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 language | English |
|---|---|
| Pages (from-to) | 4560-4579 |
| Number of pages | 20 |
| Journal | Mathematical Methods in the Applied Sciences |
| Volume | 44 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - 2021 Apr |
Bibliographical note
Funding Information:The first author was supported by POSCO Science Fellowship of POSCO TJ Park Foundation and Yonsei University Research Fund of 2019-22-021. The second and third author were partially supported by the GNAMPA 2019 project Modelli alle derivate parziali per sistemi multi-agente (INdAM).
Funding Information:
The first author was supported by POSCO Science Fellowship of POSCO TJ Park Foundation and Yonsei University Research Fund of 2019‐22‐021. The second and third author were partially supported by the GNAMPA 2019 project Modelli alle derivate parziali per sistemi multi‐agente (INdAM).
Publisher Copyright:
© 2020 John Wiley & Sons, Ltd.
All Science Journal Classification (ASJC) codes
- Mathematics(all)
- Engineering(all)