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We consider the freeway network control problem where the aim is to optimizethe operation of traffic networks modeled by the Cell Transmission Model viaramp metering and partial mainline demand control. Optimal control problemsusing the Cell Transmission Model are usually non-convex, due to the nonlinearfundamental diagram, but a convex relaxation in which demand and supplyconstraints are relaxed is often used. Previous works have establishedconditions under which solutions of the relaxation can be made feasible withrespect to the original constraints. In this work, we generalize theseconditions and show that the control of flows into merging junctions issufficient to do so if the objective is to minimize the total time spent intraffic. We derive this result by introducing an alternative systemrepresentation. In the new representation, the system dynamics are concave andstate-monotone. We show that exactness of the convex relaxation of finitehorizon optimal control problems follows from these properties. Deriving themain result via a characterization of the system dynamics allows one to treatarbitrary monotone, concave fundamental diagrams and several types of controlfor merging junctions in a uniform manner. The derivation also suggests astraightforward method to verify if the results continue to hold for extensionsor modifications of the models studied in this work.

An Exact Convex Relaxation of the Freeway Network Control Problem with Controlled Merging Junctions