The Closed Orbit Controllability Criterium

TL;DR

This paper establishes a criterion for controllability of control systems on manifolds based on the properties of closed trajectories and the Lie algebra rank condition, providing a practical test for controllability.

Contribution

It introduces a new controllability criterion based on closed trajectories and the Lie algebra rank condition, with necessary examples and applications to systems on manifolds.

Findings

Controllability on a manifold is characterized by the existence of closed trajectories through every point.

The Lie algebra rank condition at some point on a trajectory implies local controllability.

The criterion is both necessary and sufficient for systems with an open control set on compact connected manifolds.

Abstract

We prove that every closed "general" trajectory of the control system $\Sigma_M$ has an open neighborhood on which $\Sigma_M$ is controllable if 1) this orbit contains some point where the Lie algebra rank condition ($LARC$) is satisfied, and 2) the set of control vectors is "involved" at $q$. In particular, for the control systems $\Sigma_M$ on the compact connected manifold $M^n$ with an open control set this gives the following "Closed Orbit Controllability Criterium": The dynamical system $\Sigma_M$ of the considered type is controllable on $M^n$ if and only if for an arbitrary point $q$ of $M^n$ there exists a closed trajectory of the control system going through this point. We also present examples which show that our conditions are necessary.