Finite Time and Exact Time Controllability on Compact Manifolds

TL;DR

This paper proves that smooth controllable systems on compact manifolds are finite time controllable, introduces new elementary proofs for controllability equivalences, and provides conditions for exact time controllability with applications to Lie groups.

Contribution

It presents a novel proof technique for finite time controllability on compact manifolds and establishes new conditions for exact time controllability, with applications to linear systems on Lie groups.

Findings

Smooth controllable systems on compact manifolds are finite time controllable

Controllability for essentially bounded and piecewise constant inputs are equivalent

Provides conditions for exact time controllability on compact manifolds

Abstract

It is first shown that a smooth controllable system on a compact manifold is finite time controllable. The technique of proof is close to the one of Sussmann's orbit theorem, and no rank condition is required. This technique is also used to give a new and elementary proof of the equivalence between controllability for essentially bounded inputs and for piecewise constant ones. Two sufficient conditions for controllability at exact time on a compact manifold are then stated. Some applications, in particular to linear systems on Lie groups, are provided.