On the existence and uniqueness properties of the Hoogendoorn–Bovy pedestrian flow model

Abstract

This study addresses the existence and uniqueness of solutions to the Hoogendoorn–Bovy (HB) pedestrian flow model, which describes the dynamic user-optimal pedestrian flow assignment problem in continuous space and time. The HB model consists of a forward conservation law (CL) equation that governs density and a backward Hamilton–Jacobi–Bellman (HJB) equation that contains a maximum admissible speed constraint (MASC), in which the flow direction is determined by the path-choice strategy. The existence and uniqueness of solutions are significantly more difficult to determine when the HJB equation contains an MASC; however, we prove that the HB model can be formulated as a forward CL equation and backward Hamilton–Jacobi (HJ) equation in which the MASC is non-binding if suitable model parameters are selected. This model is formulated as a fixed-point problem upon the simultaneous satisfaction of both equations. To verify the existence and uniqueness results, we first demonstrate the existence and uniqueness of the solutions to the CL and HJ equations, and then show that the coupled HB model is well-posed and has a unique solution. A numerical example is presented to illustrate the properties of the HB model.