Mathematics of Traffic Flow Networks: Modeling, Simulation and Optimization

Traffic flow has been a continuous source of challenging mathematical
problems. The following work is dedicated to recent questions in
modeling, simulation and optimization of traffic flow networks.

Mathematics can help to solve traffic problems in different
ways. Modelling provides fundamental understanding of
traffic dynamics and behaviour. Optimization yields solutions for
complex situations and helps to organize traffic flow. During the last
decade there has been intensive research in different fields of and related
to traffic flow. One of the primary research activities focus on the
development of new and more realistic models for traffic flow on a
single road. Our work's primary
focus is on models for networks.
We provide new ideas on modelling flow in networks and solve different
optimization problems analytically and numerically.

The main result is the derivation of a hierarchy of
models treating different situations with suitable
traffic flow models.
To each level of modeling we consider the optimal control
problems and present techniques to address those problems.
Furthermore, we
derive an adjoint calculus for scalar hyperbolic equations with
nonlinear boundary controls.


The derived concepts fit for general network problems
as well as they do for traffic flow issues. The principles of
modeling and simplification can be applied to all kinds of network
flows, like fluid flow in
open channels or gas networks.