2002/08/31 by Stephane Gaubert, Ricardo Katz
Mathematics · #math.OC #msc:93B27 #msc:06F05
published as Kybernetika -- volume 40 (2004), number 2, pages 153--180. · 24 pages, 9 postscript figures; example in section 4.3 expanded
arxiv created 2002/11/13 · arxiv updated 2009/11/30
We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring, in order to extend the geometric approach of linear control to discrete event systems. We say that a subsemimodule of the free semimodule Sn over a semiring S is rational if it has a generating family that is a rational subset of Sn, Sn being thought of as a monoid under the entrywise product. We show that for various semirings of max-plus type whose elements are integers, rational semimodules are stable under the natural algebraic operations (union, product, direct and inverse image, intersection, projection, etc). We show that the reachable and observable spaces of max-plus linear dynamical systems are rational, and give various examples.