  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.
