Optimal control of day-to-day traffic dynamics under user learning with dynamic congestion pricing

Abstract A continuous day-to-day dynamic traffic assignment model is proposed in this study, where several typical day-to-day models are integrated into a more general form. The effects of past information on travelers’ routing behavior are characterized by the second-order derivatives of the traffic flow. Thereupon, the day-to-day dynamic takes the form of a second-order nonhomogeneous ordinary differential equation with constant coefficients. We prove that the dynamical process is globally asymptotically stable at the user equilibrium with the existence of travelers’ route-swapping behavior. Moreover, we prove that there always exists a limited time point, after which the nonnegative flows can be guaranteed in the proposed model. The optimal control of the system is realized by utilizing dynamic congestion pricing. Different optimization objectives are investigated, including cost, time, revenue, and combined minimizations. Necessary conditions for optimal congestion prices are analyzed to uncover the bang-bang charging strategy. Numerical examples are provided to illustrate the trajectory of the flow evolution under the optimal control.