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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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