Coupled intervals for discrete symplectic systems
type: Item
instance of: scholarly article
title: Coupled intervals for discrete symplectic systems
zbMATH Open document ID: 1115.39019
author: Vera Zeidan
author: Roman Šimon Hilscher
published in: Linear Algebra and its Applications
publication date: 2007-02-14T00:00:00Z
review text: Consider discrete symplectic systems of the form \(Z_{k+1}=s_{k}z_{k} , k\in[0, N]\) where \(z_{k}=(x_{k}, u_{k})\) is a real \(2n\)-vector, \(s_{k}=(A_{k}, B_{k}, C_{k}, D_{k})\) is a real symplectic \(2n\times 2n\)-matrix and definiteness of the corresponding quadratic functional \(F_{0}(z)=\sum_{k=0}^{N} Z_{k}^{T}{S_{k}^{T}HS_{k}-H}z_{k}\), where \(H=(0, 0, I, 0)\) is a \(2n\times 2n\)-matrix and \([0, N]\) is a discrete interval with indicated endpoints. It was shown that the oscillation theory of discrete Hamiltonian systems \(\Delta x_{k}=A_{k}X_{k+1}+B_{k}u_{k}, \Delta u_{k}=C_{k}X_{k+1}-A_{k}^{T}u_{k}\), \(k\in[0, N]\) where \(B_{k}\) and \(C_{k}\) are symmetric and \(I-A_{k}\) is invertible, extends naturally to discrete sympletic system. Recently, the authors derived new characterizations of the definiteness of the Hamiltonian quadratic functional associated with the discrete Hamiltonian systems via the motion of a (strict) coupled interval. In this paper the authors extend the above mentioned Hamiltonion coupled interval notion to the discrete symplectic system and relate it to the definiteness of the corresponding quadratic functional. The main difficulty in this generalization lies in the fact that the evolution matrix \(\Phi_{k+1}=A_{k}A_{k-1}\dots A_{j}\) may be singular in the general symplectic case, as apposed to the Hamiltonian case where the matrices \(A_{k}=(I-A_{k})^{-1}\) are invertible.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 37J05
zbMATH DE Number: 5126042
zbMATH Keywords: Discrete symplectic systems
zbMATH Keywords: Quadratic functional
zbMATH Keywords: Coupled interval
zbMATH Keywords: Conjugate interval
zbMATH Keywords: Conjoined basis
zbMATH Keywords: Discrete Hamiltonian systems
reviewed by: Ahmed Hegazi
MaRDI profile type: Publication
full work available at URL: j.laa.2006.06.016
OpenAlex ID: W2027170788
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DOI: 10.1016/J.LAA.2006.06.016
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description: scientific article; zbMATH DE number 5126042