Asymptotic Stabilizability of Underactuated Hamiltonian Systems With Two Degrees of Freedom

TL;DR

This paper establishes that a specific condition for stabilizing underactuated Hamiltonian systems with two degrees of freedom is both necessary and sufficient for asymptotic stabilization without extra assumptions.

Contribution

It proves the necessity of a previously sufficient condition for asymptotic stabilizability in underactuated Hamiltonian systems with two degrees of freedom.

Findings

The stabilizability condition is necessary and sufficient.

No additional assumptions are needed for asymptotic stabilizability.

Extends previous results by removing extra assumptions.

Abstract

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 above mentioned condition is not only sufficient, but also a necessary one. And, more importantly, we show that no additional assumption is needed to ensure asymptotic stabilizability.