Approximating the solutions of differential inclusions driven by measures
type: Item
instance of: scholarly article
title: Approximating the solutions of differential inclusions driven by measures
zbMATH Open document ID: 1440.34005
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2019-11-27T00:00:00Z
review text: The paper focuses on measure differential inclusions of the form \[ \mathrm{d}x(t)\in F(t,x(t))\mathrm{d}\mu(t),\quad t\in[0,1], \] where \(F\) is defined on \([0,1]\times\mathbb R^d\), its values are compact convex subsets of \(\mathbb R^d\), and \(\mu\) is a Stieltjes measure generated by a left-continuous nondecreasing function. A solution of the inclusion is a function \(x:[0,1]\to\mathbb R^d\) for which there exists a \(g:[0,1]\to\mathbb R^d\) such that \(g(t)\in F(t,x(t))\) \(\mu\)-almost everywhere in \([0,1]\), and \[ x(t)=x(0)+\int_0^t g(s)\,\mathrm{d}\mu(s),\quad t\in[0,1]. \] The main results are a global existence theorem yielding the existence of a bounded variation solution, and two theorems dealing with continuous dependence of solutions with respect to the choice of the measure \(\mu\). The results are based on the concept of a bounded \(\varepsilon\)-variation introduced in [\textit{D. Fraňková}, Math. Bohem. 116, No. 1, 20--59 (1991; Zbl 0724.26009)]. In particular, it is assumed that the right-hand side \(F\) satisfies a condition that corresponds to bounded \(\varepsilon\)-variation for multifunctions.
reviewed by: Antonín Slavík
Mathematics Subject Classification ID: 34A06
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 26A42
zbMATH DE Number: 7137055
zbMATH Keywords: measure differential inclusion
zbMATH Keywords: regulated function
zbMATH Keywords: Kurzweil-Stieltjes integral
zbMATH Keywords: existence
zbMATH Keywords: continuous dependence
zbMATH Keywords: \(\varepsilon\)-variation
author: Luisa Di Piazza
author: Valeria Marraffa
author: Bianca Satco
MaRDI profile type: Publication
full work available at URL: s10231-019-00857-6
OpenAlex ID: W2940165449
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DOI: 10.1007/S10231-019-00857-6
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description: scientific article; zbMATH DE number 7137055