The Closed Orbit Controllability Criterium

Details The Closed Orbit Controllability Criterium The Closed Orbit Controllability Criterium

2010-03-05

arxiv.org/abs/1003.1246v1

Mathematics

Dynamical Systems

16 page

Scientific paper

We prove that every closed "general" trajectory of the control system $\Sigma_M$ has an open neighborhood on which $\Sigma_M$ 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 $\Sigma_M$ on the compact connected manifold $M^n$ with an open control set this gives the following "Closed Orbit Controllability Criterium": The dynamical system $\Sigma_M$ 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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