Piecewise structure of Lyapunov functions and densely checked decrease conditions for hybrid systems

We propose a class of locally Lipschitz functions with piecewise structure for use as Lyapunov hybrid dynamical systems. Subject to some regularity the dynamics, we show that inequalities can be checked only on dense set and thus avoid checking them at points nondifferentiability function. Connections other classes or regular are also discussed, applications systems included.