A probabilistic argument for the controllability of conservative systems

TL;DR

This paper proves that divergence-free systems with a conserved quantity satisfying H"ormander's condition are controllable, using a novel probabilistic approach that leverages invariant measures, with implications for statistical mechanics and fluid dynamics.

Contribution

It introduces a probabilistic proof of controllability for conservative systems under H"ormander's condition, extending geometric control theory results with new analytic and probabilistic techniques.

Findings

Controllability follows from H"ormander's condition and invariant measure existence.

Probabilistic proof offers a new perspective on controllability in conservative systems.

Applications include ergodicity in statistical mechanics and Euler equation approximations.

Abstract

We consider controllability for divergence-free systems that have a conserved quantity and satisfy a H\"ormander condition. It is shown that such systems are controllable, provided that the conserved quantity is a proper function. The proof of the result combines analytic tools with probabilistic arguments. While this statement is well-known in geometric control theory, the probabilistic proof given in this note seems to be new. We show that controllability follows from H\"ormander's condition, together with the a priori knowledge of an invariant measure with full topological support for a diffusion that `implements' the control system. Examples are given that illustrate the relevance of the assumptions required for the result to hold. Applications of the result to ergodicity questions for systems arising from non-equilibrium statistical mechanics and to the controllability of Galerkin approximations to the Euler equations are also given.