Relaxed sufficient conditions for asymptotic stability for a class of underdamped nonautonomous Hamiltonian systems

The asymptotic stability to the origin for a class of underdamped, nonautonomous, nonlinear, Hamiltonian systems is investigated. For this class of systems, the step from stability to uniform asymptotic stability needs some hard additional conditions to hold true. In principle, these conditions are of observability type, that are difficult to be checked due to the nonlinearities and nonautonomous nature of the system. In this paper, in Proposition 1 and Corollary 1, it is proven that the required uniform observability condition can be expressed as a sufficient, time-running, matrix-rank condition. However, since to check this condition is not always an easy task, a simple but much more constrained sufficient, time-invariant rank condition can be obtained as presented in Proposition 2. An illustrative example is performed and simulated to verify the theoretical analysis.