Approximate Dynamic Programming for a Mean-field Game of Traffic Flow: Existence and Uniqueness

TL;DR

This paper develops a mean-field game approach for traffic flow control, deriving a dynamic programming PDE system, and uses approximate dynamic programming to obtain adaptive, sub-optimal traffic density controls with proven existence and uniqueness.

Contribution

It introduces a novel ADP method for mean-field traffic control, establishing existence and uniqueness of solutions for the associated PDE system.

Findings

ADP yields effective sub-optimal traffic control strategies.

Proven existence and uniqueness of the PDE system solutions.

Numerical simulations demonstrate practical applicability.

Abstract

Highway vehicular traffic is an inherently multi-agent problem. Traffic jams can appear and disappear mysteriously. We develop a method for traffic flow control that is applied at the vehicular level via mean-field games. We begin this work with a microscopic model of vehicles subject to control input, disturbances, noise, and a speed limit. We formulate a discounted-cost infinite-horizon robust mean-field game on the vehicles, and obtain the associated dynamic programming (DP) PDE system. We then perform approximate dynamic programming (ADP) using these equations to obtain a sub-optimal control for the traffic density adaptively. The sub-optimal controls are subject to an ODE-PDE system. We show that the ADP ODE-PDE system has a unique weak solution in a suitable Hilbert space using semigroup and successive approximation methods. We additionally give a numerical simulation, and interpret the results.