Flocking with finite communication speed: a Doppler-inspired Lyapunov approach
Abstract
We study the Cucker--Smale model with state-dependent transmission delays arising from finite-speed information propagation. Our main contribution is a Lyapunov functional inspired by the Doppler effect, combining a delay potential with a logarithmic correction involving the receiver velocity. Its dissipation yields uniform delay bounds without relying on velocity contraction estimates. For two agents in one spatial dimension, we prove asymptotic flocking for arbitrary compatible subluminal initial histories, assuming only that the influence function has a divergent integral at infinity. A bounded geometric correction extends this result to arbitrary spatial dimension when all values of the initial velocity histories lie on one common affine line. For multiple agents in one dimension, we obtain uniform delay bounds, and thus flocking, provided that a fixed agent is leftmost for all sufficiently large times. In all these flocking regimes, the spatial diameter remains uniformly bounded and the velocity diameter decays exponentially. We also derive a conditional criterion involving logarithmic sums when the leftmost agent changes infinitely often. Finally, we construct a multidimensional analogue of the Doppler-inspired functional and identify an additional assumption needed to obtain delay bounds in the general case.
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BibTeX
@misc{haskovec2026flocking,
title = {{Flocking with finite communication speed: a Doppler-inspired Lyapunov approach}},
author = {Jan Haskovec and Mauro Rodriguez Cartabia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33170},
eprint = {2610.33170},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33170},
}
RIS
TY - UNPB
TI - Flocking with finite communication speed: a Doppler-inspired Lyapunov approach
AU - Jan Haskovec
AU - Mauro Rodriguez Cartabia
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33170
UR - https://arxiv.org/abs/2610.33170
AB - We study the Cucker--Smale model with state-dependent transmission delays arising from finite-speed information propagation. Our main contribution is a Lyapunov functional inspired by the Doppler effect, combining a delay potential with a logarithmic correction involving the receiver velocity. Its dissipation yields uniform delay bounds without relying on velocity contraction estimates. For two agents in one spatial dimension, we prove asymptotic flocking for arbitrary compatible subluminal initial histories, assuming only that the influence function has a divergent integral at infinity. A bounded geometric correction extends this result to arbitrary spatial dimension when all values of the initial velocity histories lie on one common affine line. For multiple agents in one dimension, we obtain uniform delay bounds, and thus flocking, provided that a fixed agent is leftmost for all sufficiently large times. In all these flocking regimes, the spatial diameter remains uniformly bounded and the velocity diameter decays exponentially. We also derive a conditional criterion involving logarithmic sums when the leftmost agent changes infinitely often. Finally, we construct a multidimensional analogue of the Doppler-inspired functional and identify an additional assumption needed to obtain delay bounds in the general case.
ER -