On the stability of unforced nonautonomous underdamped dissipative Hamiltonian systems

A sufficient rank condition that guarantees uniform asymptotic stability at the origin for a class of underdamded dissipative nonautonomous Hamiltonian system is derived. To this end, first it is shown that this kind of systems require an observability condition to hold true for asymptotic stability. However, since observability cannot be easily checked for nonautonomous systems, this condition is suitably transformed to a simpler rank constraint. Simultaneously, it is proven that for this class of systems, the time varying eigenvalues should remain on the left half of the complex plane. The theoretical results are finally verified by a simulated example.