Application of operator theory for the representation of continuous and discrete distributed parameter systems [Elektronische Ressource] / vorgelegt von Vitali Dymkou

Sujets

• Mathematics

Informations

• Publié par: friedrich-alexander-universitat_erlangen-nurnberg
• Publié le: 01 janvier 2006
• Nombre de lectures: 18
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Application of Operator Theory for the
Representation of
Continuous and Discrete
Distributed Parameter Systems
Der Technischen Fakulta¨t der
Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg
zur Erlangung des Grades
Doktor-Ingenieur
vorgelegt von
Vitali Dymkou
Erlangen, 2006Als Dissertation genehmigt von
der Technischen Fakult¨at der
Friedrich-Alexander-Universitat¨
Erlangen-Nurnberg¨
Tag der Einreichung: 22. Dezember 2005
Tag der Promotion: 23. March 2006
Dekan: Prof. Dr.-Ing. Alfred Leipertz
Berichterstatter: Apl. Prof. Dr.-Ing. habil. Peter Steffen
Apl. Prof. Dr.-Ing. habil. Krzysztof GalkowskiAcknowledgements
I would like to thank my supervisors, Apl. Prof. Dr.-Ing. habil. Peter Steffen and Priv.
Doz. Dr.-Ing. habil. Rudolf Rabenstein, who introduced me to the subject of engineering
problems and gave me the opportunity to work in their group. I would like to express
them my gratitude for the excellent supervision and support and of course for the very
warm atmosphere during my work and life in Erlangen. Also, I would like to thank Apl.
Prof. Dr.-Ing. habil. Krzysztof Galkowski, who wisely advised me to start my scientific
work in the Telecommunications Laboratory in Erlangen and for reviewing my thesis.
Iwouldliketothank”GraduiertenkollegDreidimensionaleBildanalyseund-Synthese”
and especially Prof. Dr. Gunther Greiner for their financial support.¨
I am deeply thankful to all my colleagues from the Telecommunications Laboratory
for their patience and support over many months. I wish to thank Ursula Arnold for her
help with all my administrative questions, Wolfgang Preiss for his software support and
Manfred Lindner for his wonderful refrigerator. Especially I would like to thank Stefan
Petrausch, who was always ready to translate, explain and answer all my private and
scientific questions.
I am deeply grateful to all my old friends in Russia and Belarus for their thousand
calls and mails. They did not forget me. I would also like to thank my new friends in
Germany Hamza Amasha, Juliane Gebhardt, Nael and Larissa Popova for always being
there for me.
Finally, and most importantly, I wish to thank my supportive family who accepted
my time away from them. A special thanking word goes to my first teachers to my father
Michael Dymkov and to my mother Raisa Dymkova. Also, of course, I want to thank my
older brother Dymkou Siarhei, his wife Irina and their son Alexei.
This work is dedicated to my family.v
Contents
Abbreviations and Acronyms ix
List of mathematical symbols ix
Variables xi
1 Introduction 1
2 Basic Notions from Functional Analysis 5
2.1 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Linear operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Unbounded operators . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Linear differential forms . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.3.1 Homogeneous boundary conditions . . . . . . . . . . . . . 10
2.2.3.2 Green’s formula and associated forms . . . . . . . . . . . . 11
2.2.4 Linear differential operators . . . . . . . . . . . . . . . . . . . . . . 12
2.2.4.1 Adjointhomogeneousboundaryconditionsandtheadjoint
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.4.2 Nonhomogeneous boundary conditions . . . . . . . . . . . 15
2.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Spectral Theory of Operators 17
3.1 The resolvent operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.1 Canonical systems of the prime and the adjoint operator . . . . . . 20
3.1.2 Expansion of the resolvent operator by the canonical system . . . . 24
3.1.3 Sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.3.1 Definition of sectorial operators . . . . . . . . . . . . . . . 26
3.1.3.2 Examples of sectorial operators . . . . . . . . . . . . . . . 26
3.1.4 Semigroups and canonical systems . . . . . . . . . . . . . . . . . . . 28
3.1.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 28vi Contents
3.1.4.2 Connection between the semigroup of the operator, the
resolvent and the Laplace transformation . . . . . . . . . . 29
3.1.5 Expansion of the semigroup by the canonical system . . . . . . . . 30
3.2 The C−resolvent operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Canonical systems of the prime and the adjoint operator . . . . . . 37
3.2.2 Expansion of the C−resolvent operator by the canonical systems . . 39
3.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Mathematical Modelling of Physical Processes 43
4.1 Ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.1 Classification of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.2 State-Space model . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 Classification of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2 Initial-boundary-value problems . . . . . . . . . . . . . . . . . . . . 48
4.2.3 Solution of initial-boundary-value problems in the Laplace domain . 49
4.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Description of Multidimensional Systems 53
5.1 Multi-functional transformation (MFT) . . . . . . . . . . . . . . . . . . . . 54
5.1.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1.2 Inverse MFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Application of the MFT method to initial-boundary-value problems with
homogeneous boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.1 Initial-boundary-valueproblem with homogeneous boundarycondi-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.2 Laplace transformation . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.3 Spatial transformation (MFT) . . . . . . . . . . . . . . . . . . . . . 59
5.2.4 Inverse MFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2.5 Discretization of the MFT model . . . . . . . . . . . . . . . . . . . 61
5.2.5.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . 62
5.2.5.2 Time discretization . . . . . . . . . . . . . . . . . . . . . . 62
5.2.6 Inverse z−transformation . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.7 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Application of the MFT method to initial-boundary-value problems with
nonhomogeneous boundary conditions . . . . . . . . . . . . . . . . . . . . . 64
5.3.1 Initial-boundary-value problems with nonhomogeneous boundary
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.2 Laplace transformation . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3.3 Spatial transformation (MFT) . . . . . . . . . . . . . . . . . . . . . 66Contents vii
5.3.4 Inverse MFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.5 Discretization of the MFT model . . . . . . . . . . . . . . . . . . . 69
5.4 Application of the MFT method to general vector initial-boundary-value
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4.2 General vector initial-boundary value problem . . . . . . . . . . . . 70
5.4.3 Laplace transformation . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4.4 Spatial transformation (MFT) . . . . . . . . . . . . . . . . . . . . . 72
5.4.5 Inverse MFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4.6 Discretization of the MFT model . . . . . . . . . . . . . . . . . . . 74
5.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 Examples of the MFT Simulations 77
6.1 Heat flow equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.1.1 Prime and adjoint operators . . . . . . . . . . . . . . . . . . . . . . 78
6.1.2 Eigenproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.1.3 Biorthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.1.4 The general solution . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.1.5 MFT Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2 Heat flow through a wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2.1 Adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.2.2 Eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.2.3 Associated vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2.4 Biorthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.5 The general solution . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2.6 MFT simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Telegraph equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3.1 Adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . .