Necessary and sufficient conditions for local invariance for semilinear differential inclusions
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions for local invariance for semilinear differential inclusions
zbMATH Open document ID: 1257.34045
author: Victor Postolache
author: Ovidiu Cârjă
published in: Set-Valued and Variational Analysis
publication date: 2012-03-09T00:00:00Z
review text: The authors study local invariance of a set \(K\subseteq X\), where \(X\) is a separable Banach space, for the autonomous semilinear differential inclusion \[ {u^{\prime }}(t) \in Au(t)+F(u(t)), \quad {u(0) = \xi }.\tag{1} \] The assumptions generally include that \(A:D(A)\subseteq X\rightarrow X\) is the infinitesimal generator of a \(C_{0}\)-semigroup \(\left\{ S(t):X\rightarrow X\mid t\geq 0\right\} \), \(F:X\rightarrow 2^{X}\) is closed and bounded-valued, and satisfies a Lipschitz condition, and \(K\) is locally closed. The authors make use of a tangency concept in which a function \(f\in L_{\text{loc}}^{1}\left({R}_{+},X\right) \) is said to be \({A}\)-tangent to the set \({K}\) at the point \({\xi \in K}\) if \[ \lim \inf_{h\downarrow 0}\frac{1}{h}\operatorname{dist}\left( S(h)\xi +\int_{0}^{h}S\left( h-s\right) f(s)ds,K\right) =0. \] Both necessary and sufficient conditions are specified. The final section gives an application to an optimal control problem.
reviewed by: Daniel C. Biles
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 49K21
Mathematics Subject Classification ID: 49K27
zbMATH DE Number: 6013490
zbMATH Keywords: invariance
zbMATH Keywords: differential inclusion
zbMATH Keywords: Lipschitz
zbMATH Keywords: tangency
zbMATH Keywords: optimal control
zbMATH Keywords: semilinear
MaRDI profile type: Publication
full work available at URL: s11228-011-0173-0
OpenAlex ID: W2077934454
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DOI: 10.1007/S11228-011-0173-0
description: scientific article; zbMATH DE number 6013490