Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves
[Transformation de backstepping de type Fredholm pour des opérateurs critiques et application à la stabilisation rapide de l’équation linéarisée des « water waves »]
Gagnon, Ludovick1 ; Hayat, Amaury2 ; Xiang, Shengquan3 ; Zhang, Christophe1
1 Université de Lorraine, CNRS, Inria équipe SPHINX, F-54000 Nancy (France)
2 CERMICS, École des Ponts ParisTech, 6 - 8, Avenue Blaise Pascal, Cité Descartes-Champs sur Marne, 77455 Marne la Vallée (France)
3 School of Mathematical Sciences, Peking University, 100871, Beijing, (P. R. China)
Annales de l'Institut Fourier, Online first, 78 p.

La transformation de backstepping de type Fredholm, introduite par Coron et Lü, s’est rapidement développée au cours de la dernière décennie et est devenue un outil puissant pour montrer la stabilisation rapide d’une équation. Sa force réside dans son approche systématique, permettant de déduire la stabilisation rapide à partir de la contrôlabilité approchée. L’existence d’une telle transformation, reposant de façon clé sur l’existence d’une base de Riesz associée, est cependant limitée à des opérateurs de la forme |D x | α pour α>3/2. Nous présentons ici une nouvelle méthode de compacité/dualité, s’appuyant sur l’alternative de Fredholm, pour franchir ce seuil α=3/2 dans le cas d’opérateurs anti-adjoint satisfaisait α>1. L’illustration de cette nouvelle méthode est démontrée pour obtenir la stabilisation rapide de l’équation linéarisée des water waves présentant un opérateur d’ordre critique α=3/2.

Fredholm-type backstepping transformation, introduced by Coron and Lü, intensively developed over the last decade and has become a powerful tool for rapid stabilization. Its strength lies in its systematic approach, allowing to deduce rapid stabilization from approximate controllability. But limitations to prove the existence of a Fredholm backstepping transformation exist with the current approach for operators of the form |D x | α for α∈(1,3/2]. We present here a new compactness/duality method which hinges on Fredholm’s alternative to overcome the α=3/2 threshold. More precisely, the compactness/duality method allows to prove the existence of a Riesz basis for the backstepping transformation for skew-adjoint operators satisfying α>1, a key step in the construction of the Fredholm backstepping transformation, where the usual methods only work for α>3/2. The illustration of this new method is shown on the rapid stabilization of the linearized capillary-gravity water waves equation exhibiting an operator of critical order α=3/2.

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DOI : 10.5802/aif.3690
Classification : 35S50, 76B15, 93B05
Keywords: Water waves, compactness/duality method, Fredholm transformation, backstepping, rapid stabilization
Mots-clés : Capillarité/gravité, méthode de dualité/compacité, transformation de Fredholm, backstepping, stabilisation rapide

Gagnon, Ludovick 1 ; Hayat, Amaury 2 ; Xiang, Shengquan 3 ; Zhang, Christophe 1

1 Université de Lorraine, CNRS, Inria équipe SPHINX, F-54000 Nancy (France)
2 CERMICS, École des Ponts ParisTech, 6 - 8, Avenue Blaise Pascal, Cité Descartes-Champs sur Marne, 77455 Marne la Vallée (France)
3 School of Mathematical Sciences, Peking University, 100871, Beijing, (P. R. China) 
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Gagnon, Ludovick; Hayat, Amaury; Xiang, Shengquan; Zhang, Christophe. Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves. Annales de l'Institut Fourier, Online first, 78 p.

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