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.

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 conditions...... 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 of linear operator L rank(L) rankoftheoperator L ρ(L) resolvent set of the operator L Variables xi

σ(L) spectrumoftheoperator L R(s, L) resolvent of the operator L empty set ∅

Variables

λ eigenvalue x continuous space variable x vector of continuous spatial coordinates z complex frequency variable of the z transformation − s complex frequency variable of the Laplace transformation t continuous time variable i, m, p, indices µ, ν xii Variables 1 1 Introduction

The development of computer technologies, the availability of high-speed processors and various programming languages allow nowadays the researchers in different areas of science to investigate and design numerous algorithms to solve physical phenomena on the computer. However, to construct high precision models of a real process one has to begin with its mathematical description and analysis in order to obtain specific characteristics of the considered problem. This helps to design very efficient numerical methods which can be implemented directly on the computer. The past decades, in particular, have seen a continually growing interest in the application of functional analysis and especially general operator theory to engineering problems. This development is clearly related to the wide variety of applications of both practical and theoretical interests. Many physical and information processes in various fields possess identical mathematical structures, that can be described in a common operator form. Such a generalization allows to construct general algorithms to solve a wide class of problems. It is, however, not sufficient to analyze a given problem on a pure theoretical basis. The practical application may pose additional constraints like real-time performance of the system, low-delay requirements or restrictions on computing power or memory. Therefore, it is necessary to develop a discrete model for the solution of the problem which is suitable for computer implementation. The general steps in such a development are illustrated in Figure 1.1. The presented thesis is oriented towards the development of algorithms based on spectral operator theory and their application for solving practical problems arising in the discrete simulation of continuous systems governed by partial differential equations with unbounded and non-self-adjoint operators, in general. We deal with the theory of well- posed problems arising in systems simulation. The need to consider the case of unbounded operators has been stimulated by numerous engineering applications where system models of physical processes yield this class of operators. In spite of the large popularity of these processes, relatively few results of the rigorous mathematical theory have been applied to practice. The main problem is that a number of results obtained in the classical case of bounded and self-adjoint operators is no longer valid or is incomplete in the case of unboundedness and non-self-adjointness, in general. Moreover, as is well known, there is no general spectral theory for unbounded and non-self-adjoint operators in a Hilbert space. Spectral theory treats only some special classes of such operators (see [GK69]). 2 1. Introduction

Problem Mathematical Physical Description Set of PDEs

Discretization and the Model Transformations in Discrete Approximation Continuous Domain

Simulation

Figure 1.1: The general procedure between the formulation of the physical problems and their computer realization. Here the physical problem is a distributed parameter system, described by a set of partial diﬀerential equations (PDEs).

The most elementary case of finite dimensional spaces and matrices as operators, which we assume to be known to the reader, is well established (see [Gan59]). It has a set of powerful tools which also might be useful to have in the more general case of infinite dimensional spaces and general operators. Of particular interest is the problem of a canonical form for arbitrary linear transformations in a Hilbert space as a generalization of the Jordan form for matrices (some texts refer to it as the Jordan canonical form or the Jordan normal form). The Jordan structure for matrices is obtainable by using the special basis, which in the most simple case is formed by the set of linear independent eigenvectors, and in the more general case the associated vectors (also known as generalized eigenfunctions) are to be added to form such a basis. The case of bounded operators in a general Banach space can be treated essentially in the same way. M.V. Keldysh was the first who generalized the notion of the Jor- dan chain of vectors [Gan59] to a wide class of non-self-adjoint bounded operators (see [Kel51],[Kel71]). For that reason it was called the Keldysh chain. Further results were obtained e.g. in the work of M.G. Krein [Kre59], I. Gohberg [IGK90], V.B. Lidskii [Lid59], A.S. Markus [Mar88], M.A. Naimark [Nai62], A.V. Fursikov [Fur01b],[Fur01a] respectively. For further references see [GK69]. In this work, methods of spectral analysis are applied to the class of unbounded and non-self-adjoint operators with compact resolvent. It summarizes, extends and generalizes recent research, performed by the author in [DRS04a],[DRS04b],[DRS05],[DRS]. The basic principles are based on the method of Keldysh. The intention is to obtain a spectral decomposition of the solution of initial-boundary-value problems, which is adapted to the structure of the spatial differential operator which leads to its representation with respect to its canonical system and its adjoint. It is a consequence of the spatial separation that the resulting structure is well adapted to the simulation of the complete system with 1. Introduction 3 distributed parameters, i.e. the initial-boundary-value problem. The desired discrete simulation can be finally done by standard approximation and simulation techniques. To this end, one needs to counteract the non-regularity of the system operators by exploiting properties of the bases in suitable functional spaces formed by the eigenfunctions and associated functions of certain operators. There are two representations of the solution in the case of bounded operators, a series expansion and an integral formula with a complex contour integral. However, in the case of unbounded operators, the contour integral admits a generalization to sectorial operators. If an operator is sectorial, the solution of the corresponding initial-boundary-value problem is given by an holomorphic semigroup. Sectorial operators and holomorphic semigroups are basic tools in the theory of abstract parabolic problems, and in the solution of systems of partial differential equations of parabolic type. They will be introduced and investigated in the following chapters. The main goals of this work are: (i) to present the state of the art of the theory of unbounded and non-self-adjoint operators in the context of holomorphic semigroups, (ii) to expand the solution of initial-boundary-value problems with respect to the canonical systems of the sectorial operators, (iii) to design functional transformations for a generalized frequency domain representation, and for the subsequent discretization and (iv) to apply the presented methods to selected differential equations of technical interest. These four goals are reflected by Chapter 2 through Chapter 6. Chapter 2 contains the basic concepts of functional analysis such as spaces and operators. This section is important in allowing us to be consistent with a rigorous mathematical formulation of the problem. In the sense that for each problem one needs to define the corresponding space in which the problem is considered and the corresponding operators which reflect the physical behavior or relations. Moreover, the problem can possess some additional constraints such as boundary or initial conditions. Chapter 3 provides a summary of spectral theory of unbounded operators with compact resolvent. This detailed overview is included there because it is not easily accessible in the engineering literature and also some fundamental results are presented only in the Russian literature. The presentation is centered around general linear operators in Hilbert spaces and their canonical systems, i.e. their eigenvalues, eigenvectors and associated vectors. An important subclass form the so-called sectorial operators. They describe the evolution of dynamical systems through an appropriate holomorphic semigroup. This section closes with new results in the representation of a certain holomorphic semigroup by the canonical system of the underlying sectorial operator and the expansion of the generalized resolvent operator by its canonical system. Chapter 4 introduces initial-boundary-value problems which are defined by a sectorial differential operator with compact resolvent. It is shown how the methods compiled in Chapter 3 allow to represent the solution of initial-boundary-value problems by the 4 1. Introduction canonical systems of the corresponding spatial differential operators. In short, Chapter 4 presents a representation of the general solution of a wide class of problems frequently encountered in physics and engineering. Chapter 5 presents the main results of this contribution. It shifts the focus from mathematics to multidimensional systems theory. This shift is subtle, but important. Rather than emphasizing the solution of an initial-boundary-value problem, i.e. a function of time and space, we consider the initial-boundary-value problem as the description of a multidimensional system. The representation of all functions under consideration by the canonical system of its spatial differential operator is then formalized through the introduction of a rather general functional transformation, the so-called multi-functional transformation (MFT). It describes the multidimensional system in the temporal and the generalized spatial frequency domain by an infinite set of one-dimensional systems. In turn, each one-dimensional system has a very simple structure. Its details are determined by the structure of the canonical system. The corresponding blocks are given in terms of the temporal and generalized spatial frequency variables. The uniform nature of these blocks makes it easy to apply standard discretization methods well-known for one-dimensional systems. This section concludes with a discrete-time, discrete-space approximation of multidimensional systems under consideration. The new results presented in this section may be summarized as follows: The structure of initial-boundary-value problems is mapped to the block structure of discrete-time, discrete-space multidimensional systems. These results provide a mathematically rigorous link between the theory of multidimensional systems (e.g. [Bos82],[DM84]) and their discrete computer implementation. The presented procedure is explained by examples of physical problems in Chapter 6. 5 2 Basic Notions from Functional Analysis

In this chapter we describe the appropriate context in which one can deﬁne and analyze the spectral properties of unbounded linear operators, particularly those which are closed and non-self-adjoint. The precise description of the operators will be the main focus of attention throughout this chapter. The reader is supposed to be familiar with the fundamental tools of applied functional analysis which will be used systematically throughout the text, otherwise we refer to [Bal76],[Yos80],[Gol66],[Has00].

2.1 Spaces

Before one can start to study the behaviour of operators one has to choose an appropriate space on which they act. It turns out that some properties of the operator can change depending upon the spaces on which it acts.

Definition 1 A set X is called a linear space (or vector space) over a field R (C) if the following conditions are satisfied:

1. an addition + is deﬁned: for every elements x, y X there exists an associated ∈ element z of X, such that z = x + y;

2. x + y = y + x;

3. x +(y + z)=(x + y)+ z;

4. there exists a zero element of X, denoted by 0, such that x +0= x;

5. for every x X there exists an element x X such that x +( x)=0; ∈ − ∈ − 6. a scalar multiplication is deﬁned: for every element x Xand each α R(C) there ∈ ∈ exists an associated element y of X, such that y = αx;

7. α(βx)=(αβ)x;

8. 1 x = x; · 6 2. Basic Notions from Functional Analysis

9. α(x + y)= αx + αy;

10. (α + β)x = αx + βx;

Definition 2 A linear space X is called a normed linear space, if for every x X, there ∈ is associated a real number x , the norm of the element x, such that: || ||X 1. x 0 and x =0 iff x = 0; || ||X ≥ || ||X 2. αx = α x , α R(C); || ||X | |·|| ||X ∈ 3. x + y x + y (triangle inequality); || ||X ≤ || ||X || ||X Definition 3 A sequence x in a normed linear space X converges to the element { n} x X if x x 0, n . ∈ || n − ||X → →∞ Definition 4 A sequence x in a normed linear space X is said to be a Cauchy sequence { n} if ε> 0 N n>N, p N : x x <ε. ∀ ∃ ∀ ∀ ∈ || n+p − n||X Definition 5 If every Cauchy sequence is convergent, a normed linear space is said to be complete. A Banach space is a complete normed linear space.

Deﬁnition 6 A subset S of a normed linear space X is dense in X if the closure of S is the entire space X (S¯ = X).

Deﬁnition 7 A subset S of a normed linear space X is called compact if every (inﬁnite) sequence in X has a convergent subsequence.

Definition 8 An inner product (or scalar product) on a linear space X defined over the field R (C) is a map ( , ) : X X R(C) such that · · × −→ 1. (x, x) 0 and (x, x)=0 iff x = 0; ≥

2. (x, y)∗ =(y, x) for all x, y X; ∈ 3. (αx + βy,z)= α(x, z)+ β(y, z) for all x, y, z X, α,β R(C); ∈ ∈ A linear space X with an inner product ( , ) is called an inner product space. · · We can consider any inner product space as a normed linear space (X, ) by || · ||X deﬁning the norm as = (x, x). ||·||X p Deﬁnition 9 A Hilbert space is an inner product space which is complete as a normed space under the induced norm. 2.1. Spaces 7

The spaces taken into consideration are the usual spaces of complex valued functions deﬁned on RM or on an open set Ω of RM . The following two classes of Banach spaces are widely used in functional analysis and the theory of diﬀerential equations.

Deﬁnition 10 [Ada75] The Lebesgue space L (Ω), 1 p< + , is the set of all complex- p ≤ ∞ valued functions y(x)= y(x1, x2,...,xM ) deﬁned in Ω and such that

1 p p y Lp(Ω) = y(x) dx < , (2.1) k k | | ∞ ZΩ where the integral is taken in the Lebesgue-sense.

Deﬁnition 11 [Ada75] Let k be a positive integer, and 1 p< + . The Sobolev space ≤ ∞ Wk p(Ω) is the set of all complex-valued functions y(x) = y(x1, x2,...,xM ) deﬁned on Ω and such that

1 p a p y Wk(Ω) = D y(x) dx < , (2.2) k k p | | ∞ Z a k Ω |X|≤ where a = (α1,...,αM ) is a vector of nonnegative integers, a = α1 + + αM and a ∂α1 ∂αM | | ··· D = α . . . α is the weak or distributional partial derivative of order a . ∂x 1 ∂x M 1 M | | The following inequalities will be useful in the next chapters. Let 1

fgdx fg dx f L g L (2.4) ≤ | | ≤k k p(Ω) ·k k p′ (Ω) Z Z Ω Ω

holds for any functions f L (Ω) and g L ′ (Ω). ∈ p ∈ p Minkowski’s inequality

f + g L f L + g L (2.5) k k p(Ω) ≤k k p(Ω) k k p(Ω) holds for any f,g L (Ω). ∈ p Minkowski’s integral inequality

1 1 p p p p dx f(x1, x2)dx dx f(x1, x2) dx (2.6)  2 1  ≤ 1  | | 2 Z Z Z Z Ω2 Ω1 Ω1 Ω2     holds for any integrable function f(x , x ) deﬁned on the set Ω Ω (x Ω RM1 , x 1 2 1× 2 1 ∈ 1 ⊂ 2 ∈ Ω RM2 ) and such that integral in the right part of (2.6) is ﬁnite. 2 ⊂ 8 2. Basic Notions from Functional Analysis

2.2 Linear operators

Let X,Y be Banach spaces over the complex field C and A be a map acting from X to Y. Denote by D(A) the set of elements from X where the mapping A is defined or its domain, by Ran(A) = y Y : Ax = y, x D(A) its image or range and by { ∈ ∈ } Ker(A)= x D(A) : Ax =0 its kernel. { ∈ } Definition 12 The mapping A : X Y is a linear operator if → i) the set D(A) is linear subspace of X; ii) the mapping A is linear on D(A), i. e.

A(α x + α x )= α Ax + α Ax , x , x D(A), α , α C. (2.7) 1 1 2 2 1 1 2 2 ∀ 1 1 ∈ 1 2 ∈ Note that D(A) is a linear subspace of X, while Ran(A) is a linear subspace of Y . The case D(A) = X and/or Ran(A) = Y may occur. 6 6

2.2.1 Unbounded operators

One of the basic instruments in functional analysis is a bounded linear operator. The need to consider the case of unbounded operators has been stimulated by numerous engineering applications where system models of the physical process yield this class of operators. Moreover, the most diﬀerential operators are unbounded when considered as acting on any usual Banach or Hilbert spaces.

Definition 13 The linear operator A : X Y is bounded if → i) D(A)= X; ii) there exists a constant c > 0 such that Ax c x for all x X. The norm k kY ≤ k kX ∈ A of the linear bounded operator A is defined as k k Ax A = sup k kY , (2.8) k k x=0 x X 6 k k Remark 1 The infimum of all such constants c is equal to the norm of the operator A, inf(c)= A . k k Thus, the linear operator A can be unbounded if the condition i) or/and the condition ii) of Definition 13 are violated. It is known that in the case where X and Y are normed linear spaces the boundedness of the linear operator A is equivalent to its continuity. It is said that the linear operator A : X Y is continuous at the point x D(A) if x x 0 leads to Ax → 0 ∈ k n − 0kX → k n − Ax 0. The linear operator A is continuous if it is continuous at an arbitrary point 0kY → x D(A). 0 ∈ 2.2. Linear operators 9

Example 1 [KF81] Unbounded operator

Denote by C([0, 2π]) the Banach space of all continuous complex valued functions defined on the compact interval [0, 2π] R with the norm x C = max x(t) ( x C = x ). 0 t 2π ∞ ∈ k k ≤ ≤ | | k k k k Define the differential operator A : C([0, 2π]) C([0, 2π]) by the formula Af = f ′, → with D(A) = C1([0, 2π]) C([0, 2π]), where C1([0, 2π]) is the set of all continuously ⊂ differentiable functions on [0, 2π]. It is obvious that C1([0, 2π]) is a dense subspace in C([0, 2π]). This operator is linear since

d dx dx (α x + α x )= α 1 + α 2 . dt 1 1 2 2 1 dt 2 dt

This operator is not bounded. To show this consider, for example, the sequences xn(t)= dxn dxn sin nt, xn C 1, C = max n cos nt = n. Hence sup C =+ , which proves dt 0 t 2π dt k k ≤ k k ≤ ≤ | | n k k ∞ the unboundedness of the operator A.

Example 2 [KF81] Bounded operator

In contrast to the case above, consider the diﬀerential operator A acting from the Banach 1 space C ([0, 2π]) with the norm x C1 = max x(t) + max x′(t) to the Banach space 0 t 2π 0 t 2π k k ≤ ≤ | | ≤ ≤ | | C([0, 2π]) by the same formula: Af = f ′. This operator is linear and bounded since

1 Ax C = x′ C = max x′(t) max x(t) + max x′(t) = x C1 , x C ([0, 2π]). 0 t 2π 0 t 2π 0 t 2π k k k k ≤ ≤ | | ≤ ≤ ≤ | | ≤ ≤ | | k k ∀ ∈

Remark 2 The given examples show that the operator given by the same formula can be bounded or unbounded with respect to diﬀerent norms associated with the underlying Banach spaces.

2.2.2 Adjoint operators

Now we consider Hilbert spaces H1 and H2 over the ﬁeld C with inner products denoted as ( , ) and ( , ) , respectively. L is assumed to be a linear operator acting from H · · H1 · · H2 1 into H with the domain D(L) H . 2 ⊂ 1

Deﬁnition 14 Let f ∞ be a sequence of elements f D(L). If from f f, n { n}n=1 n ∈ n → →∞ and Lf g, n it follows that f D(L) and Lf = g then the operator L is called n → →∞ ∈ a closed operator.

Deﬁnition 15 A bounded operator is called a compact operator if it maps each bounded

subset of H1 into a compact subset of H2. 10 2. Basic Notions from Functional Analysis

Deﬁnition 16 Let the operator L have a dense domain D(L) in H1. The adjoint operator

L† is that operator with domain D(L†) containing all those elements g H for which ∈ 2 there exists an element h H such that the following equality holds: ∈ 1 f D(L) (Lf,g) =(f, h) . (2.9) ∀ ∈ H2 H1

In this case, by deﬁnition, L†g = h.

Equivalently we can write:

f D(L) , g D(L†) (Lf,g) =(f, L†g) . (2.10) ∀ ∈ ∀ ∈ H2 H1

Lemma 1 [Yos80] Let the operator L have a dense domain D(L) in H1. Assume that 1 1 1 the operator L− exists and its domain D(L− ) is dense in H2. Then (L− )† exists and satisﬁes

1 1 (L− )† =(L†)− . (2.11)

In the following we assume that the Hilbert spaces Hi are identical, i.e H1 = H2 = H. The scalar product and its corresponding induced norm will now be simply denoted by ( , ) and , respectively. · · k·k

2.2.3 Linear differential forms In this section we introduce the basic concept of linear differential forms and their adjoints. We will consider the set C(n)([a, b]) of continuously differentiable functions up to the order n over finite interval [a, b]. An arbitrary linear differential form l applied to an arbitrary element y C(n)([a, b]) ∈ is defined as n (n) (n 1) (ν) l(y)(x) := pn(x)y (x)+ pn 1(x)y − (x)+ + p0(x)y(x)= pν(x)y (x), (2.12) − ··· ν=0 X where the functions p (x), ν =0, , n are continuously differentiable up to the order ν ν ··· over the given interval [a, b].

2.2.3.1 Homogeneous boundary conditions

′ (n 1) ′ (n 1) Denote by y ,y , ,ya − and y ,y , ,y − the values of the function y(x) and its a a ··· b b ··· b derivatives at the points x = a and x = b, respectively. Let m linear independent forms be given by

µ µ ′ µ (n 1) Uµ(y) :=α0 ya + α1 ya + + αn 1ya − + ··· − (2.13) µ µ ′ µ (n 1) µ µ C +β0 yb + β1 yb + + βn 1yb − , αi , βi . ··· − ∈ 2.2. Linear operators 11

For an arbitrary number m 2n of such kind of forms we say that the equalities ≤

Uµ(y)=0, µ =1,...,m (2.14) represent homogeneous boundary conditions. If m = 2n, it is obvious, that from (2.14) ′ (n 1) follow the equalities y =0,y =0, ,ya − = 0. a a ···

2.2.3.2 Green’s formula and associated forms

Let us introduce the following vectors with n components

ya yb za zb  (1)   (1)   (1)   (1)  ya yb za zb y =  (2)  , y =  (2)  , z =  (2)  , z =  (2)  . (2.15) a  ya  b  yb  a  za  b  zb                           ·   ·   ·   ·   (n 1)   (n 1)   (n 1)   (n 1)   ya −   y −   za −   z −     b     b          b (ν) We consider the integral pν (x)y (x)z(x)∗dx for ν =0, , n. Repeated integration by a ··· parts shows that for any Ry, z C(n)([a, b]) and p (x) C(ν)([a, b]) the equality ∈ ν ∈ b (ν) (ν 1) ′ (ν 2) p (x)y (x)z(x)∗dx = p (x)z∗(x)y − (x) (p (x)z∗(x)) y − (x)+ ν ν − ν ··· Za x=b b ν 1 (ν 1) ν (ν) +( 1) − (p (x)z∗(x)) − y(x) +( 1) y(x) p (x)z(x)∗ dx ··· − ν − ν x=a Za (2.16) holds. The sum of all equalities (2.16) over the indices ν = 0, , n gives us that the ··· diﬀerential form l(y) satisﬁes the following identity

(l(y), z)= P (η,ζ)+(y, l†(z)). (2.17)

Here n n (n) n 1 (n 1) ν (ν) l†(z)=( 1) (pn∗ z) +( 1) − (pn∗ 1z) − + + p0∗z = ( 1) (pν∗z) (2.18) − − − ··· − ν=0 X represents an associated (formally adjoint) diﬀerential form to l(y) and

n ν x=b ν i (ν i) (i 1) P (η,ζ)= ( 1) − (p z∗) − y − (2.19) − ν ν=1 i=1 x=a X X is a bilinear form of the variables η = (y , y ) and ζ = (z , z ). The relation (2.17) is a b a b called the Green’s formula. 12 2. Basic Notions from Functional Analysis

If we introduce the matrix Q as follows,

Q =

(1) n 1 (n 1) n 2 n 1 p1 p2 + +( 1) − pn − p2 p3 p4 ( 1) − pn 1 ( 1) − pn − ··· − − − ··· − − − (1) n 2 (n 2) n 2  p p + +( 1) − pn − p p p ( 1) − p 0  2 − 3 ··· − − 3 4 − 5 ··· − n  (1) n 3 (n 3)   p3 p4 + +( 1) − pn − p4 p5 p6 0 0   − ··· − − − ··· (2.20)      ··· ······ · ·   (1) (2)   pn 2 pn 1 + pn pn 1 pn 0 0 0   − − − − − ···   (1)   pn 1 pn pn 0 0 0 0   − − − ···     p 0 0 0 0 0   n   ···    the bilinear form P (η,ζ) can be written in compact form as

P (η,ζ)= yT Q(b)z yT Q(a)z . (2.21) b b − a a

2.2.4 Linear differential operators Define the differential operator L : C(n)([a, b]) C([a, b]) as follows: → i) the domain D(L) of the operator is

D(L)= y C(n)([a, b]) : U (y)=0, µ =1,...,m ; (2.22a) ∈ µ ii) the operator is given by the formula

Ly = l(y), y D(L). (2.22b) ∈ Assigning different boundary conditions (2.14) to a fixed differential form l will generate different differential operators L with different domains, in general.

Remark 3 The precise specification of the domain of the operator is very important since it turns out that the same linear differential form (2.12) with different boundary conditions (2.14) produces different differential operators L, in general. For these reasons when using the term ”differential operator” we shall understand that we have already chosen the boundary conditions if we are thinking in more applied terms, or that we have already chosen the precise domain of definition of the operator if we are thinking in abstract terms.

It is obvious that D(L) is a linear subspace of C(n)([a, b]), and D(L) = C(n)([a, b]) if there are no restrictions of the form (2.22a). Therefore, the linear diﬀerential form l itself is a linear diﬀerential operator with domain D(l)= C(n)([a, b]). Also, we always have the inclusion D(L) D(l). ⊂ 2.2. Linear operators 13

2.2.4.1 Adjoint homogeneous boundary conditions and the adjoint operators

We consider m linear independent forms U1, , Um of the variables ′ (n 1) ′ (n 1) ··· y ,y , ,ya − ,y ,y , ,y − given by a a ··· b b ··· b

ya 1 1 1 1 1 1 (1) α0 α1 αn 1 β0 β1 βn 1  ya  U ··· − ··· − 1 2 2 2 2 2 2  α0 α1 αn 1 β0 β1 βn 1     U  ··· − ··· −   2  ·     (n 1)       ya −     · · ··· · · · ··· ·    =  ·  .          yb     · · ··· · · · ··· ·     ·     (1)       yb     · · ··· · · · ··· ·     ·   m m m m m m      α0 α1 αn 1 β0 β1 βn 1 Um  ··· − ··· −     m 1  m 2n  ·    ×   ×  (n 1)     y −   b 2n 1   × (2.23) Using block-matrices A , B and the vector U of appropriate dimensions we rewrite 1 1 1 (2.23) in more compact form as

y a A B = U1 . (2.24) 1 1 " y # h i b h i

If m < 2n we can add 2n m arbitrary linear independent forms U , , U in − m+1 ··· 2n such a way that the resulting forms U , , U are linear independent as well. Then we 1 ··· 2n can write

1 1 1 1 1 1 α0 α1 αn 1 β0 β1 βn 1 ya U1 ··· − ··· − 2 2 2 2 2 2 (1)  α0 α1 αn 1 β0 β1 βn 1   ya   U2  ··· − ··· −              · · ··· · · · ··· ·   ·   ·     (n 1)       ya −     · · ··· · · · ··· ·    =  ·  ,          yb     · · ··· · · · ··· ·     ·   m m m m m m   (1)     α0 α1 αn 1 β0 β1 βn 1   yb     ··· − ··· −     ·               · · ··· · · · ··· ·   ·   ·   2n 2n 2n 2n 2n 2n   (n 1)     α0 α1 αn 1 β0 β1 βn 1   yb −   U2n   ··· − ··· − 2n 2n  2n 1  2n 1   ×   ×  (2.25)× or equivalently 14 2. Basic Notions from Functional Analysis

A B y U1 1 1 a = , (2.26) A B y U " 2 2 # " b # " 2 # where A and B are block-matrices of appropriate dimensions and U is a vector of 2 2 2 length 2n m. − In this case, by inverting the matrix in 2.26, each y and y can be represented as a a b linear combination of these forms

y C D U1 a = 1 1 . (2.27) y C D U " b # " 2 2 # " 2 # Thus y = C U + D U , (2.28a) a 1 1 1 2 y = C U + D U . (2.28b) b 2 1 2 2 Substitution of these representations into P (η,ζ) yields

P (η,ζ)=(C U + D U )T Q(b)z (C U + D U )T Q(a)z 2 1 2 2 b − 1 1 1 2 a = UT [CT Q(b)z CT Q(a)z ]+ UT [DT Q(b)z DT Q(a)z ] (2.29) 1 2 b − 1 a 2 2 b − 1 a T T = U1 V2 + U2 V1, where

T T T V2 = [V2n,V2n 1,...,V2n m] = C Q(b)zb C Q(a)za, (2.30a) − − 2 − 1 T T T V1 = [V2n m 1,V2n m 2,...,V1] = D Q(b)zb D Q(a)za, (2.30b) − − − − 2 − 1 and Vi, i =1,..., 2n are linear independent forms of the variables za and zb, too. Now we rewrite the bilinear form (2.29) in terms of Ui and Vi

P (η,ζ)= U1V2n + + UmV2n m+1 + Um+1V2n m + + U2nV1. (2.31) ··· − − ··· Substitution of the prime boundary conditions U =0, U =0, , U = 0 yields 1 2 ··· m

P (η,ζ)= Um+1V2n m + + U2nV1. (2.32) − ··· We say that the expressions

V1 =0, V2 =0, ,V2n m =0 (2.33) ··· − are the adjoint boundary conditions associated with the prime boundary conditions: U1 = 0, U =0, , U =0. The purpose of the special partitioning into two sets U and V 2 ··· m i i 2.2. Linear operators 15 is that under conditions (2.14) and (2.33) the expression P (η,ζ) will vanish, no matter how the remaining conditions are chosen.

