Diagonal stability of interval matrices and applications
type: Item
instance of: scholarly article
title: Diagonal stability of interval matrices and applications
zbMATH Open document ID: 1204.15028
author: Octavian Pastravanu
author: Mihaela-Hanako Matcovschi
published in: Linear Algebra and its Applications
publication date: 2010-10-22T00:00:00Z
review text: An interval matrix \({\mathcal A}^I= \{A\in\mathbb{R}^{n\times n}\mid A^-\leq A\leq A^+\}\) is called Schur stable, respectively, Hurwitz stable if the eigenvalues of all matrices \(A\in{\mathcal A}^I\) lie in the open unit disc of the complex plane, respectively, in its open left half plane. It is called Schur, respectively, Hurwitz diagonally stable relative to a Hölder \(p\)-norm \(\|\cdot\|_p\) \((1\leq p\leq\infty)\) if there exists a positive definite diagonal matrix \(D\) such that \[ \| A\|^D_p:=\| D^{-1} AD\|_p< 1, \] respectively \[ m^D_p(A):= \lim_{h\downarrow 0}\,h^{-1}(\| I+ hA\|^D_p- 1)< 0 \] holds for all matrices \(A\in{\mathcal A}^I\). The first part of the paper provides criteria for these latter types of stability. It presents methods for finding \(D\), analyses the robustness and investigates the connection with the standard concept of Schur and Hurwitz stability for interval matrices. The second part considers an equivalence of Schur respectively Hurwitz diagonal stability of \({\mathcal A}^I\) with some properties of a discrete- or continuous-time dynamical interval system whose motion is described by \({\mathcal A}^I\).
reviewed by: Günter Mayer
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93C41
Mathematics Subject Classification ID: 93D20
Mathematics Subject Classification ID: 65G30
zbMATH DE Number: 5804418
zbMATH Keywords: interval matrix
zbMATH Keywords: diagonal stability
zbMATH Keywords: Stein/Lyapunov matrix inequality
zbMATH Keywords: Gershgorin's disks
zbMATH Keywords: interval dynamical systems
zbMATH Keywords: invariant sets
zbMATH Keywords: (diagonally invariant) exponential stability
zbMATH Keywords: Schur stability
zbMATH Keywords: eigenvalues
zbMATH Keywords: positive definite diagonal matrix
zbMATH Keywords: Hurwitz stability
MaRDI profile type: Publication
full work available at URL: j.laa.2010.06.016
OpenAlex ID: W1995091508
cites work: Exact Maximum Singular Value Calculation of an Interval Matrix
cites work: A new vertex result for robustness problems with interval matrix uncertainty
cites work: Positive diagonal solutions to the Lyapunov equations
cites work: Matrix Diagonal Stability and Its Implications
cites work: Set-theoretic methods in control
cites work: Sufficient conditions on stability of interval matrices: connections and new results
cites work: Comments on "A necessary and sufficient condition for the stability of nonnegative interval discrete systems" by B. Shafai et al
cites work: Q5533446
cites work: Robust stability conditions for polytopic systems
cites work: Q4040931
cites work: Q4937699
cites work: Assessing the stability of linear time-invariant continuous interval dynamic systems
cites work: Dynamic interval systems analysis and design
cites work: Criteria for robust absolute stability of time-varying nonlinear continuous-time systems
cites work: Quadratic stability and stabilization of dynamic interval systems
cites work: Robust absolute stability of time-varying nonlinear discrete-time systems
cites work: Necessary and Sufficient Conditions for Componentwise Stability of Interval Matrix Systems
cites work: Generalized matrix diagonal stability and linear dynamical systems
cites work: On stability of interval matrices
cites work: Necessary and sufficient conditions for stability of a class of interval matrices
cites work: On stability of discrete-time interval matrices
cites work: Matrix measure and application to stability of matrices and interval dynamical systems
DOI: 10.1016/J.LAA.2010.06.016
description: scientific article; zbMATH DE number 5804418