Semianalytical Solutions to the Lighthill–Whitham–Richards Equation With Time-Switched Triangular Diagrams: Application to Variable Speed Limit Traffic Control

Abstract

This article proposes a new approach for computing a semiexplicit form of the solution to a class of traffic flow problems encoded by a Hamilton-Jacobi (HJ) partial differential equation (PDE), with time-switched Hamiltonian. Using a characterization of the problem derived from viability theory, we show that the solution associated with the problem can be formulated as a minimization problem involving the trajectory of an auxiliary dynamical system. A generalized Lax-Hopf formula for the switched Hamiltonian problem is derived, which enables us to compute the solution associated with affine initial or boundary conditions as a linear program involving the control function of the auxiliary dynamical system. This formulation allows us to compute the solution to the original problem exactly, unlike dynamic programming methods. In addition, this method allows one to very efficiently recompute the boundary conditions associated with an initial condition problem, allowing large-scale variable speed limit traffic control problems to be solved.