Lyapunov density for almost attraction of nonlinear time-varying systems: A condition without assuming local stability

Stability analysis of nonlinear systems based on Lyapunov densities has been receiving a considerable amount of attention after the original work of Rantzer for almost attraction of nonlinear time-invariant systems. Though a stability criterion has been proposed also for nonlinear time-varying systems, the assumption of local stability of the equilibrium is needed in the previous result. The purpose of this paper is to provide a condition in terms of Lyapunov densities under which the nonlinear time-varying system is almost attractive, i. e. almost all of the trajectories of the system converge to the equilibrium, where the proposed condition does not need an assumption on local stability of the system.