Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime
Jan Haskovec, Ioannis Markou
TL;DR
This work analyzes a Cucker-Smale flocking model with reaction-type delay $\tau>0$ and establishes conditions under which velocity alignment occurs in a non-oscillatory regime. The authors develop backward-forward analytical estimates and a Lyapunov functional to prove a uniform bound on the velocity fluctuations $V(t)$ and a decay mechanism driven by a delayed interaction functional $D(t)$, leading to monotone convergence of $V(t)$ to zero for delays below a computable threshold $\tau_c$ that depends on initial data via $L^0$ and $M^0$. In the two-agent, constant-communication-rate reduction, the method yields a sharp non-oscillatory threshold $0<\lambda\tau<e^{-1}$ (with exponential decay for a subrange) and illuminates the delicate balance between delay and feedback strength. The paper also discusses formal mean-field limits in the presence of delay, highlighting potential ill-posedness and noting a variant with present-time interaction that yields a standard kinetic description. Overall, the results delineate a rigorous non-oscillatory regime for delayed flocking and provide a constructive pathway to compute the critical delay from initial data.
Abstract
We study a variant of the Cucker-Smale system with reaction-type delay. Using novel backward-forward and stability estimates on appropriate quantities we derive sufficient conditions for asymptotic flocking of the solutions. These conditions, although not explicit, relate the velocity fluctuation of the initial datum and the length of the delay. If satisfied, they guarantee monotone decay (i.e., non-oscillatory regime) of the velocity fluctuations towards zero for large times. For the simplified setting with only two agents and constant communication rate the Cucker-Smale system reduces to the delay negative feedback equation. We demonstrate that in this case our method provides the sharp condition for the size of the delay such that the solution be non-oscillatory. Moreover, we comment on the mathematical issues appearing in the formal macroscopic description of the reaction-type delay system.