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For an underactuated (simple) Hamiltonian system with two degrees of freedomand one degree of underactuation, a rather general condition that ensures itsstabilizability, by means of the existence of a (simple) Lyapunov function, wasfound in a recent paper by D.E. Chang within the context of the energy shapingmethod. Also, in the same paper, some additional assumptions were presented inorder to ensure also asymptotic stabilizability. In this paper we extend theseresults by showing that above mentioned condition is not only sufficient, butalso a necessary one. And, more importantly, we show that no additionalassumption is needed to ensure asymptotic stabilizability.

Asymptotic Stabilizability of Underactuated Hamiltonian Systems With Two Degrees of Freedom