Necessary conditions for a nonlinear control system
type: Item
instance of: scholarly article
title: Necessary conditions for a nonlinear control system
zbMATH Open document ID: 0642.49013
DOI: 10.1016/0022-0396(85)90157-3
author: B. George
published in: Journal of Differential Equations
publication date: 1985-01-01T00:00:00Z
review text: The authors prove that if x(t) is a boundary trajectory of a Lipschitz continuous differential inclusion \(x\in F(t,x)\), \(x(0)=0\), the tangent cone to F(t,x(t)) at x(t) and that of the attainable set E(t) at x(t) coincide for almost every t provided that the boundary of F(t,x) is smooth. This is a noteworthy extension of \textit{H. Hermes} [J. Differ. Equations 3, 256-270 (1967; Zbl 0155.426)] and the first author [Astérisque 75-76, 187-197 (1980; Zbl 0459.49028)]. They also prove that the outward normal to these cones along the trajectory is Lipschitz continuous in t. Using the lower one-sided directional derivative the paper obtains first order necessary conditions which are stronger than existing ones. Two examples are given.
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 4046502
zbMATH Keywords: differential inclusion
zbMATH Keywords: tangent cone
zbMATH Keywords: outward normal
zbMATH Keywords: directional derivative
zbMATH Keywords: first order necessary conditions
reviewed by: Ethelbert Nwakuche Chukwu
MaRDI profile type: Publication
full work available at URL: 0022-0396(85)90157-3
OpenAlex ID: W2139594913
cites work: Q3752985
cites work: Attainable sets and generalized geodesic spheres
cites work: Q5563478
cites work: On Continuous and Measurable Selections and the Existence of Solutions of Generalized Differential Equations
cites work: Control problems of contingent equation
cites work: On some fundamental theorems of contingent equations in connection with the control problems
cites work: On contingent equations satisfying the Carathéodory type conditions
cites work: Q3909602
cites work: Q3283026
cites work: Q5513273
cites work: Uniqueness of optimal trajectories for non-linear control systems
cites work: Q5513899
cites work: Q5342650
Wikidata QID: Q116201200
description: scientific article; zbMATH DE number 4046502