Thus, we have the adjoint linear differential form l†(z) and the adjoint boundary conditions V1, ,V2n m. ··· − Since the adjoint operator L† to the operator L can be defined only in some Hilbert space (see Definition 16), let C(n)([a, b]) H, where H is the Hilbert space with the scalar b ⊂ product (y, z)= y(x)z∗(x)dx (for example we can set H = L2([a, b])). a Therefore, theR adjoint linear differential form l†(z) and the adjoint boundary conditions

V1, ,V2n m generate the adjoint operator L† deﬁned as: ··· − i’) the domain D(L†) is

(n) D(L†)= z C ([a, b]) : V (z)=0, µ =1,..., 2n m ; (2.34a) { ∈ µ − } ii’) the operator acts by the formula

L†z = l†(z), z D(L†). (2.34b) ∈

2.2.4.2 Nonhomogeneous boundary conditions

In the case of nonhomogeneous boundary conditions m

Uµ(y)= φµ, φµφµ∗ > 0 (2.35) µ=1 X the diﬀerential operator L is deﬁned by the following conditions i) the domain D(L) of the operator is

D(L)= y C(n)([a, b]) : U (y)= φ , µ =1,...,m ; (2.36a) ∈ µ µ ii) the operator is given by the formula

Ly = l(y), y D(L). (2.36b) ∈ When no confusion may arise, we introduce some new notations. In particular, denote by

l the diﬀerential form of the operator under consideration, by L0 the operator with ho-

mogeneous boundary conditions and by Lφ the operator with nonhomogeneous boundary conditions. Thus

L : D(L )= y C(n)([a, b]) : U (y)=0, µ =1,...,m , (2.37) 0 0 { ∈ µ } and L : D(L )= y C(n)([a, b]) : U (y)= φ , µ =1,...,m . (2.38) φ φ { ∈ µ µ }

Remark 4 Note that due to the nonhomogeneous boundary condition the operator Lφ is no longer linear. 16 2. Basic Notions from Functional Analysis

2.3 Chapter summary

This chapter described briefly the basic notions and facts from functional analysis and accompanied by the appropriate author’s interpretation. More exactly, the main focus was on linear operators in Hilbert spaces. It has been shown that to define an operator in rigorous mathematical terms, in addition to its formal definition it is necessary to define the space where operator acts and to specify its domain. The corresponding space contains those elements which satisfy the natural constraints such as continuity, differentiability or integrability. In turn, the domain of the operator is a subspace of this abstract or general space, and consists of such a set of elements which satisfy the desired properties or more specific restrictions generated in many cases by the nature of physical processes. Also, since we have defined the Hilbert space, it is possible to consider some relations between the elements such as scalar product and norm. And of course, as we will see in the next chapters, the notion of the adjoint operator is a very powerful tool in the investigation of the prime operator. In this chapter we have restricted our detailed consideration to the case of ordinary differential operators of order n which act on the subspace C(n)([a, b]) of scalar functions of the Hilbert space L2([a, b]). But in a similar way the case of spaces with functions f : R CM and ordinary differential operators on it can be treated (see [Naj68]). → Moreover, the case of partial differential operators can be also considered analogously. However, since partial differential operators include more complicated boundary conditions (due to the multidimensional argument), the generalization to a such compact presentation described above involves the corresponding generalization of the 1D Green’s formula (2.17) to its MD analogy (see [Tay], [Sho77]). Also, the strong requirements to have a continuous partial derivative for functions in a usual (classical) sense (C(n)([a, b])), can be substantially weakened by introducing the corresponding Sobolev spaces and weak or distributional partial derivatives. For further details about Sobolev spaces for use in some application we refer to [Ada75], [Eva02]. On the one hand, the main attempt has been made in this chapter to present all necessary basic material of operator theory in sufficient generality, in order to be able to manipulate in abstract terms. On the other hand, it was intended to cover in detail the procedure of defining the operators in special cases, in order to understand the applied aspects of the presented theory. 17 3 Spectral Theory of Operators

In the theory of linear and time-invariant systems and network theory one of the fundamental concepts is the system function or transfer function. It helps to describe the system, its behaviour and its transfer properties from the input to the output (see [GRS01]). Most applications lead to models with meromorphic transfer functions. This means such functions are completely determined by all their poles and zeroes up to a multiplicative constant. We remark that the number of poles and zeroes might be inﬁnite. In any case, the class of rational functions is included in the more general class of meromorphic functions. Since prescribed poles and zeroes are easy to implement in realizing structures, frequency-domain models are often used. In principle, there is a similar situation in the general operator theory, as will become obvious in this chapter. The application of spectral methods in the presence of general operators provides similar advantages as the introduction of transfer function above. In this chapter we will presuppose a closed linear operator L : H H, the domain D(L) of 7→ which is dense in the Hilbert space H. In addition we will consider the operator

L =(sI L) : H H, s C, s − 7→ ∈

where I is the identity operator. The investigation of the operator Ls is called the spectral theory for the operator L. It includes the characterization of the distribution of the values

of s for which Ls has an inverse and the properties of this inverse when it exists. Similar

to the former case, the general theory of the inverse of Ls leads to spectral decomposition

based on a certain set of poles. As we shall see later, the inverse of the operator Ls plays a similar role in obtaining the solution of initial-boundary-value problems as the system function in common systems.

3.1 The resolvent operator

Deﬁnition 17 If s0 is such that the image Ran(Ls0 ) is dense in H and Ls0 has a bounded 1 inverse (s0I L)− , we say that s0 is a regular point of L and we denote this inverse 1 − (s I L)− by R(s , L) and call it the resolvent operator of L at s . 0 − 0 0 Deﬁnition 18 The set of all regular points of the operator L is called the resolvent set, and is denoted by ρ(L). The set C ρ(L) is called the spectrum of the operator L and is \ denoted by σ(L). 18 3. Spectral Theory of Operators

In the following we assume that neither the spectrum nor the resolvent set is empty. Thus, for any s ρ(L) the resolvent operator is deﬁned as ∈

1 R(s, L)=(sI L)− , s ρ(L). (3.1) − ∈

Let L† : H H be the adjoint operator of L. Since the operator L is assumed to be 7→ closed with dense domain D(L) in H, it follows that its adjoint L† is also closed, and its

domain D(L†) is dense in H, too (see e.g. [Bal76],[Yos80],[Gol66]). From the equality

1 1 R(s, L†)=(sI L†)− = [(s∗I L)− ]† = R(s∗, L)†, s ρ(L†) (3.2) − − ∀ ∈ it follows immediately that

ρ(L†)= ρ(L)∗, and σ(L†)= σ(L)∗. (3.3)

In addition, any resolvent satisﬁes the Hilbert identity (see e.g. [Bal76],[Yos80])

R(s, L) R(w, L)=(w s)R(s, L)R(w, L), s,w ρ(L), (3.4) − − ∈ which can easily be veriﬁed by use of (3.1).

Deﬁnition 19 The complex number λ σ(L) is called an eigenvalue of the operator L, ∈ if Ker(λI L) = 0 . (3.5) − 6 { } Any vector e = e(λ) Ker(λI L) 0 , is called an eigenvector of the operator L ∈ − \ { } corresponding to the eigenvalue λ.

It will turn out for all relevant cases that the eigenspaces Ker(λI L) have ﬁnite dimension, − i.e. P = P (λ) = dim Ker(λI L) < . Hence, we can assume that a basis B of this − ∞ 0 eigenspace is given by the set B = B (λ)= e , e , , e (3.6) 0 0 1,0 2,0 ··· P,0 of the linear independent eigenvectors ep,0 corresponding to the eigenvalue λ. For later purposes we have introduced the subscript 0. Using the simple property (3.4) of the resolvent operator of L we are already able to obtain a useful characterization of the spectrum of L.

Lemma 2 [Bal76] Let L be a closed operator with dense domain in H. If there exists s ρ(L) such that the resolvent operator R(s , L) is compact, then 0 ∈ 0 R(s, L) is compact in any point of the resolvent set ρ(L); moreover, the spectrum σ(L) consists of a discrete set of points and hence is denumerable. 3.1. The resolvent operator 19

The important statement of this lemma is the fact that it guaranties that the spectrum of L consists of a countable set of points. Consequently, all those values λ are isolated points. They are the poles of the resolvent operator R(s, L). It can therefore be represented in a certain neighborhood of such a singular point λ by the following Laurent series expansion

+ ∞ R(s, L)= (s λ)νR , (3.7) − ν ν= X−∞ where the coeﬃcients Rν are operators and can be represented by the formula

1 ν 1 R = (s λ)− − R(s, L)ds (3.8) ν 2πj − s Iλ =r | − | for suﬃciently small r> 0. Here j denotes the imaginary unit.

The next two lemmas contain some properties of the operators Rν in the series expansion (3.7).

Lemma 3 [Yos80]

a) the operators Rν in (3.7) commute with each other and the operator L;

2 b) the operator R 1 is a projection operator, i.e. R 1 = R 1; − − − c) the following formula is valid

Rν 1 +(λI L)Rν = δ0,νI, ν Z. (3.9) − − ∈

Here δµ,ν is the standard Kronecker symbol deﬁned as

1, µ = ν δ = . (3.10) µ,ν 0, µ = ν ( 6 Lemma 4 [Fur01a][Fur01b] Let L be a closed operator with dense domain and compact resolvent operator and λ σ(L), then ∈ d) the Laurent series (3.7) in the neighborhood of the eigenvalue λ has a ﬁnite number M of terms with negative powers of (s λ) − + ∞ R(s, L)= (s λ)νR , (3.11) − ν ν= M X− i.e.

R =0, ν< M; (3.12) ν − 20 3. Spectral Theory of Operators

e) the operator R 1 can be characterized by the condition −

M Ran(R 1) =Ker([λI L] ); (3.13a) − −

the operators R ν , ν> 0 have ﬁnite dimensional images −

dim Ran(R ν ) < , ν> 0 (3.13b) { − } ∞ and satisfy the relations

ν ν R ν 1 =( 1) (λI L) R 1, ν =1,...,M 1. (3.13c) − − − − − −

It is obvious that the Laurent series expansion of the adjoint resolvent operator R(s, L†)

in the neighborhood of λ∗ σ(L†) is ∈ + ∞ ν R(s, L†)= (s λ∗) R† . (3.14) − ν ν= M X−

In particular, it has the same order M of the pole λ∗ and the corresponding operators Rν† are adjoint to the operators Rν. Thus, by analogy with Lemma 3 and Lemma 4 we have

Z Rν† 1 +(λ∗I L†)Rν† = δ0,νI, ν , − − ∈

dim Ran(R† ν ) = dim Ran(R ν) < , ν> 0, { − } { − } ∞ (3.15) M Ran(R† 1) = Ker([λ∗I L†] ), − − ν ν R† ν 1 =( 1) (λ∗I L†) R† 1, ν =1,...,M 1. − − − − − −

3.1.1 Canonical systems of the prime and the adjoint operator

An important part of matrix theory is the transformation which leads to Jordan form of a matrix and the corresponding Jordan chains of vectors (the sets of eigenvectors and generalized eigenvectors) (see [Gan59]). There is a similar situation in the general spectral theory of operators. M.V. Keldysh was the ﬁrst who generalized the notions of the Jordan chains of vectors to a more general classes of operators (see [Kel51],[Kel71]). He has considered the canonical systems for the operator pencil of the form L(λ) = n 1 n 1 n n A0 + λBA1 + + λ − B − An 1 + λ B in the Hilbert space H, where A0, A1,...,An 1 ··· − − are arbitrary compact operators, B is a self-adjoint operator and Bf = 0, if f = 0. 6 6 Here we reformulate the deﬁnitions introduced by Keldysh in a context which is suitable for our purpose. We consider a closed linear operator L : H H, the domain D(L) 7→ of which is dense in the Hilbert space H. 3.1. The resolvent operator 21

Definition 20 For a fixed eigenvalue λ and a fixed eigenvector e , where 1 p P (λ) p,0 ≤ ≤ (P (λ) is the dimension of the eigenvector space belonging to the eigenvalue λ, see (3.6)), we say that the vector e = 0 is an associated vector (or generalized eigenvector) of p,m 6 order m of the eigenvector ep,0, if it satisfies the following equations

(λI L)e =0, − p,0 ep,0 +(λI L)ep,1 =0, − (3.16) , ········· ep,m 1 +(λI L)ep,m =0. − −

The maximum possible number m of associated vectors is denoted by Mp = Mp(λ) and

(Mp + 1) is called the multiplicity of the eigenvector ep,0. The set

E = E (λ)= e , e , , e (3.17) p p p,0 p,1 ··· p,Mp is called the complete chain (or Keldysh chain) of associated vectors that belongs to the eigenvector ep,0 of λ (note that the eigenvector itself is also included).

Remark 5 When no confusion may arise, we will write P instead of P (λ).

It will be shown later that all multiplicities Mp + 1 are less or equal to the order M of the corresponding pole in the Laurent series expansion (3.7) for the considered eigenvalue λ, M +1 M. p ≤ Deﬁnition 21 For each ﬁxed eigenvalue λ we consider the union

P (λ)

E(λ)= Ep(λ) (3.18) p=1 [ over the set of linear independent eigenvectors ep,0, p = 1,...,P (λ). E(λ) is called a canonical system of the operator L corresponding to the ﬁxed eigenvalue λ. P (λ) The number N(λ)= M1 +1+ M2 +1+ + MP (λ) +1= P (λ)+ Mp is called the ··· p=1 multiplicity of the considered eigenvalue λ. P

In the following we assume without loss of generality that the sequence of eigenvectors is ordered with respect to their multiplicities (M 1) = M M M 0. − 1 ≥ 2 ≥···≥ P (λ) ≥ To illustrate the notion used in (3.16)-(3.18), the canonical system in (3.18) can be arranged as 22 3. Spectral Theory of Operators

E(λ)= E (λ), E (λ), , E (λ), , E (λ) 1 2 ··· p ··· P (λ) e e e e 1,0 2,0 ··· p,0 ··· P (λ),0  e1,1 e2,1 ep,1 eP (λ),1  ··· ···      · · ···· ····   e1,m e2,m ep,m eP (λ),m  (3.19)  ··· ···  =   .  · · ···· ····   e  · · ···· ··· P (λ),MP (λ) e  p,Mp   · · ···   e2,M2   ·   e   1,M1      The ﬁrst row contains the set of P (λ) eigenvectors, and the p th column shows the − complete chain of associated vectors corresponding to the eigenvector ep,0 as deﬁned in (3.16).

In addition to the canonical system E(λ) with the elements ep,m in (3.18) we deﬁne

another canonical system E† consisting of elements ǫp,m, that corresponds to the eigen-

value λ∗ of the adjoint operator L†. This collection of eigenvectors together with the corresponding set of associated vectors satisﬁes the following equations

(λ∗I L†)ǫ =0, − p,0 ǫ +(λ∗I L†)ǫ =0, p,0 − p,1 , ········· (3.20) ǫp,m 1 +(λ∗I L†)ǫp,m =0, − − , ········· ǫ +(λ∗I L†)ǫ =0, p,Mp−1 − p,Mp for each p =1, ,P (λ∗). ··· By analogy with (3.16)-(3.18), we introduce the complete chains

E† = E†(λ∗)= ǫ , ǫ , , ǫ (3.21) p p p,0 p,1 ··· p,Mp and the canonical system of the operator L† for a ﬁxed eigenvalue λ∗ as

P (λ∗)

E†(λ∗)= Ep†(λ∗). (3.22) p=1 [ In addition, we will use the following notations :

E = E(λi) (3.23) λi ∪σ(L) ∈ 3.1. The resolvent operator 23 is called the canonical system of the operator L;

E† = E†(λi∗) (3.24) λ∗ ∪σ(L†) i ∈

is called the canonical system of the adjoint operator L†.

We have to mention here that in the general case neither E nor E† will be an orthogonal set. However, the following theorem is true

Theorem 1 After proper normalization of ǫp,Mp for each eigenvalue the introduced canonical systems E and E† are biorthogonal

(ep,m(λi), ǫl,n(λw∗ )) = δi,wδp,lδm,Mp n. (3.25) −

Proof. see Remark 18 in Appendix. To illustrate the biorthogonality relations of the canonical systems we assume that it has for example only 4 (Mp = 3) elements in the chain Ep(λ). In Figure 3.1 solid lines mean nonorthogonal relations, and dotted lines show orthogonality.

ep,0 ǫp,3

ep,1 ǫp,2

ep,2 ǫp,1

ep,3 ǫp,0

Figure 3.1: Biorthogonality relations of a canonical systems

Deﬁnition 22 A system A = ai i Z, ai H is called minimal if and only if none of { } ∈ ∈ the elements a A is contained in the closed linear span of the remaining elements i ∈ a / span(A a ). (3.26) i ∈ \{ i}

From the biorthogonality-relation we can deduce that the canonical systems E and E† are both minimal systems. The eigenvectors and associated vectors of the canonical system are linear independent.

Remark 6 Note, that for an arbitrary linear independent system it does not necessary follow the existence of the corresponding biorthogonal system. For example, the sequence i a = x ∞ is linear independent system in L ([0, 1]) and has no biorthogonal system. { i }i=0 2 24 3. Spectral Theory of Operators

For each ﬁxed eigenvalue the corresponding elements of the canonical system form a subspace and its dimension is equal to the order of the eigenvalue. Therefore, one can set the natural question about the basis property of such a system.

Remark 7 Currently, the development of so-called completeness and basis properties of the introduced canonical systems for the considered case of operators is a subject of future work. But we note that the fundamental theorem concerning completeness of the canonical system was presented by Keldysh (see [Kel71]). That the canonical system constitutes a

basis was established by Ilyin for the case of diﬀerential operators of order n in L2 (see [Il’76]).

3.1.2 Expansion of the resolvent operator by the canonical system We consider closed operators L with dense domain and compact resolvent. Lemma 2 states that in this case the spectrum σ(L) consists only of a discrete set of eigenvalues and the resolvent operator can be represented by the expansion (3.11). In the following we will formulate some of these important properties of the resolvent operator which are needed below.

Agreement 1 We introduce the following notation: for u H and v H the symbol ∈ ∈ [u, v] represents the operator S : H H deﬁned on functions f H by the following → ∈ formula:

[u, v]f = Sf =(f, v)u, (3.27) where ( , ) is the inner product in H. · · Obviously, the operator S in (3.27) projects the whole space H onto the onedimensional space spanned by the vector u. Consequently the operator S has rank 1. Since for f,g H the following equality holds ∈

(Sf,g)=((f, v)u,g)=(f, v)(u,g)=(f, (u,g)∗v)=(f, (g,u)v), (3.28) the adjoint operator S† acts on element g H as ∈

[u, v]†g = S†g =(g,u)v. (3.29)

The expansion of the resolvent operator by the canonical system was established ﬁrst by M. Keldysh [Kel51] for the case of a compact operator pencil. But for our purpose we extend his results to the class of unbounded and non-self-adjoint operators L with compact resolvent. We use the fact that the class of compact operators excludes the class of operators with compact resolvent. Lemma 5 states this fact. 3.1. The resolvent operator 25

Lemma 5 [Bal76] A compact operator cannot have a compact resolvent.

Using the presented results of lemmas 2-4 we are now able to give an exact representation of the resolvent comprised in theorem 2. Theorem 2 [Fur01b][Fur01b] Let L be a closed operator with dense domain, compact resolvent operator R(s, L), and non-empty spectrum σ(L). For an arbitrary pole s = λ of the resolvent R(s, L) of order M the following statements are valid:

i) the pole λ is an eigenvalue of L, and λ∗ is an eigenvalue of L†. Moreover, the

eigenvectors corresponding to these eigenvalues λ and λ∗ have identical multiplicities; ii) in a suﬃciently small neighborhood of λ, the main part of the Laurent series expansion for the resolvent R(s, L) can be represented as follows

1 P (λ) − ν [ep,0, ǫp,0] [ep,0, ǫp,1] + [ep,1, ǫp,0] (s λ) Rν = + + . . . − (s λ)Mp+1 (s λ)Mp ν= M p=1 X− X − − (3.30)

[ep,0, ǫp,Mp ] + [ep,1, ǫp,Mp 1]+ + [ep,Mp , ǫp,0] + − ··· , s λ −

and hence

P (λ) Mp+1 ν −

R ν = [ep,µ, ǫp,Mp+1 ν µ], ν =1, , M, (3.31) − − − ··· p=1 µ=0 X X where ep,m(λ) are the elements from an arbitrary canonical system E(λ) corresponding to

the eigenvalue λ of the operator L, and ǫp,n(λ∗) are elements from the canonical system

E†(λ∗) corresponding to the eigenvalue λ∗ of the operator L†, which uniquely follows from E(λ). Proof. See Section A.1 in Appendix. This theorem states that each pole of the resolvent is an eigenvalue of L. Moreover, the expression (3.30) shows how the main part of the resolvent R(s, L) at a pole λ of order M can be determined by means of the corresponding canonical systems E(λ) and

E†(λ∗).

3.1.3 Sectorial operators Now we will introduce the subclass of those operators termed as sectorial operators [Hen81, LLMP05]. This class is suitable for the mathematical description of many engineering problems. An essential feature of these operators is that the corresponding initial-boundary-value problem admits an analytical representation of the solution by means of holomorphic semigroups. 26 3. Spectral Theory of Operators

3.1.3.1 Deﬁnition of sectorial operators

Deﬁnition 23 [Hen81] A linear operator L on a Hilbert space H is called a sectorial operator, if i) it is closed and has a dense domain D(L) in H; ii) for some ϕ (0,π/2) and a R the sector (see Figure 3.2) ∈ ∈ S = s C ϕ arg(s a) π, s = a (3.32) a,ϕ { ∈ | ≤| − | ≤ 6 } belongs to the resolvent set ρ(L) and iii) for some C > 0 the following estimate

1 (sI L)− C/ s a , s S (3.33) k − k ≤ | − | ∀ ∈ a,ϕ holds.

Im(s)

S a,ϕ ϕ a Re(s)

Figure 3.2: The sector Sa,ϕ in the complex plane

3.1.3.2 Examples of sectorial operators

Example 3 Sectorial operator

We denote by H the Hilbert space of square integrable functions on the interval [0, 1], i.e. H = L2([0, 1]), by AC the set of absolutely continuous functions on the interval [0, 1], 1 1 1 by AC = y AC : y′ AC and by H = y AC : y,y′,y′′ H . { ∈ ∈ } { ∈ ∈ } 1 We consider the operator L : H H, Ly = y′′ with domain D(L) = y H : → − ∈ y(0) = 0, y′(1) = 0 . n It can be easilyo shown that the operator L is not bounded, but closed and has dense domain D(L) L2([0, 1]). To show that this operator is sectorial we need to check only ⊂ (2i+1)2π2 the conditions ii) and iii). The spectrum σ(L) consists of the values λi = 4 , i = 0, 1,.... Thus, the condition ii) holds for any ϕ (0,π/2) and a = 0 (see Figure 3.3). ∈ 3.1. The resolvent operator 27

Im(s)

Sa,ϕ ϕ Re(s)

Figure 3.3: The sector Sa,ϕ belongs to the resolvent set ρ(L)

To show that the estimate iii) holds, let f =(sI L)y. Then − 2 2 2 2 2 2 f y ((sI L)y,y) = s y y′ . || || · || || ≥ − || || − || ||

For s = σ + jω we obtain that

2 2 2 2 2 2 2 2 2 s y y′ = (σ + jω) y y′ =(σ y y′ ) + || || − || || || || − || || || || − || || 2 4 2 4 2 2 4 2 4 + ω y = σ y 2σ y y′ + y′ + ω y . || || || || − || || || || || || || || Thus, if Re s = σ 0, we have { } ≤ f 2 y 2 σ2 y 4 + ω2 y 4 = s 2 y 4. || || · || || ≥ || || || || | | || || For the case Re s = σ > 0 { } f y ((sI L)y,y) Im ((sI + L)y,y) = ω y 2. || ||·|| || ≥ − ≥ { } || ||

Now we choose ϕ = π/4, it means that for all s = σ + jω S 0,π/ 4 such that Re s > 0, ∈ { } we have that ω σ > 0 and | | ≥ f y ω y 2 σ y 2. || ||·|| || ≥ || || ≥ || ||

Thus

2 f 2 y 2 s 2 y 4, || || ·|| || ≥ || ||

√2 f y s y 2 || ||·|| || ≥ · || || and, therefore, the desired inequality

√2 y f . || || ≤ s || || | | holds. 28 3. Spectral Theory of Operators

The required estimate follows immediately as

1 √2 (sI L)− , s S . k − k ≤ s ∀ ∈ 0,π/4 | | With the constants ϕ (0,π/2), a = 0 and C = √2 we conclude that L is a sectorial ∈ operator.

Example 4 Non-sectorial operator

We consider the operator Py = Ly = y′′, where L is from the previous example. It is − obvious that for this case the condition ii) is not satisﬁed. Hence, P cannot be a sectorial operator.

Example 5 Sectorial operator with mixed Dirichlet and periodic Neumann boundary conditions.

We consider the second order derivative operator in H = Lp([0, 1]) with mixed Dirichlet and periodic Neumann boundary conditions. d2 L L Operator L = dx2 : p([0, 1]) p([0, 1]), 1 p< + with domain D(L)= y 2 − → ≤ ∞ { ∈ W ([0, 1]) : y(0) = 0, y′(0) y′(1) = 0 L ([0, 1]) is sectorial. p − } ⊂ p Proof. See Section A.2 in the Appendix.

3.1.4 Semigroups and canonical systems

To any sectorial operator L one can assign a holomorphic semigroup which describes the evolution of a system described by L along a certain parameter (see [Hen81, HP57, Kat66]). In many applications this parameter is the continuous-time variable t and the systems described by the operator L arise from initial-boundary-value problems.

3.1.4.1 Deﬁnitions

Lt This section investigates the properties of the holomorphic semigroup e 0. The most { }≥ important one is that each operator eLt can be expressed by the canonical system of the operator L.

