Approximate weak invariance and relaxation for fully nonlinear differential inclusions
type: Item
instance of: scholarly article
title: Approximate weak invariance and relaxation for fully nonlinear differential inclusions
zbMATH Open document ID: 1272.34085
author: I. Căpraru
published in: Mediterranean Journal of Mathematics
publication date: 2013-03-20T00:00:00Z
review text: The first part of the paper is devoted to establishing necessary and sufficient conditions for the approximate weak invariance of a set with respect to fully nonlinear differential inclusions. The criterion for weak invariance is given in terms of new tangency condition involving the concept of \(A\)-quasi tangency, introduced in [\textit{O. Cârja} et al., Set-Valued Anal. 16, No. 5--6, 701--731 (2008; Zbl 1179.34068)]. In the second part, the author proves a relaxation type theorem, without assuming that the multifunction is Lipschitz. To be more precise, it is shown that the set of approximate solutions to the considered differential inclusion is dense in the set of solutions to the relaxed one. The proof is in the spirit of \textit{F. H. Clarke} et al. [J. Dyn. Control Syst. 3, No. 4, 493--518 (1997; Zbl 0951.49007)]. The problem is treated as a viability problem for a suitably constructed set. The main tools are viability results adapted from [O. Cârja et al., loc. cit.], and an interesting-by-itself result, which states that a subset of a separable Banach space with uniformly convex dual is \(A\)-quasi tangent if and only if its convex hull is \(A\)-quasi tangent.
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 6147182
zbMATH Keywords: approximate weak invariance
zbMATH Keywords: approximate solutions
zbMATH Keywords: relaxed nonlinear differential inclusion
reviewed by: Ovidiu Cârjă
MaRDI profile type: Publication
full work available at URL: s00009-012-0194-4
OpenAlex ID: W2133690445
cites work: Q3441590
cites work: Necessary and sufficient conditions for viability for nonlinear evolution inclusions
cites work: Approximate invariance and differential inclusions in Hilbert spaces
cites work: A priori estimates for operational differential inclusions
cites work: Q3904867
cites work: Approximate weak invariance for semilinear differential inclusions in Banach spaces
cites work: Q4842674
DOI: 10.1007/S00009-012-0194-4
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description: scientific article; zbMATH DE number 6147182