Controllability of control systems simple Lie groups and the topology of flag manifolds

TL;DR

This paper establishes conditions under which a subsemigroup of a complex simple Lie group equals the entire group, using topological and geometric properties of flag manifolds and orbit structures.

Contribution

It proves that containing a specific subgroup generated by root spaces ensures the subsemigroup is the whole group, improving controllability results.

Findings

Subsemigroup with a subgroup G(α) equals G.

Invariant control set is contractible in flag manifold.

Several orbits are 2-spheres not null homotopic.

Abstract

Let $S$ be subsemigroup with nonempty interior of a complex simple Lie group $G$. It is proved that $S=G$ if $S$ contains a subgroup $G(\alpha) \approx \mathrm{Sl}(2,\mathbb{C}) $ generated by the $\exp \mathfrak{g}_{\pm \alpha}$, where $\mathfrak{g}%_{\alpha}$ is the root space of the root $\alpha $. The proof uses the fact, proved before, that the invariant control set of $S$ is contractible in some flag manifold if $S$ is proper, and exploits the fact that several orbits of $G(\alpha)$ are 2-spheres not null homotopic. The result is applied to revisit a controllability theorem and get some improvements.