Abstract

This paper aims to unify and extend existing techniques for deriving upper bounds on the transient of max-plus matrix powers. To this aim, we introduce the concept of weak CSR expansions: At=CStR⊕Bt. We observe that most of the known bounds (implicitly) take the maximum of (i) a bound for the weak CSR expansion to hold, which does not depend on the values of the entries of the matrix but only on its pattern, and (ii) a bound for the CStR term to dominate.To improve and analyze (i), we consider various cycle replacement techniques and show that some of the known bounds for indices and exponents of digraphs apply here. We also show how to make use of various parameters of digraphs. To improve and analyze (ii), we introduce three different kinds of weak CSR expansions (named after Nachtigall, Hartman–Arguelles, and Cycle Threshold). As a result, we obtain a collection of bounds, in general incomparable to one another, but better than the bounds found in the literature.

Highlights

  • Max-plus algebra is a version of linear algebra developed over the max-plus semiring, which is the set Rmax = R ∪ {−∞} equipped with the multiplication a ⊗ b = a + b and the addition a ⊕ b = max(a, b)

  • We investigate the sequence of max-plus matrix powers At =

  • As a common ground of transience bounds and CSR decomposition, we propose the new concept of weak CSR expansions

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Summary

Introduction

The smallest T that can be chosen in (1.2) is called the transient of A; we denote it by T (A) Since it satisfies x(t) = Atv every max-plus linear dynamical system, i.e., every sequence x(t) satisfying (1.1) is periodic in the same sense whenever A is irreducible. Bounds on the transients were obtained by Hartmann and Arguelles [14], Bouillard and Gaujal [4], Soto y Koelemeijer [27], Akian et al [2], and Charron-Bost et al [8] Those bounds are incomparable because they depend on different parameters of A or assume different hypotheses.

Preliminaries
Weak CSR Schemes
Main Results
Comparison to Previous Transience Bounds
Proof Strategy
Cycle Removal
11. Cycle Insertion
12. Local Reductions
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