Invariance Entropy of Control Sets

Invariance entropy for continuous-time control systems measures how often open-loop controls have to be updated in order to render a compact and controlled invariant subset of the state space invariant. A special type of a controlled invariant set for a control-affine system is the closure of a control set, i.e., a maximal set of approximate controllability. In this paper, we investigate the properties of the invariance entropy of such sets. Our main result gives an upper bound of this quantity in terms of the positive Lyapunov exponents of a periodic solution in the interior of the control set. Moreover, for one-dimensional systems with a single control vector field we provide an analytical formula for the invariance entropy of a control set in terms of the drift vector field, the control vector field and their derivatives. As an application, we study a controlled linear oscillator