A few results concerning the Hurwitz stability of polytopes of complex polynomials
type: Item
instance of: scholarly article
title: A few results concerning the Hurwitz stability of polytopes of complex polynomials
zbMATH Open document ID: 1246.30011
author: Stanisław Białas
author: Michał Góra
published in: Linear Algebra and its Applications
publication date: 2012-03-07T00:00:00Z
review text: Let \(f_1,f_2,\dots, f_m\) be complex polynomials with \[ n-1 = \mathrm{deg} f_1 =\dots =\mathrm{deg}f_k< \mathrm{deg}f_{k+1}= \dots = \mathrm{deg}f_m =n. \] Let further \[ C_{f_1,f_2,\dots, f_m}=\left\{\sum\limits_{i=1}^n \alpha_i f_i\;: \alpha_i \geq 0 \; (i=1, \dots , m),\; \sum\limits_{i=1}^m \alpha_i=1 \right\} \] be the polytope of the polynomials \(f_1,f_2,\dots ,f_m\). Let \(H\) be a nontrivial supporting hyperplane of \(C_{f_1,f_2,\dots, f_m}\). The one-dimensional polytopes \(C_{f_1,f_2,\dots, f_m}\cap H\) are called exposed edges. Under the above assumptions it is proved that the polytope \(C_{f_1,f_2,\dots, f_m}\) is Hurwitz stable if and only if all its exposed edges are stable. In the case of two complex polynomials of arbitrary degrees also a necessary and sufficient condition is given for the (Hurwitz) stability of segments.
reviewed by: Piroska Lakatos
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 26C10
Mathematics Subject Classification ID: 93D09
Mathematics Subject Classification ID: 12D10
zbMATH DE Number: 6012749
zbMATH Keywords: zeros of polynomials
zbMATH Keywords: polytopes of polynomials
zbMATH Keywords: robust stability
zbMATH Keywords: Hurwitz stability
MaRDI profile type: Publication
full work available at URL: j.laa.2011.07.042
OpenAlex ID: W2061843682
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DOI: 10.1016/J.LAA.2011.07.042
description: scientific article; zbMATH DE number 6012749