Conservation laws with moving constraints arising in traffic modeling

We consider the Cauchy problem for a strongly coupled PDE-ODE system modeling the influence of a slow vehicle on the surrounding road traffic. The model consists of a conservation law describing the main traffic evolution and an ODE accounting for the trajectory of the slower vehicle that depends on the downstream traffic density. The moving constraint is operated by an inequality on the flux, which accounts for the bottleneck created on the road by the presence of the slower vehicle. We present the proof of existence of weak entropy solutions obtained via the wave-front tracking method, and a finite volume scheme able to capture exactly the non-classical discontinuities that may arise at the constraint position. Optimal control problems for traffic management are also addressed numerically.