A sampling approach to finding Lyapunov functions for nonlinear discrete-time systems

This paper considers the problem of stability verification for discrete–time 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 approach to stability analysis of nonlinear systems. 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 exploiting continuity properties. This result involves no apriori analytic description of the continuity property and it is based on multi–resolution sampling, to perform efficient state–space exploration. Moreover, the stability verification is decentralized in the sampling points, which makes the method parallelizable. The proposed methodology is illustrated on two examples.