The Benjamin-Ono equation with 2D control input: approximate controllability and its application

TL;DR

This paper proves approximate controllability for the nonlinear Benjamin-Ono equation on a torus using 2D control inputs and explores the long-term behavior of solutions under random forcing.

Contribution

It adapts geometric control methods to establish controllability for a nonlinear PDE and analyzes the unboundedness of trajectories under stochastic forcing.

Findings

Approximate controllability in L^2 for the Benjamin-Ono equation established.

Trajectories are almost surely unbounded in Sobolev norms under nondegenerate random forcing.

Abstract

We establish the approximate controllability in $L^2$ for the nonlinear Benjamin-Ono equation on torus via two-dimensional control input. Our proof is based on adaptations of geometric control approach introduced by Agrachev and Sarychev. As an application of this control result, we study long-time dynamics of a randomly forced equation. It is proved that the trajectories are unbounded in Sobolev norms almost surely, when the random force is nondegenerate and statistically periodic in time.