Discrete-Time System Optimal Dynamic Traffic Assignment (SO-DTA) with Partial Control for Physical Queuing Networks

We consider the System Optimal Dynamic Traffic Assignment (SO-DTA) problem with Partial Control for general networks with physical queuing. Our goal is to optimally control any subset of the networks agents to minimize the total congestion of all agents in the network. We adopt a flow dynamics model that is a Godunov discretization of the Lighthill–Williams–Richards partial differential equation with a triangular flux function and a corresponding multicommodity junction solver. The partial control formulation generalizes the SO-DTA problem to consider cases where only a fraction of the total flow can be controlled, as may arise in the context of certain incentive schemes. This leads to a nonconvex multicommodity optimization problem. We define a multicommodity junction model that only requires full Lagrangian paths for the controllable agents, and aggregate turn ratios for the noncontrollable (selfish) agents. We show how the resulting finite horizon nonlinear optimal control problem can be efficiently solved using the discrete adjoint method, leading to gradient computations that are linear in the size of the state space and the controls. The online appendix is available at https://doi.org/10.1287/trsc.2017.0800.