Global Approximate Controllability of the Camassa-Holm Equation by a Finite Dimensional Force

TL;DR

This paper demonstrates that the Camassa-Holm equation on a periodic domain can be approximately controlled globally using a three-dimensional external force, employing geometric control theory methods.

Contribution

It establishes the global approximate controllability of the Camassa-Holm equation with a finite-dimensional force, a novel result in control theory for this PDE.

Findings

Global approximate controllability achieved with a 3D force

Uses Agrachev-Sarychev geometric control approach

Applicable for solutions in $H^s$ with $s > 3/2$

Abstract

In this paper, we consider the Camassa-Holm equation posed on the periodic domain $\mathbb{T}$. We show that Camassa-Holm equation is globally approximately controllable by three dimensional external force in $H^s(\mathbb{T})$ for $ s > \frac{3}{2}$ . The proof is based on Agrachev-Sarychev approach in geometric control theory.