Lagrange-type extremal trajectories in differential inclusions
type: Item
instance of: scholarly article
title: Lagrange-type extremal trajectories in differential inclusions
zbMATH Open document ID: 0773.49002
DOI: 10.1016/0167-6911(92)90119-D
author: Stanisław jun. Łojasiewicz
author: Barbara Kaśkosz
published in: Systems \& Control Letters
publication date: 1993-01-04T00:00:00Z
review text: The authors consider the differential inclusion \((*)\) \(x'\in F(t,x)\), where the multifunction \(F\) is defined for \(t\in I=[0,1]\) and \(x\in \Omega\) when \(\Omega\) is an open subset of \(\mathbb{R}^ n\). Denote by \(B(F)\) the set of all boundary trajectories of the inclusion \((*)\). Let also \(H(F)\) the family of all Hamiltonian extremals of the inclusion \((*)\), \(L(F)\) the family of all Lagrangian extremals of the inclusion \((*)\) and \(L_{co}(F)\) the class of all relaxed Lagrangian extremals of \((*)\). The authors give in the first part of the paper a comparison of the above three classes of extremals \(H(F)\), \(L(F)\) and \(L_{co}(F)\) presenting seven examples of autonomous, two-dimensional versions of differential inclusion \((*)\). Another approach of comparing types of extremality conditions was given by \textit{P. D. Loewen} and \textit{R. T. Rockafellar} [Trans. Am. Math. Soc. 325, No. 1, 39-72 (1991; Zbl 0734.49009)]. In the second part they prove, without any convexity hypotheses, that every boundary trajectory is a Lagrangian extremal, that is \(B(F)\subset L(F)\).
Mathematics Subject Classification ID: 49J24
Mathematics Subject Classification ID: 34A60
zbMATH DE Number: 86072
zbMATH Keywords: extremal trajectories
zbMATH Keywords: maximum principle
zbMATH Keywords: differential inclusion
zbMATH Keywords: boundary trajectories
zbMATH Keywords: extremality conditions
MaRDI profile type: Publication
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full work available at URL: 0167-6911(92)90119-d
OpenAlex ID: W2035277526
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description: scientific article; zbMATH DE number 86072