Regularity of solution maps of differential inclusions under state constraints
type: Item
instance of: scholarly article
title: Regularity of solution maps of differential inclusions under state constraints
zbMATH Open document ID: 1126.34011
author: Piernicola Bettiol
author: Hélène Frankowska
published in: Set-Valued Analysis
publication date: 2007-04-26T00:00:00Z
review text: The paper concerns the Lipschitzian properties of the set-valued map \({\mathcal S}^K_{[t_0,T]}\) which assigns to every point \(\xi_0\in K\) the set of solutions \(x:[t_0,T]\to\mathbb R^n\) to the system \[ x'(t)\in F(t,x(t)), \quad x(t)\in K,\;x(t_0)=\xi_0. \] Here, the \(t\)-measurable, \(x\)-lipschitzean set-valued map \(F:[t_0,T]\times\mathbb R^n\to\mathbb R^n\) has (possibly unbounded) closed convex images and satisfies an inward pointing condition on the sufficiently smooth boundary of the closed set \(K\subseteq\mathbb R^n\). The paper extends and simplifies the approach developed in \textit{P. Bettiol, P. Cardaliaguet,} and \textit{M. Quincampoix} [Int. J. Game Theory 34, No.4, 495--527 (2006; Zbl 1103.49019)]. The result is applied to investigate Lipschitzian properties of the value function for the constrained Bolza problem of optimal control theory.
reviewed by: Corneliu Ursescu
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 49K24
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 5146248
zbMATH Keywords: differential inclusions
zbMATH Keywords: state constraints
zbMATH Keywords: Lipschitz dependence
MaRDI profile type: Publication
full work available at URL: s11228-006-0018-4
OpenAlex ID: W2072575566
cites work: Continuity of admissible trajectories for state constraints control problems
cites work: Q3976548
cites work: Q3782303
cites work: Zero-sum state constrained differential games: Existence of value for Bolza problem
cites work: Some Characterizations of Optimal Trajectories in Control Theory
cites work: Hamilton-Jacobi Equations with State Constraints
cites work: Invariant Solutions of Differential Games and Hamilton--Jacobi--Isaacs Equations for Time-Measurable Hamiltonians
cites work: Shape analysis via oriented distance functions
cites work: Oriented Distance Function and Its Evolution Equation for Initial Sets with Thin Boundary
cites work: Q4086303
cites work: Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation
cites work: Measurable viability theorems and the Hamilton-Jacobi-Bellman equation
cites work: A measurable upper semicontinuous viability theorem for tubes
cites work: Filippov's and Filippov-Ważewski's theorems on closed domains
cites work: Q4322122
cites work: Convex Analysis
cites work: Optimal Control with State-Space Constraint I
DOI: 10.1007/S11228-006-0018-4
description: scientific article; zbMATH DE number 5146248