Control and stabilization of the Benjamin-Ono equation on a periodic domain

Abstract

It was proved by Linares and Ortega that the linearized Benjamin-Ono equation posed on a periodic domain T \mathbb {T} with a distributed control supported on an arbitrary subdomain is exactly controllable and exponentially stabilizable. The aim of this paper is to extend those results to the full Benjamin-Ono equation. A feedback law in the form of a localized damping is incorporated into the equation. A smoothing effect established with the aid of a propagation of regularity property is used to prove the semi-global stabilization in L 2 ( T ) L^2(\mathbb {T}) of weak solutions obtained by the method of vanishing viscosity. The local well-posedness and the local exponential stability in H s ( T ) H^s(\mathbb {T}) are also established for s > 1 / 2 s>1/2 by using the contraction mapping theorem. Finally, the local exact controllability is derived in H s ( T ) H^s(\mathbb {T}) for s > 1 / 2 s>1/2 by combining the above feedback law with some open loop control.