Uniqueness of solutions of a \(H^\infty\) optimization problem and complex geometric convexity
type: Item
instance of: scholarly article
title: Uniqueness of solutions of a \(H^\infty\) optimization problem and complex geometric convexity
zbMATH Open document ID: 0965.93051
DOI: 10.1007/BF02923093
author: Andrei Vityaev
published in: The Journal of Geometric Analysis
publication date: 2000-12-11T00:00:00Z
review text: Denote by \(H^\infty_N\) the space of vector-valued functions \(f= (f_1,f_2,\dots, f_N)\) defined on the unit circle \(T= \{z\in\mathbb{C}, |z|=1\}\) such that each coordinte \(f_j\) extends to be analytic on the entire unit disc. Let \(\Gamma: T\times \mathbb{C}^N\to \mathbb{R}\) be a continuous and nonnegative function. In this paper is considered the following extremal problem (OPT-problem): Find \(f^*\in H^\infty_N\) such that \[ \inf_{f\in H^\infty_N} \sup_{\theta\in \mathbb{R}} \Gamma(e^{i\theta}, f(e^{i\theta}))= \sup_{\theta\in\mathbb{R}} \Gamma(e^{i\theta}, f^*(e^{i\theta})). \] The OPT-problem is central to frequency domain system design problems and is important to the area of \(H^\infty\)-control. In the particular case when \(\Gamma(e^{i\theta}, z)=|g(e^{i\theta})- z|\), the OPT-problem reduces to the Nehari one. In this paper the author proved a new uniqueness result for a type of convexity which is strictly between geometric convexity and pseudoconvexity.
reviewed by: Stanislaw Walczak
Mathematics Subject Classification ID: 93B36
Mathematics Subject Classification ID: 32F99
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 1542083
zbMATH Keywords: Nahari type problem
zbMATH Keywords: extremal problem
zbMATH Keywords: frequency domain system
zbMATH Keywords: \(H^\infty\)-control
zbMATH Keywords: convexity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1542083