Archive for Rational Mechanics and Analysis

, Volume 218, Issue 3, pp 1531–1575 | Cite as

Control and Stabilization of the Benjamin-Ono Equation in \({L^2({\mathbb{T})}}\)

Article

Abstract

We study the control and stabilization of the Benjamin-Ono equation in \({L^2(\mathbb{T})}\), the lowest regularity where the initial value problem is well-posed. This problem was already initiated in Linares and Rosier (Trans Am Math Soc 367:4595–4626, 2015) where a stronger stabilization term was used (that makes the equation of parabolic type in the control zone). Here we employ a more natural stabilization term related to the L2–norm. Moreover, by proving a theorem of controllability in L2, we manage to prove the global controllability in large time. Our analysis relies strongly on the bilinear estimates proved in Molinet and Pilod (Anal PDE 5:365–395, 2012) and some new extension of these estimates established here.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Camille Laurent
    • 1
  • Felipe Linares
    • 2
  • Lionel Rosier
    • 3
  1. 1.Laboratoire Jacques-Louis LionsCNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598Paris Cedex 05France
  2. 2.Instituto de Matematica Pura e AplicadaRio de JaneiroBrazil
  3. 3.Centre Automatique et SystèmesMINES ParisTech, PSL Research UniversityParis Cedex 06France

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