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

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 $\mathfrak{g}$ 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.

18 pages