# American Institute of Mathematical Sciences

June  2020, 7(1): 57-81. doi: 10.3934/jcd.2020003

## Approximation of Lyapunov functions from noisy data

 1 Department of Mathematics, University of Sussex, Falmer, BN1 9QH, UK 2 Department of Mathematics, Imperial College London, London SW7 2AZ, UK

Received  May 2017 Revised  October 2019 Published  December 2019

Fund Project: B. Hamzi was supported by Marie Curie Fellowships. M. Rasmussen was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1 and K.N. Webster was supported by the EPSRC Grant EP/L00187X/1 and a Marie Skł odowska-Curie Individual Fellowship Grant Number 660616

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.

Citation: Peter Giesl, Boumediene Hamzi, Martin Rasmussen, Kevin Webster. Approximation of Lyapunov functions from noisy data. Journal of Computational Dynamics, 2020, 7 (1) : 57-81. doi: 10.3934/jcd.2020003
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Domains and sets used in the statement and proof of Theorem 2.1. The dotted lines show the boundary of the set $\mathcal{D} = \Omega\setminus B_{\varepsilon}(\overline{x})$ where the Lyapunov functions are approximated. We also have $\Omega_V,\Omega_T\subset A(\overline{x})$
Errors between $f_1$ and ${f}^1_{{\bf z},\lambda}$
Errors between $f_2$ and ${f}^2_{{\bf z},\lambda}$
Lyapunov function using Algorithm 2 with 360 points
Lyapunov function using Algorithm 2 with 1520 points
Orbital derivative of the Lyapunov function with respect to the original system using Algorithm 2 with 360 points
Orbital derivative of the Lyapunov function with respect to the original system using Algorithm 2 with 1520 points
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