A uniqueness theorem for differential inclusions
type: Item
instance of: scholarly article
title: A uniqueness theorem for differential inclusions
zbMATH Open document ID: 0705.34015
DOI: 10.1016/0022-0396(90)90132-9
author: P. R. Wolenski
published in: Journal of Differential Equations
publication date: 1990-01-01T00:00:00Z
review text: The author considers the autonomous differential inclusion problem \(\dot x\in F(x(t))\) a.e. \(t\in [0,T]\), \(x(0)=\xi\), where x: [0,T]\(\to {\mathbb{R}}^ n\) is an absolutely continuous vector function, F a multifunction assigning to each point in an open subset X of \({\mathbb{R}}^ n\) a nonempty compact subset of \({\mathbb{R}}^ n\), and \(\xi\) a given vector in \({\mathbb{R}}^ n\). By the reachable set \(R^{(T)}(\xi)\) is understood the set of vectors x(T) for x(t) being a solution of the above problem. He observes that the family \(\{R^ T\}_{T\geq 0}\) of multifunctions is a semigroup since \(R^{(s+t)}(\xi)=\cup_{\eta \in R^{(t)}(\xi)}R^{(s)}\eta =:(R^ s\circ R^{(t)})\xi\) for s,t\(\geq 0\) and that the Hausdorff distance between the set \((R^{(t)}(\xi)-\xi)/t\) and F(\(\xi\)) tends to zero as \(t\to 0\). Thus, the semigroup \(R^{(t)}\) possesses an infinitesimal generator. These facts enable him to show that if F is locally Lipschitzian on X then the reachable sets \(R^{(t)}(\xi)\) exhibit the same uniqueness properties as the solutions of initial value problems of ordinary differential equations.
Mathematics Subject Classification ID: 34A60
zbMATH DE Number: 4156936
zbMATH Keywords: autonomous differential inclusion
zbMATH Keywords: reachable set
reviewed by: Ludvík Janoš
MaRDI profile type: Publication
full work available at URL: 0022-0396(90)90132-9
OpenAlex ID: W1979500420
cites work: Q3324260
cites work: Q4000274
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cites work: Stability in general control systems
cites work: On generalized dynamical systems defined by contingent equations
cites work: The Exponential Formula for the Reachable Set of a Lipschitz Differential Inclusion
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description: scientific article; zbMATH DE number 4156936