Approximation of Lyapunov functions from noisy data

Peter Giesl; Boumediene Hamzi; Martin Rasmussen; Kevin Webster

doi:10.3934/jcd.2020003

Article Contents

2020, Volume 7, Issue 1: 57-81. Doi: 10.3934/jcd.2020003

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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: June 2020

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

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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.

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Mathematics Subject Classification: 37B25, 65N15, 37M99.

Citation:

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Figure 1. 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}) $

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Figure 2. Errors between $ f_1 $ and $ {f}^1_{{\bf z},\lambda} $

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Figure 3. Errors between $ f_2 $ and $ {f}^2_{{\bf z},\lambda} $

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Figure 4. Lyapunov function using Algorithm 2 with 360 points

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Figure 5. Lyapunov function using Algorithm 2 with 1520 points

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Figure 6. Orbital derivative of the Lyapunov function with respect to the original system using Algorithm 2 with 360 points

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Figure 7. Orbital derivative of the Lyapunov function with respect to the original system using Algorithm 2 with 1520 points

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