Comptes Rendus
Partial Differential Equations/Optimal Control
A general method for proving sharp energy decay rates for memory-dissipative evolution equations
[Une méthode générale pour obtenir des taux de décroissance de l'énergie des équations d'évolution avec dissipation-mémoire]
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 867-872.

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.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2009.05.011

Fatiha Alabau-Boussouira 1 ; Piermarco Cannarsa 2

1 L.M.A.M. CNRS-UMR 7122 et INRIA Équipe-projet CORIDA, université Paul-Verlaine-Metz, Ile du Saulcy, 57045 Metz cedex 01, France
2 Dipartimento di Matematica, Università di Roma Tor Vergata, 00133 Roma, Italy
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Fatiha Alabau-Boussouira; Piermarco Cannarsa. A general method for proving sharp energy decay rates for memory-dissipative evolution equations. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 867-872. doi : 10.1016/j.crma.2009.05.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.05.011/

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