An exact convex relaxation of the freeway network control problem with controlled merging junctions

We consider the freeway network control problem where the aim is to optimize the operation of traffic networks modeled by the cell transmission model via ramp metering and partial mainline demand control. Optimal control problems using the cell transmission model are usually non-convex, due to the nonlinear fundamental diagram, but a convex relaxation in which demand and supply constraints are relaxed is often used. Previous works have established conditions under which solutions of the relaxation can be made feasible with respect to the original constraints. In this work, we generalize these conditions and show that the control of flows into merging junctions is sufficient to do so if the objective is to minimize the total time spent in traffic. We derive this result by introducing an alternative system representation. In the new representation, the system dynamics are concave and state-monotone. We show that exactness of the convex relaxation of finite horizon optimal control problems follows from these properties. Deriving the main result via a characterization of the system dynamics allows one to treat arbitrary monotone, concave fundamental diagrams and several types of control for merging junctions in a uniform manner. The derivation also suggests a straightforward method to verify if the results continue to hold for extensions or modifications of the models studied in this work.