Sampling–based verification of Lyapunov’s inequality for piecewise continuous nonlinear systems

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.