  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.
