B. De Schutter and B. De Moor, "On the sequence of consecutive matrix powers of boolean matrices in the max-plus algebra," in Theory and Practice of Control and Systems (Proceedings of the 6th IEEE Mediterranean Conference on Control and Systems, Alghero, Italy, June 1998) (A. Tornambè, G. Conte, and A.M. Perdon, eds.), Singapore: World Scientific, ISBN 981-02-3668-9, pp. 672-677, 1999.
In this paper we consider sequences of consecutive powers of boolean matrices in the max-plus algebra, which is one of the frameworks that can be used to model certain classes of discrete event systems. The ultimate behavior of a sequence of consecutive max-plus-algebraic powers of a boolean matrix is cyclic. First we derive upper bounds for the length of the cycles as a function of the size of the matrix. Then we study the transient part of the sequence of consecutive powers of a max-plus-algebraic boolean matrix, and we derive upper bounds for the length of this transient part. These results can then be used in the max-plus-algebraic system theory for discrete event systems.