Comptes Rendus
no. 6
Partial differential equations/Optimal control
Local indirect stabilization of N–d system of two coupled wave equations under geometric conditions
[Stabilisation locale indirecte d'un système N–d de deux équations d'ondes couplées sous conditions géométriques]

Présenté par : Jean-Michel Coron

Chiraz Kassem1, 2 ; Amina Mortada3 ; Layla Toufayli3 ; Ali Wehbe3
1 Université libanaise, EDST & Hadath, Beyrouth, Liban
2 Université Savoie Mont Blanc, Laboratoire de mathématiques LAMA, bâtiment Le Chablais, campus scientifique, 73376 Le Bourget-du-Lac cedex, France
3 Université libanaise, Faculté des sciences 1 et EDST & Hadath, Beyrouth, Liban
Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 494-512.

Nous nous intéressons à la stabilisation d'un système multidimensionnel de deux équations d'ondes couplées par les termes de vitesse, et dont l'une seulement est localement amortie. La principale nouveauté contenue dans cette note est que les ondes ne se propagent pas forcément à la même vitesse et que le coefficient de couplage n'est pas supposé être positif et petit. Nous supposons que la zone de couplage et la zone d'amortissement s'intersectent. D'abord, nous montrons que notre système est fortement stable sans conditions géométriques. Puis nous étudions le taux de décroissance de l'énergie en distinguant deux cas. Dans le premier cas, nous supposons que les ondes se propagent à la même vitesse ; nous établissons alors, sous certaines conditions géométriques, un taux de décroissance exponentiel de l'énergie du système pour des données initiales usuelles. Dans le second cas, nous montrons d'abord que l'énergie ne décroît pas exponentiellement vers 0. Ensuite, sous les mêmes conditions géométriques, nous établissons un taux de décroissance polynomial de type 1t pour des données initiales régulières. Finalement, dans le cas particulier où la dimension de l'espace est égale à 1, en utilisant la partie réelle du développement asymptotique des valeurs propres de système, nous montrons, de plus, que le taux polynomial obtenu est optimal.

The purpose of this note is to investigate the stabilization of a system of two wave equations coupled by velocities with only one localized damping. The main novelty in this note is that the waves are not necessarily propagating at same speed and the coupling coefficient is not assumed to be positive and small. Assume that the coupling region and the damping region intersect. We prove that our system is strongly stable without geometric conditions. We then study the energy decay rate by distinguishing two cases. The first one is when the waves propagate at the same speed. In this case, under appropriate geometric conditions, we establish an exponential energy decay estimate for usual initial data. For the other case, we first show that our system is not uniformly stable. Next, under the same geometric conditions, we establish a polynomial energy decay of type 1t for smooth initial data. Finally, in one space dimension, using the real part of the asymptotic expansion of eigenvalues of the system, we prove that the obtained polynomial decay rate is optimal.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.06.002
Chiraz Kassem 1, 2 ; Amina Mortada 3 ; Layla Toufayli 3 ; Ali Wehbe 3

1 Université libanaise, EDST & Hadath, Beyrouth, Liban
2 Université Savoie Mont Blanc, Laboratoire de mathématiques LAMA, bâtiment Le Chablais, campus scientifique, 73376 Le Bourget-du-Lac cedex, France
3 Université libanaise, Faculté des sciences 1 et EDST & Hadath, Beyrouth, Liban 
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Chiraz Kassem; Amina Mortada; Layla Toufayli; Ali Wehbe. Local indirect stabilization of N–d system of two coupled wave equations under geometric conditions. Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 494-512. doi : 10.1016/j.crma.2019.06.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.06.002/

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