Conditions to the density of accessible sets

TL;DR

This paper investigates conditions under which the accessible sets of a control system are dense in a manifold, using stochastic differential equations and the support theorem to relate controllability to diffusion properties.

Contribution

It provides a new characterization of density of accessible sets via the infinitesimal generator and invariant measures, and offers a different proof of Krener's theorem.

Findings

Accessible sets can be dense under certain conditions related to the diffusion's support.

The support theorem links control system properties to stochastic processes.

A novel proof of Krener's theorem is presented.

Abstract

Given a control system $\dot{p} = X_0(p) + \sum_i u_i (t)X_i(p)$ on a compact manifold M we study conditions for the foliation defined by the accessible sets be dense in M . To do this we relate the control system to a stochastic differential equation and, using the support theorem, we give a characterization of the density in terms of the infinitesimal generator of the diffusion and its invariant measures. Also we give a different proof of Krener's theorem.