Communications in Mathematical Sciences
Volume 19 (2021)
Emergence of stochastic flocking for the discrete Cucker–Smale model with randomly switching topologies
Pages: 205 – 228
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.
Cucker–Smale model, randomly switching topology, directed graphs, flocking
2010 Mathematics Subject Classification
34D05, 39A12, 68M10
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