Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime

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.