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

Camille Laurent (1) , Felipe Linares (2) , Lionel Rosier (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 IMPA - Instituto Nacional de Matemática Pura e Aplicada
3 CAS - Centre Automatique et Systèmes

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.

Domaines: Equations aux dérivées partielles [math.AP]

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⟩