Vol. 2, No. 3, 2020

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Stabilization of wave equations on the torus with rough dampings

Nicolas Burq and Patrick Gérard

Vol. 2 (2020), No. 3, 627–658

For the damped wave equation on a compact manifold with continuous dampings, the geometric control condition is necessary and sufficient for uniform stabilization. On the two-dimensional torus, in the special case where a(x) = j=1Naj1xRj (Rj are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilization. We also propose a natural generalization of the geometric control condition which makes sense for L dampings. We show that this condition is always necessary for uniform stabilization (for any compact (smooth) manifold and any L damping), and we prove that it is sufficient in our particular case on 𝕋2 (and for our particular dampings).

Pour l’équation des ondes amortie sur une variété compacte, dans le cas d’un amortissement continu, la condition de contrôle géométrique est nécessaire et suffisante pour la stabilisation uniforme. Sur le tore 𝕋2 et dans le cas où a(x) = j=1Naj1xRj (Rj sont des polygones), nous exhibons une condition géométrique nécessaire et suffisante très simple. Nous proposons aussi une généralisation naturelle de la condition de contrôle géométrique, pour un amortissement seulement L . Cette généralisation est toujours nécessaire pour la stabilisation uniforme (sur toute variété compacte régulière), et nous démontrons qu’elle est suffisante dans notre cas particulier du tore 𝕋2 (et pour nos fonctions d’amortissement particulières).

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wave equations, control, stabilization, second microlocalization, geometric control
Mathematical Subject Classification 2010
Primary: 34L20, 35L05, 35Q93, 35S05, 58J40, 93D15
Received: 9 December 2019
Revised: 2 February 2020
Accepted: 15 March 2020
Published: 17 November 2020
Nicolas Burq
Laboratoire de Mathématiques d’Orsay
Université Paris-Saclay
France CNRS, Institut Universitaire de France
Patrick Gérard
Laboratoire de Mathématiques d’Orsay
Université Paris-Saclay
CNRS, UMR 8628