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

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.