A semigroup approach to boundary feedback systems
type: Item
instance of: scholarly article
title: A semigroup approach to boundary feedback systems
zbMATH Open document ID: 1048.47054
author: Valentina Casarino
author: Klaus-Jochen Engel
author: Gregor Nickel
author: Rainer Nagel
published in: Integral Equations and Operator Theory
publication date: 2003-12-16T00:00:00Z
review text: In this paper, ``abstract boundary feedback systems'' of the following form are studied: \[ \dot{u}(t) = A_m u(t), \;\dot{x}(t) = Bx(t) + Cu(t), \;Lu(t) = x(t), \;t\geq 0, \;u(0)= f_0, \;x(0)=x_0, \] where \(A_m\) is a linear operator on the state space \(X\), \(B\) is a linear operator on the boundary space \(\partial X\), \(L\) (the boundary operator) is a linear and bounded operator from the domain of \(A_m\) to \(\partial X\), and \(C:X\rightarrow \partial X\) is the feedback operator. In order to solve this abstract boundary feedback system, the authors convert it into an abstract Cauchy problem of the form \(\dot{x}(t) = {\mathcal A}{x}(t) + F(t)\), \(t\geq 0\), \({x}(0)= {x}_0\), where \({\mathcal A}\) is an operator matrix on the product \(X\times \partial X\), and then semigroup methods are used. Further, spectral theory is applied in order to discuss stability under bounded feedback. As an example, the heat equation on a bounded domain \(\Omega\subset \mathbb R^n\) with Dirichlet and Neumann boundary conditions is studied.
reviewed by: Birgit Jacob
Mathematics Subject Classification ID: 47N70
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 93D15
zbMATH DE Number: 2018223
zbMATH Keywords: boundary feedback
zbMATH Keywords: strongly continuous semigroup
zbMATH Keywords: positivity
zbMATH Keywords: stability
zbMATH Keywords: operator matrices
MaRDI profile type: Publication
full work available at URL: s00020-002-1163-2
OpenAlex ID: W2031050023
DOI: 10.1007/S00020-002-1163-2
description: scientific article; zbMATH DE number 2018223