Control systems of zero curvature are not necessarily trivializable

TL;DR

This paper characterizes when control systems with zero curvature can be simplified to a standard form using feedback, providing intrinsic criteria based on feedback invariants and applying results to Zermelo-like problems.

Contribution

It offers a complete intrinsic characterization of trivializable control systems and those with commuting vector fields up to feedback transformations.

Findings

Characterization of trivializable control systems using feedback invariants.

Identification of systems where $f$ and $rac{ ext{d}f}{ ext{d}u}$ commute.

Application of results to Zermelo-like navigation problems on Riemannian manifolds.

Abstract

A control system $\dot{q} = f(q,u)$ is said to be trivializable if there exists local coordinates in which the system is feedback equivalent to a control system of the form $\dot{q} = f(u)$. In this paper we characterize trivializable control systems and control systems for which, up to a feedback transformation, $f$ and $\partial f/\partial u$ commute. Characterizations are given in terms of feedback invariants of the system (its control curvature and its centro-affine curvature) and thus are completely intrinsic. To conclude we apply the obtained results to Zermelo-like problems on Riemannian manifolds.