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Complete Lyapunov Functions: Determination of the Chain-Recurrent Set Using the Gradient

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Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1260))

Abstract

Complete Lyapunov functions (CLF) are scalar-valued functions, which are non-increasing along solutions of a given autonomous ordinary differential equation. They separate the phase-space into the chain-recurrent set, where the CLF is constant along solutions, and the set where the flow is gradient-like and the CLF is strictly decreases along solutions. Moreover, one can deduce the stability of connected components of the chain-recurrent set from the CLF.

While the existence of CLFs was shown about 50 years ago, in recent years algorithms to construct CLFs have been designed to determine the chain-recurrent set using the orbital derivative. These algorithms require iterative methods that constructed better and better approximations to a CLF, based on previous iterations. A drawback of these methods is the overestimation of the chain-recurrent set, which has been addressed by different methods.

In this paper, we construct a CLF using the previous method, but in contrast to previous work we will use the norm of the gradient of the computed CLF, rather than its orbital derivative, to determine the chain-recurrent set. We will show in this paper that this new approach determines the chain-recurrent set very well without the need of iterations or further methods to reduce the overestimation.

The first author in this paper is supported by the Icelandic Research Fund (Rannís), Iceland grant number 1163074-052, Complete Lyapunov functions: Efficient numerical computation.

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Authors and Affiliations

  1. Science Institute, University of Iceland, Dunhagi 5, 107, Reykjavík, Iceland

    Carlos Argáez & Sigurdur Freyr Hafstein

  2. University of Sussex, Falmer, BN1 9QH, UK

    Peter Giesl

Authors
  1. Carlos Argáez
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  2. Peter Giesl
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  3. Sigurdur Freyr Hafstein
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Corresponding author

Correspondence to Carlos Argáez .

Editor information

Editors and Affiliations

  1. Fellow of IEEE, Dean of College of Computing and Informatics, University of Sharjah, Sharjah, United Arab Emirates

    Mohammad S. Obaidat

  2. School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON, Canada

    Tuncer Ören

  3. Universität Hannover, Hannover, Germany

    Helena Szczerbicka

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Argáez, C., Giesl, P., Hafstein, S.F. (2021). Complete Lyapunov Functions: Determination of the Chain-Recurrent Set Using the Gradient. In: Obaidat, M., Ören, T., Szczerbicka, H. (eds) Simulation and Modeling Methodologies, Technologies and Applications. SIMULTECH 2019. Advances in Intelligent Systems and Computing, vol 1260. Springer, Cham. https://doi.org/10.1007/978-3-030-55867-3_6

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