AU - Alabau-Boussouira, Fatiha AU - Cannarsa, Piermarco TI - A general method for proving sharp energy decay rates for memory-dissipative evolution equations JO - Comptes Rendus. Mathématique PY - 2009 SP - 867 EP - 872 VL - 347 IS - 15-16 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.05.011/ DO - 10.1016/j.crma.2009.05.011 LA - en ID - CRMATH_2009__347_15-16_867_0 ER -
%0 Journal Article %A Alabau-Boussouira, Fatiha %A Cannarsa, Piermarco %T A general method for proving sharp energy decay rates for memory-dissipative evolution equations %J Comptes Rendus. Mathématique %D 2009 %P 867-872 %V 347 %N 15-16 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.05.011/ %R 10.1016/j.crma.2009.05.011 %G en %F CRMATH_2009__347_15-16_867_0
Alabau-Boussouira, Fatiha; Cannarsa, Piermarco. A general method for proving sharp energy decay rates for memory-dissipative evolution equations. Comptes Rendus. Mathématique, Tome 347 (2009) no. 15-16, pp. 867-872. doi : 10.1016/j.crma.2009.05.011. http://www.numdam.org/articles/10.1016/j.crma.2009.05.011/
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