The Closed Orbit Controllability Criterium

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We prove that every closed "general" trajectory of the control system Σₘ has an open neighborhood on which Σₘ 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 Σₘ on the compact connected manifold Mⁿ with an open control set this gives the following "Closed Orbit Controllability Criterium": The dynamical system Σₘ of the considered type is controllable on Mⁿ if and only if for an arbitrary point q of Mⁿ there exists a closed trajectory of the control system going through this point. We also present examples which show that our conditions are necessary.