Stability Analysis and Control Synthesis for Dynamical Transportation Networks∗

We study dynamical transportation networks in a framework that includes extensions of the classical Cell Transmission Model to arbitrary network topologies. The dynamics are modeled as systems of ordinary differential equations describing the traffic flow among a finite number of cells interpreted as links of a directed network. Flows between contiguous cells, in particular at junctions, are determined by merging and splitting rules within con-straints imposed by the cells ’ demand and supply functions as well as by the drivers ’ turning preferences, while inflows at on-ramps are modeled as exogenous and possibly time-varying. First, we analyze stability properties of dynamical transportation networks. We asso-ciate to the dynamics a state-dependent dual graph whose connectivity depends on the signs of the derivatives of the inter-cell flows with respect to the densities. Sufficient con-ditions for the stability of equilibria and periodic solutions are then provided in terms of the connectivity of such dual graph. Then, we consider synthesis of optimal control policies that use a combination of turn-ing preferences, scaling of the demand functions through speed limits, and thresholding o