Asymptotic Stabilizability of Underactuated Hamiltonian Systems With Two Degrees of Freedom

For an underactuated (simple) Hamiltonian system with two degrees of freedom and one degree of underactuation, a rather general condition that ensures its stabilizability, by means of the existence of a (simple) Lyapunov function, was found in a recent paper by D.E. Chang within the context of the energy shaping method. Also, in the same paper, some additional assumptions were presented in order to ensure also asymptotic stabilizability. In this paper we extend these results by showing that the above-mentioned condition is not only sufficient, but also necessary. And, more importantly, we show that no additional assumption is needed to ensure asymptotic stabilizability.