Consensus and flocking with transmission and reaction delays

TL;DR

This paper analyzes how transmission and reaction delays influence consensus and flocking in multi-agent systems, providing explicit stability conditions and demonstrating collective behavior emergence under certain delay constraints.

Contribution

It introduces explicit stability conditions for multi-agent consensus and flocking considering both transmission and reaction delays, using a Lyapunov functional approach.

Findings

Explicit stability conditions for two-agent systems with delays.

Sufficient conditions for consensus and flocking in multi-agent systems.

Delays must be small relative to initial data and influence decay rate.

Abstract

We investigate consensus formation and flocking behavior in multi-agent systems subject to two distinct types of delays: a transmission delay accounting for information exchange between agents, and a reaction delay representing the processing time before agents adjust their states. For a simplified linear two-agent system, we provide explicit insight into how these delays affect asymptotic stability. We then derive sufficient conditions for asymptotic consensus and flocking in the general multi-agent setting with a nonlinear, globally positive influence function. These conditions require the delays to be sufficiently small relatively to the initial data and the decay rate of the influence function. The analysis is based on a Lyapunov functional approach combined with a Halanay-type inequality. Our results establish rigorous conditions under which collective behavior emerges in delayed multi-agent systems where both communication and reaction lags are non-negligible, with applications to biological, social, and engineered systems.