Approximation algorithm for an infinite-dimensional operator equation \(XL-BX=C\)
type: Item
instance of: scholarly article
title: Approximation algorithm for an infinite-dimensional operator equation \(XL-BX=C\)
zbMATH Open document ID: 0831.93013
DOI: 10.1007/BF01211486
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 1996-02-13T00:00:00Z
review text: The operator equation \(XL- BX= C\) for operators in a separable Hilbert space is studied. The operator \(L\) is assumed to be unbounded selfadjoint with a discrete spectrum tending to infinity (the results can be extended to a certain class of spectral operators \(L\) with compact resolvent). The equation arises in a stabilization scheme of general linear parabolic systems, and the operators \(B\) and \(C\) appear in the compensator. The operator \(C\) is finite-dimensional, while \(B\) is an unbounded operator with discrete spectrum whose eigenvectors form a Riesz basis. The condition \(\sigma(L)\cap \sigma(B)=\emptyset\) guarantees existence of a unique solution \(X\) which can be expressed in terms of the eigenvalues and eigenvectors of \(B\) and \(C\) and the resolvent of \(L\) (a result proved in an earlier paper by the author). A concrete algorithm for the approximation of the solution \(X\), with an estimate of the rate of convergence, is devised. The approximation is based on solving finite- dimensional equations of the same type.
Mathematics Subject Classification ID: 93B29
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 47A62
Mathematics Subject Classification ID: 93B60
zbMATH DE Number: 700868
zbMATH Keywords: operator equation
zbMATH Keywords: stabilization
zbMATH Keywords: linear parabolic systems
author: Takao Nambu
reviewed by: Leiba Rodman
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01211486
OpenAlex ID: W2170624718
description: scientific article; zbMATH DE number 700868