A probabilistic argument for the controllability of conservative systems

Martin Hairer

Published 2005-06-24, updated 2008-02-27Version 2

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.