Verification estimates for the construction of Lyapunov functions using meshfree collocation. (English) Zbl 1462.65093
Summary: Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sub-level 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 meshfree collocation with radial basis functions (RBF). In this paper, we propose two verification estimates combined with this RBF construction method to ensure that the constructed function is a Lyapunov function. We show that this combination of the RBF construction method and the verification estimates always succeeds in constructing and verifying a Lyapunov function for nonlinear ODEs in \(\mathbb{R}^d\) with an exponentially stable equilibrium.
MSC:
65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
65L70 | Error bounds for numerical methods for ordinary differential equations |
65P40 | Numerical nonlinear stabilities in dynamical systems |