The singular case of the problem of absolute stability of systems of ordinary differential equations
type: Item
instance of: scholarly article
title: The singular case of the problem of absolute stability of systems of ordinary differential equations
zbMATH Open document ID: 0687.34040
DOI: 10.1007/BF01054233
author: M. A. Nudel'man
published in: Siberian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let \[ J(\xi (.))=\int^{\infty}_{-\infty}\xi (i\omega)^*\Pi (i\omega)\xi (i\omega)d\omega +2 Re\int^{\infty}_{- \infty}r(i\omega)^*\xi (i\omega)d\omega, \] be a functional on \(H^ 2({\mathbb{C}}^ m)\). Let \(\Pi\) (p) be an \(m\times m\) matrix with rational entries, \(\Pi (i\omega)^*=\Pi (i\omega)\), r(p) a column rational m- vector and the elements of \(\Pi\) (p) and r(p) be bounded on the imaginary axis and \(r(\infty)=0\). The following theorem is proved: under previous assumptions J(\(\xi\) (.)) is semibounded if and only if the following conditions hold: 1) for every \(\omega\in {\mathbb{R}}\), \(\Pi\) (i\(\omega)\) is hermitian and nonnegative; 2) for any \(\xi (.)\in H^ 2({\mathbb{C}}^ n)\) with \(\Pi (i\omega)\xi (i\omega)=0\) the function \(r(.)^*\xi (.)\) belongs to \(H^ 2({\mathbb{C}})\).
reviewed by: A.R.Castro Figueroa
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 93D10
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 4125849
zbMATH Keywords: Popov-type stability
MaRDI profile type: Publication
cites work: Q3955633
cites work: Q3737581
description: scientific article; zbMATH DE number 4125849