Max-plus switching systems and long max-plus matrix products

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Switching max-plus linear (SMPL) systems are used to model the propagation of delays in a railway network.Suppose that departure times of a network at a particular instant are given by a vector x(k). The departure times at instant k are given by a recurrence relation based upon the departure times at a previous instant multiplied in the sense of max-plus arithmetic by matrix A(u(k)) where u(k) is a control variable. At every instant, the network may slightly change, which is encapsulated in this control variable.This has a residual effect on the ultimate departure times, which one often seeks to optimize in some way: for instance, by minimizing the cumulative delay with respect to a given regular schedule. The main idea of the proposed research is to study the switching max-plus linear systems as vector orbits of a finitely generated max-plus matrix semigroup: the set of all possible max-plus matrix products of a given finite set of matrices.In this research we will be particularly interested in the long max-plus matrix products and developing for them an analogue of the CSR representation of max-plus matrix powers suggested by Sergeev and Schneider. We then aim to apply the new theoretical results which we obtain to switching max-plus systems and the above mentioned optimization problems over them.