Approximate weak invariance for differential inclusions in Banach spaces
type: Item
instance of: scholarly article
title: Approximate weak invariance for differential inclusions in Banach spaces
zbMATH Open document ID: 1267.34116
published in: Journal of Dynamical and Control Systems
publication date: 2013-02-14T00:00:00Z
review text: The authors consider the differential inclusion \[ x^{\prime }(t)\in F(x(t)) \] in a Banach space and study the relationship between approximate weak invariance and a tangency condition. The definition of approximate weak invariance of a set \(K\) requires that an approximate solution starting in \(K\) remains near \(K\) for sufficiently small time. In Theorem 1, sufficient conditions for a locally closed set to be approximately weakly invariant are given, namely that \(F\) is locally bounded and that \(F(x)\) is quasi-tangent to \(K\) at every point \(x\in K\). Theorem 2 specifies conditions sufficient for approximate weak invariance of a locally closed set \(K\) with the restriction that one considers only exact solutions in the definition. \(F\) is specified to be bounded-valued, satisfy a global Lipschitz condition and \(F(x)\) is quasi-tangent to \(K\) at every point \(x\in K\). As an application, the authors prove a result on Lipschitz dependence of approximate solutions on the initial state. Theorem 3 specifies that the quasi-tangency assumption is necessary for a set \(K\) to be approximately weakly invariant in a separable Banach space under the assumptions that \(F\) satisfy an upper semicontinuity condition and a growth condition. The paper is motivated by the desire to extend to Banach spaces the results in Hilbert spaces found by \textit{F. H. Clarke} et al. [J. Dyn. Control Syst. 3, No. 4, 493--518 (1997; Zbl 0951.49007)].
Mathematics Subject Classification ID: 34G25
zbMATH DE Number: 6136289
zbMATH Keywords: differential inclusion
zbMATH Keywords: invariance
zbMATH Keywords: tangency
zbMATH Keywords: approximate solution
zbMATH Keywords: Banach space
author: Ovidiu Cârjă
author: A. I. Lazu
reviewed by: Daniel C. Biles
MaRDI profile type: Publication
full work available at URL: s10883-012-9140-9
OpenAlex ID: W2017606039
cites work: Weak tangency, weak invariance, and Carathéodory mappings
cites work: Necessary and sufficient conditions for viability for semilinear differential inclusions
cites work: Approximate invariance and differential inclusions in Hilbert spaces
cites work: A priori estimates for operational differential inclusions
cites work: On the solution set of differential inclusions in Banach space
DOI: 10.1007/S10883-012-9140-9
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description: scientific article; zbMATH DE number 6136289