Computation of complete Lyapunov functions for three-dimensional systems

Complete Lyapunov functions are of much interest in control theory because of their capability to describe the long-time behaviour of nonlinear dynamical systems. The state-space of a system can be divided in two different regions determined by a complete Lyapunov function: the region of the gradient-like flow, where the Lyapunov function is strictly decreasing along solution trajectories, and the chain-recurrent set whose chain-transitive components are level sets of the Lyapunov function. There has been continuous effort to properly identify both regions and in this paper we discuss the extension of our methods to compute complete Lyapunov functions in the plane to the three-dimensional case, which is directly applicable to higher dimensions, too. When extending the methods to higher dimensions, the number of points for collocation and evaluation grows exponentially. To keep the number of evaluation points under control, we propose a new way to choose them, which does not depend on the dimension.