Measurable viability theorems and the Hamilton-Jacobi-Bellman equation
type: Item
instance of: scholarly article
title: Measurable viability theorems and the Hamilton-Jacobi-Bellman equation
zbMATH Open document ID: 0836.34016
published in: Journal of Differential Equations
publication date: 1995-03-22T00:00:00Z
review text: Let \(t \rightsquigarrow P(t) : [0,T] \rightsquigarrow R^d\) be an absolutely continuous set-valued map and \((t,x) \rightsquigarrow F(t,x) : [0,T] \times R^d \rightsquigarrow R^d\) a set-valued map with closed convex values, measurable in \(t\), continuous in \(x\), and satisfying \(\sup \{|y |; y \in F(t,x),\;x \in R^d\} \leq \mu(t)\), \(t \in [0,T]\) for some Lebesgue integrable function \(\mu\). Then for any \(t_0 \in [0,T]\) and any \(x_0 \in P(t_0)\), the dynamical system (DS) \(x(t) \in F(t,x(t))\), \(t \in [t_0,T]\), \(x(t_0) = x_0\), has a solution \(x\) such that \(x(t) \in P(t)\) for all \(t \in [t_0,T]\) iff for almost all \(t \in [0,T]\) and all \(x \in P(t)\), \(F(t,x) \cap DP (t,x) (1) \neq \emptyset\), where \(DP(t,x)\) is the contingent derivative of \(P\) at \((t,x) \). If, moreover, \(F\) has compact values and is locally Lipschitz in \(x\), then any solution \(x\) of (DS) satisfies \(x(t) \in P(t)\) for all \(t \in [t_0,T]\) iff \(F(t,x) \subset DP (t,x) (1)\). The above results are used to study semicontinuous solutions of the Hamilton-Jacobi-Bellman equation \(u_t + H(t,x,u_x) = 0\), where \(H\) is measurable in \(t\), locally Lipschitz in \(x\) and convex in the third variable.
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 93B03
zbMATH DE Number: 736405
zbMATH Keywords: viability
zbMATH Keywords: set-valued map
zbMATH Keywords: Hamilton-Jacobi-Bellman equation
author: Hélène Frankowska
author: Slawomir Plaskacz
author: Tadeusz Rzeżuchowski
reviewed by: Jan Kučera
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1036
OpenAlex ID: W2000958946
DOI: 10.1006/JDEQ.1995.1036
Wikidata QID: Q131316920
description: scientific article; zbMATH DE number 736405