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.

References in corpus (4)

• Global well-posedness of the Benjamin-Ono equation in H^1(R)
• The Cauchy problem for the Benjamin-Ono equation in $L^2$ revisited
• On the decay properties of solutions to a class of Schrödinger equations
• Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

Cited by in corpus (4)

• Stabilization and control for the biharmonic Schrödinger equation
• On the controllability and Stabilization of the Benjamin Equation
• Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems
• Global control aspects for long waves in nonlinear dispersive media