A sampling approach to constructing Lyapunov functions for nonlinear continuous-time systems

The problem of constructing a Lyapunov function for continuous-time nonlinear dynamical systems is tackled in this paper via a sampling-based approach. The main idea of the sampling-based method is to verify a Lyapunov-type inequality for a finite number of points (known state vectors) in the state-space and then to extend the validity of the Lyapunov inequality to a neighborhood around these points. In this way, the validity of a Lyapunov function candidate can be certified for a region of interest in the state-space in a systematic way. A candidate Lyapunov function is computed for each sample point using a recent converse Lyapunov theorem for continuous-time nonlinear systems. For certifying the candidate Lyapunov function on a neighborhood of the sampling point we propose both a deterministic and a probabilistic approach. The deterministic approach provides a formal guarantee at the cost of verifying a more conservative Lyapunov inequality, which is not valid in a neighborhood of the origin. The probabilistic approach verifies the original Lyapunov inequality and provides a probabilistic guarantee in terms of a reliability estimate. An example from the literature illustrates the proposed sampling-based approach.