Strong invariance and one-sided Lipschitz multifunctions
type: Item
instance of: scholarly article
title: Strong invariance and one-sided Lipschitz multifunctions
zbMATH Open document ID: 1068.34010
author: V. Ríos
author: T. Donchev
author: P. R. Wolenski
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-02-22T00:00:00Z
review text: The paper is devoted to the study of some viability issues concerning differential inclusions in the Euclidean space \(\mathbb{R}^n\). The authors study the Cauchy problem \[ x'\in F(t, x), \quad x\in S, \quad t\in I= [t_0, t_1), \quad x(t_0)= x_0\in S,\tag \(*\) \] where \(S\subset\mathbb{R}^n\) is closed, \(F\) has convex compact values, is almost upper semicontinuous and integrably bounded. In order to establish the strong invariance (strong viability), that is to show that any solution to \((*)\) with \(x_0\in S\) remains in \(S\), they introduce the max-Hamiltonian \(H_F(t,x,y)= \sup_{z\in F(t,x)}\langle z, y\rangle\) and, instead of the Lipschitz continuity of \(F\) which has been usually assumed to get the strong invariance, they assume that \(H_F\) satisfies the so-called one sided Lipschitz estimates of the form \(H_F(t,x,x- y)- H_F(t,y,x- y)\leq k(t)\| x- y\|\) where \(k\) is locally integrable. They further prove that, under these assumptions, \(S\) is strongly invariant, provided that the for almost all \(t\), asymptotically \(H(t,x,y)\leq 0\) for all vectors \(y\) in the proximal normal cone \(N^P_S(x)\) at \(x\in S\). The authors study also the situation of time dependent state constraints \(S(t)\), \(t\in I\), and show conditions equivalent to the weak and strong invariance. These conditions are stated in terms of Hamilton-Jacobi-type inequalities.
reviewed by: Wojciech Kryszewski
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 49L99
zbMATH DE Number: 2136692
zbMATH Keywords: Strong invariance
zbMATH Keywords: One-sided Lipschitz maps
zbMATH Keywords: Differential inclusions
zbMATH Keywords: Hamilton-Jacobi inequality
MaRDI profile type: Publication
full work available at URL: j.na.2004.09.050
OpenAlex ID: W2016930623
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DOI: 10.1016/J.NA.2004.09.050
description: scientific article; zbMATH DE number 2136692