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Rational Semimodules over the Max-Plus Semiring and Geometric Approach of Discrete Event Systems

Abstract : We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring. We say that a subsemimodule of the free semimodule S^n over a semiring S is rational if it has a generating family that is a rational subset of S^n, S^n 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). Rational semimodules are a tool to extend the geometric approach of linear control to discrete event systems. In particular, we show that the reachable and observable spaces of max-plus linear dynamical systems are rational.
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https://hal.inria.fr/inria-00072069
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Submitted on : Tuesday, May 23, 2006 - 7:42:21 PM
Last modification on : Thursday, February 3, 2022 - 11:13:56 AM
Long-term archiving on: : Sunday, April 4, 2010 - 10:51:38 PM

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  • HAL Id : inria-00072069, version 1

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Stéphane Gaubert, Ricardo David Katz. Rational Semimodules over the Max-Plus Semiring and Geometric Approach of Discrete Event Systems. [Research Report] RR-4519, INRIA. 2002. ⟨inria-00072069⟩

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