Asymptotic methods in the theory of differential equations with discontinuous and multi-valued right-hand sides
type: Item
instance of: scholarly article
title: Asymptotic methods in the theory of differential equations with discontinuous and multi-valued right-hand sides
zbMATH Open document ID: 0872.34003
DOI: 10.1007/BF02363930
author: V. A. Plotnikov
published in: Journal of Mathematical Sciences (New York)
publication date: 1997-10-12T00:00:00Z
review text: The averaging Bogolyubov theorem is generalized for differential inclusions. It is shown that for every solution of the averaged differential inclusion \(\xi'\in\varepsilon \overline{X}(\xi)\), \[ \overline{X}(x)= 1/T\lim_{T\to\infty} \int_0^T X(t,x)dt \] there exists a solution of the original differential inclusion \(x'\in X(t,x)\) such that the difference between these two solutions is small on the time scale \(1/\varepsilon\). This implies that the attainable set \(R(t,\varepsilon)\) of the original differential inclusion is upper semicontinuous with respect to \(\varepsilon\) at \(\varepsilon=0\), if \(R(t,0)\) denotes the attainable set of the averaged differential inclusion. If, for example, \(F(t,x)\) is piecewise continuous, then it is proved that the map \(R(t,\varepsilon)\) is continuous at \(\varepsilon=0\). If the differential inclusion \(x'\in F(t,x)\) is obtained as a Filippov regularization of a discontinuous differential equation, then solutions may move along the discontinuity manifold. An averaging principle for such sliding regime is also discussed in the present paper.
reviewed by: Vlastimil Křivan
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34C29
zbMATH DE Number: 993758
zbMATH Keywords: averaging Bogolyubov theorem
zbMATH Keywords: differential inclusions
zbMATH Keywords: Filippov regularization
zbMATH Keywords: discontinuous differential equation
MaRDI profile type: Publication
cites work: Q3688668
cites work: Q3726880
cites work: Q4302119
cites work: N. N. Bogolyubov's theorem for hyperbolic differential inclusions
cites work: Mutual \(\epsilon\)-approximation of solutions of a system of differential inclusions and of the averaged inclusion
cites work: Averaging of differential inclusions with ``fast'' and ``slow'' variables
cites work: Q5513216
cites work: Q4884302
cites work: Q3934212
cites work: Q3967817
cites work: Q4692459
cites work: Q4341867
description: scientific article; zbMATH DE number 993758