Analysis of Positive Invariance and Almost Regional Attraction Via Density Functions With Converse Results

This note is concerned with analysis of positive invariance of nonlinear systems and convergence of trajectories in a region of the state-space via density functions. If there exists a density function that is positive inside a set containing an equilibrium point and tends to zero as approaches to the boundary, the set is positively invariant and almost all of the trajectories starting from there converge to the equilibrium. Converse results are also provided to prove the existence of such density functions.