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A State Space Approach to Canonical Factorization with Applications

  • Book
  • © 2010

Overview

Authors:
  1. Harm Bart

    1. Econometrisch Instituut, Erasumus Universiteit Rotterdam, Rotterdam, The Netherlands

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  2. Marinus A. Kaashoek

    1. Department of Mathematics, FEW, Vrije Universiteit, Amsterdam, The Netherlands

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    You can also search for this author in PubMed   Google Scholar

  3. André C. M. Ran

    1. Department of Mathematics, FEW, Vrije Universiteit, Amsterdam, The Netherlands

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  • The state space factorization method is systematically used and developed further for various classes of matrix and operator functions
  • Emphasis is put on canonical factorization problems, including spectral and J-spectral factorizations problems and related Ricatti equations
  • Elements of H-infinity control theory and the related Nehari approximation problem are covered
  • Applications concern elements of H-infinity control theory and the related Nehari approximation problem, problems in mathematical analysis (inversion problems for singular integral equations and Wiener-Hopf integral equations, related Riemann Hilbert problems), and problems from mathematical physics (linear transport theory)
  • A large part the book deals with rational matrix functions only
  • Includes supplementary material: sn.pub/extras

Part of the book series: Operator Theory: Advances and Applications (OT, volume 200)

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Table of contents (21 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    1. Introduction

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 1-6

  3. Convolution equations, canonical factorization and the state space method

    1. Front Matter

      Pages 7-7
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    2. The role of canonical factorization in solving convolution equations

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 9-17

    3. The state space method and factorization

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 19-33

  4. Convolution equations with rational matrix symbols

    1. Front Matter

      Pages 35-35
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    2. Explicit solutions using realizations

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 37-56

    3. Factorization of non-proper rational matrix functions

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 57-74

  5. Equations with non-rational symbols

    1. Front Matter

      Pages 75-75
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    2. Factorization of matrix functions analytic in a strip

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 77-113

    3. Convolution equations and the transport equation

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 115-142

    4. Wiener-Hopf factorization and factorization indices

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 143-167

  6. Factorization of selfadjoint rational matrix functions

    1. Front Matter

      Pages 169-169
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    2. Preliminaries concerning minimal factorization

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 171-179

    3. Factorization of positive definite rational matrix functions

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 181-196

    4. Pseudo-spectral factorizations of selfadjoint rational matrix functions

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 197-209

    5. Review of the theory of matrices in indefinite inner product spaces

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 211-216

  7. Riccati equations and factorization

    1. Front Matter

      Pages 217-217
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    2. Canonical factorization and Riccati equations

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 219-231

    3. The symmetric algebraic Riccati equation

      • Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 233-247
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Keywords

About this book

The present book deals with canonical factorization problems for di?erent classes of matrix and operator functions. Such problems appear in various areas of ma- ematics and its applications. The functions we consider havein common that they appear in the state space form or can be represented in such a form. The main results are all expressed in terms of the matrices or operators appearing in the state space representation. This includes necessary and su?cient conditions for canonical factorizations to exist and explicit formulas for the corresponding f- tors. Also, in the applications the entries in the state space representation play a crucial role. Thetheorydevelopedinthebookisbasedonageometricapproachwhichhas its origins in di?erent ?elds. One of the initial steps can be found in mathematical systems theory and electrical network theory, where a cascade decomposition of an input-output system or a network is related to a factorization of the associated transfer function. Canonical factorization has a long and interesting history which starts in the theory of convolution equations. Solving Wiener-Hopf integral equations is closely related to canonical factorization. The problem of canonical factorization also appears in other branches of applied analysis and in mathematical systems theory, in H -control theory in particular.

Reviews

From the reviews:

“This monograph develops a theory of canonical factorizations for various classes of matrix and operator functions, where the functions are represented in the form of a transfer function of an input-state-output linear system. … The book is self-contained and accessible for specialists in different areas of mathematics, science, and engineering.” (Dmitry Kaliuzhnyi-Verbovetskyi, Zentralblatt MATH, Vol. 1203, 2011)

Authors and Affiliations

  • Econometrisch Instituut, Erasumus Universiteit Rotterdam, Rotterdam, The Netherlands

    Harm Bart

  • Department of Mathematics, FEW, Vrije Universiteit, Amsterdam, The Netherlands

    Marinus A. Kaashoek, André C. M. Ran

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