Approximate observability of abstract evolution equation with unbounded observation operator
type: Item
instance of: scholarly article
title: Approximate observability of abstract evolution equation with unbounded observation operator
zbMATH Open document ID: 1029.93006
DOI: 10.1016/S0895-7177(02)00170-X
published in: Mathematical and Computer Modelling
publication date: 2003-10-14T00:00:00Z
review text: The standard assumptions in this paper are: (i) \(A\) is the infinitesimal generator of a strongly continuous semigroup on the Banach space \(X\), (ii) \(A\) has a complete basis of (generalized) eigenfunctions, and (iii) the sequence \(\{f_{jk}(t):= t^k e^{\lambda_j t}\), \(j\in\mathbb{N}\), \(k\in\{0,1,\dots, \alpha_j- 1\}\}\) is minimal on \(L^2(0, t_1)\) for some \(t_1> 0\), where \(\lambda_j\) is the \(j\)th eigenvalue of \(A\) with multiplicity \(\alpha_j\). Under these assumptions the author shows that the system \(\dot x(t)= Ax(t)\), \(y(t)= Cx(t)\) is approximately observable if and only if there exists no eigenvector of \(A\) in the kernel of \(C\). Unfortunately, the author seems to be unaware of the results obtained by \textit{S. A. Avdonin} and \textit{S. A. Ivanov} as presented in their book [Families of exponentials. The method of moments in controllability problems for distributed parameter systems, Cambridge Univ. Press (1995; Zbl 0866.93001)].
Mathematics Subject Classification ID: 93B07
Mathematics Subject Classification ID: 93C25
zbMATH DE Number: 1992507
zbMATH Keywords: evolution systems
zbMATH Keywords: strongly continuous semigroup
zbMATH Keywords: unbounded observation operator
zbMATH Keywords: unobservable set
zbMATH Keywords: minimal sequences of functions
zbMATH Keywords: approximate observability
author: Benzion Shklyar
reviewed by: Hans Zwart
MaRDI profile type: Publication
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full work available at URL: s0895-7177(02)00170-x
OpenAlex ID: W2059377639
description: scientific article; zbMATH DE number 1992507