  This paper considers a sampling-based approach to stability verification for
piecewise continuous nonlinear systems via Lyapunov functions. Depending on the
system dynamics, the candidate Lyapunov function and the set of initial states
of interest, one generally needs to handle large, possibly non-convex or
non-feasible optimization problems. To avoid such problems, we propose a
constructive and systematically applicable sampling-based method to Lyapunov's
inequality verification. This approach proposes verification of the decrease
condition for a candidate Lyapunov function on a finite sampling of a bounded
set of initial conditions and then it extends the validity of the Lyapunov
function to an infinite set of initial conditions by automatically exploiting
continuity properties. This result is based on multi-resolution sampling, to
perform efficient state- space exploration. Using hyper-rectangles as basic
sampling blocks, to account for different constraint scales on different
states, further reduces the amount of samples to be verified. Moreover, the
verification is decentralized in the sampling points, which makes the method
scalable. The proposed methodology is illustrated through examples.
