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# Modeling, Analysis, Synthesis, and Simulation of Time-Optimal Train Traffic in Large Networks

## Bavafa-Toosi, Yazdan; Blendinger, Christoph; Mehrmann, Volker; Steinbrecher, Andreas; Unger, Roman

## Preprint-Reihe des Instituts fÃ¼r Mathematik, Technische UniversitÃ¤t Berlin

From a system-theoretic standpoint, a constrained state-space model for train traffic in a large railway network is developed. The novelty of the work is the transformation or rather reduction of the directed graph of the network to some parallel lists. Mathematization of this sophisticated problem is thus circumvented. All the aspects of a real network (such as that of the German Rail) are completely captured by this model. Some degrees of freedom as well as some robustness can be injected into the operation of the system. The problem of time-optimal train traffic in large networks is then defined and solved using the maximum principle. The solution is obtained by reducing the boundary value problem arising from the time-optimality criterion to an initial value problem for an ordinary differential equation. A taxonomy of all possible switching points of the control actions is presented. The proposed approach is expected to result in faster-than-real-time simulation of time-optimal traffic in large networks and thus facilitation of real-time control of the network by dispatchers. This expectation is quantitatively justified by analysis of simulation results of some small parts of the German Rail Network.