Deﬁnition 24 The set of linear bounded operators T (t) t 0 on a Hilbert space H is called a holomorphic semigroup if ≥ i) T (0) = I, T (t + s)= T (t)T (s) for t, s 0; ≥ ii) T (t)f f, t +0 for f H → → ∈ ii) the mapping t T (t)f is holomorphic on 0

Definition 25 The infinitesimal generator L of T (t) is defined by T (t)f f Lf = lim − , (3.34) t +0 → t with the domain T (t)f f D(L)= f H : − converges for t +0 . (3.35) { ∈ t → } It is obvious that L is a linear operator, and it can be shown that the domain of this operator is dense in H (see [Bal76]).

3.1.4.2 Connection between the semigroup of the operator, the resolvent and the Laplace transformation

To show the connection between the semigroup and the resolvent let us consider the following simple case. We take the space of complex numbers and consider linear transformations on it. Here

T (t)y = eaty, t 0, a C. (3.36) ≥ ∈ is a semigroup of linear transformations. Then the corresponding inﬁnitesimal transformation L is T (t)y y Ly = lim − = ay (3.37) t +0 → t and with (3.1) the corresponding resolvent is given by 1 R(s, L)y = y. (3.38) s a − 1 at It is easy to check that the function s a is the Laplace transform of e : − ∞ at st 1 e e− dt = , for Re s > Re a . (3.39) s a { } { } Z0 − Hence the inverse Laplace transformation yields 1 1 est ds = eat, t 0, (3.40) 2πi s a ≥ Zγ − for a suitable choice of the path of integration γ. The correspondence principle a L, eat T (t) and 1 R(s, L) gives us the → → s a → following relations −

∞ st R(s, L)= T (t)e− dt, (3.41)

Z0 30 3. Spectral Theory of Operators

1 T (t)= R(s, L)estds. (3.42) 2πj Zγ

Thus, we would expect the resolvent R(s, L) of the generator L to be the Laplace transform of the semigroup operator T (t), and conversely the semigroup operator should be obtainable from the resolvent by the inversion of the Laplace integral. The following theorem states this fact for the general operator case.

Theorem 3 [Hen81] If ( L) is a sectorial operator, then L is the generator of the holo- − morphic semigroup T (t) t 0 and ≥ I, t =0 Lt T (t)= e = 1 1 st , (3.43)  2πj (sI L)− e ds, t> 0  γ − R where γ is a counterclockwise contour lying in the resolvent set ρ(L) such that arg(s) θ → ∓ if s for some θ (π/2, π). | |→∞ ∈ Remark 8 Note that the required contour γ exists because ( L) is a sectorial operator − and we can take it lying in the sector S (see Figure 3.4). − a,ϕ

Im(s)

S a,ϕ − θ ϕ Re(s) γ

Figure 3.4: The sector S in the complex plane and the contour γ − a,ϕ

3.1.5 Expansion of the semigroup by the canonical system Now, our aim is to express (3.43) using the results of Theorem 2. For this purpose let σ 0 be any real number such that 0 ≥ σ(L) s C : Re(s) = σ = . (3.44) ∩{ ∈ − 0} ∅ Consider the case when some elements of the set σ(L) are at the right side of the line Re(s) = σ (note that the number of such elements is always ﬁnite). The integral { − 0} 3.1. The resolvent operator 31

(3.43) can then be rewritten as

Lt 1 1 st 1 1 st e = (sI L)− e ds + (sI L)− e ds, 2πj − 2πj − (3.45) i γZσ s λZi =r 0 X | − | where the contour γ coincides with the original path γ for Re s σ and that σ0 { { } ≤ − 0} segment of the line Re s = σ which connects the two branches of γ (see Figure { { } − 0} 3.5). Summation is taken over all points λ σ(L) which are at the right side of the line i ∈ Re(s) = σ . { − 0} Im(s)

S a,ϕ − σ0 Re(s) − γσ0

Figure 3.5: The contour γσ0

Applying the residue theorem and using Theorem 2 the corresponding element of the semigroup is obtained as follows

Lt 1 1 st e = (sI L)− e ds+ 2πj − γZσ0

P (λi) Mp λit t + e [ep,0(λi), ǫp,0(λi∗)]+ Mp! Re(λi)> σ0 p=1 (3.46) X− X Mp 1 t − + [e (λ ), ǫ (λ∗)]+[e (λ ), ǫ (λ∗)] + . . . + (M 1)! p,0 i p,1 i p,1 i p,0 i p −

+ [e (λ ), ǫ (λ∗)] + + [e (λ ), ǫ (λ∗)] . p,0 i p,Mp i ··· p,Mp i p,0 i Since the operator ( L) was assumed to be sectorial, it can easily be shown (see − [Fur01a, Fur01b, Hen81, Kat66]) that the integral on the right hand side of (3.46) can be estimated as

1 1 st σ0t (sI L)− e ds C e− , t> 0. (3.47) k2πj − kH ≤ 1 γZσ0 32 3. Spectral Theory of Operators

Now it follows that for each ﬁxed t> 0

1 1 st (sI L)− e ds 0 for σ + . (3.48) k2πj − kH → 0 → ∞ γZσ0 Finally, using (3.46) and (3.48) we can formulate the basic result concerning the representation of the semigroup eLt . { } Theorem 4 The elements eLt, t> 0 of the holomorphic semigroup eLt with the gen- { } erator L are represented in the following form

P (λi) Mp Lt λit t e = e [ep,0(λi), ǫp,0(λi∗)]+ Mp! λi σ(L) p=1 X∈ X

Mp 1 t − (3.49) + [e (λ ), ǫ (λ∗)]+[e (λ ), ǫ (λ∗)] + + M 1! p,0 i p,1 i p,1 i p,0 i ··· p −

+ [ep,0(λi), ǫp,Mp (λi∗)] + . . . + [ep,Mp (λi), ǫp,0(λi∗)] . Thus, we have shown that the semigroup eLt generated by L can be expressed in terms { } of the canonical systems for the prime operator and its adjoint. To write the last expression in more compact form the following abbreviation is used m

Kp,m(λi)= [ep,µ(λi), ǫp,m µ(λi∗)]. (3.50) − µ=0 X Then we can write instead of (3.49)

P (λi) Mp m Lt λit t e = e Kp,Mp m(λi). (3.51) m! − λi σ(L) p=1 m=0 X∈ X X 3.2 The C resolvent operator − In this section we extend the above results to the general case when some singularity eﬀects can be taken into account. In the following we assume that L is a closed operator that acts in the Hilbert space H (with the scalar product ( , ) and the norm ) and its domain D(L) is dense in H. · · k·k Also, let C be a linear bounded operator acting on H (D(C)= H).

Deﬁnition 26 If for some complex number s C there exists the bounded operator ∈ 1 R (s, L)=(sC L)− : H H (3.52) C − 7→ then this operator R (s, L) is called the C resolvent operator for the operator L and s is C − called a regular point of L. 3.2. The C resolvent operator 33 −

By setting the operator C equal to the identity operator I, we have the usual definition of the common resolvent operator R(s, L). Next, we will generalize the definitions and results which were obtained in previous sections to the case of C resolvent operator. − Definition 27 The set of all regular points of the operator L is called the C resolvent − set, and is denoted by ρ (L). The set C ρ (L) is called the C spectrum of the operator C \ C − L and is denoted by σC (L). Thus, for any s ρ (L) the resolvent operator is defined as ∈ C

1 R (s, L)=(sC L)− , s ρ (L). (3.53) C − ∈ C

The following lemma is obvious Lemma 6 If Ker(L) Ker(C) = 0 , then ρ (L)= . ∩ 6 { } C ∅ Let L† : H H be the adjoint operator for the operator L. Since the operator L is 7→ closed and the domain D(L) is dense in H, then the adjoint L† is also closed, and its

domain D(L†) is dense in H, too (see e.g. [Bal76],[Yos80]). From the equality

1 1 R † (s, L†)=(sC† L†)− = [(s∗C L)− ]† = [R (s∗, L)]†, s ρ(L†) (3.54) C − − C ∀ ∈ it follows immediately that

ρC† (L†)= ρC (L)∗, and σC† (L†)= σC (L)∗. (3.55) In addition, for any C resolvent the analog of Hilbert identity holds − 1 1 1 1 (sC L)− (ˆsC L)− =(s sˆ)(sC L)− C(ˆsC L)− , (3.56) − − − − − − or equivalently

R (s, L) R (ˆs, L)=(s sˆ)R (s, L)CR (ˆs, L). (3.57) C − C − C C Deﬁnition 28 The complex number λ σ (L) is called an C eigenvalue of the operator ∈ C − L, if Ker(λC L) = 0 , (3.58) − 6 { } and any vector e Ker(λC L) 0 , p = 1, ,P (λ) is called a C eigenvector of p,0 ∈ − \{ } ··· − the operator L corresponding to the given C eigenvalue λ. Here P = P (λ) denotes the − total number of linear independent eigenvectors corresponding to the C eigenvalue λ. − Lemma 7 Let L be a closed operator with dense domain in H, C is a bounded linear operator in H and Ker(L) Ker(C)= 0 . If there exists sˆ ρ (L) such that the resolvent ∩ { } ∈ C operator RC (ˆs, L) is compact, then RC (s, L) is compact in any point of the resolvent set

ρC (L); moreover, the spectrum σC (L) consists of a discrete set of C-eigenvalues which has no point of accumulation except possibly inﬁnity. 34 3. Spectral Theory of Operators

Proof. The ﬁrst part of the Lemma is clear from (3.57) (the product of a compact operator with a bounded linear operator is compact (see [Yos80])). To prove the second part of the lemma, lets ˆ ρ (L) be ﬁxed and s C. Then the ∈ C ∈ equality

1 (sC L)=((s sˆ)C(ˆsC L)− + I)(ˆsC L) (3.59) − − − − holds. If s = λ is a C eigenvalue of the operator L and e is a corresponding C eigenvector, 0 − 0 − then it follows from (3.59) that

1 (λ C L)e = ((λ sˆ)C(ˆsC L)− + I)(ˆsC L)e . (3.60) 0 − 0 0 − − − 0 Let us introduce the following notation g = (ˆsC L)e . Thus, from (3.60)we have the 0 − 0 sequence of equalities

1 0=((λ sˆ)C(ˆsC L)− + I)g , 0 − − 0 1 g =(λ sˆ)C(ˆsC L)− g , (3.61) − 0 0 − − 0 1 1 (ˆs λ )− g = C(ˆsC L)− g . − 0 0 − 0

1 Therefore, we conclude that λ˜ = (ˆs λ )− , λ˜ = 0 is the usual eigenvalue of the com- 0 − 0 0 6 pact operator CRC (ˆs, L) (C is bounded, RC (ˆs, L) is compact, the product of a compact operator with a bounded linear operator is compact (see [Yos80])). It is well known (see [Bal76],[Yos80]) that the spectrum of a compact operator consists of an at most countable set of points of the complex plane which has no point of accumulation except possibly zero and every non-zero element of the spectrum is an eigenvalue of ﬁnite multiplicity. Due to this fact, the C spectrum of the operator L is also discrete. This proves the second part − of Lemma 7. The important statement of this lemma is the fact that it guaranties that the C spectrum of L consists of a countable set of points λ. Consequently all those val- − ues λ are isolated points. They are the poles of the resolvent operator RC (s, L). It can therefore be represented in a certain neighborhood of such a singular point λ by the following Laurent series expansion

+ ∞ R (s, L)= (s λ)νR , (3.62) C − ν ν= X−∞ where the coeﬃcients Rν are operators and can be represented by the formula

1 ν 1 R = (s λ)− − R (s, L)ds (3.63) ν 2πj − C s Iλ =r | − | 3.2. The C resolvent operator 35 − for suﬃciently small r > 0 such that the circle s λ r does not contain other | − | ≤ singularities then s = λ and the integration is performed counter-clockwise. The next lemma contains some properties of the operators in the series expansion (3.62).

Lemma 8 a) the Laurent series (3.62) in the neighborhood of the eigenvalue λ has a ﬁnite number M of elements with negative powers of (s λ); − b) the dimensions of the images of the operators R are ﬁnite, dim Ran(R ) < , ν< ν { ν } ∞ 0; c) the following formula is valid

CR ν 1 +(λC L)R ν =0, ν 1. (3.64) − − − − ≥ Proof. Lets ˆ ρ (L) be ﬁxed and s ρ (L) arbitrary, then the following equality holds ∈ C ∈ C 1 sC L = ((s sˆ)C(ˆsC L)− + I)(ˆsC L). (3.65) − − − − By simple manipulation of (3.65) we obtain

1 1 1 (ˆsC L)(sC L)− = ((s sˆ)C(ˆsC L)− + I)− , (3.66) − − − − or equivalently

1 1 1 1 (s sˆ)(ˆsC L)(sC L)− =(C(ˆsC L)− +(s sˆ)− I)− . (3.67) − − − − − Let us introduce the notations 1 w = (ˆs s)− , − (3.68) 1 1 B(w)= w− (ˆsC L)(sC L)− . − − In terms of (3.68), the equality (3.67) has the form

1 1 1 B(w)=(wI C(ˆsC L)− )− =(wI CR (ˆs, L))− = R(w,CR (ˆs, L)). (3.69) − − − C C

Since C is bounded, RC (ˆs, L) is compact and the product of a compact operator with a bounded linear operator is compact (see [Yos80]), therefore, B(w) is a usual resolvent

operator of the compact operator CRC (ˆs, L). If λ is a C eigenvalue of the operator L, 1 − then ρ = (ˆs λ)− , ρ = 0 is an eigenvalue of the operator CR (ˆs, L). The operator − 6 C B(w) is the resolvent of a compact operator, and therefore can be represented in the neighborhood of the isolated singular point ρ by the following Laurent series expansion (see [Yos80])

+ ∞ B(w)= (w ρ)kB , (3.70) − k k= M1 X− 36 3. Spectral Theory of Operators

where the coeﬃcients Bk are the operators determined by

1 k 1 B = (w ρ)− − B(w)dw (3.71) k 2πj − w Iρ =r | − |

for suﬃciently small r> 0. It is known that in this case dim Ran(B 1) < , and ρ is a { − } ∞ pole of ﬁnite order of the operator B(w) and dim Ran(B k) < , k> 0 ([Yos80]). Let { − } ∞ us now show that dim Ran(R 1) < . { − } ∞ From (3.67),(3.69) follows

1 R (s, L)= wR (ˆs, L)(wI CR (ˆs, L))− . (3.72) C C − C Substituting the Laurent series expansions (3.62), (3.70) into (3.72), we get the formula

+ + ∞ ∞ (s λ)lR = wR (ˆs, L) (w ρ)kB , (3.73) − l C − k l= k= M1 X−∞ X− where

1 1 w = (ˆs s)− , ρ = (ˆs λ)− . (3.74) − − For s λ < sˆ λ we get the expansion | − | | − | 1 1 s λ − 1 ∞ (s λ)i w = 1 − = − . (3.75) sˆ λ − sˆ λ sˆ λ (ˆs λ)i i=0 − − − X − For k N we have the following expansions ∈ k k k k (ˆs λ) (ˆs λ + λ s) k m k m 2k m (w ρ)− = − − − = C (s λ) − ( 1) (ˆs λ) − , (3.76) − (s λ)k m − − − m=0 − X

k (s λ)k (s λ)k ∞ (s λ)m (w ρ)k = − = − − . (3.77) − (ˆs λ)k(ˆs λ + λ s)k (ˆs λ)2k (ˆs λ)m m=0 − − − − X − Substituting the expressions (3.75),(3.76),(3.77) into the right part of (3.73), we obtain

+ ∞ (s λ)lR = − l l= X−∞ 1 k ∞ i − 1 (s λ) k m k m 2k m = − R (ˆs, L) C (s λ) − ( 1) (ˆs λ) − B (3.78) sˆ λ (ˆs λ)i C m − − − k i=0 k= M1 m=0 − X − X− X + k ∞ (s λ)k ∞ (s λ)m + − − B . (ˆs λ)2k (ˆs λ)m k m=0 Xk=0 − X − 3.2. The C resolvent operator 37 −

Finally, according to the equality (3.78), which has on the right-hand side a ﬁnite number of negative powers of (s λ), we conclude that there exists a constant 0 M < such − ≤ ∞ that Rl =0, l< M. The statement a) is proven. − 1 In particular, we see that (s λ)− has a coeﬃcient which is expressed as − 1 − R 1 = alBl, (3.79) − l= M1 X− where a are some constants. Then, since dim Ran(B ) < , l< 0 we obtain that l { l } ∞ dim Ran(R 1) < . Similar conclusions can be stated for all operators in the series { − } ∞ expansion (3.62), dim RanR < , M ν < 1. It concludes the statement b). { ν } ∞ ≤ For the proof of the statement c) see Proof of Theorem 5 in Appendix.

It is obvious that the Laurent series expansion of the adjoint resolvent operator RC† (s, L†) in the neighborhood of λ∗ σ(L†), ∈ + ∞ ν R † (s, L†)= (s λ∗) R† , (3.80) C − ν ν= M X−

has the same order M of the pole λ∗ and the operators Rν† are adjoint to the operators Rν.

3.2.1 Canonical systems of the prime and the adjoint operator In analogy with section (3.1.1) we deﬁne the C canonical systems of the prime operator − and its adjoint.

Definition 29 For a fixed C eigenvalue λ and a fixed C eigenvector e , where 1 − − p,0 ≤ p P (λ), we say that the vector e = 0 is an C associated vector of order m of the ≤ p,m 6 − C eigenvector e , if it satisfies the following equations − p,0

(λC L)e =0, − p,0 Cep,0 +(λC L)ep,1 =0, − (3.81) , ········· Cep,m 1 +(λC L)ep,m =0. − −

The maximum possible number m of associated vectors is denoted by Mp and (Mp +1) is called the multiplicity of the C eigenvector e . − p,0 The set

E = E (λ)= e , e , , e (3.82) p p p,0 p,1 ··· p,Mp 38 3. Spectral Theory of Operators is called the complete chain of C associated vectors that belongs to the C eigenvector − − ep,0 of λ (note that the eigenvector is also included).

It will be shown later that all multiplicities Mp + 1 are less or equal to the order M of the corresponding pole in the Laurent series expansion (3.62) for the considered eigenvalue λ, M +1 M. p ≤ Deﬁnition 30 For each ﬁxed C eigenvalue λ we consider the union − P (λ)

E(λ)= Ep(λ) (3.83) p=1 [ over the set of linear independent C eigenvectors e , p =1,...,P (λ). E(λ) is called a − p,0 C canonical system of the operator L corresponding to the ﬁxed C eigenvalue λ. − − P (λ) The number N(λ)= M1 +1+ M2 +1+ + MP (λ) +1= P (λ)+ Mp is called the ··· p=1 multiplicity of the considered C eigenvalue λ. P − In the following we assume without loss of generality that the sequence of C eigenvectors − is chosen such that their multiplicities satisfy (M 1) = M M M 0. − 1 ≥ 2 ≥···≥ P (λ) ≥ To illustrate the notion used in (3.81)-(3.83), the canonical system in (3.83) can be arranged as

E(λ)= E (λ), E (λ), , E (λ), , E (λ) 1 2 ··· p ··· P (λ) e e e e 1,0 2,0 ··· p,0 ··· P (λ),0  e1,1 e2,1 ep,1 eP (λ),1  ··· ···      · · ···· ····   e1,m e2,m ep,m eP (λ),m  (3.84)  ··· ···  =   .  · · ···· ····   e  · · ···· ··· P (λ),MP (λ) e  · · ··· p,Mp   e   2,M2   ·   e1,M1      The ﬁrst row contains the set of P (λ) C eigenvectors, and thep th column shows − − the complete chain of C associated vectors corresponding to the C eigenvector e as − − p,0 deﬁned in (3.81).

In addition to the C canonical system E(λ) in (3.83) we deﬁne another C† canonical − − system E† that corresponds to the C† eigenvalue λ∗ of the adjoint operator L†. This − collection of C† eigenvectors together with the set of C† associated vectors satisﬁes the − − equations for each p =1, ,P (λ∗) ··· 3.2. The C resolvent operator 39 −

(λ∗C† L†)ǫ =0, − p,0 C†ǫ +(λ∗C† L†)ǫ =0, p,0 − p,1 , ········· (3.85) C†ǫp,m 1 +(λ∗C† L†)ǫp,m =0, − − , ········· C†ǫ +(λ∗C† L†)ǫ =0. p,Mp−1 − p,Mp In analogy to (3.81)-(3.83), we introduce the complete chains

E† = E†(λ∗)= ǫ , ǫ , , ǫ (3.86) p p p,0 p,1 ··· p,Mp and the C† canonical system of the operator L† for a ﬁxed eigenvalue λ∗ as − P (λ∗)

E†(λ∗)= Ep†(λ∗). (3.87) p=1 [ In addition, we will use the following notations :

E = E(λi) (3.88) λi ∪σ(L) ∈ is called the C canonical system of the operator L; −

E† = E†(λi∗) (3.89) λ∗ ∪σ(L†) i ∈

is called the C† canonical system of the operator L†. −

Remark 9 We mention that in the general case neither E nor E† will be an orthogonal

basis. However, it can be shown that after proper normalization of ǫp,Mp for each eigenvalue the introduced canonical systems are biorthogonal with respect to weighting operator C (see Remark 19 in Appendix)

(Cep,m(λi), ǫl,n(λw∗ )) = δi,wδp,lδm,Mp n. (3.90) −

3.2.2 Expansion of the C resolvent operator by the canonical − systems

As well as the usual resolvent, the C resolvent operator can be represented by the ex- − pansion of the corresponding canonical systems. The following theorem states this fact. 40 3. Spectral Theory of Operators

Theorem 5 Let L be a closed operator with dense domain, compact C resolvent operator − RC (s, L), and non-empty spectrum σ(L). The linear operator C is bounded. For an arbitrary pole s = λ of the resolvent RC (s, L) of order M the following statements are valid:

i) the pole λ is an C eigenvalue of L, and λ∗ is an C† eigenvalue of L†. Moreover, − − the eigenvectors corresponding to these eigenvalues λ and λ∗ have identical multiplicities; ii) in a suﬃciently small neighborhood of λ, the main part of the Laurent series expansion for the resolvent RC (s, L) can be represented as follows

1 P (λ) − ν [ep,0, ǫp,0] [ep,0, ǫp,1] + [ep,1, ǫp,0] (s λ) Rν = + + . . . − (s λ)Mp+1 (s λ)Mp ν= M p=1 X− X − − (3.91)

[ep,0, ǫp,Mp ] + [ep,1, ǫp,Mp 1]+ + [ep,Mp , ǫp,0] + − ··· , s λ −

and hence

P (λ) Mp+1 ν −

R ν = [ep,µ, ǫp,Mp+1 ν µ], ν =1, , M, (3.92) − − − ··· p=1 µ=0 X X where ep,m(λ) are the elements from an arbitrary canonical system E(λ) corresponding to

the eigenvalue λ of the operator L, and ǫp,n(λ∗) are elements from the canonical system

E†(λ∗) corresponding to the eigenvalue λ∗ of the operator L†, which uniquely follows from E(λ).

Proof. See Section A.3 in Appendix

3.3 Chapter summary

In this chapter we have developed some new results for spectral analysis of unbounded and non-self-adjoint operators with a discrete spectrum. The notions of resolvent operators, canonical systems, sectorial operators and semigroups were introduced. They play a key role for the adequate mathematical modelling of real physical problems. An essential feature of the considered subclass of sectorial operators is that the corresponding initial- boundary-value problem admits an analytical representation of the solution by means of holomorphic semigroups. The analytical formulae for the required elements of this semigroup was established. It was shown that under appropriate conditions on the operator the corresponding resolvent operator admits the spectral decomposition in terms of canonical systems. More- over, the associated semigroup of operators can be also decomposed by such systems. As 3.3. Chapter summary 41 we shall see later, such expansions are very useful in the representation of the solutions of initial-boundary-value problems. It should be emphasized that one of the advantages of the introduced canonical systems of the prime and the adjoint operators is that they are biorthogonal. It was shown that such a property is equivalent to the minimality of the canonical systems. Therefore, with some additional conditions they can be considered as a basis in the corresponding Hilbert space. The generalization of the concept of resolvent operator was also introduced. It was shown that the fundamental result about its spectral decomposition in terms of canonical systems can be also obtained in this case. The extension of these results to the more general case was given, too. The singularity eﬀects were taken into consideration. For this purpose the concept of C resolvent − operator was introduced. It was shown that the fundamental result about its spectral decomposition in terms of canonical systems can be also obtained in this case, too. 42 3. Spectral Theory of Operators 43 4 Mathematical Modelling of Physical Processes

Engineers have a variety of reasons for studying differential equations: almost all the elementary and numerous of the advanced parts of system theory are posed mathematically in terms of differential equations. A fundamental problem in science is that of obtaining a mathematical model of a physical system or process. Usually a mathematical model is determined by the theoretician postulating, based perhaps on some physical laws, and then this model is checked against experimental evidence until the model reasonably represents the process under consideration. It is then possible to make predictions from the model, or to design controls for the process so that it works in some desirable manner. By far the largest class of mathematical models are those given in terms of differential equations, and in this chapter we will consider a variety of types including ordinary and partial ones. Our aim will be to show how the solution of various types of linear differential equations fit into the general formulation, and how the solution may be obtained in terms of semigroups.

4.1 Ordinary diﬀerential equations

A diﬀerential equation is an equation that contains derivatives of an unknown function which expresses the relationship we seek. If there is only one independent variable and, as a consequence, total derivatives like dx/dt, the equation is called an ordinary diﬀerential equation (ODE).

