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.
Author(s): Harm Bart, Marinus A. Kaashoek, André C. M. Ran (auth.)
Series: Operator Theory: Advances and Applications 200
Edition: 1
Publisher: Birkhäuser Basel
Year: 2010
Language: English
Pages: 432
Tags: Operator Theory; Linear and Multilinear Algebras, Matrix Theory; Operations Research, Mathematical Programming; Functions of a Complex Variable
Front Matter....Pages i-xii
Introduction....Pages 1-6
Front Matter....Pages 7-7
The role of canonical factorization in solving convolution equations....Pages 9-17
The state space method and factorization....Pages 19-33
Front Matter....Pages 35-35
Explicit solutions using realizations....Pages 37-56
Factorization of non-proper rational matrix functions....Pages 57-74
Front Matter....Pages 75-75
Factorization of matrix functions analytic in a strip....Pages 77-113
Convolution equations and the transport equation....Pages 115-142
Wiener-Hopf factorization and factorization indices....Pages 143-167
Front Matter....Pages 169-169
Preliminaries concerning minimal factorization....Pages 171-179
Factorization of positive definite rational matrix functions....Pages 181-196
Pseudo-spectral factorizations of selfadjoint rational matrix functions....Pages 197-209
Review of the theory of matrices in indefinite inner product spaces....Pages 211-216
Front Matter....Pages 217-217
Canonical factorization and Riccati equations....Pages 219-231
The symmetric algebraic Riccati equation....Pages 233-247
J -spectral factorization....Pages 249-288
Front Matter....Pages 289-289
Factorization of positive real rational matrix functions....Pages 291-300
Contractive rational matrix functions....Pages 301-312
J -unitary rational matrix functions....Pages 313-345
Front Matter....Pages 347-347
Application to the rational Nehari problem....Pages 349-370
Review of some control theory for linear systems....Pages 371-378
Front Matter....Pages 347-347
H-infinity control applications....Pages 379-404
Back Matter....Pages 405-423
