First-Order Continuation for Heterogeneous Multi-Agent Systems on a Ring: From PDE Modeling to Feedback Control

In large-scale multi-agent systems, analyzing individual dynamics quickly becomes impractical, which motivates the use of continuous models to describe collective behavior. We introduce a continuation method on a ring that systematically derives macroscopic partial differential equations (PDEs) from microscopic dynamics, while allowing for heterogeneous interaction structures between agents. Control laws designed at the PDE level can then be consistently transferred to the underlying microscopic system, rendering the agents effectively autonomous under feedback. The approach is illustrated on the first-order Optimal Velocity traffic model, where the vehicle distribution is successfully steered toward a prescribed stationary density profile despite the presence of non-uniform interactions.