Lyapunov functions are functions with negative orbital derivative, whose existence guarantee the stability of an equilibrium point of an ODE. Moreover, sub-level sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. In this thesis, we improve an established numerical method to construct Lyapunov functions using the radial basis functions (RBF) collocation method. The RBF collocation method approximates the solution of linear PDE's using scattered collocation points, and one of its applications is the construction of Lyapunov functions. More precisely, we approximate Lyapunov functions, that satisfy equations for their orbital derivative, using the RBF collocation method. Then, it turns out that the RBF approximant itself is a Lyapunov function.
Our main contributions to improve this method are firstly to combine this construction method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of collocation points needed to construct Lyapunov functions. Moreover, we design two modified refinement algorithms to deal with the issue of the early termination of the original refinement algorithm without constructing a Lyapunov function. These algorithms uses cluster centres to place points where the Voronoi vertices failed to do so.
Secondly, we derive two verification estimates, in terms of the first and second derivatives of the orbital derivative, to verify if the constructed function, with either a regular grid of collocation points or with one of the refinement algorithms, is a Lyapunov function, i.e., has negative orbital derivative over a given compact set. Finally, the methods are applied to several numerical examples up to 3 dimensions
