Density Functions Characterizing Positive Invariance and Regional Almost Attraction with Converse Results

This paper is concerned with analysis of regional almost attraction and positive invariance of nonlinear systems by using density functions. If there exists a density function that is positive inside a set containing an equilibrium point and tends to zero as it approaches the boundary, the set is positively invariant and almost all of the trajectories starting from there converge to the equilibrium. This can be applied to synthesis of nonlinear static state feedback gains under state and input constraints via convex optimization. A weak derivative of measure is introduced to allow a wider class of density functions. Moreover, converse results are also provided to prove the existence of density functions