Abstract
Let S be a subsemigroup with nonempty interior of a connected complex simple Lie group G. It is proved that S = G if S contains a subgroup G(α) ≈ Sl (2, \( \mathbb{C} \)) generated by the exp \( {{\mathfrak{g}}_{{\pm \alpha }}} \), where \( {{\mathfrak{g}}_{\alpha }} \) is the root space of the root α. 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(α) are 2-spheres not null homotopic. The result is applied to revisit a controllability theorem and get some improvements.
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dos Santos, A.L., Martin, L.A.B.S. Controllability of control systems on complex simple lie groups and the topology of flag manifolds. J Dyn Control Syst 19, 157–171 (2013). https://doi.org/10.1007/s10883-013-9168-5
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DOI: https://doi.org/10.1007/s10883-013-9168-5