On distributed computation of optimal control of traffic flow over networks

In this paper, we adapt the Alternating Direction Method of Multipliers (ADMM) to develop a distributed algorithm for computing finite horizon optimal control for traffic flow over networks, where the cost is in terms of the density and flow vectors. The dynamics is described by the Cell Transmission Model. While the problem is non-convex in general, recent work has shown the existence of an equivalent convex relaxation, which we adopt in this paper. An equivalent finite dimensional representation for constraints is developed when the external inflows and control are piecewise constant. The auxiliary variables for ADMM implementation consist of a time shifted copy of the density vector, and two copies of the flow vector. A particular division of these variables into two blocks is provided which facilitates distributed implementation, as well as convergence of the ADMM iterates to an optimal solution, if the initial density on every link is strictly positive and strictly below the jam capacity, and the cost function is proper, closed, convex, and separable over density and flow variables, and over links. We also examine the necessary condition, as given by the maximum principle, to provide sufficient conditions under which the optimal control is unique for non-strict convex cost functions, and of bang-bang type, for a linear network.