A Lyapunov-density criterion for almost everywhere stability of a class of Lipschitz continuous and almost everywhere C1 nonlinear systems

This paper is concerned with the analysis of almost everywhere stability (AES) of a class of nonlinear systems via Lyapunov densities. While many of the previous results on Lyapunov densities assume that the vector field is continuously differentiable, this paper deals with nonlinear systems that can have points of non-differentiability. A key to prove AES is monotonicity of a measure defined with a Lyapunov density to deduce that the set of non-convergent initial values has zero Lebesgue measure. The monotonicity condition of a measure involves a change of variables in integrals of the Lyapunov density over a set. Such a change of variables along trajectories is not available for vector fields that are not globally continuously differentiable, even though they are smooth at almost all points of the state space. In this paper, an alternative inequality is presented that proves monotonicity of a measure along trajectories of a class of nonlinear systems that are Lipschitz continuous and almost everywhere continuously differentiable. Based on this inequality, we provide a sufficient condition for AES while guaranteeing positive invariance of a specified region.