Global Controllability of the Kawahara Equation at Any Time

TL;DR

This paper proves that the nonlinear Kawahara equation on a periodic domain can be approximately controlled at any time using a two-dimensional control force, employing geometric control theory methods.

Contribution

It establishes the global approximate controllability of the nonlinear Kawahara equation at any time with a novel application of geometric control theory.

Findings

Proves global approximate controllability of Kawahara equation.

Uses Agrachev-Sarychev approach in geometric control theory.

Achieves controllability with a two-dimensional control force.

Abstract

In this article, we prove that the nonlinear Kawahara equation on the periodic domain \(\mathbb{T}\) (the unit circle in the plane) is globally approximately controllable in \(H^s(\mathbb{T})\) for \(s \in \mathbb{N}\), at any time \(T > 0\), using a two-dimensional control force. The proof is based on the Agrachev-Sarychev approach in geometric control theory.