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

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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 L 2–norm. Moreover, by proving a theorem of controllability in L 2, 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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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Boite courrier 187, 75252, Paris Cedex 05, France

    Camille Laurent

  2. Instituto de Matematica Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil

    Felipe Linares

  3. Centre Automatique et Systèmes, MINES ParisTech, PSL Research University, 60 boulevard Saint-Michel, 75272, Paris Cedex 06, France

    Lionel Rosier

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  1. Camille Laurent
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  2. Felipe Linares
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  3. Lionel Rosier
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Correspondence to Felipe Linares.

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Communicated by A. Bressan

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Laurent, C., Linares, F. & Rosier, L. Control and Stabilization of the Benjamin-Ono Equation in \({L^2({\mathbb{T})}}\) . Arch Rational Mech Anal 218, 1531–1575 (2015). https://doi.org/10.1007/s00205-015-0887-5

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  • DOI: https://doi.org/10.1007/s00205-015-0887-5

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