A general formula for decay rates of nonlinear dissipative systems

Fatiha Alabau-Boussouira

doi:10.1016/j.crma.2003.10.024

Optimal Control

A general formula for decay rates of nonlinear dissipative systems
[Une formule générale pour le taux de décroissance des systèmes dissipatifs non linéaires]

Présenté par : Philippe G. Ciarlet

Fatiha Alabau-Boussouira ¹

¹ LMAM, CNRS-UMR 7122, Université de Metz, Ile du Saulcy, 57045 Metz cedex 01, France

Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 35-40.

Résumé
Abstract

On étudie le problème de la stabilisation des équations de type hyperbolique par un feedback qui peut être frontière ou bien localement distribué. L'objet de cette Note est de montrer qu'il existe une formule générale qui permet d'obtenir un taux de décroissance de l'énergie en fonction du comportement au voisinage de zéro du terme de dissipation non linéaire. Cette formule permet d'unifier tous les cas et notamment ceux pour lesquels le feedback croı̂t polynomialement et ceux pour lesquels il s'écrase exponentiellement en zéro. On donne aussi deux autres exemples significatifs de croissance non polynomiale. On montre pour tous ces exemples que l'on retrouve ou obtient de meilleurs taux de décroissance que ceux de Lasiecka et Tataru (Differential Integral Equations 8 (1993) 507–533) et Martinez (Rev. Mat. Comput. 12 (1999) 251–283).

This work is concerned with stabilization of hyperbolic systems by a nonlinear feedback which can be localized on part of the boundary or locally distributed. We present here a general formula which gives the energy decay rates in terms of the behavior of the nonlinear feedback close to the origin. This formula allows us to unify for instance the cases where the feedback has a polynomial growth at the origin, with the cases where it goes exponentially fast to zero at the origin. We give also two other significant examples of nonpolynomial growth at the origin. We also show that we either obtain or improve significantly the decay rates of Lasiecka and Tataru (Differential Integral Equations 8 (1993) 507–533) and Martinez (Rev. Mat. Comput. 12 (1999) 251–283).

Reçu le : 2003-08-06
Accepté le : 2003-10-13
Publié le : 2003-12-09

DOI : 10.1016/j.crma.2003.10.024

Affiliations des auteurs :

Fatiha Alabau-Boussouira ¹

¹ LMAM, CNRS-UMR 7122, Université de Metz, Ile du Saulcy, 57045 Metz cedex 01, France

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Fatiha Alabau-Boussouira. A general formula for decay rates of nonlinear dissipative systems. Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 35-40. doi : 10.1016/j.crma.2003.10.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.024/

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