Stability analysis of nonlinear systems via locally defined density functions

This paper considers local stability analysis of nonlinear systems with deriving a positively invariant set based on Rantzer´s stability theory by using density functions. We define notion of locally defined density functions around an equilibrium that give monotonously increasing positive measures near the equilibrium of a nonlinear system. Under several assumptions, it is shown that some level set of a locally defined density function is a positively invariant set where almost all of the system trajectories converge to the equilibrium.