# Construction of a Complete Lyapunov Function using Quadratic Programming

### Peter Giesl, Carlos Argáez, Sigurdur Hafstein, Holger Wendland

#### 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.

Download#### 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