Absolute stability in delay equations
type: Item
instance of: scholarly article
title: Absolute stability in delay equations
zbMATH Open document ID: 0636.34063
DOI: 10.1016/0022-0396(87)90116-1
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: An equilibrium of a delay equation is said to be absolutely stable if it is asymptotically stable for all delays. As is well known, this is equivalent to the characteristic equation having all its roots in the left half-plane for all delays. The author considers delay equations where the characteristic equation is of the form \(P(z)+Q(z)e^{-\tau z}=0\) (\(\tau\) is the delay, p,Q are square matrices) and proves the following result. Theorem. Let P and Q be analytic in some open set containing Re \(z\geq 0\) and let P(z)\(\neq 0\), Re \(z\geq 0\), \(P(-iy)=P(iy)\), \(Q(-iy)=Q(iy)\), \(| Q(iy)| <| P(iy)|,\) \(0\leq y<\infty\), \(\lim_{| z| \to \infty,Re z\geq 0}| Q(z)P^{-1}(z)| =0.\) Then the equation is absolutely stable.
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 4036258
zbMATH Keywords: delay equation
zbMATH Keywords: characteristic equation
author: Fred Brauer
reviewed by: Stig-Olof Londen
MaRDI profile type: Publication
full work available at URL: 0022-0396(87)90116-1
OpenAlex ID: W2072493736
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description: scientific article; zbMATH DE number 4036258