Well-posedness for scalar conservation laws with moving flux constraints

We consider a strongly coupled ODE-PDE system representing moving bottlenecks immersed in vehicular traffic. The systems can be used to represent the effect of autonomous vehicles immersed in the bulk traffic. The PDE consists of a scalar conservation law modeling the traffic fow evolution and the ODE models the trajectory of a slow moving vehicle. The moving bottleneck influences the bulk traffic flow via a point flux constraint, which is given by an inequality on the flux at the slow vehicle position. We prove uniqueness and continuous dependence of solutions with respect to initial data of bounded variation. The proof is based on a new backward in time method established to capture the values of the norm of generalized tangent vectors at every time. The results are the first step to study control problems for traffic fow via the action of autonomous vehicles.