Control and Stabilization of the Benjamin-Ono Equation in ${L^2({\mathbb{T})}}$ - Archive ouverte HAL Access content directly
Journal Articles Archive for Rational Mechanics and Analysis Year : 2015

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

Abstract

We prove the control and stabilization of the Benjamin-Ono equation in $L^2(\T)$, the lowest regularity where the initial value problem is well-posed. This problem was already initiated in \cite{LinaresRosierBO} 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 \cite{MolinetPilodBO} and some new extension of these estimates established here.
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Dates and versions

hal-00945105 , version 1 (11-02-2014)

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Camille Laurent, Felipe Linares, Lionel Rosier. Control and Stabilization of the Benjamin-Ono Equation in ${L^2({\mathbb{T})}}$. Archive for Rational Mechanics and Analysis, 2015, 218 (3), pp.1531 - 1575. ⟨10.1007/s00205-015-0887-5⟩. ⟨hal-00945105⟩
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