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Invariant control sets on flag manifolds and ideal boundaries of symmetric spaces. (English) Zbl 1030.22005
The concept of invariant control sets of a semigroup is transferred to the case where $$G$$ is a semisimple real Lie group of non-compact type with finite center and $$S\subseteq G$$ a semigroup with nonempty interior. If $$P\subseteq G$$ is a parabolic subgroup, the homogeneous space $$G/P$$ is a compact manifold called generalized flag manifold. The analysis of invariant control sets for the left action of $$S$$ on $$G/P$$ relies on a basic result insuring the existence of a unique invariant control set defined by the points fixed by elements in the interior of $$S$$. In this paper the invariant control set is determined of a semigroup $$S\subseteq G$$ in the ideal boundary $$\partial_\infty (Sx_0)$$ consisting of the points in the ideal boundary $$\partial_\infty (G/K)$$ contained in the closure on any orbit $$Sx_0$$, where $$x_0$$ is an arbitrary point of the symmetric space $$G/K$$. Two main theorems are proved and two examples are added.
##### MSC:
 22E46 Semisimple Lie groups and their representations 93B29 Differential-geometric methods in systems theory (MSC2000) 49K15 Optimality conditions for problems involving ordinary differential equations
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