4.1.1 Classification of ODEs Physical problems are governed by many different ordinary differential equations. There are two different types, or classes, of ordinary differential equations, depending on the type of auxiliary conditions specified. If all the auxiliary conditions are specified at the same value of the independent variable and the solution is to be marched forward from that initial point, the differential equation constitutes an initial-value-problem. If the auxiliary conditions are specified at two different values of the independent variable, the end points or boundaries of the domain of interest, the differential equation defines a boundary-value-problem. 44 4. Mathematical Modelling of Physical Processes

4.1.2 State-Space model

The simplest system of diﬀerential equations in the nonhomogeneous case can be represented in following state-space form

z˙ = Az + Bx, (4.1) y = Cz, where x is a column vector of M inputs, z is a column vector of N state variables and y is a vector of K outputs. A, B, C are the matrices of size N N, N M, K N × × × respectively. If there is direct inﬂuence of the input on the output, then the state-space model has the form

z˙ = Az + Bx, (4.2) y = Cz + Dx, where D is a matrix of the size K M. × The general solution of the problem (4.1) with initial condition z(0) = z0 is

t At A(t τ) y(t)= Ce z0 + C e − Bx(τ)dτ. (4.3)

For each arbitrary matrix A exists the exponential function eAt, which can be obtained with help of the power series

k A ∞ t T (t)= e t = Ak. (4.4) k! Xk=0 This series is convergent for all t C, especially for t R . ∈ ∈ + Then, we can consider the matrix A as a bounded operator A : CN CN . Thus, the → operator A is a generator of the semigroup T (t)= eAt, t 0. ≥ However, direct application of the formula (4.4) is inconvenient for the calculation of the semigroup’s operators T (t). In the matrix case it can be found as follows (see [Gan59]): Let Q be a constant, nonsingular matrix. The change of variables according to

z = Qw (4.5) 4.1. Ordinary diﬀerential equations 45 transforms (4.1) into

w˙ = Jw + B′x, (4.6) y = C′w,

1 1 where J = Q− AQ, B′ = Q− B, C′ = CQ. It is easy to verify that

At Jt 1 e = Qe Q− . (4.7)

Let Q be chosen so that J is in a Jordan normal form, i.e.,

J = diag[J(1),..., J(k)], (4.8)

where J(i), i =1,...,k is a square matrix of size h(i) h(i) with all its diagonal elements × equal to a number λ = λ(i) and if h = h(i) > 1, its subdiagonal elements equal to 1 and its other elements are equal to 0:

λ 0 0 0 0 0 ··· 1 λ 0 0 0 0  ···   0 1 λ 0 0 0  J(i)=  ···  = λI + K , (4.9)   h h    · · ····· · ·   0 0 0 1 λ 0   ···     0 0 0 0 1 λ   ···    where Kh is the nilpotent square matrix of the form

0 0 0 0 0 0 ··· 1 0 0 0 0 0  ···   0 1 0 0 0 0  K =  ···  (4.10) h      · · ····· · ·   0 0 0 1 0 0   ···     0 0 0 0 1 0   ···    and h(1) + h(2) + + h(k)= N. J(i) is a scalar λ = λ(i) and K =0 if h(i)=1. ··· h From J = diag[J(1),..., J(k)] follows that Jn = diag[Jn(1),..., Jn(k)], hence

eJt = diag[eJ(1)t,...,eJ(k)t]. (4.11)

J(i)t λt Kht 2 Moreover, e = e e where λ = λ(i), h = h(i). Note that Kh is obtained from Kh 3 by moving the 1 from the subdiagonal to the diagonal below this; Kh is obtained by the 46 4. Mathematical Modelling of Physical Processes

h moving 1 to the next lower diagonal, etc. In particular, Kh = 0. Hence

1 0 0 0 00 ··· t 1 0 0 00  ···  t2 J  2! t 1 0 00  e (i)t = eλt  ···  , h = h(i) and λ = λ(i). (4.12)    · · · ······   th−2 th−3 th−4   (h 2)! (h 3)! (h 4)! t 1 0   − − − ···   th−1 th−2 th−3 t2   (h 1)! (h 2)! (h 3)! 2! t 1   − − − ···    Thus the problem of determining the solutions of (4.1) can be reduced to the algebraic one 1 of determining the Jordan normal form J of A and a matrix Q such that J = Q− AQ.

4.2 Partial diﬀerential equations

Most differential equations of physics involve quantities depending on both space and time. Inevitably they involve partial derivatives, and so are partial differential equations (PDEs). Partial differential equations arise in all fields of engineering and science. Most real physical processes are governed by partial differential equations. In many cases, sim- plifying approximations are made to reduce the governing PDEs to ordinary differential equations (ODEs) or even to algebraic equations. However, because of the ever increasing requirement for more accurate modelling of physical processes, engineers and scientists are more and more required to solve the actual PDEs that govern the physical problem being investigated. Chapter 5 is devoted to the solution of partial differential equations by multi-functional transformation method (MFTM).

4.2.1 Classification of PDEs To show the main principle of the classification, let us focus on second order equations in two variables x1 and x2 such as ∂2y ∂2y ∂2y a(x1, x2) 2 +2b(x1, x2) + c(x1, x2) 2 + (lower orders) = f(x1, x2). (4.13) ∂x1 ∂x1∂x2 ∂x2 A linear partial differential equation is said to be homogeneous or non-homogeneous depending on whether the right-hand member f(x1, x2) is equal to zero or is not identically equal to zero.

The equation is said to be hyperbolic, elliptic, or parabolic at a point (x1, x2) if

a(x1, x2) b(x1, x2) 2 = a(x1, x2)c(x1, x2) b (x1, x2) (4.14) b(x1, x2) c(x1, x2) −

4.2. Partial differential equations 47 is less than, greater than, or equal to zero, respectively. This classification helps to understand what sort of initial or boundary data we have to specify in order to guarantee a unique solution. To determine some physical process completely one needs the equations which describe this process. And it is also necessary to have the initial data of the process (initial conditions) and to specify the conditions on the boundary of the domain where this process is defined (boundary conditions). Mathematically, it is connected with not-uniqueness of the solution of differential equations. For example even for ordinary differential equations of the order n the general solution depends on n arbitrary constants. For partial differential equations the solution depends on arbitrary functions, for example the general solution of the equation ∂y(x1,x2) = 0 has a form y(x , x )= f(x ), where f is a arbitrary function. ∂x1 1 2 2 Thus, to find the solution which describes the real physical process it is necessary to assign additional conditions. There are three broad classes of boundary conditions: a) Dirichlet boundary conditions: The value of the dependent variable is specified on the boundary. b) Neumann boundary conditions: The normal derivative of the dependent variable is specified on the boundary. c) Cauchy boundary conditions: Both the value and the normal derivative of the dependent variable are specified on the boundary. Less commonly met with are: d) Robin boundary conditions: The value of a linear combination of the dependent variable and the normal derivative of the dependent variable is specified on the boundary. Cauchy boundary conditions are analogous to the initial conditions for a second-order ordinary differential equation. These are given at one end of the interval only. The other three classes of boundary condition are higher-dimensional analogues of the conditions we impose on an ODE at both ends of the interval. Each class of PDEs requires a different class of boundary conditions in order to have a unique, stable solution. 1) Elliptic equations require either Dirichlet or Neumann boundary conditions on a closed boundary surrounding the region of interest. Other boundary conditions are either insufficient to determine a unique solution, overly restrictive, or lead to instabilities. 2) Hyperbolic equations require Cauchy boundary conditions on a open surface. Other boundary conditions are either too restrictive for a solution to exist, or insufficient to determine a unique solution. 3) Parabolic equations require Dirichlet or Neumann boundary conditions on a open surface. Other boundary conditions are too restrictive. When we derive partial differential equations proceeding from general laws governing the natural phenomena in question, there arise some additional conditions imposed on the sought-for solutions. The proof of the existence and the uniqueness of the solutions 48 4. Mathematical Modelling of Physical Processes satisfying the additional conditions plays an important role in the theory of partial differential equations. When it turns out that small variations of the data contained both in the equations and in the additional conditions produce small variations of the solutions satisfying them or, as we say, when the sought-for solutions are stable, we speak of well- posed (or correctly set) problems; if otherwise, the problems in question are referred to as improperly posed (not well-posed). It should also be noted that the conditions of the problems which must be satisfied by the wanted solutions are essentially dependent on the type of the equations under consideration.

4.2.2 Initial-boundary-value problems

The previous section showed that operators and their associated semigroups can be used to obtain the solution of the ODEs. This representation is now exploited for the solution of initial-boundary-value problems. It is shown that for certain classes of problems it also can be expressed by the corresponding semigroup. We consider the following non-homogeneous initial-boundary-value problem

∂ y(t, x)= Ly(t, x)+ v(t, x), t> 0, x Ω ∂t ∈ y(t, x) = y (x), x Ω (4.15) |t=0 a ∈ U y(t, x) =0, t> 0, x ∂Ω, µ =1, , m. µ ∈ ··· The set Ω denotes a domain in RN and ∂Ω its boundary. The functions v(t, ) and • y(t, ) are elements of H (or H(Ω)) for each fixed value of t 0, where H is a Hilbert space • ≥ with the scalar product ( , ). The operator L is assumed to be a differential operator with · · respect to the N dimensional variable x which acts on properly elected elements of H. − We presuppose that ( L) is a sectorial operator on the Hilbert space H such that L has − a compact resolvent and the spectrum σ(L) is not empty. The condition Uµ y(t, x) =0 represents the collection of all homogeneous boundary conditions which are necessary to yield a unique solution of the initial-boundary-value problem. Now, we restrict the domain of the operator L to that set of such functions which satisfy the homogeneous boundary conditions generated by m linear independent forms U , µ = 1,...,m (see (2.13)-(2.14)), i.e. D(L) = y(t, ) H : U y(t, x) = 0, t> µ { • ∈ µ 0, x ∂Ω, µ = 1,...,m . We also introduce the short form notation for the initial ∈ } condition u = u(t) = y (x). a |t=0 a If the functions f = v (t, ), f = v (t, ) H depend on t , t , respectively we will 1 1 • 2 1 • ∈ 1 2 4.2. Partial differential equations 49 write the scalar product more precisely as

(f1(t1), f2(t2)) = v1(t1, x)v2∗(t2, x)dx, (4.16) ZΩ

which in general depends on the time instants t1 and t2. Next, we write the problem (4.15) in abstract form for elements in H as follows

∂u(t) = Lu(t)+ f(t), t> 0, ∂t (4.17) u(t) = u , u(t) D(L). |t=0 a ∈

Lt In this case the operator L is the generator of the holomorphic semigroup e t 0 { } ≥ and the solution of the problem (4.17) can be represented by means of this semigroup as

t Lt L(t τ) u(t)= e u + e − f(τ)dτ, t 0. (4.18) a ≥ Z0

Thus, the solution of the problem (4.17) and hence equivalently of (4.15) is completely determined by (4.18). However, the direct realization of such a formula on a digital computer is not possible for arbitrary inputs f.

4.2.3 Solution of initial-boundary-value problems in the Laplace domain The important tool in our further proceeding consists in applying the one-sided Laplace transformation [Yos80, GRS01] with respect to the time variable t + ∞ st u(t) = U(s)= e− u(t)dt, (4.19) L{ } Z0 where s is the complex temporal frequency variable. The Laplace transform U(s) is deﬁned for such s that the integral in (4.19) is ﬁnite. Obviously, if u(t) < Ceσt, t 0, where C and σ are constant, then the corresponding | | ≥ Laplace transform U(s) is analytic in the half-plane Re s > σ. { } Finally, the Laplace transform of the representation (4.18) with respect to the time variable t is given as

t . Lt L(t τ) u(t) = U(s)= e u + e − f(τ)dτ . (4.20) L{ } L{ a} L Z0 50 4. Mathematical Modelling of Physical Processes

Using the representation (3.51) of the operator eLt in terms of the canonical systems, the Laplace transform in (4.20) can be obtained by piece-wise Laplace transform of each term in (3.51). Therefore, we have

P (λi) Mp m Lt λit t e ua = e Kp,Mp m(λi) ua = L{ } L{ m! − } λi σ(L) p=1 m=0 X∈ X X

P (λi) Mp eλittm = Kp,Mp m(λi)ua = (4.21) L{ m! } − λi σ(L) p=1 m=0 X∈ X X P (λ ) M i p 1 = m+1 Kp,Mp m(λi)ua. (s λi) − λi σ(L) p=1 m=0 X∈ X X − Next we determine t t ∞ L(t τ) st L(t τ) e − f(τ)dτ = e− e − f(τ)dτdt = L Z0 Z0 Z0

t ∞ ∞ ∞ st Lτ Lτ st = e− e f(t τ)dτdt = e f(t τ)e− dtdτ = − − Z0 Z0 Z0 Zτ (4.22) ∞ ∞ ∞ ∞ Lτ s(t+τ) Lτ sτ st = e f(t)e− dtdτ = e e− f(t)e− dtdτ =

Z0 Z0 Z0 Z0

∞ P (λi) Mp Lτ sτ Lt 1 = e e− F (s)dτ = e F (s) = m+1 Kp,Mp m(λi)F (s). L{ } (s λi) − Z λi σ(L) p=1 m=0 0 X∈ X X −

Denoting by F¯p,l(s,λi) andu ¯a,p,l(λi) the following inner products

F¯p,l(s,λi)=(F (s), ǫp,l(λ∗)), i (4.23) u¯a,p,l(λi)=(ua, ǫp,l(λi∗)) and with (3.27), the Laplace transform of the desired solution is represented as

u(t) = U(s)= L{ } P (λi) Mp m (4.24) F¯p,m−µ(s,λi)+¯ua,p,m−µ(λi) = µ+1 ep,Mp m(λi). (s λi) − λi σ(L) p=1 m=0 µ=0 − ∈P P P P Now, we can express the Laplace transform of the solution of the initial-boundary- value problem (4.15) in terms of the excitation function v(t, x) and the initial value ya(x), both expressed by the canonical system of the operator L 4.3. Chapter summary 51

y(t, x) = Y (s, x)= L{ } P (λi) Mp m (4.25) V¯p,m−µ(s,λi)+¯ya,p,m−µ(λi) = µ+1 ep,Mp m(x,λi), (s λi) − λi σ(L) p=1 m=0 µ=0 − ∈P P P P where

V¯p,l(s,λi)=(V (s, ), ǫp,l( ,λ∗)), • • i (4.26) y¯ (λ )=(y ( ), ǫ ( ,λ∗)). a,p,l i a • p,l • i The concept of canonical systems of an operator and its adjoint introduced in Chapter 2 is obviously a suitable tool for expressing the solution of initial-boundary-value problems in a rather general way. This property is further exploited in the next chapter.

4.3 Chapter summary

One of the proper ways of handling of physical systems is to make use of differential equations. For one-dimensional (1D) systems, these are usually ordinary differential equations (ODEs), for multidimensional (MD) systems, however, partial differential equations (PDEs). This chapter is restricted to the simplest classification of differential equations and presents a brief introduction to the possible approaches for the solution representation of differential equations by help of holomorphic semigroups of operators. For this purpose, it was demonstrated first by the simple case of the ordinary state-space model where the associated semigroup is detailed via Jordan normal form of the matrix. For a rather general class of PDEs, the complete discussion and precise description of this method will be done in the next chapters. Also this chapter concentrates on the Laplace transformation with respect to the time variable. A key feature and principal importance of this approach for wide engineering practice is that it leads to algebraic methods of studying ordinary linear state-space system via transfer functions. These function are very important in classical theory and practice, since they have the advantage of being directly realizable. The application of this idea to the general case of multidimensional partial differential equations is a central contribution of the next chapters. 52 4. Mathematical Modelling of Physical Processes 53 5 Description of Multidimensional Systems

The previous chapter dealt with the analytical solution of initial-boundary-value problems, i.e. with a function in time and space and with its Laplace transform, respectively. The key moment was that the solution of the considered problems can be expressed in terms of the corresponding semigroup. This semigroup, in turn, can be represented by chains of eigenfunctions and associated functions generated by the spatial diﬀerential operator

L and its adjoint operator L†. These operators are unbounded and non-self-adjoint, in general. Together with the Laplace transformation applied to the time variable t, the L obtained decomposition of the semigroup gives the solution of the problem in the Laplace domain. In this chapter, we shift the focus from functions to multidimensional systems which produce such functions as an output signal. To this end, we consider an initial-boundary- value problem as the description of a multidimensional system with the excitation function v(t, x) as input signal and the solution y(t, x) as output signal. When such a system is implemented, it will produce the proper solution to any permissible input signal. This property holds not only for continuous-time, continuous-space-systems, but approx- imately also for discrete-time, discrete-space counterparts, typically used for computer- implementation. This chapter shows how to construct these discrete counterparts in such a way that the structure of the canonical system and hence the properties of the continuous system are preserved. This chapter introduces a physical modelling method for the simulation of initial- boundary-value problems. This method is a generalization of the functional transformation method (FTM) introduced by R. Rabenstein [Rab99] and later developed by L. Trautmann [RT02] and S. Petrausch [PR04]. Figure 5.1 illustrates the main concept of the application of the FTM to the initial-boundary-value problems. At ﬁrst, the corresponding PDE with its initial and boundary conditions is transformed by the Laplace transformation with respect to time. The Laplace transformation removes the time L derivatives, they are turned into multiplications of the directly transformed solution with the complex temporal frequency variable. The Laplace transformation includes the initial conditions as additive terms. Thus, the resulting equation contains only spatial derivatives of the transformed solution. Similar to Laplace transform, the transformation T for the spatial variables turns the spatial derivatives into multiplications with a spatial 54 5. Description of Multidimensional Systems frequency variable and it includes the boundary conditions as additive terms into the resulting algebraic equation. Therefore, successive application of both transformations yields a solution that can be calculated by purely algebraic operations in the multidimensional frequency domain, or by using terminology of R. Rabenstein in the multidimensional (MD) transfer function model (TFM). Then, MD TFM is discretized by well known time discretization methods from classical theory on continuous and discrete-time systems [GRS01, Kow93, Sch81]. Finally, after application of the inverse z-transformation 1 1 − and the inverse spatial transformation − we have the discretized approximative Z T solution in the time and space domain.

PDE L{·}PDE T{·} continuous IC, BC BC MD TFM

discretization

1 1 − − discrete T {·} recursive Z {·} discrete solution systems MD TFM

Figure 5.1: General procedure of the FTM for solving initial-boundary-value problems in form of PDEs with initial- (IC) and boundary (BC) conditions.

The FTM can be applied to problems with the simplest structure of the canonical system of the spatial operator, namely, it contains only the set of simple eigenvectors (no associated vectors). In contrast to the FTM the proposed multi-functional transformation (MFT) method can be applied to a more wide class of problems with unbounded and non- self-adjoint operators which are allowed to have a more general spectral structure with multiple eigenvectors (with associated vectors).

5.1 Multi-functional transformation (MFT)

In the following we presuppose a sectorial operator L : H H on the Hilbert space H, 7→ with compact resolvent operator R(s, L) and non-empty spectrum σ(L). The elements f of the Hilbert space H are deﬁned on a ﬁnite region Ω RN . Therefore, we will use more ⊂ precise notation of this space as H(Ω). In addition, E and E† are the canonical systems

of the prime operator L and its adjoint L†, respectively. 5.1. Multi-functional transformation (MFT) 55

5.1.1 Deﬁnition and properties

In this section the canonical system E† with the elements ǫp,m(x,λi∗) is used to introduce the multi-functional transformation (MFT) with respect to the space variable x RN . ∈ Deﬁnition 31 The Multi-Functional Transformation is the set of transformations T (λ ) , which are constructed by the canonical system E† of the adjoint spatial oper- {Tp,m i } ator L† and act on elements f H as ∈

(λ ) f( ) = f¯ (λ )=(f( ), ǫ ( ,λ∗)), (5.1) Tp,m i { • } p,m i • p,m • i where p = 1,...,P = P (λ ), m = 0,...,M are deﬁned for each λ σ(L) by the i i p i ∈ canonical system and ( , ) denotes the inner product in the Hilbert space H. · · From the deﬁnitions of the MFT and the adjoint operator it is obvious that

p,m(λi) Lf( ) =(Lf( ), ǫp,m( ,λi∗)) T { • } • • (5.2) =(f( ), L†ǫ ( ,λ∗)). • p,m • i Therefore, by deﬁnition of the canonical system (see (3.20)) we have for m = 0 : •

p,0(λi) Lf( ) =(f( ), L†ǫp,0( ,λi∗)) T { • } • • (5.3a) = λ (f( ), ǫ ( ,λ∗)) = λ f¯ (λ ), i • p,0 • i i p,0 i and for any m =1, , M : • ··· p

p,m(λi) Lf( ) = λif¯p,m(λi)+ f¯p,m 1(λi). (5.3b) T { • } − Thus, the following theorem is true:

Theorem 6 ﬀ

p,m(λi) Lf( ) = λif¯p,m(λi)+ f¯p,m 1(λi), m =0,...,Mp, (5.4) T { • } − where we use the agreement f¯ (λ ) 0, µ< 0. p,µ i ≡ Theorem 6 is a generalization of the so called ”diﬀerentiation theorem” well known from Laplace-, Fourier-, and other functional transformations. In the image domain, the eﬀect

of the operator L is expressed in the transformed domain by a multiplication with λi, which takes the role of the discrete frequency variable. The addition of the term with index m 1 is due to the general structure of the canonical system. Therefore, application − of the MFT to Lf( ) replaces the image of the operator L on the element f H by its • ∈ direct transformations in the frequency domain (see Figure 5.2). 56 5. Description of Multidimensional Systems

(λ ) f = f¯ (λ ) λ p,0(λi) Lf Tp,0 i { } p,0 i i T { }

(λ ) f = f¯ (λ ) λi + p,1(λi) Lf Tp,1 i { } p,1 i T { }

(λ ) f = f¯ (λ ) λi + p,m(λi) Lf Tp,m i { } p,m i T { }

Figure 5.2: The ”diﬀerentiation theorem” of the MFT.

5.1.2 Inverse MFT

To deﬁne the inverse transformation we assume that the canonical systems E and E† form a basis in the Hilbert space H. Then, the inverse transformation is given by the following series

Pi Mp 1 ¯ ¯ − fp,m(λi) = f( )= fp,m(λi)ep,Mp m( ,λi). (5.5) T { } • − • λi σ(L) p=1 m=0 X∈ X X

Remark 10 The MFT f¯ (λ ) of the element f H is equal to the corresponding coef- p,m i ∈ ﬁcient in its basis expansion by the canonical system E.

Next we use the following notations ¯ fp,m( ,λi)= fp,m(λi)ep,Mp m( ,λi), (5.6a) • − •

Mp f ( ,λ )= f ( ,λ ), (5.6b) p • i p,m • i m=0 X Pi f( ,λ )= f ( ,λ ). (5.6c) • i p • i p=1 X f ( ,λ ) is a projection on the subspace E (λ ), which is generated by the eigenvector p • i p i e ( ,λ ) and its associated vectors e ( ,λ ), m = 0, , M corresponding to the p,0 • i p,m • i ··· p eigenvalue λi (see Figure 5.3). 5.2. Application of the MFT method to initial-boundary-value problems with homogeneous boundary conditions 57

¯ f (λ ) λ ep,Mp ( ,λi) p,0 i i •

¯ fp,1(λi) λi + ep,Mp 1( ,λi) + − •

f¯ (λ ) λi + ep,Mp ( ,λi) + fp( ,λi) p,Mp i • •

Figure 5.3: The inverse MFT.

f( ,λ ) is a projection on the subspace E(λ ), generated by the eigenvectors e ( ,λ ) • i i p,0 • i and the associated eigenvectors e ( ,λ ), p = 1, ,P , m = 0, , M corresponding p,m • i ··· i ··· p to the eigenvalue λi. Now (5.5) can be rewritten in shorter way

f( )= f( ,λ ). • • i (5.7) λi σ(L) X∈ 5.2 Application of the MFT method to initial- boundary-value problems with homogeneous boundary conditions

5.2.1 Initial-boundary-value problem with homogeneous boundary conditions We consider the following non-homogeneous initial-boundary-value problem

∂ y(t, x)= Ly(t, x)+ v(t, x), t> 0, x Ω, ∂t ∈ y(t, x) = y (x), x Ω, (5.8) |t=0 a ∈ U y(t, x) =0, t> 0, x ∂Ω, µ =1, , m. µ ∈ ··· 58 5. Description of Multidimensional Systems

The set Ω denotes a domain in RN and ∂Ω its boundary. The functions v(t, ) and • y(t, ) are elements of H (or H(Ω)) for each ﬁxed value of t 0, where H is a Hilbert space • ≥ with the scalar product ( , ). The operator L is assumed to be a diﬀerential operator with · · respect to the N dimensional variable x which acts on properly elected elements of H. − We presuppose that ( L) is a sectorial operator on the Hilbert space H such that L has − a compact resolvent and the spectrum σ(L) is not empty. The condition U y(t, x) = 0 represents the collection of all homogeneous boundary conditions which are necessary to yield a unique solution of the initial-boundary-value problem. Now, we restrict the domain of the operator L to that set of such functions which satisfy the homogeneous boundary conditions generated by m linear independent forms U , µ = 1,...,m (see (2.13)-(2.14)), i.e. D(L) = y(t, ) H : U y(t, x) = 0, t> µ { • ∈ µ 0, x ∂Ω, µ =1,...,m . ∈ }

5.2.2 Laplace transformation

Applying the Laplace transform with respect to the variable t directly to the problem (5.8) gives

sY (s, x) = LY (s, x)+ V (s, x)+ ya(x), s C, x Ω, ∈ ∈ (5.9) U Y (s, x) =0, s C, x ∂Ω, µ =1, , m. µ ∈ ∈ ··· Here we assume that the Laplace transformation commutes with the operator L.

Lemma 9 If f(t, ) H, then f(t, ) H. • ∈ L{ • } ∈

Remark 11 Application of the Laplace transform to the solution of differential equations is one of the most important reasons for the widespread use of operational calculus (see [DP62]). Laplace transform reduces linear ordinary differential equations with constant coefficients to algebraic equations. In some special cases this technique can be applied for the solution of partial differential equations (two-dimensional Laplace transform [DP62]). Here we use the one-dimensional Laplace transform with respect to time, since its differentiation theorem is appropriate for the consideration of time derivatives and initial conditions at t = 0. For the space variable, we use the MFT, since its property according to Theorem 6 is appropriate for spatial operators with boundary conditions on finite regions. 5.2. Application of the MFT method to initial-boundary-value problems with homogeneous boundary conditions 59 5.2.3 Spatial transformation (MFT) Evaluating the MFT for m = 0 to (5.9) yields

sY¯p,0(s,λi)= λiY¯p,0(s,λi)+ V¯p,0(s,λi)+¯ya,p,0(λi), (5.10)

where ¯ Vp,0(s,λi)=(V (s, x), ǫp,0(x,λi∗)), (5.11)

y¯a,p,0(λi)=(ya(x), ǫp,0(x,λi∗)). Thus, the transform (λ ) of the output in the s domain is given by Tp,0 i − V¯ (s,λ ) y¯ (λ ) Y¯ (s,λ )= p,0 i + a,p,0 i . (5.12) p,0 i s λ s λ − i − i Evaluating the MFT for m> 0 to the boundary value problem (5.9) leads to

sY¯p,m(s,λi)= λiY¯p,m(s,λi)+ Y¯p,m 1(s,λi)+ V¯p,m(s,λi)+¯ya,p,m(λi), (5.13) − where ¯ Vp,m(s,λi)=(V (s, x), ǫp,m(x,λi∗)), (5.14)

y¯a,p,m(λi)=(ya(x), ǫp,m(x,λi∗). Hence, in this case the transform (λ ) of the output in the s domain is represented Tp,m i − as

V¯p,m(s,λi) y¯a,p,m(λi) Y¯p,m 1(s,λi) Y¯ (s,λ )= + + − p,m i s λ s λ s λ − i − i − i m (5.15) V¯p,m µ(s,λi) y¯a,p,m µ(λi) = − + − . (s λ )µ+1 (s λ )µ+1 µ=0 i i X − −

Figure 5.4a shows a block diagram of the initial-boundary-value problem after transformation with the MFT. The corresponding multidimensional transfer function model (MD TFM) is represented through the blocks 1 . Both, the transformed excitation function s λi − V¯p,m(s,λi) and the transformed initial conditiony ¯a,p,m(λi) act as input for MD TFM. And, since both transformed excitation function and transformed initial condition is input for equivalent transfer function blocks, they can be combined to a single input of the form

V¯p,m(s,λi)+¯ya,p,m(λi) (see Figure 5.4b). Also, we emphasize that the transformed output variable Y¯p,m(s,λi) can be calculated from the previously transformed output Y¯p,m 1(s,λi), − see (5.15). In Figure 5.4b dotted lines show the corresponding previously transformed outputs. 60 5. Description of Multidimensional Systems

¯ 1 Vp,0(s,λi) s λi −

V¯ (s,λ ) 1 p,1 i + s λi −

¯ Vp,m(s,λi) + 1 s λi + − 1 y¯a,p,0(λi) s λi −

1 y¯a,p,1(λi) + s λi −

1 y¯a,p,m(λi) + Y¯p,m(s,λi) s λi + −

Figure 5.4a: Block diagram of the MFT for calculating Y¯p,m(s,λi)

By collecting the blocks 1 together, the MD FTM illustrated in Figure 5.4b can be s λi − 1 replaced by equivalent block diagram with the transfer functions of the form µ (see (s λi) Figure 5.4c). −

5.2.4 Inverse MFT The resulting formula for the inverse MFT transformation is

Pi Mp ¯ Y (s, x)= Yp,m(s,λi)ep,Mp m(x,λi). (5.16) − λi σ(L) p=1 m=0 X∈ X X More precisely we have

Pi Mp m V¯p,m µ(s,λi)+¯ya,p,m µ(λi) − − Y (s, x)= µ+1 ep,Mp m(x,λi). (5.17) (s λi) − λi σ(L) p=1 m=0 µ=0 X∈ X X X −

Figure 5.5 shows a block diagram for calculating the projection Yp(s, x,λi). Remark 12 The MFT reduces the diﬀerential operators of boundary value problems to algebraic equations, similar to the Laplace transformation approach used for the time variable. 5.2. Application of the MFT method to initial-boundary-value problems with homogeneous boundary conditions 61

¯ 1 Y¯p,0(s,λi) Vp,0(s,λi)+¯ya,p,0(λi) s λi −

1 ¯ V¯p,1(s,λi)+¯ya,p,1(λi) Yp,1(s,λi) + s λi −

¯ 1 Vp,m(s,λi)+¯ya,p,m(λi) + Y¯p,m(s,λi) s λi −

Figure 5.4b: Block diagram of the MFT for calculating Y¯p,m(s,λi)

1 ¯ ¯ m+1 Yp,m(s,λi) Vp,0(s,λi)+¯ya,p,0(λi) (s λi) + −

1 ¯ 2 Vp,1(s,λi)+¯ya,p,1(λi) (s λi) + −

¯ 1 Vp,m(s,λi)+¯ya,p,m(λi) s λi + −

Figure 5.4c: Block diagram of the MFT for calculating Y¯p,m(s,λi)

Remark 13 Note, that the structure of Fig. 5.5 closely resembles the structure of the canonical system of the operator L according to (3.19).

