Some properties of the attainable set for the abstract control problem with application to controllability
type: Item
instance of: scholarly article
title: Some properties of the attainable set for the abstract control problem with application to controllability
zbMATH Open document ID: 0826.93005
DOI: 10.4171/ZAA/682
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1995-11-27T00:00:00Z
review text: For linear differential control systems described by the following evolution equation: \({dx\over dt} = Ax(t) + Bu(t)\), \(x(0) = x_0\), \(y(t) = Cx(t)\) the attainable set \[ K(t) := \{x \in X: x = \int^t_0 S(t - \tau) Bu (\tau) d\tau,\quad u(\cdot) \in L_2([0,t]), U)\} \] and the unobservable set \[ N(t) = \{x \in X : CS(\tau) x = 0,\quad \forall \tau \in [0,t]\} \] where \(A\) is a linear operator generating a strongly continuous semigroup \(\{S(t)\}_{t \geq 0}\) on a Banach space \(X\), \(B: U \to X\) and \(C : X \to Y\) are linear bounded operators and \(Y\), \(U\) are Banach spaces, are investigated. Conditions of independence of \(t\) for the closure \(\overline {K(t)}\) and \(N(t)\) have been established. Those conditions are useful for investigations of approximate controllability and observability of control systems under consideration.
reviewed by: Wiesław Kotarski
Mathematics Subject Classification ID: 93B03
Mathematics Subject Classification ID: 93C25
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93C99
zbMATH DE Number: 761130
zbMATH Keywords: linear
zbMATH Keywords: attainable set
zbMATH Keywords: unobservable set
zbMATH Keywords: operator
zbMATH Keywords: approximate controllability
author: Benzion Shklyar
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/682
OpenAlex ID: W2055254171
description: scientific article; zbMATH DE number 761130