Approximate controllability on the group of volume-preserving diffeomorphisms

TL;DR

This paper investigates the approximate controllability of volume-preserving diffeomorphisms on the torus, focusing on systems driven by fixed divergence-free fields and constant translation controls in two and three dimensions.

Contribution

It introduces new results on controllability for volume-preserving diffeomorphisms on the torus, specifically for systems with translation controls in 2D and 3D.

Findings

Controllability results established for 2D and 3D cases.

Analysis of translation-based control systems on the torus.

Insights into the structure of volume-preserving diffeomorphisms.

Abstract

We study controlability issues for the group of volume-preserving diffeomorphisms of the torus $\mathbb T^d$ for system $\dot x=f(x)+u(t)$, where $f$ is a fixed divergence free vector field on $\mathbb T^d$ and $u(t)$ are constant vector fields which generate translations of the torus. Main results concern $d$ equals two or three.