Viability for differential inclusions on graphs
type: Item
instance of: scholarly article
title: Viability for differential inclusions on graphs
zbMATH Open document ID: 1182.34083
author: Mihai Necula
author: Marius Popescu
author: Ioan I. Vrabie
published in: Set-Valued Analysis
publication date: 2009-03-16T00:00:00Z
review text: Let \(X\) be a Banach space, \(I\) a nonempty bounded interval and let \(K:I \to X\) and \(F:\mathcal{K}\) \(\to X\) be two multifunctions with nonempty values, where \(\mathcal{K}\) is the graph of \(K\). The authors consider the following Cauchy problem \[ u'(t)\in F(t,u(t)), \quad u(\tau)=\xi. \] A solution to the previous problem on \([\tau,T]\subset I\) is a function \(u \in W^{1,1}([\tau,T],X)\) satisfying \((t,u(t)) \in \mathcal{K}\), \(u(\tau)=\xi\) and \(u'(t)\in F(t,u(t))\) for a.e. \(t \in [\tau,T]\). The authors prove several necessary and sufficient conditions in order that \(\mathcal{K}\) be viable with respect to \(F\), which means that for each \((\tau, \xi) \in \mathcal{K}\) there exists \(T> \tau\) such that \([\tau,T]\subset I\) and the problem above has at least a solution \(u\) on \([\tau,T]\).
reviewed by: Francesca Papalini
Mathematics Subject Classification ID: 34G25
zbMATH DE Number: 5529276
zbMATH Keywords: Differential inclusion
zbMATH Keywords: locally closed graph
zbMATH Keywords: tangent set
zbMATH Keywords: tangency condition
zbMATH Keywords: multivalued mapping
zbMATH Keywords: viability
MaRDI profile type: Publication
full work available at URL: s11228-008-0090-z
OpenAlex ID: W2073508677
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DOI: 10.1007/S11228-008-0090-Z
description: scientific article; zbMATH DE number 5529276