On the existence of optimal spaces for nonlinear functional equations
type: Item
instance of: scholarly article
title: On the existence of optimal spaces for nonlinear functional equations
zbMATH Open document ID: 1043.46057
DOI: 10.1023/A:1020382900997
author: V. G. Zamuraev
published in: Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: The author deals with nonlinear equations for (linear) functionals and not with nonlinear functional equations. The topic of the paper is connected with optimal control problems. Let \(C_{\text{ad}}\) be a metric space (of admissible controls), let \(F\) be a (real and separable) Hilbert space with inner product \([\cdot,\cdot]\) and let \(\{F_c\}_{c\in C_{\text{ad}}}\) be a family of closed subspaces of \(F\). Furthermore, let \(W: F\to F\) be a nonlinear operator with continuous inverse \(W^{-1}\) such that for all \(c\) the conditions \(v\in F_c\) and \(W(v)\in F_c\) are equivalent. Given a continuous linear functional \(l: F\to \mathbb{R}\), let \(u^0_c\in F_c\) be such that \([W(u^0_c), v]= l(v)\) for all \(v\in F_c\). Then, given some functional \(J: C_{\text{ad}}\times F\to \mathbb{R}\), the author shows that under some assumptions on \(C_{\text{ad}}\), the family of \(F_c\)'s, and on \(J\), there is (at least) one \(c^*\in C_{\text{ad}}\) such that \(J(c^*, u^0_{c^*})\leq J(c, u^0_c)\) for all \(c\in C_{\text{ad}}\). \(c^*\) is called an optimal control and \(F_{c^*}\) an optimal space (for \(l\) with respect to \(J\)).
reviewed by: Jens Schwaiger
Mathematics Subject Classification ID: 46N10
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 1916116
zbMATH Keywords: optimal control equations for functionals
zbMATH Keywords: nonlinear functional analysis
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1916116