A regularized solution to the Birkhoff interpolation problem
type: Item
instance of: scholarly article
title: A regularized solution to the Birkhoff interpolation problem
zbMATH Open document ID: 1091.41011
author: Clyde Martin
author: Yishao Zhou
published in: Communications in Information and Systems
publication date: 2006-03-16T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.cis/1119639957
review text: The authors study a relationship between the Birkhoff interpolation problem and a problem of linear control theory. Namely they consider the system \[ \dot{x}=Ax+bu,x(0)=x_0, \qquad y=cx, \] where \(A\) is \(n\times n,\) \(b\) is \(n\times 1,\) \(c\) is \(1\times n,\) \((x_0,u)\) supposed to be in \(\mathcal H=\mathbb R^n\times L_2[0,T].\) The basis functions \(l_{(i,j)}(s)\) are defined as \(cA^{j-1}e^{A(t_I-s)b}\chi_{(-\infty,t_I)}(s).\) As for the Birkhoff interpolation problem an incidence \((n\times n)\) matrix \(J\) with \(n\) unit entries \((i,j)\in E\) gives \(n\) constraints (for \((i,j)\in E\)) \[ \alpha_{(i,j)}=cA^{j-1}e^{At_I}x_0+\int_0^Tl_{(i,j)}(s)u(s)\,ds. \] The optimal problem is to find \[ \min_{(x_0,u)\in\mathcal H}I(x_0,u) \] subject to the \(n\) constraints from above, where \(I(x_0,u)\) is the cost function \(I(x_0,u)=\lambda x_0^{\prime}x_0+\int_0^Tu^2(t)\,dt.\) The main result: Birkhoff interpolation problem with incidence matrix \(J\) has a solution if and only if the solutions of the optimal problem from above are uniformly bounded for \(\lambda\in (0,T).\)
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 49N05
Mathematics Subject Classification ID: 93C05
zbMATH DE Number: 5012659
zbMATH Keywords: Birkhoff interpolation
zbMATH Keywords: optimal control systems
zbMATH Keywords: control theoretic splines
reviewed by: Alexei Lukashov
MaRDI profile type: Publication
OpenAlex ID: W2074228252
DOI: 10.4310/CIS.2004.V4.N1.A5
description: scientific article; zbMATH DE number 5012659