5.2.5 Discretization of the MFT model

The representation of the multidimensional system in the form of Fig. 5.5 is possible due to the discrete spectrum of the operator L. The properties of this operator are reflected by the overall structure of the block diagram. The time derivative appears in the form of the Laplace frequency variable s. However, s is only found inside the blocks with 1 transfer functions of the form µ . To represent this continuous system by a discrete- (s λi) space, discrete-time system it is− thus sufficient to replace these blocks by discrete-time counterparts. For this purpose, we refer to well-known computationally efficient numerical algorithms from the theory of digital signal processing [GRS01, Kow93, Sch81, Sch90]. 62 5. Description of Multidimensional Systems

1 ¯ ep,Mp Y (s,λ , x) Vp,0 +¯ya,p,0 s+λi + p i

1 ¯ ep,Mp 1 Vp,1 +¯ya,p,1 + s+λi − +

¯ 1 Vp,Mp +¯ya,p,Mp ep,0 + + s+λi

Figure 5.5: Block diagram for calculating the projection Yp(s, x,λi)

5.2.5.1 Spatial discretization

As it has already been shown (5.15), the MFT turns an initial-boundary-value problem into the spatial frequency domain in the form of a discrete spectrum σ(L).

The discrete realization is performed as a parallel arrangement of outputs Yp(s, x,λi) particular for each eigenvalue λi (see (5.16)). Also, as it does not involve any operations with respect to the space variable x, spatial discretization is performed simply by evaluating (5.16) at the discrete spatial points x = xn of interest, with some spatial grid

xn n J , where J is an appropriately chosen set of indices. { } ∈ There are no restrictions with respect to a maximum spacing of the discrete grid points xn, since no discretization errors occur as in other numerical methods. As an extreme case, if for some reason the solution is only required at one single spatial grid point x1,

then Y (s, x1) can be calculated from (5.16) without any approximation by evaluating

Y (s, x1)= Yp(s, x1,λi). (5.18)

λi σ(L) p=1 X∈ X

5.2.5.2 Time discretization

1 In the realizing structure time dependence is represented through the blocks µ . We (s λi) can apply well known time discretization methods from classical theory on continuous− and discrete-time systems [GRS01, Kow93, Sch81]. As an example consider the impulse invariant transformation (IIT)

1 µ 1 Pµ(z, zi) T − (5.19) (s λ )µ −→ (z z )µ − i − i 5.2. Application of the MFT method to initial-boundary-value problems with homogeneous boundary conditions 63

λiT with zi = e , containing the temporal sampling interval T. Here Pµ(z, zi) is a polynomial of order µ 1,µ> 1, which can be calculated recursively by −

z dPµ 1(z, zi) Pµ(z, zi)= (z zi) − (µ 1)Pµ 1(z, zi) (5.20) −µ 1 − dz − − − − starting with P (z, z )= z (see [Sch81]). Then we have for the transformation of the 1 i Tp,m output (5.15)

m ¯ (d) µ Pµ+1(z, zi) ¯ (d) (d) Yp,m(z,λi)= T Vp,m µ(z,λi)+¯ya,p,m µ(λi) . (5.21) (z z )µ+1 − − µ=0 i X −

(d) V¯p,m(z,λi) denotes the z-transform (see [GRS01]) of the sampled time function corre-

sponding to V¯p,m(s,λi) according to

(d) 1 V¯ (z,λ )= − V¯ (s,λ ) (5.22) p,m i Z{L { p,m i }|t=kT } (d) and similar for Y¯p,m(z,λi). 1 Here, means z-transformation, − means inverse Laplace transformation, Z{·} L {·} and means sampling of a time function with temporal sampling interval T and ·|t=kT discrete time index k. IIT gives exact results for the response with respect to the initial conditions. Also other conventional discretization schemes like e.g. bilinear transformation can be applied (see [Sch90]).

5.2.6 Inverse z transformation − Finally, inverse z-transformation turns (5.21) into a diﬀerence equation for the output sequence 1 (d) y¯ (k,λ )= − Y¯ (z,λ ) . (5.23) p,m i Z { p,m i } The discrete-time approximation of the solution is given by a parallel arrangement

of all blocks realizing these diﬀerence equations for each eigenvalue λi weighted with the corresponding eigenfunction or associated eigenfunction

Pi Mp (d) (d) y (kT, xn)= y¯p,m(k,λi)ep,Mp m(xn,λi). (5.24) − λi σ(L) p=1 m=0 X∈ X X

(d) y (kT, xn) is an approximation to the exact solution of the initial-boundary value (d) (d) problem, since the sequencesy ¯p,m(k,λi) give approximations to Y¯p,m(z,λi) due to discrete- time processing. 64 5. Description of Multidimensional Systems

Finally, also the structure of the discrete system is determined by the canonical system of the operator L. This canonical system is obviously the backbone of the procedure to obtain a discrete-time, discrete-space approximation of the multidimensional system for the solution of an initial-boundary-value problem.

5.2.7 Approximation

The impulse-invariant transformation produces exactly a sampled version of the continuous system when excited with an impulse or more general by impulse train.

However, since the inﬁnite sum given in (5.24) over λi can not be realized on the computer it must be truncated to a ﬁnite sum with the following criterion

λ σ(L) : Re λ > σ , (5.25) i ∈ { i} − 0

where σ0 is a properly chosen positive constant (see also Section 3.1.5). Finally, we get

Pi Mp (d) (d) yσ (kT, xn)= y¯p,m(k,λi)ep,Mp m(xn,λi). 0 − (5.26) λi σ(L) p=1 m=0 ∈ Re{λXi}>−σ0 X X

The resulting system can be implemented directly on the computer.

5.3 Application of the MFT method to initial- boundary-value problems with nonhomogeneous boundary conditions

5.3.1 Initial-boundary-value problems with nonhomogeneous boundary conditions

For the given domain Ω in RN with boundary ∂Ω we consider the following initial- boundary-value problem

∂ y(t, x)= l(y(t, x)+ v(t, x), x Ω, t> 0, ∂t ∈ y(t, x) = y (x), x Ω, (5.27) |t=0 a ∈ U y(t, x) = φ (t), t> 0, x ∂Ω, µ =1, , m. µ µ ∈ ··· 5.3. Application of the MFT method to initial-boundary-value problems with nonhomogeneous boundary conditions 65 where l is a diﬀerential form (for details see Chapter 2), the functions v(t, ),t > 0 and • y(t, ), t 0 are elements of the Hilbert space H. The linear forms U represent the m • ≥ µ nonhomogeneous boundary conditions, while the initial condition is given by ya(x). Let us consider the following function z (t, x)= y(t, x) z (t, x), where the function 0 − φ zφ(t, x) is so chosen that

U z (t, x) = φ (t), t> 0, x ∂Ω, µ =1, , m. (5.28) µ φ µ ∈ ···

Therefore, z0(t, x) must satisfy the homogeneous boundary conditions of the form

U z (t, x) =0, t> 0, x ∂Ω, µ =1, , m. (5.29) µ 0 ∈ ···

With y(t, x)= zφ(t, x)+ z0(t, x) we rewrite the problem (5.27) in the form ∂ z0(t, x)= l(z0(t, x)) + v(t, x) z˙φ(t, x)+ l(zφ(t, x)), x Ω, t> 0, ∂t − ∈ (5.30a) z (0, x)= y (x) z (0, x), x Ω, 0 a − φ ∈ with the corresponding set of boundary conditions for the function z0(t, x)

U z (t, x) =0, t> 0, x ∂Ω, µ =1, , m, (5.30b) µ{ 0 } ∈ ··· Finally, using the notations of the Chapter 2, we have the problem in operator form ∂ z (t, x)= L z (t, x)+ v(t, x) z˙ (t, x)+ l(z (t, x)), x Ω, t> 0, ∂t 0 0 0 − φ φ ∈ z (0, x)= y (x) z (0, x), x Ω, (5.31) 0 a − φ ∈ U z (t, x) =0, t> 0, x ∂Ω, µ =1, , m, µ{ 0 } ∈ ···

where the operator L0 is given by the formula L0z0(t, x)= l(z0(t, x)) and its domain is

D(L )= z (t, ) H : U z (t, x) =0, t> 0, x ∂Ω, µ =1, , m . 0 { 0 • ∈ µ 0 ∈ ··· } To construct the MFT, the operator L has to be a sectorial operator L : H H on 0 0 7→ the Hilbert space H, with compact resolvent operator R(s, L0) and non-empty spectrum

σ(L0). Now, for simplicity, we consider the case of an one-dimensional spatial variable and a diﬀerential operator of the form n ∂ν l(y(t, x)) = p (x) y(t, x), (5.32) ν ∂xν ν=0 X where the functions p (x), ν =0, , n are continuously diﬀerentiable up to the order ν ν ··· over the given interval Ω = [a, b]. The Hilbert space H is the Lebesgue space L2([a, b]), b with the scalar product (f,g)= f(x)(g(x))∗dx, f,g H. a ∈ R 66 5. Description of Multidimensional Systems

5.3.2 Laplace transformation

The application of LT to the PDE (5.31) gives the equation

sZ (x, s)= L Z (x, s)+ V (x, s)+ y (x) sZ (x, s)+ l Z (x, s) (5.33) 0 0 0 a − φ { φ } with the boundary conditions of the form U Z (x, s) = 0, µ = 1, 2, , m. And also µ 0 ··· we have transformed boundary values for function Z (x, s) : U Z (x, s) = Φ (s), µ = φ µ φ µ 1, 2, , m. ···

5.3.3 Spatial transformation (MFT)

The corresponding eigenproblem for the prime operator L0 can be summarized in the form

σ(L )= λ the set of eigenvalues 0 { i}i − L0 : (5.34a)  E = e (x, λ ) the canonical system.  { p,m i }i,p,m −  The associated eigenproblem for the adjoint operator L0† is summarized as follows

σ(L0† )= λi∗ i the set of eigenvalues L† : { } − (5.34b) 0  E† = ǫ (x, λ∗) the canonical system.  { p,m i }i,p,m −

Remark 14 It is important to note that all elements ep,m(x, λi) from the canonical system

E should satisfy the homogeneous boundary conditions Uµ ep,m(x, λi) =0, µ =1,...,m (as the elements from the domain of the operator L ). And all elements ǫ (x, λ ) from 0 p,m i∗ the adjoint canonical system E† should satisfy the corresponding adjoint homogeneous

boundary conditions V ǫ (x, λ∗) =0, µ =1,..., 2n m. µ p,m i − The canonical system of the adjoint operator L0† is used as kernels of the required spatial transformation:

(λ ) Z (s, x) = Z¯ (s,λ ) Tp,m i { 0 } 0,p,m i b (5.35) =(Z0(s, x), ǫp,m(x, λi∗)) = Z0(x, s)ǫp,m∗ (x, λi∗)dx. Za Using the properties of the MFT it can be shown that when we apply the MFT to the

image of the operator L0 the result is represented directly by the transform of the Z¯0 or

by linear combinations of transforms of Z¯0 with corresponding indices 5.3. Application of the MFT method to initial-boundary-value problems with nonhomogeneous boundary conditions 67

p,m(λi) L0Z0(s, x) = λiZ¯0,p,m(s,λi)+ Z¯0,p,m 1(s,λi), m =0,...,Mp, (5.36) T { } − where we use the agreement Z¯ (s,λ ) 0, µ< 0. 0,p,µ i ≡ For the MFT of the diﬀerential form l we have the following expression

p,m(λi) l Zφ(s, x) = λiZ¯φ,p,m(s,λi)+ Z¯φ,p,m 1(s,λi)+ Pi(η,ζp,m), m =0,...,Mp, T { } − (5.37) with the agreement Z¯ (s,λ ) 0, µ< 0. The term P (η,ζ ) is a bilinear form φ,p,µ i ≡ p,m of the variables η = (Zφ(s)a, Zφ(s)b) and ζ = (ǫp,m,a(λi∗), ǫp,m,b(λi∗)) (see (2.21)). And

Zφ(s)a, Zφ(s)b, ǫp,m,a(λi∗), ǫp,m,b(λi∗) are the following vectors with n components

Zφ(s, x) Zφ(s, x)  ∂Zφ(s,x)   ∂Zφ(s,x)  Z (s) = ∂x , Z (s) = ∂x , (5.38) φ a   φ b    ·   ·   n−1   n−1   ∂ Zφ(s,x)   ∂ Zφ(s,x)   n−1   n−1   ∂x x=a  ∂x x=b    

ǫp,m(x, λi∗) ǫp,m(x, λi∗) ∗ ∗  ∂ǫp,m(x,λi )   ∂ǫp,m(x,λi )  ∂x ∂x ǫ (λ∗)= , ǫ (λ∗)= . (5.39) p,m,a i   p,m,b i    ·   ·   n−1 ∗   n−1 ∗   ∂ ǫp,m(x,λi )   ∂ ǫp,m(x,λi )   n−1   n−1   ∂x x=a  ∂x x=b     Now the boundary value problem under consideration is represented with help of the constructed spatial transformation (λ ) in the following form Tp,m i for the case m =0, , M • ··· p

sZ¯0,p,m(s,λi) λiZ¯0,p,m(s,λi) Z¯0,p,m 1(s,λi)= − − − = V¯ (s,λ )+¯y (λ ) sZ¯ (s,λ )+ λ Z¯ (s,λ )+ (5.40) p,m i a,p,m i − φ,p,m i i φ,p,m i

+ Z¯φ,p,m 1(s,λi)+ Pi(η,ζp,m), − where we use both previous agreements Z¯ (s,λ ) 0 and Z¯ (s,λ ) 0 for µ< 0. 0,p,µ i ≡ φ,p,µ i ≡ Since z (t, x)= y(t, x) z (t, x) then the Laplace transform leads us 0 − φ Z (s, x)= Y (s, x) Z (s, x) (5.41) 0 − φ and the spatial transformation yields Tp,m Z¯ (s,λ )= Y¯ (s,λ ) Z¯ (s,λ ). (5.42) 0,p,m i p,m i − φ,p,m i 68 5. Description of Multidimensional Systems

Hence, the algebraic equation (5.40) for the transformed output Y¯p,m(s,λi) can be rewritten as for the case m =0, , M • ··· p

sY¯p,m(s,λi) λiY¯p,m(s,λi) Y¯p,m 1(s,λi)= − − − (5.43) V¯p,m(s,λi)+¯ya,p,m(λi)+ Pi(η,ζp,m) with the agreement Y¯ (s,λ ) 0, µ< 0. p,µ i ≡ The ﬁnal formula for calculating transformed function Y¯p,m can be written as

V¯p,m(s,λi) y¯a,p,m(λi) Y¯p,m 1(s,λi) Pi(η,ζp,m) Y¯p,m(s,λi)= + + − + λi + s λi + s λi + s λi + s

m (5.44) V¯p,m q(s,λi) y¯a,p,m q(λi) Pi(η,ζp,m) = − + − + . (λ + s)q+1 (λ + s)q+1 (λ + s)q+1 q=0 i i i X The ﬁrst two terms of the expression (5.44) represent the transformed excitation func-

tion Vp,m(s,λi) and the transformed initial conditions ya,p,m(λi). Next, we will show that the transformed boundary conditions are also included into the representation of the

solution Y¯p,m(s,λi) through the bilinear form Pi(η,ζp,m).

Here, we remind the reader that the bilinear form Pi(η,ζp,m) can be expressed in terms of the prime and the adjoint boundary linear forms (see 2.31) as

Pi(η,ζp,m)= U1V2n + + UmV2n m+1 + Um+1V2n m + + U2nV1, (5.45) ··· − − ··· where U Y (s, x) = Φ (s), µ = 1, 2, , m is a set of transformed nonhomogeneous µ µ ··· boundary conditions for the solution, and V ǫ (x, λ∗) = 0, µ = 1, 2, , 2m m is µ p,m i ··· − a corresponding set of homogeneous boundary conditions for the elements of the adjoint canonical system. Therefore,

Pi(η,ζp,m) = Φ1(s)V2n + + Φm(s)V2n m+1, (5.46) ··· − where the forms V , µ =2m m , 2n are obtainable from the formulas (2.30a). µ − ··· Next we use the following notation Φ¯ p,m(s,λi)= Pi(η,ζp,m) for the transformed boundary conditions. Figure 5.6 shows a block diagram of the initial-boundary-value problem after transformation with the MFT. Note, that not only the transformed initial conditions and the transformed excitation function but also the transformed boundary conditions act as input for similar transfer function blocks. Remark 15 To construct the MFT model in the case of nonhomogeneous boundary conditions we have to ﬁnd the eigenfunctions and associated functions for the spatial operator with homogeneous conditions. It is also important to note that the boundary functions are involved into the construction of the spatial transformation. 5.3. Application of the MFT method to initial-boundary-value problems with nonhomogeneous boundary conditions 69 Transformed Transformed Transformed Excitation Function Initial Condition Boundary Conditions

¯ 1 1 ¯ 1 Vp,0 y¯a,p,0 Φp,0 s+λi s+λi s+λi

¯ V¯ p,1 + y¯a,p,1 + Φp,1 +

1 1 1 s+λ s+λi s+λi i

¯ V¯ p,m + y¯a,p,m + Φp,m +

1 1 1 s+λi s+λi s+λi

Y¯p,m(s,λi)

Figure 5.6: Block diagram of the MFT for calculating Y¯p,m(s,λi)

5.3.4 Inverse MFT

Using the inverse MFT we can obtain the solution of the problem in the Laplace domain as P Mp 1 ¯ ¯ − Yp,m(s,λi) = Y (s, x)= Yp,m(s,λi)ep,Mp m(x, λi). (5.47) T { } − λi σ(L0) p=1 m=0 ∈X X X Thus the solution in the Laplace domain is

P Mp m V¯p,m q +¯ya,p,m q + Φ¯ p,m q − − − Y (s, x)= q+1 ep,Mp m(x, λi). (5.48) (s + λi) − λi σ(L0) p=1 m=0 q=0 ∈X X X X

Figure 5.7 shows a block diagram for calculating the projection Yp(s, x,λi). Taking the P summation over the index p we receive the projection Y (s,λi, x)= Yp(s,λi, x) summa- p=1 tion over all spectrum leads to the solution in the Laplace domain YP(s, x)= Y (s,λi, x). λi P 5.3.5 Discretization of the MFT model

The discretization of the solution (5.48) of the initial-boundary-value problem (5.27) with nonhomogeneous boundary conditions can be performed by the same procedure as for the initial-boundary-value problem (5.8) with homogeneous boundary conditions. 70 5. Description of Multidimensional Systems

1 ¯ ¯ ep,Mp Y (s,λ , x) Vp,0 +¯ya,p,0 + Φp,0 s+λi + p i

1 ¯ ¯ ep,Mp 1 Vp,1 +¯ya,p,1 + Φp,1 + s+λi − +

¯ ¯ 1 Vp,Mp +¯ya,p,Mp + Φp,Mp ep,0 + + s+λi

Figure 5.7: Block diagram for calculating the projection Yp(s, x,λi)

5.4 Application of the MFT method to general vector initial-boundary-value problems

5.4.1 Notation

For two given integers M and N we consider functions h which are deﬁned on a subset A of RN

h : A RN CM . (5.49) ⊂ → For x A we have the image ∈ h(x) = [h (x), h (x), , h (x)]T . (5.50) 1 2 ··· M

We will only consider functions h, whose components hν are elements of the space H deﬁned by

H= L2(A), (5.51)

where L2(A) is the usual space of quadratically integrable functions on A. We deﬁne the space = HM = H H H, (5.52) H × ×···× and instead of (5.49)-(5.52) we can say

h . (5.53) ∈ H

5.4.2 General vector initial-boundary value problem

Let us consider the following homogeneous Cauchy problem in the Hilbert space H 5.4. Application of the MFT method to general vector initial-boundary-value problems 71

Cy˙ (t, x)= Ly(t, x)+ v(t, x), x A,t> 0, ∈ y(0, x)= y (x), x A, (5.54) a ∈ U y(t, x) = 0, x ∂A, t> 0, µ =1,...,m. µ ∈ The matrix C is a quadratic matrix of dimension M M and the operator L is assumed × to be a diﬀerential operator with respect to the N dimensional variable x which acts − on properly elected elements of . We presuppose that ( L) is a C sectorial operator H − − 1 on the Hilbert space such that L has a compact C resolvent RC(s, L)=(sC L)− H − − and the spectrum σ (L) is not empty. The conditions U y(t, x) = 0, x ∂A, t> C µ ∈ , µ ,...,m 0 = 1 represents the collection of all homogeneous boundary conditions which are necessary to yield a unique solution of the initial-boundary-value problem. Note that the matrix C is now allowed to be singular, i.e. we admit

RankC M. (5.55) ≤ 1 For RankC = M, the matrix C− exists and the problem can be reduced to an equivalent problem where C is not present anymore. The given function v(t, x) has to satisfy the condition

v(t, ) (5.56) • ∈ H for each ﬁxed t> 0. The desired solution y(t, x) must satisfy

y(t, ) , ˙y(t, ) (5.57) • ∈ H0 • ∈ H for each ﬁxed t > 0. Additionally we assume that y(t, ) as well as ˙y(t, ) depends • • continuously on t > 0. Here we have introduced the subspace of which contains H0 H only all those functions which satisfy a set of linear homogeneous boundary conditions, speciﬁed by the linear operator Uµ

U y(t, x) = 0, t> 0, x ∂A, µ =1,...,m. (5.58) µ ∈ 5.4.3 Laplace transformation The first step in obtaining the solution of the problem consists in the application of the Laplace transformation with respect to the time-variable t. The definition of the one-sided Laplace transformation and its differentiation theorem are given below

∞ st y(t, x) = Y(s, x)= y(t, x)e− dt, L{ } 0 (5.59) ˙y(t, x) = sY(s, xR) y(0, x), L{ } − 72 5. Description of Multidimensional Systems where the complex temporal frequency variable is denoted by s. The application of (5.59) to the initial-boundary-value problem (5.54) results in

sCY(s, x)= LY(s, x)+ V(s, x)+ Cy (x), a (5.60) Uµ Y(s, x) = 0. Here we have assumed that L and commute. L

5.4.4 Spatial transformation (MFT) To construct the MFT we need the corresponding C eigenfunctions and C associated − − functions of the operators L and L† (i.e. the C canonical systems ep,m(x,λi) p,m,i and − H { } ǫ (x,λ∗) ) with respect to the matrix C and C , respectively. Here we assume { p,m i }p,m,i that the corresponding C spectrum and C resolvent sets are not empty. Eigen- and − − associated functions satisfy the following equations, where both arguments have been omitted [λC L]e = 0, − p,0 (5.61a) p =1,...,P (λ), and [λC L]ep,m = Cep,m 1, − − (5.61b) m =1,...,Mp, where P (λ) is the maximum number of C eigenfunctions corresponding to the considered − C eigenvalue λ, and M is the maximum number of C associated eigenfunctions corre- − p − sponding to each C eigenfunction e . Note that all elements e of the C canonical − p,0 p,m − system belong to the domain of the operator L. Hence, they have to satisfy the boundary conditions (5.58), too.

Analogously, for the adjoint operator L† we have the problem

H [λ∗C L†]ǫ = 0, − p,0 (5.62a) p =1,...,P (λ∗) and

H H [λ∗C L†]ǫp,m = C ǫp,m 1, − − (5.62b) m =1,...,Mp.

We remark as before that all elements ǫp,m of the adjoint canonical system belong to the domain of the operator L†. This means that they have to satisfy the adjoint homogeneous boundary conditions which can be obtained from the deﬁnition of the adjoint operator L†.

