  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.
