Time-dependent optimal assignment control for traffic flow at network junctions

The advancing interconnection and automation technologies have led to the emergence of new topics in traffic management, driving innovations in various control strategies for macroscopic traffic planning. This paper focuses on time-dependent assignment control for traffic flow at network junctions, which can be formulated as the optimal boundary control problem with constraints based on conservation laws. By considering block multiconvexity, we establish the convexity of the optimal assignment control problem and the existence of the optimal controls. However, we demonstrate that the problem can be ill-posed, and the assignment controls can be ineffective under certain conditions with the objective functionals utilized in the literature. To address this issue, we apply Tikhonov regularization with $ H^{1} $-seminorm to regularize the model problem. A block proximal-linear iteration method is proposed to solve the regularized optimal assignment control problem with adaptive step sizes. By extensive numerical experiments, we illustrate the ill-posedness of the original problem and the effectiveness of the regularized model, as well as the efficiency of the proposed iteration method.