Complete controllability of spectrum assignment in infinite dimensional spaces
type: Item
instance of: scholarly article
title: Complete controllability of spectrum assignment in infinite dimensional spaces
zbMATH Open document ID: 0820.47076
DOI: 10.1007/BF01322549
published in: Integral Equations and Operator Theory
publication date: 1995-09-06T00:00:00Z
review text: The authors address the problem of complete controllability in infinite dimensions: Let \(U\), \(X\) be Banach spaces and form with \(A\in {\mathcal L}(X)\), \(B\in {\mathcal L}(U, X)\) the initial value problem \[ x'(t)= Ax(t)+ Bu(t),\quad x(0)= x_ 0, \] where \(u\) is piecewise continuous. This problem is called completely controllable if, for each \(x_ 0\in X\), one can find \(u\) and \(T> 0\) such that the unique solution \(x\) satisfies \(x(T)= 0\). The authors consider six conditions for complete controllability, all of which were already discussed in the literature (at least in finite dimensions). The 6th condition is treated in infinite dimensions here for the first time but seems to clarify the picture somewhat. It is shown as the main result that all conditions are equivalent in Hilbert spaces (Theorem 10). The dual problem of observability is also discussed, and some relation with the theory of holomorphic operator functions is established.
Mathematics Subject Classification ID: 47N70
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 93B28
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 515538
zbMATH Keywords: holomorphic operator functions
zbMATH Keywords: controllability in infinite dimensions
zbMATH Keywords: dual problem of observability
author: A. Markus
author: V. R. Ol'shevskij
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01322549
OpenAlex ID: W2091346653
description: scientific article; zbMATH DE number 515538