Approximation of Lyapunov Functions from Noisy Data

TL;DR

This paper introduces a method to approximate Lyapunov functions from noisy data by first estimating the underlying vector field and then constructing the Lyapunov function, blending machine learning with classical theory.

Contribution

It presents a novel algorithm for Lyapunov function approximation from noisy data without known evolution equations, including error estimates.

Findings

Effective approximation of vector fields from noisy data.

Successful construction of Lyapunov functions with error bounds.

Applicable to systems with unknown dynamics.

Abstract

Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume that the evolution equations are known. We consider the problem of approximating a given Lyapunov function using radial basis functions where the evolution equations are not known, but we instead have sampled data which is contaminated with noise. We propose an algorithm in which we first approximate the underlying vector field, and use this approximation to then approximate the Lyapunov function. Our approach combines elements of machine learning/statistical learning theory with the existing theory of Lyapunov function approximation. Error estimates are provided for our algorithm.