This Note is concerned with the links between nonlinear stabilization of hyperbolic systems and linear observability for the unforced corresponding linear system, for locally distributed and boundary feedbacks as well. We show that if the linear system is observable through a locally distributed (resp. boundary) observation, then any dissipative nonlinear feedback locally distributed (resp. active only on a part of the boundary) stabilize the system and we give a general energy decay formula. Our results generalize previous results by Haraux (1989) and Ammari and Tucsnak (2001) for linear feedbacks. We show by this way that for the locally distributed case, one can combine the optimal geometric conditions of Bardos et al. (1992) and the method of Alabau-Boussouira (2005) to deduce energy decay rates for nonlinear damped systems.
On étudie dans cette Note, le problème de la stabilisation par rétro-action non linéaire localement distribuée ou frontière d'équations abstraites, comme conséquence d'une inégalité d'observabilité pour le problème linéaire associé sans rétro-action. On montre sous des hypothèses très générales sur le feedback, notamment sans hypothèse de croissance à l'origine, que si le système conservatif linéaire est observable par une observation localement distribuée (resp. par une observation frontière), dans l'espace d'énergie naturelle (resp. dans le domaine de l'opérateur), alors tout feedback non linéaire localement distribué (resp. frontière) stabilise le système et on donne un taux de décroissance de l'énergie quasi-optimal. On donne des exemples d'application de ces résultats à des EDP. On montre ainsi, pour le cas localement distribué, que l'on peut combiner les hypothèses géométriques optimales de Bardos et al. (1992) et la méthode de Alabau-Boussouira (2005) pour déduire des résultats de stabilisation non linéaire.
Accepted:
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Fatiha Alabau-Boussouira 1; Kaïs Ammari 2
@article{CRMATH_2010__348_3-4_165_0, author = {Fatiha Alabau-Boussouira and Ka{\"\i}s Ammari}, title = {Nonlinear stabilization of abstract systems via a linear observability inequality and application to vibrating {PDE's}}, journal = {Comptes Rendus. Math\'ematique}, pages = {165--170}, publisher = {Elsevier}, volume = {348}, number = {3-4}, year = {2010}, doi = {10.1016/j.crma.2009.12.009}, language = {en}, }
TY - JOUR AU - Fatiha Alabau-Boussouira AU - Kaïs Ammari TI - Nonlinear stabilization of abstract systems via a linear observability inequality and application to vibrating PDE's JO - Comptes Rendus. Mathématique PY - 2010 SP - 165 EP - 170 VL - 348 IS - 3-4 PB - Elsevier DO - 10.1016/j.crma.2009.12.009 LA - en ID - CRMATH_2010__348_3-4_165_0 ER -
%0 Journal Article %A Fatiha Alabau-Boussouira %A Kaïs Ammari %T Nonlinear stabilization of abstract systems via a linear observability inequality and application to vibrating PDE's %J Comptes Rendus. Mathématique %D 2010 %P 165-170 %V 348 %N 3-4 %I Elsevier %R 10.1016/j.crma.2009.12.009 %G en %F CRMATH_2010__348_3-4_165_0
Fatiha Alabau-Boussouira; Kaïs Ammari. Nonlinear stabilization of abstract systems via a linear observability inequality and application to vibrating PDE's. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 165-170. doi : 10.1016/j.crma.2009.12.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.12.009/
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