Optimal strategies for the control of a train
A train modelled by a point mass is subject to a friction force, the effect of gravitation which depends on the track inclination which is constant on each subinterval \(I_k\) of a subdivision of the total interval corresponding to the travel, and controls which take discrete values (traction, coast and breaking). The dynamics are written such that speed and time are the states which depend on the distance along the track. It is assumed that the speed increases (resp. decreases) in a traction (resp. coast or breaking) mode, that traction and coast modes alternate followed by a final coast-breaking phase and that two successive switching points do not occur in \(I_k\). The train has to travel a certain distance in a certain time (the constraints) while minimizing fuel consumption which is proportional to the traction force; the cost is expressed as a function of the switching points. Quantities and relations which arise as necessary conditions for optimality under constraints are introduced. This allows to construct iteratively a switching point from the next one on the track. There are no proofs but the derivation of the necessary conditions should be available in a technical report of the author (1993) referenced in the article. This approach and related works of the author have resulted in up to twenty percent savings in fuel consumption of metropolitan trains in Australia. Let us remark that the Appendix A5 states that a measurable optimal control can be approximated by a switching type control provided there are enough switchings but this contradicts the hypothesis (which by the way is not clearly stated) that at most one switching occurs in \(I_k\). Thus, the switching strategies of the article are suboptimal with respect to an optimization problem involving a larger set of admissible controls.
scientific article; zbMATH DE number 885861