Train control technology enhances the safety and efficiency of railroad operation by safeguarding the motion of trains to prevent them from leaving designated areas of operation and colliding with other trains. It is crucial for safety that the trains engage their brakes early enough in order to make sure they never leave the safe part of the track. Efficiency considerations, however, also require that the train does not brake too soon, which would limit operational suitability. It is surprisingly subtle to reach the right tradeoffs and identify the right control conditions that guarantee safe motion without being overly conservative. In pursuit of an answer, we develop a hybrid system model with discrete control decisions for acceleration, brakes, and with continuous differential equations for their physical effects on the motion of the train. The resulting hybrid system model is systematically derived from the Federal Railway Administration model for flat terrain by conservatively neglecting minor forces. The main contribution of this paper is the identification of a controller with control constraints that we formally verify to always guarantee collision freedom in the FRA model. The safe braking behavior of a train is influenced not only by the train configuration (e.g., train length and mass), but also by physical characteristics (e.g., brake pressure propagation and reaction time). We formalize train control safety properties in differential dynamic logic and prove the correctness of the train control models in the theorem prover KeYmaera X.
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