Analytical Optimal Solution of Perimeter Traffic Flow Control Based on MFD Dynamics: A Pontryagin's Maximum Principle Approach

Perimeter traffic flow control, based on the macroscopic fundamental diagram (MFD), has been introduced for traffic control and congestion management in large-scale networks. The perimeter controller is a set of traffic signals on the border between the regions manipulating the transfer flows with the aim to maximize the number of trips that reach their destinations. This paper tackles the optimal perimeter control of MFD systems for two-region urban networks which model heterogeneously congested cities. The modeling of the system results in nonlinear state dynamics, a non-quadratic cost function, and constraints on control actions and traffic states. We prove the existence of the optimal controller, analytically derive the optimal control policy, and introduce a numerical method to solve the optimal control policy. Based on the indirect optimal approach, HJB equation, and Pontryagin's maximum principal, we demonstrate that the optimal controller is in the form of Bang-Bang control. We apply the Chebyshev pseudospectral method to solve the two-point boundary value problem (TPBVP) for the proposed constrained optimal control problem. Consequently, the TPBVP is reduced to determination of the solution of a nonlinear system with algebraic equations. A numerical study is performed to measure the effectiveness of the proposed method.