The adjoint operator L† is the unique operator with domain D(L†) containing those ⊂ H elements z for which there exists an element y D(L) such that the following equality ∈ holds (Ly, z)=(y, L†z), (5.63) 5.4. Application of the MFT method to general vector initial-boundary-value problems 73 where ( , ) denotes the inner product in the Hilbert space . · · H Finally, the required spatial transformation (MFT) of the element h is given by ∈ H (λ ) h(x) = h¯ (λ )=(Ch(x), ǫ (x,λ∗)), (5.64) Tp,m i { } p,m i p,m i where p =1,...,Pµ, m =1,...,Mp are deﬁned by the canonical system. Biorthogonality. It has already been shown (see Remark 19 in the Appendix) that after appropriate scaling of ǫ the canonical systems e (x,λ ) and ǫ (x,λ∗) are p,Mp { p,m i } { p,m i } biorthogonal with respect to the weighting matrix C

(Cep,m(x,λi), ǫl,n(x,λj∗)) = δp,lδi,jδm,Mp n. (5.65) −

Diﬀerentiation theorem. The scalar product (Lh(x), ǫp,m(x,λi∗)) can be expressed by the MFT of h(x). The relation between this scalar product and (λ ) h(x) has Tp,m i { } a similar form as the well-known diﬀerentiation theorems of the Laplace-, Fourier-, or related transforms. The exact form of this relation follows from (5.62a)-(5.64) as for the case m =0, , M • ··· p

(Lh(x), ǫp,m(x,λi∗)) = λi(Ch(x), ǫp,m(x,λi∗))+(Ch(x), ǫp,m 1(x,λi∗)) − (5.66) = λih¯p,m(λi)+ h¯p,m 1(λi), − where we use the agreement h¯ (λ ) 0, µ< 0. p,µ i ≡ With the MFT the problem (5.60) is transferred into the equation ¯ ¯ sYp,m(s,λi)=(LY(s, x), ǫp,m(x,λi∗)+ Vp,m(s,λi)+¯ya,p,m(λi). (5.67) Using the diﬀerentiation theorem we can write the transformation of the output signal in the form

V¯p,m(s,λi) y¯a,p,m(λi) Y¯p,m 1(s,λi) Y¯ (s,λ )= + + − p,m i s λ s λ s λ − i − i − i m (5.68) V¯p,m q(s,λi) y¯a,p,m q(λi) = − + − , (s λ )q+1 (s λ )q+1 q=0 i i X − − where

¯ Vp,m(s,λi)=(V(s, x), ǫp,m(x,λi∗)), (5.69a)

y¯a,p,m(λi)=(Cya(x), ǫp,m(x,λi∗)). (5.69b)

Thus, with the MFT the boundary value problem (5.60) is transformed into the spatial frequency domain by the same procedure as in previous sections. 74 5. Description of Multidimensional Systems

5.4.5 Inverse MFT The inverse MFT transformation is given by the following series

Pi Mp ¯ Y(s, x)= Yp,m(s,λi)ep,Mp m(x,λi). (5.70) − λi σ(L) p=1 m=0 X∈ X X Finally, the output signal can be represented in the following form

Y(s, x)= Yint(s, x)+ Yext(s, x), (5.71) where

Pi Mp m ¯ Yint(s, x)= Gq(s,λi)ya,p,m q(λi)ep,Mp m(x,λi), (5.72a) − − λi σ(L) p=1 m=0 q=0 X∈ X X X

Pi Mp m ¯ ¯ Yext(s, x)= Gq(s,λi)Vp,m q(x,λi)ep,Mp m(x,λi) (5.72b) − − λi σ(L) p=1 m=0 q=0 X∈ X X X and

1 G¯ (s,λ )= . (5.73) q i (s λ )q+1 − i

Yint(s, x) is the contribution of the initial conditions, while Yext(s, x) represents the response to the source function v(t, x).

5.4.6 Discretization of the MFT model The discretization of the solution (5.70) of the general vector initial-boundary value problem (5.54) with nonhomogeneous boundary conditions can be performed by the same procedure as for the scalar initial-boundary-value problem (5.8) with homogeneous boundary conditions.

5.5 Chapter summary

This chapter presented some new elements of operational calculus. Here we have introduced the multi-functional transformation (MFT) for the space variable and established its basic properties such as diﬀerentiation theorem and inverse MFT. The introduced transformation is addressed to the spatial operators with the boundary conditions given on ﬁnite space regions. 5.5. Chapter summary 75

One of the reasons for the development of the multi-functional transformation was that its application together with Laplace transformation turns the initial-boundary-value problems into the block diagram model representations which are used widely in engineering practice and which are available for the well known computationally eﬃcient numerical methods from digital signal processing. In this chapter the precise description and the step-by-step procedure of the application of the MFT method were given for the diﬀerent cases of the boundary conditions (homogeneous and non-homogeneous), as well for the general vector initial-boundary value case. 76 5. Description of Multidimensional Systems 77 6 Examples of the MFT Simulations

In the previous more theoretically oriented chapters we have already shown that in addition to the eigenfunctions, the so called associated functions can appear to form the canonical system of the operator. However, to construct examples with associated functions, is not so straightforward. The well-known heat ﬂow and wave equations with classical boundary conditions lead to canonical systems with only eigenfunctions as elements. Therefore, the main aim of this chapter is to apply the MFT methods to examples with more complicated eigenstructure, namely, with associated elements. In this chapter the MFT is applied to spatially 1-D initial-boundary-value problems given in form of scalar and vector PDEs. The corresponding theoretical background concerning the application of the MFT method has already been derived in Chapter 5.

6.1 Heat ﬂow equation

To illustrate the proposed approach we consider the well-known heat ﬂow equation in the region = R Ω= (t, x) : t> 0, 0

y(0, x)= ya(x),

U1(y)= y(t, 0)=0, (6.1b)

U (y)= y (t, x) y (t, x) =0, 2 x |x=0 − x |x=1 where y (x) L ([0, 1]) and y (t, x)= ∂ y(t, x). a ∈ 2 x ∂x Upon integrating (6.1a) with respect to the spatial variable x over the interval [0, 1] we obtain that

1 1 ∂ ∂2 y(t, x)dx = y(t, x)dx, ∂t ∂x2 Z Z 0 0 (6.2) 1 ∂ y(t, x)dx = y (t, x) y (t, x) . ∂t x |x=1 − x |x=0 Z0 78 6. Examples of the MFT Simulations

Thus, the boundary condition of the form yx(t, x) x=0 yx(t, x) x=1 = 0 is equivalent to 1 | − | the integral condition y(t, x)dx = ϕ(t) = Const. 0 Problems with suchR conditions arise very often from plasma physic. For example, the diﬀusion process of charged particles in a turbulent, strongly magnetized plasma belongs to this class of problems. In our example, the problem (6.1a)-(6.1b) is heat propagation in a thin rod in which the law ϕ(t) of variation of the total quantity of heat in the rod is equal to constant, while its left end point is kept at zero temperature. The initial heat distribution is given by ya(x).

6.1.1 Prime and adjoint operators

∂2 1 1 The spatial operator L = ∂x2 acts on the subspace H = AC ([0, 1]) (see Example 3 on page 26) of the Hilbert space H = L2([0, 1]) with the usual scalar product (f,g) = 1 f(x)(g(x))∗dx. Its domain is given by 0 R D(L)= y(t, ) H1 : y(t, 0)=0, y (t, x) y (t, x) =0 . (6.3a) · ∈ x |x=0 − x |x=1 n o ∂2 Then, it can be shown that the adjoint operator is L† = ∂x2 with the domain

1 D(L†)= z(t, ) H : z (t, 1)=0, z(t, x) z(t, x) =0 . (6.3b) · ∈ x |x=0 − |x=1 n o

Remark 16 Note, in spite of the fact that the linear differential forms l and l† are identical, i.e. l = l†, the operators L and its adjoint L† are different, since their domains are different D(L) = D(L†). Therefore, the operator L is non-self-adjoint. 6

6.1.2 Eigenproblem

The solution of the eigenproblem for the prime operator L can be summarized in the form

λ =0, e (x, λ )= c x, e (x, λ ) = c 1 0 1,0 0 0 k 1,0 0 k 0 √3  λ πi 2,  i = (2 )  −  1 L :  e1,0(x, λi)= ci sin(2πix), e1,0(x, λi) = ci . (6.4a)  k k √2   2 e (x, λ )= ci x cos(2πix), e (x, λ ) = c 1 1+ (4πi) 1,1 i − 4πi k 1,1 i k i 4πi 6   q  i =1, 2, 3,  ···   6.1. Heat ﬂow equation 79

The associated eigenproblem for the adjoint operator L† leads to the solution

λ∗ =0, ǫ (x, λ∗)= d , ǫ (x, λ∗) = d 0 1,0 0 0 k 1,0 0 k 0  2 λ∗ = (2πi) ,  i −   ǫ (x, λ )= d cos(2πix), ǫ (x, λ ) = d 1 L† :  1,0 i∗ i 1,0 i∗ i √2 . (6.4b)  k k  2 di √(4πi) 6 ǫ (x, λ∗)= (1 x) sin(2πix), ǫ (x, λ∗) = d − 1,1 i − 4πi − k 1,1 i k i 4πi√6    i =1, 2, 3,  ···  Similar to the method which was considered in Example 5 on page (see also an alternative approach in [LLMP05]) it can be shown that the operator L is sectorial and has a − compact resolvent operator.

6.1.3 Biorthogonality

It can be shown that for the obtained elements from the canonical systems the following property holds 2 (e1,m(x, λi), ǫ1,n(x, λw∗ )) = ai,nδi,wδm,1 n, (6.5) − where

1 2 c0d0 a =(e (x, λ ), ǫ (x, λ∗)) = c d xdx = , 0,0 1,0 0 1,0 0 0 0 2 Z0 1 2 cidi 2 cidi a =(e (x, λ ), ǫ (x, λ∗)) = x cos (2πix)dx = − , (6.6) i,0 1,1 i 1,0 i − 4πi 16πi Z0 1 2 cidi 2 cidi a =(e (x, λ ), ǫ (x, λ∗)) = (1 x) sin (2πix)dx = − . i,1 1,0 i 1,1 i − 4πi − 16πi Z0

By setting the coeﬃcients c =1, i =0, 1,..., d = 2 and d = 16πi, i =1, 2,... we get i 0 i − the biorthogonal canonical systems of the prime and adjoint operators (see Theorem 1).

6.1.4 The general solution

Finally, we can apply the MFT-method to represent the solution in the Laplace domain as

y¯a,1,0 y¯a,1,1 y¯a,1,0 Y (s, x)= e1,1(x, λi)+ e1,0(x, λi)+ 2 e1,0(x, λi) , (6.7) (s λi) (s λi) (s λi) λi σ(L) X∈ − − − 80 6. Examples of the MFT Simulations where

y¯a,1,m(λi)= ya(x)ǫ1,m(x, λi)dx. (6.8) Z0

6.1.5 MFT Simulation

After applying the discretization methods described in Chapter 5 the solution (6.7) of the heat ﬂow deﬁned by the initial-boundary-value problem (6.1a)-(6.1b) can be simulated in the computer. The MFT model has been discretized by impulse invariant transformation in the temporal frequency domain and the direct discretization of the spatial domain (see Chapter 5, section 5.2.5 ). The temporal sampling interval is T = 0.0005, the spatial sampling interval is h =1/70, the total number of eigenvalues N = 100 and the number of time iteration is 100. The initial condition is given by y (x)=10x(x 1) sin(3πx). a − The results of the simulation with the MFT is illustrated in Figure 6.1.

2.5

1.5

0.5

−0.5

−1

−1.5

Temperature y(t,x) −2 6 5 1 4 0.8 3 −4 0.6 x 10 2 0.4 1 0.2 0 0 t, time x, space

Figure 6.1: The heat propagation y(t, x) in a thin rod. 6.2. Heat ﬂow through a wall 81

6.2 Heat ﬂow through a wall

Consider the system of partial diﬀerential equations in the following domain = R Ω= G +× (t, x) : t> 0, 0

ri(t, x)+ Dxu(t, x)=0, (6.9a)

cDtu(t, x)+ Dxi(t, x)=0, (6.9b)

where i(t, x) denotes the scalar heat ﬂux and u(t, x) is a temperature, with heat capacity c

and r as the inverse of the thermal conductivity. Dt and Dx describe the partial derivatives with respect to the time and space variables, respectively. The quantities in this example are assumed to be normalized, i.e. their physical dimension is unity. Rewrite the given equations in matrix form as

CDty(t, x)= Ly(t, x), (6.10a)

where the operator L is given as

D r L = A + BD = − x − (6.10b) x 0 D " − x # and 0 r 1 0 0 0 A = − , B = − , C = , " 0 0 # " 0 1 # " c 0 # − (6.10c) u(t, x) y(t, x)= . " i(t, x) # Note that the matrix C is singular. The initial and boundary conditions are given by

y1(x) y(0, x)= ya(x)= . (6.10d) " y2(x) #

and

U1(y)= u(t, 0)=0, (6.10e) U (y)= i(t, 0) i(t, 1)=0, 2 − respectively. To apply the MFT we have to ﬁnd the corresponding canonical systems for the prime

and adjoint operators L and L†. 82 6. Examples of the MFT Simulations

6.2.1 Adjoint operator First we determine the adjoint operator:

1 1 (Ly, z)= zH Lydx = zH (A + BD )ydx =(y, (AH BH D )z)+ zH By x=1, (6.11) x − x |x=0 Z0 Z0 where z (t, x) z(t, x)= 1 . (6.12) " z2(t, x) # Equation (6.11) complies with the deﬁnition of the adjoint operator only if the expression zH By x=1 vanishes, |x=0 zH (t, x)y(t, x) x=1 = zH (t, 1)y(t, 1) zH (t, 0)y(t, 0) = |x=0 − (6.13) = z (t, 1)u(t, 1)+ z (t, 1)i(t, 1) z (t, 0)u(t, 0) z (t, 0)i(t, 0)=0. 1∗ 2∗ − 1∗ − 2∗ Using the boundary conditions (6.10e) this equation can be rewritten as

[z∗(t, 1) z∗(t, 0)]i(t, 0) + z∗(t, 1)u(t, 1)=0. (6.14) 2 − 2 1 The conditions for z (t, x) and z (t, x) at the boundary x = 0, 1 have to be chosen such 1 2 { } that this expression becomes zero. Thus, the adjoint operator L† is given by

H H Dx 0 L† = A B D = , (6.15) − x r D " − x # subject to the boundary conditions of the form

V1(z)= z1(t, 1)=0, (6.16) V (z)= z (t, 0) z (t, 1)=0. 2 2 − 2 6.2.2 Eigenvalue problems Prime operator. The eigenvalue problem of the operator L is [λC L]e (x, λ)= 0, − p,0 (6.17) p =1,...,P (λ),

1 2 T where ep,0(x, λ) = [fp,0(x, λ), fp,0(x, λ)] and P (λ) is the unknown number of all linear independent eigenvectors corresponding to the eigenvalue λ. Let us rewrite equation (6.17) in the following form

0 0 f 1 D r f 1 p,0 − x − p,0 = 0, (6.18) λc 0 f 2 − 0 D f 2 " # " p,0 # " − x # " p,0 # 6.2. Heat ﬂow through a wall 83

1 2 then we have the following linear system for fp,0 and fp,0

′ ′ 1 f 1 + rf 2 =0 2 fp,0 p,0 p,0 fp,0 = − r . (6.19) ′  1 2 →  1′′ 1 λcfp,0 + fp,0 =0 f λrcf =0   p,0 − p,0 By the superprimes we have denoted the derivatives with respect to the spatial variable x. Let us introduce the temporary notation β = λrc. From the theory of linear diﬀer- − 1 2 ential equations we get from (6.19) the formulas for fp,0(x, λ) and fp,0(x, λ)

1 j√βx j√βx fp,0(x, λ)= c1e + c2e− . (6.20)  2 1 j√βx j√βx f (x, λ)= [ c j√βe + c j√βe− ]  p,0 r − 1 2 Due to the boundary conditions (6.10e), we obtain that

1 fp,0(0,λ)=0 : c2 = c1 − β =2πi, i =0, 1, . (6.21)  2 2 j√β j√β ⇒ ··· f (0,λ) f (1,λ)=0 : 2=e + e−  p,0 − p,0 p

Finally, the eigenvalues λi are (2πi)2 λ = , i =0, 1, , (6.22) i − rc ··· and the corresponding eigenvectors are given by

1 rx fp,0(x, λ0) λ0 = 0 : e1,0(x, λ0)= 2 = c0 , (6.23a) " fp,0(x, λ0) #  1  −   and

1 sin(2πix) fp,0(x, λi) λi, i =1, 2,... : e1,0(x, λi)= 2 = ci . (6.23b) " fp,0(x, λi) #  2πi  −r cos(2πix)   Therefore, we conclude that P (λi)= Pi = 1.

Adjoint operator. The eigenvalue problem of the operator L† has the form

T [λ˜C L†]ǫ (x, λ˜)= 0, (6.24) − 1,0 ˜ 1 ˜ 2 ˜ T where ǫ1,0(x, λ) = [z1,0(x, λ), z1,0(x, λ)] . By analogy with the previous problem, we rewrite (6.24) as

0 λc˜ z1 D 0 z1 1,0 x 1,0 = 0, (6.25) 0 0 z2 − r D z2 " # " 1,0 # " − x # " 1,0 # 84 6. Examples of the MFT Simulations and obtain the linear system of diﬀerential equations

′′ 2 1′ 2 ˜ 2 λcz˜ z =0 z1,0 λrcz1,0 =0 1,0 − 1,0 − ′ . (6.26) ′ 2  1 2 →  1 z1,0 rz1,0 z1,0 =0 z =  −  1,0 r Now we introduce the temporary notationβ = λrc˜ . And from (6.26) we have the 1 ˜ 2 ˜ − formulas for z1,0(x, λ) and z1,0(x, λ)

2 ˜ j√βx j√βx z1,0(x, λ)= c1e + c2e− . (6.27)  1 1 j√βx j√βx z (x, λ˜)= [c j√βe c j√βe− ]  1,0 r 1 − 2 With the boundary conditions (6.16), we have the following conditions

1 ˜ j√β j√β z1,0(1, λ)=0 : c1e c2e− =0 − (6.28)  2 2 j√β j√β z (0, λ˜) z (1, λ˜)=0 : c (1 e )+ c (1 e− )=0  1,0 − 1,0 1 − 2 − and thus 

j√β j√β e e− det − =0. (6.29) j√β j√β 1 e 1 e− − −

From the equation (6.29), we derive that √β =2πi, i =0, 1, and c2 = c1. Thus, the (2πi)···2 eigenvectors corresponding to the eigenvalues λ˜ = λ∗ = are i i − rc 1 0 z1,0(x, λ0∗) λ0∗ = 0 : ǫ1,0(x, λ0∗)= 2 = d0 , (6.30a) " z1,0(x, λ0∗) # " 1 # where d0 is an arbitrary nonzero constant and

2πi 1 − sin(2πix) z1,0(x, λi∗) r λi∗, i =1, 2,... : ǫ1,0(x, λi∗)= 2 = di , (6.30b) " z1,0(x, λi∗) # " cos(2πix) # where now di, i =1, 2,... are constants not equal to zero. For the index i = 0 we have the following property

1 H (Ce1,0(x, λ0), ǫ1,0(x, λ0∗)) = ǫ1,0(x, λ0∗) Ce1,0(x, λ0)dx Z 0 (6.31) 1 0 0 rx rc = c0d0 0 1 dx = c0d0 . " c 0 # " 1 # 2 Z0 h i − 6.2. Heat ﬂow through a wall 85

2 By setting c0 = 1 and d0 = rc we normalize the last expression. Also, it can be easily shown that for each i = 0 the eigenvector e (x, λ ) of the prime 6 1,0 i operator L is orthogonal to any eigenvector ǫ1,0(x, λw∗ ) of the adjoint operator L† with respect to the weighting matrix C

1 H (Ce1,0(x, λi), ǫ1,0(x, λw∗ )) = ǫ1,0(x, λw∗ ) Ce1,0(x, λi)dx =0. (6.32) Z0

Remark 17 To construct the canonical systems E, E† of the prime operator L and its

adjoint L†, respectively, the ﬁrst step is to obtain the corresponding C eigenvectors − e1,0(x, λi) and ǫ1,0(x, λi∗). If they are orthogonal (Ce1,0(x, λi), ǫ1,0(x, λi∗)) = 0 it means

that there exist associated vectors e1,1(x, λi) and ǫ1,1(x, λi∗). The procedure continues until the (M + 1) th step the C associated vector ǫ (x, λ∗) will be not orthogonal to the 1 − − 1,M1 i C eigenvector e (x, λ ) (or equivalently the C associated vector e (x, λ∗) will be not − 1,0 i − 1,M1 i orthogonal to the C eigenvector ǫ (x, λ )) (see Figure 3.1 and Remark 9). − 1,0 i

Finally, we conclude that the canonical systems E, E† of both operators L and L† possess

valid associated vectors for all eigenvalues λi except for λ0.

6.2.3 Associated vectors

1 2 T Prime operator. The associated vector e1,1 = [f1,1, f1,1] of order 1 can be obtained from the equation [λ C L]e = Ce . (6.33) i − 1,1 1,0 With (6.10c) we have

0 0 f 1 D r f 1 0 0 f 1 1,1 − x − 1,1 = 1,0 , (6.34) λ c 0 f 2 − 0 D f 2 c 0 f 2 " i # " 1,1 # " − x # " 1,1 # " # " 1,0 # or ′ ′ f 1 f 1 + rf 2 =0 2 1,1 1,1 1,1 f1,1 = − r . (6.35) ′  1 2 1 →  1′′ 1 1 λicf1,1 + f1,1 = cf1,0 f λ rcf = rcf   1,1 − i 1,1 − 1,0 Substituting (6.23b) into (6.35), we have 

1′ 2 f1,1 f1,1 = − r . (6.36)  ′′ f 1 + (2πi)2f 1 = rc sin(2πix)  1,1 1,1 − 1 From the theory of linear nonhomogeneous diﬀerential equations we get that f1,1(x, λi) is given as

1 f1,1(x, λi)= c1 cos(2πix)+ c2 sin(2πix)+ fpart(x, λi), (6.37) 86 6. Examples of the MFT Simulations where

fpart(x, λi)= c3x cos(2πix)+ c4x sin(2πix). (6.38)

Substituting (6.38) into second equation in (6.36) we obtain

2 2c3(2πi) sin(2πix)+2c4(2πi) cos(2πix) c3(2πi) x cos(2πix) − − − (6.39) c (2πi)2x sin(2πix)+(2πi)2c x cos(2πix)+(2πi)2c x sin(2πix)= rc sin(2πix). − 4 3 4 − We derive

2c (2πi) sin(2πix)+2c (2πi) cos(2πix)= rc sin(2πix). (6.40) − 3 4 − rc Therefore, c4 = 0 and c3 = 4πi and we obtain the associated function as

e1,1(x, λi)=

c1 cos(2πix)+ c2 sin(2πix)+ c3x cos(2πix), (6.41) = .  1 2πic cos(2πix) 2πic sin(2πix)+ c cos(2πix) 2πic x sin(2πix)  − r 2 − 1 3 − 3   Since it should satisfy the boundary conditions (6.10e), it follows

1 f1,1(0,λi)=0 : c1 =0 . (6.42)  f 2 (0,λ ) f 2 (1,λ )=0 : 2πic + c =2πic + c  1,1 i − 1,1 i 2 3 2 3

In this case we can set c2 = 0. Finally, we have

r 4πi x cos(2πix) e1,1(x, λi)= − . " 1 cos(2πix) 2πix sin(2πix) # 4πi − 1 2 T Adjoint operator. The associated vector ǫ1,1 = [z1,1, z1,1] of order 1 we can obtain from the equation T T [λ∗C L†]ǫ = C ǫ , (6.43) i − 1,1 1,0 or equivalently

0 λ c z1 D 0 z1 0 c z1 i∗ 1,1 x 1,1 = 1,0 . (6.44) 0 0 z2 − r D z2 0 0 z2 " # " 1,1 # " − x # " 1,1 # " # " 1,0 # The corresponding system is

′′ 2 1′ 2 2 2 2 λ∗cz z = cz z1,1 λi∗rcz1,1 = rcz1,0 i 1,1 − 1,1 1,0 − − ′ . (6.45) ′ 2  1 2 →  1 z1,1 rz1,1 z1,1 =0 z =  −  1,1 r   6.2. Heat ﬂow through a wall 87

Substituting (6.30b) into (6.45), we have

2′′ 2 2 z1,1 + (2πi) z1,1 = rcdi cos(2πix) − . (6.46) 2′  1 z1,1  z1,1 = r

Thus, the solution has a form

2 z1,1(x, λi∗)= c1 cos(2πix)+ c2 sin(2πix)+ zpart(x, λi∗), (6.47) where

zpart(x, λi∗)= c3x cos(2πix)+ c4x sin(2πix). (6.48)

Substituting (6.48) into the second equation in (6.46) we obtain

2c (2πi) sin(2πix)+2c (2πi) cos(2πix)= rcd cos(2πix). (6.49) − 3 4 − i Thus, c = 0 and c = rcdi and the associated function results as 3 4 − 4πi

ǫ1,1(x, λi∗)=

(6.50) 1 2πic cos(2πix) 2πic sin(2πix)+ c sin(2πix)+2πic x cos(2πix) = r 2 − 1 4 4 .   c1 cos(2πix)+ c2 sin(2πix)+ c4x sin(2πix)   Since it should satisfy the boundary conditions (6.16), we have

1 z1,1(1,λi∗)=0 : c2 = c4 − . (6.51)  2 2 z (0,λ∗) z (1,λ∗)=0 : c = c  1,1 i − 1,1 i 1 1

In this case we set c1= 0. Finally, we get

1 4πi 2πi(x 1) cos(2πix) + sin(2πix) ǫ (x, λ∗)= d − . 1,1 i i   r (1 x) sin(2πix) − 4πi −   6.2.4 Biorthogonality

It can be shown that for the obtained vectors with indices i, κ =1, 2,... and m, n =0, 1 the following property holds

2 (Ce1,m(x, λi), ǫ1,n(x, λκ∗ )) = ai,nδi,κδm,1 n, (6.52) − 88 6. Examples of the MFT Simulations with

1 2 rc 2 rc a =(Ce (x, λ ), ǫ (x, λ∗)) = d x cos (2πix)dx = d , i,0 1,1 i 1,0 i −4πi i −16πi i Z 0 (6.53) 1 2 rc 2 rc a =(Ce (x, λ ), ǫ (x, λ∗)) = d (1 x) sin (2πix)dx = d . i,1 1,0 i 1,1 i −4πi i − −16πi i Z0

Hence, we have obtained the biorthogonal canonical systems of the operators L and L† with the normalization factors d = 16πi (see Remark 17). i − rc

6.2.5 The general solution

The corresponding system for the prime operator L can be summarized in the form

rx λ =0, e (x, λ )= ,  0 1,0 0    1  −     2  λi = (2πi) , i =1, 2, 3,  − ··· L :  (6.54a)   sin(2πix)  e1,0(x, λi)= ,  2πi  − cos(2πix)  r   r x cos(2 πix)  4πi  e1,1(x, λi)= − .  1  " 4πi cos(2πix) 2πix sin(2πix) #  −   The associated eigenproblem for the adjoint operator L† is summarized as follows

0 λ0∗ =0, ǫ1,0(x, λ0∗)= ,   2   rc      2  λi∗ = (2πi) , i =1, 2, 3,  − ··· L† :  2πi (6.54b)  − sin(2πix)  16πi r  ǫ1,0(x, λi∗)= rc , − " cos(2πix) #   1  16πi 4πi 2πi(x 1) cos(2πix) + sin(2πix)  ǫ1,1(x, λi∗)= rc − .  −  r   (1 x) sin(2πix)  − 4πi −     6.3. Telegraph equation 89

The components of the general solution now take the form

Y(s, x)= 1 1 = y¯a,1,0(λi)e1,1(x, λi)+ y¯a,1,1(λi)e1,0(x, λi) (s λi) (s λi) (6.55) Xλi − − 1 + y¯ (λ )e (x, λ ) , (s λ )2 a,1,0 i 1,0 i − i where

y¯a,p,m(λi)=(Cya(x), ǫp,m(x,λi∗)). (6.56) And 0 0 y1(x) 0 Cya = = . (6.57) " c 0 # " y2(x) # " cy1(x) #

6.2.6 MFT simulation

After applying the discretization methods described in Chapter 5 the solution (6.55) of the heat ﬂow deﬁned by the initial-boundary-value problem (6.10a),(6.10d) and (6.10e) can be simulated in the computer. The MFT model has been discretized by impulse invariant transformation in the temporal frequency domain and the direct discretization of the spatial domain (see Chapter 5, section 5.2.5). The initial condition is given by u(0, x)= x cos(2pix). And the material parameters, such as heat capacity c and the inverse of the thermal conductivity r, has been chosen for sand-lime brick, i.e. c =1.6 MJ/(m3K) and 1/r =0.79 W/(mK). Temporal sampling interval is T =0.001, spatial sampling interval is h = 1/70, the total number of eigenvalues N = 150, the number of time iteration is 150. The results of the simulation with the MFT is illustrated in Figures 6.2a-6.2b.

