On the well-posedness of deterministic queuing networks with feedback control

We study the well-posedness of a class of dynamical flow network systems describing the dynamical mass balance among a finite number of cells exchanging flow of traffic between themselves and with the external environment. Dynamical systems in the considered class are described as differential inclusions whereby the routing matrix is constant and the outflow from each cell in the network is limited by a control that is a Lipschitz continuous function of the state of the network. This framework finds application in particular within traffic signal control, whereby it is common that an empty queue can be allowed to have more outflow than vehicles in the queue. While models for this scenario have previously been presented for open-loop outflow controls, our result ensures the existence and uniqueness of solutions for the network flow dynamics in the case Lipschitz continuous feedback controllers.