Controllability up to renormalization and small control operators
type: Item
instance of: scholarly article
title: Controllability up to renormalization and small control operators
zbMATH Open document ID: 0910.93012
DOI: 10.1016/S0021-7824(98)80027-5
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1999-04-07T00:00:00Z
review text: In this elaborated and well written paper the author presents results on various controllability types for systems described by the equation \(x'=Ax+Bu\), where \(A\) generates a continuous semigroup \(S(t)\) on a Banach space \(X\) and \(B\) is a bounded control operator. In the first part of the paper the author gives necessary or/and sufficient conditions for exact controllability and null controllability, involving \(B\) and \(S(t)\), by using results from linear operator theory. Although most of these results are not new (for example the lack of exact controllability with compact \(B\) is an old result of Triggiani), this part is a good background for the main part of the paper where the author goes deeply into comparison with the standard operator methods. Here \(X\) is a Hilbert space and \(A\) has a Riesz basis of eigenvectors, \(A\phi_n =-\lambda_n \phi_n\), \(n\geq 1\). The problem is to describe the space of reachable points, which is more general than that of exact or null controllability. In this context the author defines the notion of controllability up to renormalization (CR), i.e., the existence of weights \(w_n >0\) such that the restriction of \((A,B)\) on the space \[ H(w_n)= \left\{ \sum_{n\geq 1} a_n \phi_n ; \sum_{n\geq 1}| a_n | ^2 \| \phi_n \| ^2 w_n ^2 <\infty\right\} \] is well defined and exactly controllable. A spectral necessary condition for \(A\) in order to have (CR) is given and then, assuming that, necessary and sufficient conditions involving \(B\) are derived. The author obtains the following result: for every quasi-normed ideal of compact operators \({\mathcal I}\) (but not for the ideal of finite rank operators) there exists \(B\in {\mathcal I}\) such that the system \((A,B)\) enjoys the (CR) property. Necessary and sufficient conditions for null controllability involving \(\lambda_n\) and \(B\) are also presented and applications to both hyperbolic and parabolic equations are given. Finally, a link between control problems and…
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B28
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 30B50
Mathematics Subject Classification ID: 47B06
zbMATH DE Number: 1187616
zbMATH Keywords: exact controllability
zbMATH Keywords: null controllability
zbMATH Keywords: linear systems
zbMATH Keywords: spectral conditions
zbMATH Keywords: controllability up to renormalization
zbMATH Keywords: ideal of compact operators
author: N. K. Nikol'skij
reviewed by: Ovidiu Cârjă
MaRDI profile type: Publication
cites work: On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems
cites work: Q4856165
cites work: Mappings Between Function Spaces
cites work: Q4848744
cites work: Q3250836
cites work: A General Theory of Observation and Control
cites work: Q3254057
cites work: Q5512384
cites work: Q3918553
cites work: Q3713006
cites work: A function model and some problems in the spectral theory of functions
cites work: A General Necessary Condition for Exact Observability
description: scientific article; zbMATH DE number 1187616