The Bogolyubov theorem with a differential inclusion as constraint
type: Item
instance of: scholarly article
title: The Bogolyubov theorem with a differential inclusion as constraint
zbMATH Open document ID: 0858.49004
DOI: 10.1007/BF02106624
author: S. I. Suslov
published in: Siberian Mathematical Journal
publication date: 1997-03-19T00:00:00Z
review text: This paper deals with the problem of the calculus of variations in \(\mathbb{R}^n\), \[ I_F(x)=\min\Biggl[\int^1_0 f_F(t,x(t),\dot x(t));\;x(0)=x_0,\;x(1)= x_1\Biggr]\tag{\(*\)} \] and its convex relaxation \[ I^{**}_F(x)=\min\Biggl[\int^1_0 f^{**}_F(t,x(t),\dot x(t));\;x(0)=x_0,\;x(1)=x_1\Biggr],\tag{\(**\)} \] where \(f^{**}_F\) is a second conjugate of the function \(f_F\) with respect to its last argument. It has been shown that if there is an inclusion \(\dot x\in F(t,x(t))\), with right-hand side \(F(t,x)\in\mathbb{R}^n\) strictly convex and nonempty interior, then the problem \((**)\) serves as a convex relaxation of the problem \((*)\), where \[ f_F(t,x,u):=\begin{cases} f(t,x,u),\quad & u\in F(t,x),\\ +\infty,\quad & u\not\in F(t,x).\end{cases} \] The above results have been obtained on the base of bang-bang form of the above-mentioned differential inclusion. A very important property of the presented investigations is that each solution of the problem \((**)\) can be approximated to any accuracy in the uniform metric by close solution to the problem \((*)\).
Mathematics Subject Classification ID: 49J24
Mathematics Subject Classification ID: 93C25
Mathematics Subject Classification ID: 34A60
zbMATH DE Number: 917101
zbMATH Keywords: optimal control
zbMATH Keywords: optimal problem of multivalued differential equations
zbMATH Keywords: differential inclusion
reviewed by: Władysław Hejmo
MaRDI profile type: Publication
cites work: Q4256552
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full work available at URL: https://doi.org/10.1007/bf02106624
OpenAlex ID: W2050857623
description: scientific article; zbMATH DE number 917101