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Control and stabilization of the Benjamin-Ono equation on a periodic domain


Authors: Felipe Linares and Lionel Rosier
Journal: Trans. Amer. Math. Soc. 367 (2015), 4595-4626
MSC (2010): Primary 93B05, 93D15, 35Q53
DOI: https://doi.org/10.1090/S0002-9947-2015-06086-3
Published electronically: March 3, 2015
MathSciNet review: 3335395
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Abstract: It was proved by Linares and Ortega that the linearized Benjamin-Ono equation posed on a periodic domain $ \mathbb{T}$ with a distributed control supported on an arbitrary subdomain is exactly controllable and exponentially stabilizable. The aim of this paper is to extend those results to the full Benjamin-Ono equation. A feedback law in the form of a localized damping is incorporated into the equation. A smoothing effect established with the aid of a propagation of regularity property is used to prove the semi-global stabilization in $ L^2(\mathbb{T})$ of weak solutions obtained by the method of vanishing viscosity. The local well-posedness and the local exponential stability in $ H^s(\mathbb{T})$ are also established for $ s>1/2$ by using the contraction mapping theorem. Finally, the local exact controllability is derived in $ H^s(\mathbb{T})$ for $ s>1/2$ by combining the above feedback law with some open loop control.


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Additional Information

Felipe Linares
Affiliation: Instituto de Matematica Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro 22460-320, Brazil
Email: linares@impa.br

Lionel Rosier
Affiliation: Institut Elie Cartan, UMR 7502 UdL/CNRS/INRIA, B.P. 70239, F-54506 Vandœuvre-lès-Nancy Cedex, France
Address at time of publication: Centre Automatique et Systemes, MINES ParisTech, PSL Research University, 60 boulevard Saint-Michel, 75272 Paris Cedex 06, France
Email: Lionel.Rosier@univ-lorraine.fr, lionel.rosier@mines-paristech.fr

DOI: https://doi.org/10.1090/S0002-9947-2015-06086-3
Keywords: Benjamin-Ono equation, periodic domain, unique continuation property, propagation of regularity, exact controllability, stabilization
Received by editor(s): September 21, 2012
Received by editor(s) in revised form: January 22, 2013
Published electronically: March 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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