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

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

2011-04-26

arxiv.org/abs/1104.5030v1

Mathematics

Optimization and Control

18 pages

Scientific paper

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.

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