Hybrid multi-population traffic flow model: Optimal control for a mean-field limit

Modeling heterogeneous and multi-lane traffic flow is essential for understanding and controlling complex transportation systems. In this work, we consider three vehicle populations: two classes of human-driven vehicles (cars and trucks) and autonomous vehicles, the latter characterized by controlled acceleration. Compared to single-population models, multi-population modeling poses greater challenges, primarily due to the increased number of parameters required to describe lane-changing behavior and the added complexity in passing to the mean-field limit. We model multi-lane traffic as a hybrid dynamical system, combining continuous dynamics within each lane and discrete events corresponding to lane-changing maneuvers. We then formulate and analyze the optimal control problem associated with such hybrid systems from both microscopic and mesoscopic perspectives. Using techniques from $\Gamma$-convergence, we prove the existence of solutions to the optimal control problem in the mean-field limit of a finite-dimensional hybrid system. Finally, we present numerical simulations illustrating the impact of trucks on overall traffic efficiency.