Micro-macro and macro-macro limits for controlled leader-follower systems

TL;DR

This paper rigorously derives hierarchical mean-field limits for controlled leader-follower particle systems, establishing stability and convergence, and supports findings with numerical simulations including complex interaction potentials.

Contribution

It introduces a two-step limiting process from micro-macro to macro-macro systems with quantitative stability estimates, advancing the theoretical understanding of controlled multi-agent dynamics.

Findings

Quantitative stability and convergence estimates are established.

Numerical simulations validate the theoretical results.

The hierarchical reduction framework applies to complex interaction potentials.

Abstract

We study a leader-follower system of interacting particles subject to feedback control and derive its mean-field limits through a two-step passage: first to a micro-macro system coupling leader particles with a follower fluid, and then to a fully continuum macro-macro system. For each limiting procedure, we establish quantitative stability and convergence estimates based on modulated energy methods and Wasserstein distances. These results provide a rigorous foundation for the hierarchical reduction of controlled multi-agent systems. Numerical simulations are presented, including examples with interaction potentials beyond the analytical class considered, to demonstrate the dynamics and support the theoretical results.