  We study a robust control problem for dynamical flow networks. In the
considered dynamical models, traffic flows along the links of a transportation
network --modeled as a capacited multigraph-- and queues up at the nodes,
whereby control policies determine which incoming queues at a node are to be
allocated service simultaneously, within some predetermined scheduling
constraints. We first prove a fundamental performance limitation by showing
that for a dynamical flow network to be stabilizable by some control policy it
is necessary that the exogenous inflows belong to a certain stability region,
that is determined by the network topology, link capacities, and scheduling
constraints. Then, we introduce a family of distributed controls, referred to
as Generalized Proportional Allocation (GPA) policies, and prove that they
stabilize a dynamical transportation network whenever the exogenous inflows
belong to such stability region. The proposed GPA control policies are
decentralized and fully scalable as they rely on local feedback information
only. Differently from previously studied maximally stabilizing control
strategies, the GPA control policies do not require any global information
about the network topology, the exogenous inflows, or the routing, which makes
them robust to demand variations and unpredicted changes in the link capacities
or the routing decisions. Moreover, the proposed GPA control policies also take
into account the overhead time while switching between services. Our
theoretical results find one application in the control of urban traffic
networks with signalized intersections, where vehicles have to queue up at
junctions and the traffic signal controls determine the green light allocation
to the different incoming lanes.
