Modified refinement algorithm to construct Lyapunov functions using meshless collocation

Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sublevel sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. One of the numerical construction methods for Lyapunov functions uses meshless collocation with radial basis functions.Recently, this method was combined with a grid refinement algorithm (GRA) to reduce the number of collocation points needed to construct Lyapunov functions. However, depending on the choice of the initial set of collocation point, the algorithm can terminate, failing to compute a Lyapunov function. In this paper, we propose a modified grid refinement algorithm (MGRA), which overcomes these shortcomings by adding appropriate collocation points using a clustering algorithm. The modified algorithm is applied to two- and three-dimensional examples.