Optimal control of network flows with convex cost and state constraints

Consider a network in which a commodity flows at a variable rate across the arcs in order to meet supply/demand at the nodes. The aim is to optimally control the rate of flow such that a convex objective functional is minimized. This is an optimal control problem with a large number of states, and with an even larger number of controls. It is also complicated by storage bounds at the nodes leading to two state constraints for each node. We show, under some mild assumptions on the problem's parameters, that an exact solution to this state-constrained optimal control problem can be found efficiently without a complete discretization of the time variable, and develop a solution algorithm, based on the active-set-on-a-graph algorithm for static convex flow problems. A brief description of a possible application as well as some numerical results are provided to illustrate the usefulness of the algorithm.