A junction condition by specified homogenization and application to traffic lights

TL;DR

This paper develops a homogenization framework for Hamilton-Jacobi equations with periodic Hamiltonians, deriving a junction condition at a discontinuity point and applying it to model traffic flow with traffic lights.

Contribution

It introduces a novel junction condition in the homogenization of Hamilton-Jacobi equations and applies it to traffic flow with traffic lights, linking microscopic traffic control to macroscopic models.

Findings

Rescaled solutions converge to an effective Hamilton-Jacobi equation with a flux limiter.

The flux limiter at the junction is explicitly characterized.

Application to traffic flow shows how traffic lights influence overall flux.

Abstract

Given a coercive Hamiltonian which is quasi-convex with respect to the gradient variable and periodic with respect to time and space at least "far away from the origin", we consider the solution of the Cauchy problem of the corresponding Hamilton-Jacobi equation posed on the real line. Compact perturbations of coercive periodic quasi-convex Hamiltonians enter into this framework for example. We prove that the rescaled solution converges towards the solution of the expected effective Hamilton-Jacobi equation, but whose "flux" at the origin is "limited" in a sense made precise by the authors in \cite{im}. In other words, the homogenization of such a Hamilton-Jacobi equation yields to supplement the expected homogenized Hamilton-Jacobi equation with a junction condition at the single discontinuous point of the effective Hamiltonian. We also illustrate possible applications of such a result by deriving, for a traffic flow problem, the effective flux limiter generated by the presence of a finite number of traffic lights on an ideal road. We also provide meaningful qualitative properties of the effective limiter.