Control and Stabilization of the Benjamin-Ono Equation on a Periodic Domain

TL;DR

This paper extends controllability and stabilization results from the linearized to the full Benjamin-Ono equation on a periodic domain, using feedback laws and regularity propagation to achieve semi-global stabilization and local controllability.

Contribution

It introduces a feedback damping law for the full Benjamin-Ono equation and proves semi-global stabilization and local controllability in Sobolev spaces.

Findings

Semi-global stabilization in L^2(T) achieved

Local exponential stability in H^s(T) for s>1/2 established

Local exact controllability in H^s(T) for s>1/2 demonstrated

Abstract

It was proved by Linares and Ortega that the linearized Benjamin-Ono equation posed on a periodic domain 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 in 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) of weak solutions obtained by the method of vanishing viscosity. The local well-posedness and the local exponential stability in H^s(T) are also established for s>1/2 by using the contraction mapping theorem. Finally, the local exact controllability is derived in H^s(T) for s>1/2 by combining the above feedback law with some open loop control.