Solution of the sweeping problem and application to a friction problem
type: Item
instance of: scholarly article
title: Solution of the sweeping problem and application to a friction problem
zbMATH Open document ID: 1009.34008
DOI: 10.12775/TMNA.2001.025
author: Mohamed Chraibi Kaadoud
published in: Topological Methods in Nonlinear Analysis
publication date: 2003-04-04T00:00:00Z
review text: The largest part of the article contains several results concerning existence and some qualitative properties of solutions \(q(\cdot)\in BV(I;H)\) (with bounded variation) to sweeping problems of the form \[ q(t)\in F(t,q(t)) \;\forall \;t\in I=[0,T], \quad -{{dq}\over{d\mu}}(t)\in N_{F(t,q(t))}q(t) \text{ a.e. }(d\mu), \quad q(0)=q_0\in F(0,q_0), \] under several types of regularity assumptions on the convex-valued multifunction \(F(\cdot,\cdot)\); here, \(H\) is either the real line (\(H=\mathbb R\)) or a finite-dimensional Hilbert space (\(H=\mathbb R^n\)), \(dq\) denotes the Stieltjes differential measure associated to the mapping \(q(\cdot)\in BV(I;H)\), \(d\mu\) is a positive measure on \(I\) for which there exists \(q'(\cdot)\in L^1(I;H)\) such that \(dq=q'd\mu\) and \(N_Aa\) denotes the normal cone in the sense of convex analysis to the convex subset \(A\subset H\) at the point \(a\in A\). Most of the results in the paper seem to be slight extensions of previous results, mainly from \textit{M. Kunze} and \textit{M. D. P. Monteiro Marques} [Topol. Methods Nonlinear Anal. 12, No. 1, 179--191 (1998; Zbl 0923.34018)]; the last part of the paper contains an application to a dry friction problem of Coulomb type.
reviewed by: Ştefan Mirică
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 70F40
zbMATH DE Number: 1751363
zbMATH Keywords: sweeping problem
zbMATH Keywords: bounded variation
zbMATH Keywords: absolute continuity
zbMATH Keywords: normal cone
zbMATH Keywords: differential inclusion
zbMATH Keywords: dry friction problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1751363