Flocking in a two-agent Cucker-Smale model with large delay

Abstract

Delay in feedback is inevitable in a multi-agent system due to time lags in information processing for self-organization. The well-known Cucker-Smale model incorporated with this information processing delay has been recently studied, and it was shown (at least for a two-agent system) that as long as the delay is below a threshold value, the system exhibits the flocking behavior where the agents ultimately reach the same velocity. Numerical studies however suggest that the threshold value established for the delay is not optimal, and the Cucke-Smale system remains to have the flocking property for large delay. Here, we rigorously prove that flocking behavior emerges for a large information processing delay, and we establish the relationship between the admissible size of time delay and the system’s parameters and initial data.

The delayed Cucker-Smale model with short range communication weights

“…Then, similar result was extended to the C−S model with distributed reaction delays in [23]. Whereafter, the flocking of two agents with large reactiontype delay was considered in [6]. For the delayed C−S model with renormalized communication weights, the flocking behavior was also studied in [8,9,21], where the convexity arguments were crucially used.…”

Collision-avoidance and flocking in the Cucker–Smale-type model with a discontinuous controller

“…Once Cucker-Smale model was put forward, it received extensive attention, and many properties of flocking behaviour were studied, see references [5,6,13,19,26,27].…”