Converse Lyapunov theorems for control systems with unbounded controls

In this paper we extend well-known relationships between global asymptotic controllability, sample stabilizability, and the existence of a control Lyapunov function to a wide class of control systems with unbounded controls, including control-polynomial systems. In particular, we consider open loop controls and discontinuous stabilizing feedbacks, which may be unbounded approaching the target, so that the corresponding trajectories may present a chattering behaviour. A key point of our results is to prove that global asymptotic controllability, sample stabilizability, and existence of a control Lyapunov function for these systems or for an impulsive extension of them are equivalent.