Proofs of averaging theorems for differential inclusions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1069.34010
publication date: 2001-01-01T00:00:00Z
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34E99
zbMATH DE Number: 1728966
zbMATH Keywords: differential inclusions with slow and fast variables
zbMATH Keywords: averaging
zbMATH Keywords: upper and lower approximation
zbMATH Keywords: small parameter
author: O. P. Filatov
MaRDI profile type: Publication
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title: Proofs of averaging theorems for differential inclusions
published in: Vestnik Samarskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Fizika, Khimiya, Biologiya
review text: The author considers the differential inclusion with slow and fast variables NEWLINE\[NEWLINE \dot{x}\in\mu F(t,x,y,\mu), \;\;\dot{y}\in G(t,x,y,\mu), NEWLINE\]NEWLINE where \(\mu\) is a small parameter. In the earlier papers [the author, Differ. Equations 31, 47--54 (1995); translation from Differ. Uravn. 31, 54--62 (1995; Zbl 0852.34015); the author and \textit{M. M. Khapaev}, Math. Notes 47, 596--601 (1990); translation from Mat. Zametki 47, No. 6, 102--109 (1990; Zbl 0723.34010) etc., see also more detailed discussion in the book by \textit{O. P. Filatov} and \textit{M. M. Khapaev}, Averaging of systems of differential inclusions (in Russian), Moscow: Moscow University Press (1998)], the author stated and studied the problem of approximation of the slow variables \(x\) by the averaged differential inclusion on the asymptotically large time interval \(O(\mu^{-1})\). It was shown that there are three different types of approximation: the upper approximation, lower one, and reciprocal one.NEWLINENEWLINENEWLINEHere, new proofs of the averaging theorems are given for approximating inclusions in all three types mentioned above. These new proofs are simpler than the previous ones. Simplification is accomplished by using the lemma on continuous dependence of the solution of the differential inclusion from the right-hand part and initial conditions. This important lemma is also completely proved in this paper.
reviewed by: Oleg Anashkin
description: scientific article; zbMATH DE number 1728966