Dissipation of Stop-and-Go Waves in Traffic Flows Using Controlled Vehicles: A Macroscopic Approach

We study the boundary stabilization of Generic Second Order Macroscopic traffic models in Lagrangian coordinates. These consist in
$2\times 2$
nonlinear hyperbolic systems of balance equations with a relaxation-type source term. We provide the existence of weak solutions of the Initial Boundary Value problem for generic relaxation terms. In particular, we do not require the sub-characteristic stability condition to hold, so that equilibria are unstable and perturbations may lead to the formation of large oscillations, modeling the appearance and persistence of stop-and-go waves. Moreover, since the largest eigenvalue of the system is null, the boundaries are characteristic, and the available results on boundary controllability do not apply. Therefore, we perform a detailed analysis of the Wave Front Tracking approximate solutions to show that weak solutions can be steered to the corresponding equilibrium state by prescribing the equilibrium speed at the right boundary. This corresponds to controlling the speed of one vehicle to stabilize the upstream traffic flow. The result is illustrated through a numerical example.