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Non-overlapping control systems on Lie groups. (English) Zbl 0819.22005

An invariant control system on a Lie group \(G\) is a family \(\Omega\) of (right) invariant vector fields on \(G\), i.e., a subset of the Lie algebra \(L(G)\) of \(G\). A trajectory of such a control system is an absolutely continuous curve \(x(t)\) in \(G\) whose derivative takes the value of an element of \(\Omega\) wherever it exists. An element \(g \in G\) is said to be attainable in time \(t\) if there exists a trajectory \(x\) with \(x(0) = e\), the identity of \(G\), and \(x(t) = g\). The system is called non- overlapping if any point in \(G\) is attainable for at most one \(t\in \mathbb{R}^ +\). The main results of the paper state that only very special systems can be non-overlapping. More precisely, very mild hypotheses suffice to show that bounded non-overlapping systems are just the ones contained in an affine subspace \(X + E \subseteq L(G)\) of codimension one such that \(X \notin E\) and \(E\) contains the commutator algebra of \(L(G)\).

MSC:

22E15 General properties and structure of real Lie groups
93B27 Geometric methods
22E60 Lie algebras of Lie groups
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References:

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