Piecewise Smooth Lyapunov Function for a Nonlinear Dynamical System

We study stability and attraction for a nonlinear dynamical system with nonsmooth Lyapunov functions. The previous results on stability and attraction with a max-type Lyapunov function are extended to the case where Lyapunov function is piecewise smooth. A condition, under which stability and attraction are guaranteed with a piecewise smooth Lyapunov function, is proposed. Taking two certain classes of piecewise smooth functions as Lyapunov functions, related conditions for stability and attraction are developed.