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arXiv · 1003.1246

The Closed Orbit Controllability Criterium

Valeri Marenitch
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Abstract

We prove that every closed "general" trajectory of the control system $Σ_M$ has an open neighborhood on which $Σ_M$ is controllable if 1) this orbit contains some point where the Lie algebra rank condition ($LARC$) is satisfied, and 2) the set of control vectors is "involved" at $q$. In particular, for the control systems $Σ_M$ on the compact connected manifold $M^n$ with an open control set this gives the following "Closed Orbit Controllability Criterium": The dynamical system $Σ_M$ of the considered type is controllable on $M^n$ if and only if for an arbitrary point $q$ of $M^n$ there exists a closed trajectory of the control system going through this point. We also present examples which show that our conditions are necessary.

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Valeri Marenitch. 2010-03-05. The Closed Orbit Controllability Criterium. https://arxiv.org/abs/1003.1246

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