Construction of a Complete Lyapunov Function using Quadratic Programming

Peter Giesl, Carlos Argáez, Sigurdur Hafstein, Holger Wendland

2018

Abstract

A complete Lyapunov function characterizes the behaviour of a general dynamical system. In particular, the state space is split into the chain-recurrent set, where the function is constant, and the part characterizing the gradient-like flow, where the function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about the stability of connected components of the chain-recurrent set and the basin of attraction of attractors therein. In a previous method, a complete Lyapunov function was constructed by approximating the solution of the PDE V0(x) = −1, where 0 denotes the orbital derivative, by meshfree collocation. We propose a new method to compute a complete Lyapunov function: we only fix the orbital derivative V0(x0) = −1 at one point, impose the constraints V0(x) ≤ 0 for all other collocation points and minimize the corresponding reproducing kernel Hilbert space norm. We show that the problem has a unique solution which can be computed as the solution of a quadratic programming problem. The new method is applied to examples which show an improvement compared to previous methods.

Paper Citation

in Harvard Style

Giesl P., Argáez C., Hafstein S. and Wendland H. (2018). Construction of a Complete Lyapunov Function using Quadratic Programming.In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics - Volume 1: CTDE, ISBN 978-989-758-321-6, pages 560-568. DOI: 10.5220/0006944305600568

in Bibtex Style

@conference{ctde18,
author={Peter Giesl and Carlos Argáez and Sigurdur Hafstein and Holger Wendland},
title={Construction of a Complete Lyapunov Function using Quadratic Programming},
booktitle={Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics - Volume 1: CTDE,},
year={2018},
pages={560-568},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0006944305600568},
isbn={978-989-758-321-6},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics - Volume 1: CTDE,
TI - Construction of a Complete Lyapunov Function using Quadratic Programming
SN - 978-989-758-321-6
AU - Giesl P.
AU - Argáez C.
AU - Hafstein S.
AU - Wendland H.
PY - 2018
SP - 560
EP - 568
DO - 10.5220/0006944305600568