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Computation and verification of Lyapunov functions

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posted on 2023-06-08, 22:40 authored by Peter GieslPeter Giesl, Sigurdur Hafstein
Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical Systems through their sublevel sets. Recently, several numerical construction methods for Lyapunov functions have been proposed, among them the RBF (Radial Basis Function) and CPA (Continuous Piecewise Affine) methods. While the first method lacks a verification that the constructed function is a valid Lyapunov function, the second method is rigorous, but computationally much more demanding. In this paper, we propose a combination of these two methods, using their respective strengths: we use the RBF method to compute a potential Lyapunov function. Then we interpolate this function by a CPA function. Checking a finite number of inequalities, we are able to verify that this interpolation is a Lyapunov function. Moreover, sublevel sets are arbitrarily close to the basin of attraction. We show that this combined method always succeeds in computing and verifying a Lyapunov function, as well as in determining arbitrary compact subsets of the basin of attraction. The method is applied to two examples.

History

Publication status

  • Published

File Version

  • Published version

Journal

SIAM Journal on Applied Dynamical Systems

ISSN

1536-0040

Publisher

Society for Industrial and Applied Mathematics

Issue

4

Volume

14

Page range

1663-1698

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2015-10-05

First Open Access (FOA) Date

2015-10-05

First Compliant Deposit (FCD) Date

2015-10-05

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