Class library in C++ to compute Lyapunov functions for nonlinear systems

Abstract

Lyapunov functions are of fundamental importance in the stability analysis of dynamical systems. Unfortunately, the construction of Lyapunov functions for nonlinear systems is in general a very dificult problem. We present software written in C++ that computes Lyapunov functions for n-dimensional, nonlinear, time-continuous dynamical systems defined through systems of autonomous differential equations. The software implements the computation of continuous and piecewise afine (CPA) Lyapunov functions through linear programming (LP) and the computation of Lyapunov functions using radial basis functions (RBF) and collocation. In the former case a common Lyapunov function for a finite set of nonlinear systems can be computed and it is guaranteed to be a true Lyapunov function, i.e. to fulfill the defining properties of a Lyapunov function exactly and rigorously (as opposed to approximately, which is so often the case with numerically contrived results). In the latter case the computed function is smooth and is guaranteed to be a Lyapunov function outside of an arbitrary small neighbourhood of the equilibrium, if the collocation points are close enough. However, it is not obvious how to determine what is close enough. A Lyapunov property test for the CPA interpolation of the computed RBF solution is therefore also part of the software.