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We propose a class of locally Lipschitz functions with piecewise structurefor use as Lyapunov functions for hybrid dynamical systems. Subject to someregularity of the dynamics, we show that Lyapunov inequalities can be checkedonly on a dense set and thus we avoid checking them at points ofnondifferentiability of the Lyapunov function. Connections to other classes oflocally Lipschitz or piecewise regular functions are also discussed andapplications to hybrid dynamical systems are included.

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