Consensus of the Hegselmann–Krause opinion formation model with time delay

Young Pil Choi, Alessandro Paolucci, Cristina Pignotti

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)

Abstract

In this paper, we study Hegselmann–Krause models with a time-variable time delay. Under appropriate assumptions, we show the exponential asymptotic consensus when the time delay satisfies a suitable smallness assumption. Our main strategies for this are based on Lyapunov functional approach and careful estimates on the trajectories. We then study the mean-field limit from the many-individual Hegselmann–Krause equation to the continuity-type partial differential equation as the number N of individuals goes to infinity. For the limiting equation, we prove global-in-time existence and uniqueness of measure-valued solutions. We also use the fact that constants appearing in the consensus estimates for the particle system are independent of N to extend the exponential consensus result to the continuum model. Finally, some numerical tests are illustrated.

Original languageEnglish
Pages (from-to)4560-4579
Number of pages20
JournalMathematical Methods in the Applied Sciences
Volume44
Issue number6
DOIs
Publication statusPublished - 2021 Apr

Bibliographical note

Publisher Copyright:
© 2020 John Wiley & Sons, Ltd.

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • General Engineering

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