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

Conference paper
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Part of the Advances in Intelligent Systems and Computing book series (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.

Keywords

Complete Lyapunov functions Chain-recurrent set Dynamical systems 

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© Springer Nature Switzerland AG 2021

Authors and Affiliations

  1. 1.Science InstituteUniversity of IcelandReykjavíkIceland
  2. 2.University of SussexFalmerUK

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