  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,
which includes 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 behavior. 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 {\em impulsive extension} of them
are equivalent.
