On the Nehari problem for a certain class of \(L^{\infty}\)-functions appearing in control theory
type: Item
instance of: scholarly article
title: On the Nehari problem for a certain class of \(L^{\infty}\)-functions appearing in control theory
zbMATH Open document ID: 0635.47027
DOI: 10.1016/0022-1236(87)90043-7
author: Allen Tannenbaum
author: Ciprian Foias
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: The authors obtain an explicit computable solution for finding \({\bar \rho}:= \inf_{q\in H^{\infty}(D)}\| W-mq\|\) where \(W\in H^{\infty}(D)\) is rational and \(m\in H^{\infty}(D)\) is an arbitrary inner function. The formula involves calculating where the function of \(\rho\) given by \(\nu(\rho)=\) trace of \((I=T^*T^{*n})P_{\rho}^{- 1}\) blows up; here T is the compression of the shift operator on \(H^ 2(D)\) (multiplication by z) to the orthogonal complement of its invariant subspace \(mH^ 2(D)\) and \(P_{\rho}= q(T)(I- \rho^{- 2}W(T)W(T)^*)q(T)^*\) where \(W=p/q\). In practice this in turn comes down to finding the roots of a transcendental equation. In the last section an explicit example is worked out for illustration of the approach. The problem is motivated by the \(H^{\infty}\)-optimization problem from control theory.
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 93B35
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 4034463
zbMATH Keywords: Nehari problem
zbMATH Keywords: compression of the shift operator
zbMATH Keywords: invariant subspace
zbMATH Keywords: roots of a transcendental equation
zbMATH Keywords: \(H^{\infty }\)-optimization problem from control theory
reviewed by: Joseph A. Ball
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90043-7
OpenAlex ID: W4205103300
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description: scientific article; zbMATH DE number 4034463