6.3 Telegraph equation

Consider the system of partial diﬀerential equations in the domain = R Ω= (t, x) : G + × { t> 0, 0

lDti(t, x)+ ri(t, x)+ Dxu(t, x)= v1(t, x), (6.58a)

cDtu(t, x)+ gu(t, x)+ Dxi(t, x)= v2(t, x), (6.58b)

where i(t, x) denotes the current and u(t, x) is the voltage. The electrical parameters are

denoted by l,c,r,g. Dt and Dx describe the partial derivatives with respect to the time and space variable, respectively. 90 6. Examples of the MFT Simulations

0.8

0.6

0.4

0.2

0 Temperature u(t,x)

−0.2

−0.4 0 0.2 0.4 1 0.8 0.6 0.6 0.8 0.4 0.2 x, space 1 0 −3 t, time x 10

Figure 6.2a: The ﬁrst component of the solution.

Rewrite the given equations in matrix form as

CDty(t, x)= Ly(t, x)+ v(t, x), (6.59a)

where the operator L is given as

D r L = A + BD = − x − (6.59b) x g D " − − x # and 0 r 1 0 0 l A = − , B = − , C = , " g 0 # " 0 1 # " c 0 # − − (6.59c) u(t, x) v (t, x) y(t, x)= , v(t, x)= 1 . " i(t, x) # " v2(t, x) # Note that the matrix C is invertible. The initial and boundary conditions are given by

y(0, x)= ya(x) (6.59d)

and U1(y)= u(t, 0) u(t, 1)=0, − (6.59e) U2(y)= i(t, 0)=0, 6.3. Telegraph equation 91

1.5

0.5

−0.5

−1

−1.5 Heat flux i(t,x)

−2

−2.5

−3 0 0.2 0.4 1 0.6 0.8 0.8 0.4 0.6 1 0 0.2 −3 x, space t, time x 10

Figure 6.2b: The second component of the solution.

respectively. To apply the MFT we have to ﬁnd the corresponding canonical systems for the prime

and adjoint operators L and L†.

6.3.1 Adjoint operator

First we determine the adjoint operator:

1 1 (Ly, z)= zH Lydx = zH (A + BD )ydx =(y, (AH BH D )z)+ zH By x=1, (6.60) x − x |x=0 Z0 Z0 where z (t, x) z(t, x)= 1 . (6.61) " z2(t, x) # Equation (6.60) complies with the deﬁnition of the adjoint operator only if the expression zH By x=1 = zH y x=1 vanishes, |x=0 − |x=0 z(t, x)H y(t, x) x=1 = zH (t, 1)y(t, 1) zH (t, 0)y(t, 0) = |x=0 − (6.62) = z (t, 1)u(t, 1)+ z (t, 1)i(t, 1) z (t, 0)u(t, 0) z (t, 0)i(t, 0)=0. 1∗ 2∗ − 1∗ − 2∗ 92 6. Examples of the MFT Simulations

Using the boundary conditions (6.59e) this equation can be rewritten as

[z∗(t, 1) z∗(t, 0)]u(t, 1)+ z∗(t, 1)i(t, 1)=0. (6.63) 1 − 1 2

With corresponding conditions for z∗(t, x) and z∗(t, x) at the boundary x = 0, 1 this 1 2 { } expression becomes zero. Thus, the adjoint operator L† is given by

H H Dx g L† = A B D = − , (6.64) − x r D " − x # subject to the boundary conditions of the form

V1(z)= z1(t, 0) z1(t, 1)=0, − (6.65) V2(z)= z2(t, 1)=0.

6.3.2 Eigenvalue problems Prime operator. The eigenvalue problem of the operator L is [λC L]e (x, λ)= 0, − p,0 (6.66) p =1,...,P (λ),

0 λl f 1 D r f 1 p,0 − x − p,0 = 0, (6.67) λc 0 f 2 − g D f 2 " # " p,0 # " − − x # " p,0 # 1 2 then we have the following linear system for fp,0 and fp,0

2 1′ 2 1′ 2 λlfp,0 + fp,0 + rfp,0 =0 fp,0 +(r + λl)fp,0 =0

 1 2′ 1 →  2′ 1  λcfp,0 + fp,0 + gfp,0 =0  fp,0 +(g + λc)fp,0 =0 ′ (6.68) f 1 2 − p,0   fp,0 = (r+λl) . →  ′′ f 1 (r + λl)(g + λc)f 1 =0  p,0 − p,0 By the superprimes we have denoted the derivatives with respect to the spatial variable x. Let us introduce the temporary notation β = (r + λl)(g + λc). From the theory of − 1 linear diﬀerential equations we get from (6.68) the explicit expressions for fp,0(x, λ) and 2 fp,0(x, λ)

1 j√βx j√βx fp,0(x, λ)= c1e + c2e− . (6.69)  2 1 j√βx j√βx f (x, λ)= [ c j√βe + c j√βe− ]  p,0 (r+λl) − 1 2  6.3. Telegraph equation 93

Due to the boundary conditions (6.59e), we obtain that

2 fp,0(0,λ)=0 : c2 = c1 β =2πi, i =0, 1, . (6.70)  1 1 j√β j√β ⇒ ··· f (0,λ) f (1,λ)=0 : 2=e + e−  p,0 − p,0 p Finally, the eigenvalues λi can be derived from the equation

2 (2πi) +(g + λic)(r + λil)=0, (6.71) and turn out to be complex

(gl + cr) (gl + cr)2 4cl(gr + (2πi)2) λi1,2 = − ± − . (6.72) p 2cl The corresponding eigenvectors are given by

1 1 r f1,0(x, λ01) λ01 = − : e1,0(x, λ01)= 2 = , (6.73a) l " f1,0(x, λ01) #  x    1 1 g f1,0(x, λ02) λ02 = − : e1,0(x, λ02)= 2 = , (6.73b) c " f1,0(x, λ02) #  0  and  

1 cos(2πix) f1,0(x, λi1,2) λi1,2 : e1,0(x, λi1,2)= 2 = . (6.73c) " f1,0(x, λi1,2) #  2πi sin(2πix)  r+λil   Therefore, we also conclude that P (λ )= P =1, i 0. i1,2 i ≥ Adjoint operator. The eigenvalue problem of the operator L† has a form

T [λ˜C L†]ǫ (x, λ˜)= 0, (6.74) − 1,0 ˜ 1 ˜ 2 ˜ T where ǫ1,0(x, λ) = [z1,0(x, λ), z1,0(x, λ)] . By analogy with the previous problem, we rewrite (6.74) as

0 λc˜ z1 D g z1 1,0 x − 1,0 = 0, (6.75) λl˜ 0 z2 − r D z2 " # " 1,0 # " − x # " 1,0 # and then obtain a linear system

λcz˜ 2 z1′ + gz2 =0 z1′ (g + λc˜ )z2 =0 1,0 − 1,0 1,0 1,0 − 1,0  λlz˜ 1 z2′ + rz1 =0 →  z2′ (r + λl˜ )z1 =0  1,0 − 1,0 1,0  1,0 − 1,0 1′ (6.76) 2 z1,0   z1,0 = (g+λc˜ ) . →  ′′ z1 (r + λl˜ )(g + λc˜ )z1 =0  1,0 − 1,0  94 6. Examples of the MFT Simulations

Again we use the temporary notation β = (r + λl˜ )(g + λc˜ ). And from (6.76) we have 1 ˜ 2 ˜ − the expressions for z1,0(x, λ) and z1,0(x, λ) 1 ˜ j√βx j√βx z1,0(x, λ)= c1e + c2e− . (6.77)  2 1 j√βx j√βx z (x, λ˜)= [c j√βe c j√βe− ]  1,0 (g+λc˜ ) 1 − 2 With boundary conditions (6.65), we obtain that 1 ˜ 1 ˜ j√β j√β z1,0(0, λ) z1,0(1, λ)=0 : c1(1 e )+ c2(1 e− )=0 − − − (6.78)  2 j√β j√β z (1, λ˜)=0 : c e c e− =0  1,0 1 − 2 and thus  j√β j√β 1 e 1 e− det − − =0. (6.79) j√β j√β e e− −

From the equation (6.79), we derive that √β =2πi, i =0, 1, and c = c . Thus, the ··· 2 1 eigenvalues λ˜i satisfy the equation

2 (2πi) +(g + λ˜ic)(r + λ˜il)=0. (6.80) ˜ Thereby we have checked the obvious property λi = λi∗ (compare (6.80) with (6.71)). The corresponding eigenvectors are given by

1 1 r z1,0(x, λ0∗1) 2 λ0∗1 = − : ǫ1,0(x, λ0∗1)= 2 = , (6.81a) l " z1,0(x, λ0∗1) # l  0 

c  1 g z (x, λ∗ ) 2 rc gl 1,0 02 − λ0∗2 = − : ǫ1,0(x, λ0∗2)= 2 = − , (6.81b) c " z1,0(x, λ0∗2) # c  x 1  − and  

1 cos(2πix) z1,0(x, λi∗1,2) λ∗ : ǫ (x, λ˜ )= = . (6.81c) i 1,2 1,0 i 2 2πi " z1,0(x, λi∗1,2) #  − ∗ sin(2πix)  g+λi 1,2c For the index i = 0 we have the following property 

(Ce1,0(x, λ01,2), ǫ1,0(x, λ0∗1,2))=1. (6.82) For each i = 0 the eigenvector e (x, λ ) of the prime operator L is orthogonal to any 6 1,0 i1,2 eigenvector ǫ1,0(x, λw∗ 1,2) of the adjoint operator L† with respect to the weighting matrix C

(Ce1,0(x, λi1,2), ǫ1,0(x, λw∗ 1,2))=0. (6.83)

Finally, with the Remark 17 we conclude that the canonical systems E, E† of both operators L and L† have associated vectors corresponding to the eigenvalues λ , i = 0. i1,2 6 6.3. Telegraph equation 95

6.3.3 Associated vectors

1 2 T Prime operator. The associated vector e1,1 = [f1,1, f1,1] of order 1 can be obtained from the equation [λ C L]e = Ce . (6.84) i − 1,1 1,0 With (6.59c) we have

0 λ l f 1 D r f 1 0 l f 1 i 1,1 − x − 1,1 = 1,0 , (6.85) λ c 0 f 2 − g D f 2 c 0 f 2 " i # " 1,1 # " − − x # " 1,1 # " # " 1,0 # or

2 1′ 2 2 1′ 2 2 λilf1,1 + f1,1 + rf1,1 = lf1,0 f1,1 +(r + λil)f1,1 = lf1,0

 1 2′ 1 1 →  2′ 1 1  λicf1,1 + f1,1 + gf1,1 = cf1,0  f1,1 +(g + λic)f1,1 = cf1,0 ′ (6.86) lf 2 f 1 2 1,0− 1,1  f1,1 =  (r+λil) . →  ′′ ′ f 1 +(r + λ l)(g + λ c)f 1 = c(r + λ l)f 1 lf 2  − 1,1 i i 1,1 i 1,0 − 1,0 Substituting (6.73c) into (6.86), we have

1′ 2 2πil sin(2πix) f1,1 f = 2 1,1 (r+λil) − (r+λil)  2 1′′ 1 (2πi) . (6.87)  f (r + λil)(g + λic)f = [ c(r + λil)+ l ] cos(2πix)  1,1 − 1,1 − r+λil  = [c(r + λ l)+ l(g + λ c)] cos(2πix) − i i   1 From the theory of linear nonhomogeneous diﬀerential equations we get that f1,1(x, λi) is given by

1 f1,1(x, λi)= c1 cos(2πix)+ c2 sin(2πix)+ fpart(x, λi), (6.88) where

fpart(x, λi)= c3x cos(2πix)+ c4x sin(2πix). (6.89)

Substituting (6.89) into second equation in (6.87) we obtain

2c (2πi) sin(2πix)+2c (2πi) cos(2πix) c (2πi)2x cos(2πix) c (2πi)2x sin(2πix) − 3 4 − 3 − 4 − (r + λ l)(g + λ c)c x cos(2πix) (r + λ l)(g + λ c)c x sin(2πix)= (6.90) − i i 3 − i i 4 = [c(r + λ l)+ l(g + λ c)] cos(2πix). − i i Using (6.71) we derive

2c (2πi) sin(2πix)+2c (2πi) cos(2πix)= [c(r + λ l)+ l(g + λ c)] cos(2πix). (6.91) − 3 4 − i i 96 6. Examples of the MFT Simulations

[c(r+λil)+l(g+λic)] − Therefore, c3 = 0 and c4 = 4πi and the associated function is

e1,1(x, λi)=

c1 cos(2πix)+ c2 sin(2πix)+ c4x sin(2πix), (6.92) = 1 2πil sin(2πix) .  +(c12πi c4) sin(2πix) (c2 c4x)2πi cos(2πix)  (r+λil) (r+λil) − − −   It should satisfy the boundary conditions (6.59e), thus

2 f1,1(0,λi)=0 : c2 =0 . (6.93)  f 1 (0,λ ) f 1 (1,λ )=0 : c = c  1,1 i − 1,1 i 1 1

In this case we put c1= 0. Finally, we have

c4 x sin(2πix)

e1,1(x, λi)= ,  1 2πil  c42πi x cos(2πix) + [ c4] sin(2πix) (r+λil) − r+λil −     c(r+λil) l(g+λic) − − where c4 = 4πi . 1 2 T Adjoint operator. The associated vector ǫ1,1 = [z1,1, z1,1] of order 1 can be obtained from the equation T T [λ∗C L†]ǫ = C ǫ , (6.94) i − 1,1 1,0 or equivalent form

1 1 1 0 λi∗c z1,1 Dx g z1,1 0 c z1,0 2 − 2 = 2 . (6.95) λ∗l 0 z − r D z l 0 z " i # " 1,1 # " − x # " 1,1 # " # " 1,0 # The corresponding system is

2 1′ 2 2 1′ 2 2 λ∗cz z + gz = cz z (g + λ∗c)z = cz i 1,1 − 1,1 1,1 1,0 1,1 − i 1,1 − 1,0  1 2′ 1 1 →  2′ 1 1 λ∗lz z + rz = lz z (r + λ∗l)z = lz  i 1,1 − 1,1 1,1 1,0  1,1 − i 1,1 − 1,0 1′ 2 (6.96) 2 z1,1+cz1,0  z1,1 = ∗  (g+λi c) . →  1′′ 1 1 2′ z (r + λ∗l)(g + λ∗c)z = l(g + λ∗c)z cz  1,1 − i i 1,1 − i 1,0 − 1,0 Substituting (6.81c) into (6.96), we have

1′ 2 z1,1 2πic sin(2πix) z1,1 = ∗ ∗ 2 (g+λi c) − (g+λi c)  1′′ 1 (2πi)2 . (6.97)  z1,1 (r + λi∗l)(g + λi∗c)z1,1 = [ l(g + λi∗c)+ c ∗ ] cos(2πix)  (g+λi c)  − − = [c(r + λ∗l)+ l(g + λ∗c)] cos(2πix) − i i   6.3. Telegraph equation 97

Thus, the solution has the form

1 z1,1(x, λi∗)= c1 cos(2πix)+ c2 sin(2πix)+ zpart(x, λi∗), (6.98) where

zpart(x, λi∗)= c3x cos(2πix)+ c4x sin(2πix). (6.99)

Substituting (6.99) into second equation in (6.97) we obtain

2c (2πi) sin(2πix)+2c (2πi) cos(2πix)= [c(r + λ∗l)+ l(g + λ∗c)] cos(2πix)(6.100) − 3 4 − i i

[c(r+λ∗l)+l(g+λ∗c)] − i i Thus, c3 = 0 and c4 = 4πi and the associated function is

ǫ1,1(x, λi∗)=

c1 cos(2πix)+ c2 sin(2πix)+ c4x sin(2πix), (6.101) = 1 2πic sin(2πix) . −  (g+λ∗c) (g+λ∗c) +(c4 c12πi) sin(2πix)+(c2 + c4x)2πi cos(2πix)  i i −   It should satisfy the boundary conditions (6.65), thus

1 1 z1,1(0,λi∗) z1,1(1,λi∗)=0 : c1 = c1 − . (6.102)  2 z (1,λ∗)=0 : c = c  1,1 i 2 − 4

In this case we put c1 = 0. Finally, we have

c (x 1) sin(2πix) 4 − ǫ1,1(x, λ∗)= , i  1 2πic  ∗ − ∗ + c4 sin(2πix)+ c42πi(x 1) cos(2πix) (g+λi c) (g+λi c) −     [c(r+λ∗l)+l(g+λ∗c)] − i i where c4 = 4πi .

6.3.4 Biorthogonality

In analogy with the previous examples, it can be shown that for the obtained vectors with indices i, κ =1, 2,... and m, n =0, 1 the following property holds

(Ce1,m(x, λi), ǫ1,n(x, λκ∗ )) = δi,κδm,1 n (6.103) − holds. Therefore, we have obtained the biorthogonal canonical systems of the operators

L and L†. 98 6. Examples of the MFT Simulations

6.3.5 The general solution The components of the general solution now take the form

1 1 Yint(s, x)= y¯a,1,0(λi)e1,1(x, λi)+ y¯a,1,1(λi)e1,0(x, λi) (s λi) (s λi) Xλi − − (6.104) 1 + y¯ (λ )e (x, λ ) , (s λ )2 a,1,0 i 1,0 i − i

1 1 Yext(s, x)= V¯1,0(λi)e1,1(x, λi)+ V¯1,1(λi)e1,0(x, λi) (s λi) (s λi) Xλi − − (6.105) 1 + V¯ (λ )e (x, λ ) . (s λ )2 1,0 i 1,0 i − i 6.4 Chapter summary

This chapter is devoted to the demonstration of the proposed approach and the presented theoretical results by three examples. These examples are of tutorial as well as of illus- trative interest. This chapter starts first with the simple case of the heat flow equation (not with classical boundary conditions) and stops with a system of partial differential equations. All key stages of the introduced MFT method were underlined,

strong mathematical formalization of the problems in terms of the operator theory; • formulation of the corresponding spectral problems for prime and adjoint operators; • search for their eigenvalues, eigenfunctions and associated eigenfunctions; • determination of the required multi-functional transformation; • applying the MFT method to calculate the ”input-output” block representation. • Further examples are contained in several publications on the application of the FTM (see [TR03], [RT02], [PR04]). The canonical systems considered previously were not of the general structure as in this contribution. However, the examples for various kinds of initial-boundary-value problems listed below serve also as an illustration of special cases of the general theory presented here. Various problems arising in physical modelling of musical instruments are covered in [TR03]. The examples include vibrations of strings, membranes, and soundboards. The telegraph equation is used as an example in [RT02]. A procedure for the determination of eigenvalues and eigenvectors in a more general context is given in [PR04]. 99 7 Conclusions

This work has presented a panorama of modern operator theory and its application. This theory has been enhanced by new developments, many of which appear here for the first time in relation to system-theory. The presentation is not encyclopedic in nature and does not cover all areas of operator theory. The text is intended to concisely present a class of linear operators, with its own character, theory, techniques and tools. Particular attention has been given to the principal important subclass of linear operators such as the sectorial differential operators. The verification that the operator is sectorial is rather difficult task. One of the possible algorithms to solve such a problem is given in Section A.2 in the Appendix. Although this subclass of the operators is restrictive, it is sufficient to solve a broad spectrum of applications as has been demonstrated in Chapters 5-6. This contribution has developed some new results on the simulation of time and space dependent phenomena governed by initial-boundary-value problems with unbounded and non-self-adjoint differential operators, in general. In particular, the fundamental solution representation for the PDEs problem under consideration was given on a strong mathematical basis by exploiting the canonical system of the underlying operator and its adjoint (see Chapter 3). The obtained solution indicates the natural way to introduce the new multi-functional transformation (MFT) with respect to the space variables. Both transformations, the Laplace transformation for the time variable and multi-functional transformation for the space variables turn the initial-boundary-value problem into an algebraic equation. Solving this equation for the frequency domain representation of the output signal results in a description of the initial-boundary-value problem by a block diagram (see Chapter 5). This block diagram resembles the structure of the original problem. It is the main advantage of this mixed frequency domain representation that a rather general class of initial-boundary-value problems can be represented in a conceptually simple fashion. This representation generalizes the transfer function approach from classical system theory. In the same way, it is the starting point for the design of suitable discrete models and their efficient computer implementation. The result of this work is a beautiful, unified and powerful theory. According to which the following steps should be done in order to model the time and space dependent phenomena on the computer

formalization of the problems in terms of the operator theory; • 100 7. Conclusions

veriﬁcation of the sectorial property of the corresponding operator; • formulation of the spectral problems for prime and adjoint operators; • search of their eigenvalues, eigenfunctions and associated eigenfunctions; • determination of the required multi-functional transformation; • applying the MFT method to calculate the ”input-output” block representation; • appropriate discretization and approximation. • With the results presented above, a computational model for general initial-boundary- value problems is derived in the same principle as for a network model of an electrical circuit. The application of this method to real-world problems has been demonstrated by the examples in Chapter 6. 101 A Appendix

A.1 Proof of Theorem 2

Statement i).

Since R(s∗, L†) = R(s, L) † is the resolvent of the adjoint operator L†, we immediately conclude that each eigenvalue λ of L will generate a corresponding eigenvalue λ of L . ∗ † This fact and lemma 4 proves the second part of the statement i). Therefore, it is obvious

that the corresponding multiplicities are invariant under the transition from L to L†. In the next steps we will state that each pole of the resolvent R(s, L) is a eigenvalue of the operator L. Statement ii).

Step 1: Recursions for the operators Rm and Rm† . It is easy to see that for the resolvent R(s, L) the identity

sR(s, L)= I + LR(s, L) (A.1)

holds. It is also known from Lemma 4 that the resolvent can be represented in the neighborhood of each isolated singular point λ by a Laurent series with a ﬁnite number M of negative powers

+ ∞ R(s, L)= (s λ)νR , (A.2) − ν ν= M X− where the coeﬃcients Rν are operators that are determined by the formula

1 ν 1 R = (s λ)− − R(s, L)ds (A.3) ν 2πj − s Zλ =r | − | for suﬃciently small r> 0. Using the representation (A.2) we rewrite (A.1) as

+ + ∞ ∞ s (s λ)νR = I + L (s λ)νR . (A.4) − ν − ν ν= M ν= M X− X− Let w = s λ, substituting s = w + λ we get − + + ∞ ∞ ν ν (w + λ) Rνw = I + L Rνw . (A.5) ν= M ν= M X− X− 102 A. Appendix

Comparison of the coeﬃcients in the main part leads to

λR M = LR M , − − λR M+1 + R M = LR M+1,  − − − (A.6)  ......  λR 1 + R 2 = LR 1. − − −   Since R(s∗, L†) = R(s, L) † is the resolvent of the adjoint operator L†, the following equalities are valid, too,

λ∗R† M = L†R† M , − − λ∗R† M+1 + R† M = L†R† M+1,  − − − (A.7)  ......  λ∗R† 1 + R† 2 = L†R† 1. − − −  From the ﬁrst equality in (A.6) it follows that for any function y H the element ∈ u0 = R M y is an eigenfunction and λ is an eigenvalue of L. Let E(λ) ia a canonical − system with the elements ep,m that correspond to the eigenvalue λ. Step 2: M +1 M, p = 1(1)P (λ). p ≤ First, we will show that the multiplicity of the eigenvector ep,0 is not greater than the order of the pole of the resolvent operator R(s, L) in the point λ, i.e. M +1 M, p = 1(1)P (λ). p ≤ For this purpose, we introduce for p = 1(1)P (λ), m = 0(1)Mp the following functions

e e e g (s)=(sI L) p,0 + p,1 + + p,m (A.8) p,m − (s λ)m+1 (s λ)m ··· s λ − − − and show that they are holomorphic with respect to s in a neighborhood of λ. This can be shown by means of the decomposition

sI L =(s λ)I +(λI L) − − − and the recurrence relations (3.16) that deﬁne the elements e of the p th chain, i.e. p,m −

(λI L)ep,0 =0 − (A.9) (λI L)ep,m = ep,m 1, m = 1(1)Mp. − − − We obtain the result

gp,m(s)= ep,m, (A.10)

stating the analyticity of gp,m. We rearrange (A.8) into the form e e e R(s, L)g (s)= p,0 + p,1 + + p,m . (A.11) p,m (s λ)m+1 (s λ)m ··· s λ − − − A.1. Proof of Theorem 2 103

By the lemma 4 we know that the resolvent R(s, L) has a pole of order M in the point λ and since g is holomorphic this order cannot increase. Hence, we conclude m +1 p,m ≤ M, p = 1(1)P (λ) and for m = M we have M +1 M, p = 1(1)P (λ). p p ≤ Step 3: Representation of R M by E(λ). − From the equalities in (A.6) it follows that for any function y H the element u0 = R M y ∈ − is an eigenfunction of multiplicity M corresponding to the eigenvalue λ of L. Next we consider the eigenfunctions e , e , , e of multiplicity M from the 1,0 2,0 ··· p0,0 canonical system E(λ) corresponding to the eigenvalue λ. Then for arbitrary y H the ∈ element R M y can be represented by the linear combination −

R M y = c1e1,0 + c2e2,0 + + cp0 ep0,0, (A.12) − ···

where ck are some coeﬃcients depending on y. Since ek,0, k = 1,...,p0, are linear

independent, each ck, k =1,...,p0, can be regarded as a linear functional on H. Therefore, there exist unique elements z H, k =1,...,p , such that k,0 ∈ 0

ck(y)=(y, zk,0), k =1,...,p0. (A.13)

Then the equality (A.12) gives

R M y =(y, z1,0)e1,0 +(y, z2,0)e2,0 + +(y, zp0,0)ep0,0 − ···

for any y H. This means that the operator R M is given as ∈ −

R M = [e1,0, z1,0] + [e2,0, z2,0]+ + [ep0,0, zp0,0]. (A.14) − ··· Since the elements e , e , , e are linear independent then y H such that 1,0 2,0 ··· p0,0 ∃ ∈ (e ,y)=1and (e ,y)=0 for l = k. Then for this y we have k,0 l,0 6

R M y =(y, zk,0)ek,0. −

Hence, for the adjoint operator R† M we have −

R† M y =(y, ek,0)zk,0 = zk,0. −

With (A.7) this means that the element zk,0 is the eigenfunction of the multiplicity M for

the adjoint operator L†.

