On the Sequence of Consecutive Matrix Powers of Boolean Matricesin

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.