Nonlinear Left and Right Eigenvectors for Max-Preserving Maps

Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 471)

Abstract

It is shown that max-preserving maps (or join-morphisms) on the positive orthant in Euclidean n-space endowed with the component-wise partial order give rise to a semiring. This semiring admits a closure operation for maps that generate stable dynamical systems. For these monotone maps, the closure is used to define suitable notions of left and right eigenvectors that are characterized by inequalities. Some explicit examples are given and applications in the construction of Lyapunov functions are described.

Keywords

Monotone systems Join-morphisms Perron-Frobenius theory Positive eigenvectors Small-gain condition Lyapunov functions 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Mathematical and Physical SciencesThe University of Newcastle (UON)CallaghanAustralia

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