Application of Operator Theory for the Representation of Continuous and Discrete Distributed Parameter Systems Der Technischen Fakult¨at der Friedrich-Alexander-Universit¨at Erlangen-Nurnberg¨ zur Erlangung des Grades Doktor-Ingenieur vorgelegt von Vitali Dymkou Erlangen, 2006 Als Dissertation genehmigt von der Technischen Fakult¨at der Friedrich-Alexander-Universit¨at 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 Galkowski Acknowledgements 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. I would like to thank ”Graduiertenkolleg Dreidimensionale Bildanalyse und -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 Linearoperators................................. 8 2.2.1 Unboundedoperators. .. .. 8 2.2.2 Adjointoperators............................ 9 2.2.3 Lineardifferentialforms . 10 2.2.3.1 Homogeneous boundary conditions . 10 2.2.3.2 Green’s formula and associated forms . 11 2.2.4 Lineardifferentialoperators . .. 12 2.2.4.1 Adjoint homogeneous boundary conditions and the adjoint operators ........................... 13 2.2.4.2 Nonhomogeneous boundary conditions . 15 2.3 Chaptersummary................................ 16 3 Spectral Theory of Operators 17 3.1 Theresolventoperator . .. .. 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 Sectorialoperators ........................... 25 3.1.3.1 Definitionofsectorialoperators . 26 3.1.3.2 Examplesofsectorialoperators . 26 3.1.4 Semigroups and canonical systems . .. 28 3.1.4.1 Definitions .......................... 28 vi 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 resolventoperator ........................... 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 Chaptersummary................................ 40 4 Mathematical Modelling of Physical Processes 43 4.1 Ordinarydifferentialequations. ..... 43 4.1.1 ClassificationofODEs . 43 4.1.2 State-Spacemodel ........................... 44 4.2 Partialdifferentialequations . ..... 46 4.2.1 ClassificationofPDEs . 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 Chaptersummary................................ 51 5 Description of Multidimensional Systems 53 5.1 Multi-functional transformation (MFT) . ....... 54 5.1.1 Definitionandproperties. 55 5.1.2 InverseMFT .............................. 56 5.2 Application of the MFT method to initial-boundary-value problems with homogeneousboundaryconditions. 57 5.2.1 Initial-boundary-value problem with homogeneous boundary condi- tions................................... 57 5.2.2 Laplacetransformation. 58 5.2.3 Spatialtransformation(MFT) . 59 5.2.4 InverseMFT .............................. 60 5.2.5 DiscretizationoftheMFTmodel . 61 5.2.5.1 Spatialdiscretization . 62 5.2.5.2 Timediscretization. 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 Laplacetransformation. 66 5.3.3 Spatialtransformation(MFT) . 66 Contents vii 5.3.4 InverseMFT .............................. 69 5.3.5 DiscretizationoftheMFTmodel . 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 Laplacetransformation. 71 5.4.4 Spatialtransformation(MFT) . 72 5.4.5 InverseMFT .............................. 74 5.4.6 DiscretizationoftheMFTmodel . 74 5.5 Chaptersummary................................ 74 6 Examples of the MFT Simulations 77 6.1 Heatflowequation ............................... 77 6.1.1 Primeandadjointoperators . 78 6.1.2 Eigenproblem .............................. 78 6.1.3 Biorthogonality ............................. 79 6.1.4 Thegeneralsolution . .. .. 79 6.1.5 MFTSimulation ............................ 80 6.2 Heatflowthroughawall ............................ 81 6.2.1 Adjointoperator ............................ 82 6.2.2 Eigenvalueproblems . 82 6.2.3 Associatedvectors ........................... 85 6.2.4 Biorthogonality ............................. 87 6.2.5 Thegeneralsolution . .. .. 88 6.2.6 MFTsimulation............................. 89 6.3 Telegraphequation ............................... 89 6.3.1 Adjointoperator ............................ 91 6.3.2 Eigenvalueproblems . 92 6.3.3 Associatedvectors ........................... 95 6.3.4 Biorthogonality ............................. 97 6.3.5 Thegeneralsolution . .. .. 98 6.4 Chaptersummary................................ 98 7 Conclusions 99 A Appendix 101 A.1 ProofofTheorem2...............................101 A.2 ProofofExample5...............................109 A.3 ProofofTheorem5...............................112 viii Contents B Titel, Inhaltsverzeichnis, Einleitung und Zusammenfassung 121 B.1 Titel .......................................121 B.2 Inhaltsverzeichnis .. .. .. 121 B.3 Einleitung ....................................124 B.4 Zusammenfassung................................128 Bibliography 130 Abbreviations and Acronyms ix Abbreviations and Acronyms FTM functional transformation method IIT impulse invariant transformation IBVP initial-boundary-value problem MD multi dimensional MFT multi-functional transformation ODE ordinary differential equation PDE partial differential equation SLT Sturm-Liouville transformation TFM transfer function model Mathematical symbols for all ∀ there exists ∃ subset of (or inclusion sign) ⊂ belongs to ∈ / does not belongs to ∈ 1 ( )− inverse operation of ( ) · · ( )H hermitian of ( ) · · ( )T transposed of ( ) · · ( )∗ conjugate complex of ( ) · · union ∪ intersection ∩ converges to → <, , >, inequality signs ≤ ≥ summation sign lim limit P + , , infinity ∞ −∞ ∞ Rank( ) rankof( ) · · Ran( ) imageorrangeof( ) · · Ker( ) kernel of ( ) · · dim dimension of ( ) {·} · Im imaginary part of ( ) {·} · Re real part of ( ) {·} · norm in H ||·||H ( , ) scalar product in H · · H span( ) linear span of ( ) · · x List of mathematical symbols inf( ) infimum of ( ) (greatest lower bound) · · sup( ) supremum of ( ) (least upper bound) · · min( ) minimum of ( ) · · max( ) maximum of ( ) · · det( ) determinant of ( ) · · k n Cn or binomial coefficient k ! π circle constant j imaginary unit Laplace transformation L{} multi-functional transformation T {} z transformation Z{} − δm,n Kronecker symbol N set of natural numbers Z set of integers R set of real numbers C set of complex numbers Cn(or Rn) set of complex (or real) n tuples − a sequence { n} (a, b) open interval [a, b] closed interval C([a, b]) set of continuous functions on [a, b] C(n)([a, b]) set of functions on [a, b] with continuous derivatives up to order n Ω subset of Rn ∂Ω boundary of Ω Ω closure of Ω Lp(Ω) Lebesgue space on Ω Wk p(Ω) Sobolev space on Ω ∂ Dt or ∂t first-order temporal derivative ∂ Dx or ∂x first-order spatial derivative y˙ first-order temporal derivative of y y′ first-order spatial derivative of y X,Y Banach spaces H, Hilbert spaces H I identity operator L, A, C linear operators L† adjoint