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 equilib-rium. 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 func-tions. 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 over-comes these shortcomings by adding appropriate collocation points using a clustering algorithm. The modified algorithm is applied to two-and three-dimensional examples.