Well-Posedness of Networked Scalar Semilinear Balance Laws Subject to Nonlinear Boundary Control Operators

Networked scalar semilinear balance laws are used as simplified macroscopic vehicular traffic models. The related initial boundary value problem is investigated, on a finite interval. The upstream boundary datum is determined by a nonlinear feedback control operator, representing the fact that traffic routing might be influenced in real time by the traffic information on the entire network. The main contribution of the present work lies in the appropriate design of nonlinear boundary control operators which meanwhile guarantee the well-posedness of the resultant systems. In detail, two different types of specific nonlinear boundary control operators are instantiated, one being Lipschitz continuous and taking into account traffic information from initial time up to present time, one using only delayed traffic information. This contribution thus presents simplified road traffic network dynamics where routing at intersections is dependent of the status of the entire network, incorporating also different classes of traffic flow.