Communications in Mathematical Sciences

Volume 19 (2021)

Number 1

Emergence of stochastic flocking for the discrete Cucker–Smale model with randomly switching topologies

Pages: 205 – 228

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n1.a9

Authors

Jiu-Gang Dong (School of Mathematical Sciences, Dalian University of Technology, Dalian, China)

Seung-Yeal Ha (Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, South Korea; and School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea)

Jinwook Jung (Research Institute of Basic Sciences, Seoul National University, Seoul, South Korea)

Doheon Kim (School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea)

Abstract

We study emergent dynamics of the discrete Cucker–Smale (in short, DCS) model with randomly switching network topologies. For this, we provide a sufficient framework leading to the stochastic flocking with probability one. Our sufficient framework is formulated in terms of an admissible set of network topologies realized by digraphs and probability density function for random switching times. As examples for the law of switching times, we use the Poisson process and the geometric process and show that these two processes satisfy the required conditions in a given framework so that we have a stochastic flocking with probability one. As a corollary of our flocking analysis, we improve the earlier result [J.-G. Dong, S.-Y. Ha, J. Jung, and D. Kim, SIAM J. Control Optim., 58(4):2332–2353, 2019] on the continuous CS model.

Keywords

Cucker–Smale model, randomly switching topology, directed graphs, flocking

2010 Mathematics Subject Classification

34D05, 39A12, 68M10

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The work of J.-G. Dong is supported by the Fundamental Research Funds for the Central Universities. The work of S.-Y. Ha is supported by the National Research Foundation of Korea (NRF-2017R1A2B2001864), the work of J. Jung is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP): NRF-2016K2A9A2A13003815, 2019R1A6A1A10073437, and the work of D. Kim is supported by a KIAS Individual Grant (MG073901) at Korea Institute for Advanced Study.

Received 20 November 2019

Accepted 14 August 2020

Published 24 March 2021