Moving bottlenecks in traffic flows: a well-posed fully coupled nonlocal PDE-ODE model with time delay

Abstract

We propose a nonlocal PDE-ODE strongly coupled system describing the interaction between traffic flow and a moving bottleneck, induced by e.g. a controlled vehicle. The bulk traffic density evolves according to a nonlocal scalar conservation law with time delay, while the bottleneck trajectory is governed by an ordinary differential equation with nonlocal velocity field depending on the downstream traffic conditions, leading to a full micro-macro coupling. We prove the well-posedness of the system by means of a fractional step finite volume approximation. We establish uniform BV estimates and \(\mathbf{L^{\infty }}\) bounds for the approximate solutions, which provide compactness of the scheme and convergence of the approximating sequence to an entropy solution, providing existence. Uniqueness follows from a \(\mathbf{L^{1}}\) stability result.