Optimization of Traffic Flow Networks

The focus of the thesis is placed on macroscopic traffic flow models on networks and corresponding optimal control problems. We present a multi-class traffic model on traffic flow networks that respects driver's destination preferences, i.e., everyone arrives at their desired destination. The result is based on and extends ideas and models devised in publications by Coclite, Garavello and Piccoli in 2005. In particular, we modify their coupling conditions at junctions and show with selected examples how to use the formulas and how they relate to existing road network models. Furthermore, we discuss the solution structure of the new multi-class model in detail and show that we can always construct a weak solution. For the special case of two classes we conduct a more detailed analysis of how to solve Riemann-problems for junctions of a particular type. Numerical simulations are carried out as well. The second aspect treated in the thesis concerns optimization. We review a general framework for PDE-constrained optimization and use it as a basis for the formal derivation of first order necessary optimality conditions for a traffic and a supply network. However, our presented formulas are not mathematically rigorous since we did not prove that all the appearing operators are well-defined in our setup. Nonetheless, the numerical results indicate that the derived equations are correct. The proposed optimization algorithm, essentially a projected gradient method, works reliably. In order to speed up the optimization process, we also investigate an instantaneous control approach in the context of traffic flow networks. Another key result is the derivation of the well-known Aw-Rascle model from a simplified kinetic model on a single road. The derivation implies that kinetic traffic flow models can contain more information than macroscopic ones. Therefore, it is worth studying them in their own right. Additionally, it might we worthwile to investigate kinetic models in a network setup.