Bilinear Systems Induced by Proper Lie Group Actions

TL;DR

This paper introduces a relaxed rank condition for bilinear systems induced by proper Lie group actions, simplifying controllability analysis by focusing on a single point rather than the entire manifold or classifying Lie algebras.

Contribution

It establishes that for such systems, the classical Lie algebra rank condition can be relaxed to a single-point check, avoiding complex classifications.

Findings

Relaxed rank condition suffices for controllability analysis.

Controllable submanifolds characterized by group orbits.

Simplifies analysis by avoiding full Lie algebra classification.

Abstract

In the study of induced bilinear systems, the classical Lie algebra rank condition (LARC) is known to be impractical since it requires computing the rank everywhere. On the other hand, the transitive Lie algebra condition, while more commonly used, relies on the classification of transitive Lie algebras, which is elusive except for few simple geometric objects such as spheres. We prove in this note that for bilinear systems induced by proper Lie group actions, the underlying Lie algebra is closely related to the orbits of the group action. Knowing the pattern of the Lie algebra rank over the manifold, we show that the LARC can be relaxed so that it suffices to check the rank at an arbitrary single point. Moreover, it removes the necessity for classifying transitive Lie algebras. Finally, this relaxed rank condition also leads to a characterization of controllable submanifolds by orbits.