Directional convexity and finite optimality conditions
type: Item
instance of: scholarly article
title: Directional convexity and finite optimality conditions
zbMATH Open document ID: 0627.49008
DOI: 10.1016/0022-247X(87)90178-8
author: Alberto Bressan
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: Given a set of directions \(\Lambda \subseteq {\mathbb{R}}^ d\), the author calls a set \(R\subseteq {\mathbb{R}}^ d\Lambda\)-convex whenever \[ r_ 0,r_ 1\in R,\quad r_ 1-r_ 0\in \Lambda,\quad \lambda \in [0,1]\quad \Rightarrow \quad (1-\lambda)r_ 0+\lambda r_ 1\in R. \] He then shows that when the reachable set R(T) of an autonomous nonlinear control system is \(\Lambda\)-convex, the transversality condition which must be satisfied by a boundary trajectory can be strengthened somewhat. The main part of the paper presents a set of sufficient conditions for the \(\Lambda\)-convexity of R(T), first in the case where the dynamics are \(\dot x=G(x)u\), and then in the fully nonlinear setting. The key hypothesis is a certain uniqueness property for solutions to the system of Pontryagin's equations; \(\Lambda\) consists of all vectors w for which every unit vector \(p(T)\in w^{\perp}\) generates a system with the uniqueness property.
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 93B03
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93C15
zbMATH DE Number: 4019848
zbMATH Keywords: reachable set
zbMATH Keywords: transversality condition
zbMATH Keywords: \(\Lambda \)-convexity
Wikidata QID: Q59264244
reviewed by: P. D. Loewen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4019848