# A convex formulation of traffic dynamics on transportation networks

## TL;DR

This paper introduces a convex optimization method for modeling traffic flow on networks, enabling efficient computation of junction flows and facilitating traffic control applications.

## Contribution

It presents a novel convex formulation that decouples PDEs on network links, simplifying traffic flow computation and control.

## Key findings

- Efficient computation of boundary flows at junctions.
- Decoupling of PDEs on network links.
- Application to ramp metering control.

## Abstract

This article proposes a numerical scheme for computing the evolution of vehicular traffic on a road network over a finite time horizon. The traffic dynamics on each link is modeled by the Hamilton-Jacobi (HJ) partial differential equation (PDE), which is an equivalent form of the Lighthill-Whitham-Richards PDE. The main contribution of this article is the construction of a single convex optimization program which computes the traffic flow at a junction over a finite time horizon and decouples the PDEs on connecting links. Compared to discretization schemes which require the computation of all traffic states on a time-space grid, the proposed convex optimization approach computes the boundary flows at the junction using only the initial condition on links and the boundary conditions of the network. The computed boundary flows at the junction specify the boundary condition for the HJ PDE on connecting links, which then can be separately solved using an existing semi-explicit scheme for single link HJ PDE. As demonstrated in a numerical example of ramp metering control, the proposed convex optimization approach also provides a natural framework for optimal traffic control applications.