To show that the elements zk,0, k =1,...,p0 are linear independent we compare the coeﬃcients in (A.11) for (s λ)M − p0

(ep,m, zk,0)ek,0 = ep,0δM 1,m, p = 1(1)P (λ), − (A.15) Xk=1 m = 0(1)Mp. 104 A. Appendix

And since ek,0, k = 1(1)P (λ) are linear independent then

(ep,m, zk,0)= δp,kδM 1,m, k p0, p = 1(1)P (λ), − ≤ (A.16) m = 0(1)Mp

and this means that zk,0, k =1,...,p0 are also linear independent. Thus, we have veriﬁed that the following representation

R M = [ek,0, zk,0], (A.17) − Xk=1 holds, where e and z , k =1, ,p are all eigenvectors of the multiplicity M for the k,0 k,0 ··· 0 operators L and L†, respectively.

Step 4: Representation of R M+1 by E(λ). − We will show that the following representation is valid

p1 p0 R M+1 = [ek,0, zk,0]+ [ek,0, zk,1] + [ek,1, zk,0] , (A.18) − k=p +1 k=1 X0 X

a) where ek,0, k = p0 +1(1)p1 are all eigenvectors of the operators L of the multiplicity (M 1); −

b) zk,0, k = p0 + 1(1)p1 are linear independent eigenvectors and zk,1, k = 1(1)p0 are

associated elements from the canonical system for the adjoint operator L†.

For this purpose introduce the following operators

Sm = R M+1+m [ek,m+1, zk,0], m = 0(1)M 2, (A.19) − − − Xk=1 where e and z , k = 1, ,p ; m = 1, 2,... are elements of the canonical systems k,m k,0 ··· 0 which were obtained in the previous step. The introduced operators act on elements y H as follows ∈

Smy = R M+1+my (y, zk,0)ek,m+1, m = 0(1)M 2. (A.20) − − − Xk=1

It is easy to show that the element S0y is an eigenvector of the operator L. Indeed, we A.1. Proof of Theorem 2 105 have

LS0y = LR M+1y L (y, zk,0)ek,1 − − k=1 X p0

= λR M+1y + R M y (y, zk,0)Lek,1 − − − k=1 p0 X

= λR M+1y + (y, zk,0)ek,0 − − Xk=1 p0 (A.21) (y, z )e + λ(y, z )e − k,0 k,0 k,0 k,1 k=1 X p0

= λR M+1y λ(y, zk,0)ek,1 − − k=1 Xp0

= λ R M+1y (y, zk,0)ek,1 − − k=1 X = λS0y.

Analogously, we can verify that the element S1y is the associated element of the order 1

LS1y =

= LR M+2y L (y, zk,0)ek,2 − − k=1 X p0

= λR M+2y + R M+1y (y, zk,0)Lek,2 − − − k=1 (A.22) Xp0

= λR M+2y + R M+1y (y, zk,0)ek,1 + λ(y, zk,0)ek,2 − − − k=1 p0 X p0

= λ R M+2y (y, zk,0)ek,2 + R M+1y (y, zk,0)ek,1 − − − − Xk=1 Xk=1 = λS1y + S0y. Here we have used the property (A.6) and the deﬁnition of the associated elements given by (3.16). Thus, in this way it can be shown that all introduced elements

S0y,S2y,S3y, ,SM 2y satisfy the sequence of equations in (3.16) . ··· − Since the eigenvector S y has the multiplicity not less then M 1=(M 2)+1, then 0 − − it can be written by the linear combination of the eigenvectors of the operator L with the multiplicities equal to M 1 and M as follows − p1

S0y = ck′ ek,0, (A.23) Xk=1 106 A. Appendix

where ck′ are some coeﬃcients. Since ek,0, k =1,...,p1 are linear independent then each c′ , k =1,...,p is a linear functional on y H. Therefore, we can rewrite (A.23) as k 1 ∈

p0 p1 S0y = (y, zk,1)ek,0 + (y, zk,0)ek,0, y H, (A.24) ∀ ∈ Xk=1 k=Xp0+1

where elements zk,0 and zk,1 are uniquely deﬁned. From this and (A.19) it follows immediately that the following representation for the

operator R M+1 −

p1 p0

R M+1 = [ek,0, zk,0]+ [ek,0, zk,1] + [ek,1, zk,0] (A.25) − k=p +1 k=1 X0 X holds.

Now we have to show that the elements zk,0 and zk,1 belong to the canonical system. Applying the second equation in (A.7) to the element x we obtain

p1 p0

λ∗ (ek,0, x)zk,0 + (ek,0, x)zk,1 +(ek,1, x)zk,0 + k=Xp0+1 Xk=1 p0

+ (ek,0, x)zk,0 (A.26) k=1 X p1 p0

= L† (ek,0, x)zk,0 + (ek,0, x)zk,1 +(ek,1, x)zk,0 , k=p0+1 k=1 X X

where we use the fact that elements zk,0, k =1,...,p0 are eigenvectors, thus

p1 p0 p0

λ∗ (ek,0, x)zk,0 + λ∗ (ek,0, x)zk,1 + (ek,0, x)zk,0 = k=p +1 k=1 k=1 X0 X X (A.27) p1 p0

= L† (ek,0, x)zk,0 + L† (ek,0, x)zk,1, k=Xp0+1 Xk=1

since ek,0, k =1,...,p1 are linear independent we ﬁnally obtain

λ∗zk,0 = L†zk,0, k = p0 + 1(1)p1 (A.28) λ∗zk,1 + zk,0 = L†zk,1, k = 1(1)p0.

Thus, zk,0, k = p0 + 1(1)p1 are eigenvectors and zk,1, k = 1(1)p0 are associated elements. M 1 Using the equalities (A.16) we compare the coeﬃcients in (A.11) for (s λ) − with − diﬀerent functions gp,m(s). A.1. Proof of Theorem 2 107

1) g (s) with indices p = 1(1)p , m = 0(1)M 1 p,m 0 p −

ep,0δM 2,m = − p1 p0

= (ep,m, zk,0)ek,0 + (ep,m, zk,1)ek,0 +(ep,m, zk,0)ek,1 (A.29) k=Xp0+1 Xk=1 p1 p0

= (ep,m, zk,0)ek,0 + (ep,m, zk,1)ek,0. k=Xp0+1 Xk=1

Since ep,0, p =1,...,p1 are linear independent we have

(ep,Mp 1, zp,1)=1, p = 1(1)p0, − (e , z )=0, p,k = 1(1)p , m = 0(1)M 2, (A.30) p,m k,1 0 p − (e , z )=0, p = 1(1)p , m = 0(1)M 1, k = p + 1(1)p . p,m k,0 0 p − 0 1

2) gp,m(s) with indices p = 1(1)p0, m = Mp

ep,1 =

p1 p0

= (ep,Mp , zk,0)ek,0 + (ep,Mp , zk,1)ek,0 +(ep,Mp , zk,0)ek,1 (A.31) k=Xp0+1 Xk=1 p1 p0

= (ep,Mp , zk,0)ek,0 +(ep,Mp , zp,0)ep,1 + (ep,Mp , zk,1)ek,0. k=Xp0+1 Xk=1

Since ep,0, p =1,...,p1 are linear independent we have

(ep,M , zp,1)=0, p = 1(1)p0, p (A.32) (ep,Mp , zk,0)=0, p = 1(1)p0, k = p0 + 1(1)p1.

3) gp,m(s) with indices p = p0 + 1(1)p1, m = 0(1)Mp

ep,0δM 2,m = − p1 p0

= (ep,m, zk,0)ek,0 + (ep,m, zk,1)ek,0 +(ep,m, zk,0)ek,1 (A.33) k=Xp0+1 Xk=1 p1 p0

= (ep,m, zk,0)ek,0 + (ep,m, zk,1)ek,0. k=Xp0+1 Xk=1

Since ep,0, p =1,...,p1 are linear independent we have

(ep,Mp , zp,0)=1, p = p0 + 1(1)p1, (e , z )=0, p,k = p + 1(1)p , m = 0(1)M 1, (A.34) p,m k,0 0 1 p − (ep,m, zk,1)=0, p = p0 + 1(1)p1, m = 0(1)Mp, k = 1(1)p0. 108 A. Appendix

Therefore, eigenvectors zp,0, p = p0 + 1(1)p1 are linear independent. Step 5: Representation of R for m> M + 1 by E(λ). m − By analogy the following representation for the others coeﬃcients of the Laurent expansion (A.2) can be stated

for the case ν = M + i, i =1,...,M 1 − − pi

R M+i = [ek,0, zk,0] − k=Xpi−1+1 pi−1

+ [ek,0, zk,1] + [ek,1, zk,0] k=Xpi−2+1 pi−2

+ [ek,0, zk,2] + [ek,1, zk,1] + [ek,2, zk,0] (A.35) k=pi +1 X−3 ········· p1

+ [ek,0, zk,i 1] + [ek,1, zk,i 2]+ + [ek,i 1, zk,0] − − ··· − k=Xp0+1 p0

+ [ek,0, zk,i]+ + [ek,i 1, zk,1] + [ek,i, zk,0] , ··· − k=1 X where

a) ek,0, k = pi 1 + 1(1)pi are the eigenvectors of the multiplicity (M i) for the − − operators L;

b) zk,0, k = pi 1 + 1(1)pi are linear independent eigenvectors and zk,m, k = − 1(1)pi 1, m = 1(1)Mp are associated elements from the canonical system for − the adjoint operator L†.

And ﬁnally, denoting the elements zp,m by ǫp,m completes the proof of the Theorem 2.

Now, consider the canonical systems e (λ ) and ǫ (λ∗ ) , corresponding to dif- { p,m i } { p,m w } ferent eigenvalues (i = w). Let us consider the following inner products 6

λw(ep,0, ǫl,0)=(ep,0,λw∗ ǫl,0)=(ep,0, L†ǫl,0)=(Lep,0, ǫl,0)= λi(ep,0, ǫl,0), (A.36)

since λ = λ then (e , ǫ )=0. Next i 6 w p,0 l,0

λw(ep,0, ǫl,1)=(ep,0,λ∗ ǫl,1)=(ep,0, L†ǫl,1 ǫl,0)=(ep,0, L†ǫl,1)= w − (A.37) =(Lep,0, ǫl,1)= λi(ep,0, ǫl,1)

and we obtain that (ep,0, ǫl,1)=0. Thus, using the same method it can be proven that in the case when λ = λ the i 6 w corresponding canonical systems are biorthogonal. A.2. Proof of Example 5 109

Remark 18 The equalities (A.16),(A.30),(A.32),(A.34) and similar equalities which can be obtained in Step 5 state the biorthogonality of the two canonical systems (see (3.25))

(ep,m(λi), ǫl,n(λw∗ )) = δi,wδp,lδm,Mp n. (A.38) −

A.2 Proof of Example 5

For s / ( , 0] set µ = √s, so that Re(µ) > 0. For every f L ([0, 1]) extend f to a ∈ −∞ ∈ p function f L (R) in a such way that f L R = f L . ∈ p || k p( ) k k p([0,1]) Let e x e + ∞ 1 µ(x τ) µ(x τ) y(x)= e− − f(τ)dτ + e − f(τ)dτ =(f h )(x), (A.39) 2µ ∗ µ Z Zx −∞ e 1 µ x e e e where hµ(x) = 2µ e− | |. It is known (see [LLMP05]) that y = y [0,1] is a solution of the equation (sI L)y = f satisfying − e f Lp([0,1]) y L k k , where θ = arg(s). (A.40) k k p([0,1]) ≤ s cos(θ/2) | | However, it may not satisfy the boundary conditions. To ﬁnd the solution that satisﬁes the boundary conditions we write it in the form

µx µx y(x)= y(x)+ c1e− + c2e , (A.41)

µx µx where e− and e are two independent solutions of the homogeneous equation sy y′′ = e − 0. We determine uniquely c1 and c2 imposing boundary conditions

y(0)=0: y(0) + c1 + c2 =0 µ µ . (A.42) y′(0) y′(1)=0: y′(0) µc + µc y′(1) + µe− c µe c =0 − − 1 2 − 1 − 2 e The unique solution exists because the following determinant is not equal to zero e e 1 1 µ µ D(µ) = det = µ(2 e− e ) =0. (A.43) µ(e µ 1) µ(1 eµ) − − 6 − − − Thus,

1 µ c = µ(e 1)y(0) y′(1) + y′(0) , 1 D(µ) − − (A.44) 1 µ c2 = µ(e− 1)ey(0) +ey′(1) ey′(0) D(µ) − − and e e e 1 µ c µ (e 1) y(0) + y′(1) + y′(0) , | 1| ≤ D(µ) | |·| − |·| | | | | | | | (A.45) 1 µ c2 µ (e− 1) ey(0) +ey′(1) +ey′(0) . | | ≤ D(µ) | |·| − |·| | | | | | | | e e e 110 A. Appendix

For suﬃciently large µ we set | | µ µ µ µ Re(µ) D(µ) = µ(2 e− e ) = µ (2 e− e ) µ e , | | | − − | | |·| − − |≈| | (eµ 1) eRe(µ), (A.46) | − | ≈ µ (e− 1) 1. | − | ≈ Also we obtain the following estimates

+ 1 + 1 ∞ p′ ∞ p′ µ τ µ τ p′ µτ p′ e− | | L R = e− | | dτ = 2 e− dτ k k p′ ( ) | | | | Z Z0 −∞ + 1 1 ∞ p′ + p′ p′Re(µ)τ 2 p′Re(µ)τ ∞ = 2 e− dτ = − e− (A.48) p Re(µ) Z ′ 0 0 1 2 p′ 2 1 1 , ≤ p′ ≤ p′ p′Re(µ) p′Re(µ) therefore

x + ∞ 1 µ(x τ) µ(x τ) y′(x)= e− − f(τ)dτ + e − f(τ)dτ . (A.50) 2 − Z Zx −∞ e e e Moreover we have

Thus, we have the following estimates

f Lp([0,1]) y′(0) , y′(1) k k 1 . (A.52) | | | | ≤ p′ p′Re(µ)

Finally, for the coeﬃcients c1e,c2 wee easily get

µx µx y(x) L ([0,1]) y(x) L ([0,1]) + c1e− L ([0,1]) + c2e L ([0,1]) k k p ≤k k p k k p k k p (A.54) µx µx = y(x) Lp([0,1]) + c1 e− Lp([0,1]) + c2 e Lp([0,1]). ke k | |·k k | |·k k 1. e 1 1 1 1 p p µx µx p p Re(µ)x e− L = e− dx = e− dx k k p([0,1]) | | Z0 Z0 (A.55) 1 1 p p Reµ p 1 p Re(µ)x x=1 1 e− = − e− = − , p Re(µ) x=0 p Re(µ) while p 1, Re(µ) > 0 we obtain ≥ µx 1 L e− p([0,1]) 1 , (A.56) k k ≤ p Re(µ) p

µx 2 f Lp([0,1]) 1 f Lp([0,1]) c1e− Lp([0,1]) k k 1 1 C1 k k . (A.57) k k ≤ µ p Re(µ) p′ · p Re(µ) p ≤ µ Re(µ) | | ′ | | 2.

1 1 1 1 p p µx µx p p Re(µ)x e L = e dx = e dx , k k p([0,1]) | | Z0 Z0 (A.58) 1 1 p Reµ 1 x=1 p e 1 p = ep Re(µ)x = − . p Re(µ) x=0 p Re(µ)

112 A. Appendix

Thus

Reµ µx e e Lp([0,1]) 1 (A.59) k k ≤ p Re(µ) p and

µx f Lp([0,1]) c e L C k k . (A.60) k 2 k p([0,1]) ≤ 2 µ Re(µ) | | If s = s eiθ with θ θ < π then Re(µ) µ cos(θ /2) and we easily get | | | | ≤ 0 ≥| | 0

f Lp([0,1]) y(x) L C k k . (A.62) k k p([0,1]) ≤ s | | for a suitable C > 0 and s R large enough. | | ≥ Since s R(s, L) is holomorphic in the resolvent set, it is continuous, hence it is 7−→ bounded on the compact set s R, args θ . {| | ≤ | | ≤ 0}

A.3 Proof of Theorem 5

It is easy to see that for the resolvent RC (s, L) the identity

sCRC (s, L)= I + LRC (s, L) (A.63)

holds. It is also known from Lemma 8 that the resolvent can be represented in the neighborhood of each isolated singular point λ by a Laurent series with a ﬁnite number M of negative powers

+ ∞ R (s, L)= (s λ)νR , (A.64) C − ν ν= M X− where the coeﬃcients Rν are operators that are determined by the formula

1 ν 1 R = (s λ)− − R (s, L)ds (A.65) ν 2πj − C s Zλ =r | − | for suﬃciently small r> 0. Using the representation (A.64) we rewrite (A.63) as

+ + ∞ ∞ sC (s λ)νR = I + L (s λ)νR . (A.66) − ν − ν ν= M ν= M X− X− A.3. Proof of Theorem 5 113

Let l = s λ, substituting s = l + λ we get: − + + ∞ ∞ ν ν (l + λ)C Rνl = I + L Rνl . (A.67) ν= M ν= M X− X− Comparison of the coeﬃcients in the main part leads to

λCR M = LR M , − − λCR M+1 + CR M = LR M+1,  − − − (A.68)  ...... ,  λCR 1 + CR 2 = LR 1. − − −   Since RC† (s∗, L†)= RC (s, L) † is the resolvent of the adjoint operator L†, the following equalities are valid, too,

λ∗C†R† M = L†R† M , − − λ∗C†R† M+1 + C†R† M = L†R† M+1,  − − − (A.69)  ......  λ∗C†R† 1 + C†R† 2 = L†R† 1. − − −  From the ﬁrst equality in (A.68) it follows that for any function y H the element ∈ u0 = R M y is an C eigenfunction and λ is an C eigenvalue of L. Let E(λ) ia a canonical − − − system with the elements e that correspond to the C eigenvalue λ. p,m − Now, we will show that the multiplicity of the C eigenvector e is not greater than − p,0 the order of the pole of the resolvent operator R (s, L) in the point λ, i.e. M +1 C p ≤ M, p = 1(1)P (λ).

For this purpose, we introduce for p = 1(1)P (λ), m = 0(1)Mp the following functions e e e g (s)=(sC L) p,0 + p,1 + + p,m (A.70) p,m − (s λ)m+1 (s λ)m ··· s λ − − − and show that they are holomorphic in a neighborhood of λ. This can be shown by means of the decomposition sC L =(s λ)C +(λC L) − − − and the recurrence relations (3.81) that deﬁne the element e of the p th chain, i.e. p,m −

(λC L)ep,0 =0 − (A.71) (λC L)ep,m = Cep,m 1, m = 1(1)Mp. − − − We obtain the result

gp,m(s)= Cep,m, (A.72)

stating the analyticity of gp,m. 114 A. Appendix

We rearrange (A.70) into the form e e e R (s, L)g (s)= p,0 + p,1 + + p,m . (A.73) C p,Mp (s λ)m+1 (s λ)m ··· s λ − − − By the lemma 8 we know that the resolvent RC (s, L) has a pole of order M in the point λ and since gp,m is holomorphic this order cannot increase. Hence, we conclude m +1 M, p = 1(1)P (λ) and for m = M we have M +1 M, p = 1(1)P (λ). ≤ p p ≤ Representation of R M by E(λ). − From the equalities in (A.68) it follows that for any function y H the element u0 = R M y ∈ − is an C eigenfunction of the multiplicity M and λ is an C eigenvalue of L. − − Next we consider the C eigenfunctions e , e , , e of the multiplicity M from − 1,0 2,0 ··· p0,0 the canonical system corresponding to the C eigenvalue λ. Then for arbitrary y H the − ∈ element R M y can be represented by the linear combination −

R M y = c1e1,0 + c2e2,0 + + cp0 ep0,0, (A.74) − ···

where ci are some coeﬃcients depending on y. Since ei,0, i =1,...,p0, are linear independent, each c , i = 1,...,p , can be regarded as a linear functional on y H. Therefore, i 0 ∈ there exist unique elements z H, i =1,...,p , such that i,0 ∈ 0

ci(y)=(y, zi,0), i =1,...,p0. (A.75) Then the equality (A.74) gives

R M y =(y, z1,0)e1,0 +(y, z2,0)e2,0 + +(y, zp0,0)ep0,0 (A.76) − ··· for any y H. This means that the operator R M is given as ∈ −

R M = [e1,0, z1,0] + [e2,0, z2,0]+ + [ep0,0, zp0,0]. (A.77) − ··· Since the elements e , e , , e are linear independent then y H such that 1,0 2,0 ··· p0,0 ∃ ∈ (e ,y)=1and (e ,y)=0 for j = i. Then for this y we have i,0 j,0 6

R M y =(y, zi,0)ei,0. (A.78) −

Hence, for the adjoint operator R† M we have −

R† M y = zi,0(ei,0,y)= zi,0. (A.79) − With (A.69) this means that the element z is the C eigenfunction of the multiplicity i,0 − M for the adjoint operator L†.

To show that the elements zk,0, k =1,...,p0 are linear independent we compare the coeﬃcients in (A.73) for (s λ)M : − p0

ep,0δM 1,m = (Cep,m, zk,0)ek,0, p = 1(1)P (λ), − (A.80) Xk=1 m = 0(1)Mp A.3. Proof of Theorem 5 115

and since ek,0, k = 1(1)P (λ) are linear independent then

(Cep,m, zk,0)= δp,kδM 1,m, k p0, p = 1(1)P (λ), − ≤ (A.81) m = 0(1)Mp

and this means that zk,0, k =1,...,p0 are also linear independent. Thus, we have veriﬁed that the following representation

R M = [ek,0, zk,0], (A.82) − Xk=1 holds, where e and z , k =1, ,p are all C eigenvectors of the multiplicity M for k,0 k,0 ··· 0 − the operators L and L†, respectively.

Representation of R M+1 by E(λ). − We will show that the following representation is valid

p1 p0 R M+1 = [ek,0, zk,0]+ [ek,0, zk,1] + [ek,1, zk,0] , (A.83) − k=p0+1 k=1 X X a) where e , k = p + 1(1)p are all C eigenvectors of the operators L of the multi- k,0 0 1 − plicity (M 1); −

b) z , k = p + 1(1)p are linear independent C† eigenvectors and z , k = 1(1)p k,0 0 1 − k,1 0 are associated elements from the canonical system for the adjoint operator L†.

For this purpose introduce the following operators

Sm = R M+1+m [ek,m+1, zk,0], m = 0(1)M 2, (A.84) − − − Xk=1 where e and z , k = 1, ,p ; m = 1, 2,... are elements of the canonical systems k,m k,0 ··· 0 which were obtained in the previous step. The introduced operators act on elements y H as follows ∈

Smy = R M+1+my (y, zk,0)ek,m+1, m = 0(1)M 2. (A.85) − − − Xk=1 It is easy to show that the element S y is an C eigenvector of the operator L. Indeed, 0 − 116 A. Appendix we have

LS0y = LR M+1y L (y, zk,0)ek,1 − − k=1 X p0

= λCR M+1y + CR M y (y, zk,0)Lek,1 − − − k=1 p0 X

= λCR M+1y + C (y, zk,0)ek,0 − − Xk=1 p0 (A.86) C (y, z )e + λ(y, z )e − k,0 k,0 k,0 k,1 k=1 X p0

= λCR M+1y C λ(y, zk,0)ek,1 − − k=1 pX0

= λC R M+1y (y, zk,0)ek,1 − − k=1 X = λCS0y.

Analogously, we can verify that the element S y is the C associated element of the 1 − order 1

LS1y =

= LR M+2y L (y, zk,0)ek,2 − − k=1 X p0

= λCR M+2y + CR M+1y (y, zk,0)Lek,2 − − − k=1 (A.87) Xp0

= λCR M+2y + CR M+1y C (y, zk,0)ek,1 + λ(y, zk,0)ek,2 − − − k=1 p0 X p0

= λC R M+2y (y, zk,0)ek,2 + C R M+1y (y, zk,0)ek,1 − − − − Xk=1 Xk=1 = CλS1y + CS0y.

Here we have used the property (A.68) and the deﬁnition of the C associated elements − given by (3.81). Thus, in this way it can be shown that all introduced elements

S0y,S2y,S3y, ,SM 2y satisfy the sequence of equations in (3.81) . ··· − Since the C eigenvector S y has the multiplicity not less then M 1=(M 2)+1, − 0 − − then it can be written by the linear combination of the C eigenvectors of the operator L − A.3. Proof of Theorem 5 117 with the multiplicities equal to M 1 and M as follows − p1

S0y = ck′ ek,0, (A.88) Xk=1

where ck′ are some coeﬃcients. Since ek,0, k =1,...,p1 are linear independent then each c′ , k =1,...,p is a linear functional on y H. Therefore, we can rewrite (A.88) as k 1 ∈

p0 p1 S0y = (y, zk,1)ek,0 + (y, zk,0)ek,0, y H, (A.89) ∀ ∈ Xk=1 k=Xp0+1

where elements zk,i are uniquely deﬁned. From this and (A.84) it follows immediately that the following representation for the

operator R M+1 −

p1 p0

R M+1 = [ek,0, zk,0]+ [ek,0, zk,1] + [ek,1, zk,0] (A.90) − k=p +1 k=1 X0 X holds.

Now we have to show that the elements zk,0 and zk,1 belong to the canonical system. Applying the second equation in (A.69) to the element x we obtain

p1 p0

λ∗C† (ek,0, x)zk,0 + (ek,0, x)zk,1 +(ek,1, x)zk,0 + k=Xp0+1 Xk=1 p0

+ C† (ek,0, x)zk,0 (A.91) k=1 Xp1 p0

= L† (ek,0, x)zk,0 + (ek,0, x)zk,1 +(ek,1, x)zk,0 , k=p0+1 k=1 X X where we use the fact that elements z , k =1,...,p are C eigenvectors, thus k,0 0 −

p1 p0 p0

λ∗C† (ek,0, x)zk,0 + λ∗C† (ek,0, x)zk,1 + C† (ek,0, x)zk,0 = k=p +1 k=1 k=1 X0 X X (A.92) p1 p0

= L† (ek,0, x)zk,0 + L† (ek,0, x)zk,1. k=Xp0+1 Xk=1

And since ek,0, k =1,...,p1 are linear independent, we ﬁnally obtain

λ∗C†zk,0 = L†zk,0, (A.93) λ∗C†zk,1 + C†zk,0 = L†zk,1. 118 A. Appendix

Thus, z , k = p + 1(1)p are C eigenvectors and z , k = 1(1)p are C associated k,0 0 1 − k,1 0 − elements. M 1 Using the equalities (A.81) we compare the coeﬃcients in (A.73) for (s λ) − with − diﬀerent functions gp,m(s)= Cep,m. 1) g (s) with indices p = 1(1)p , m = 0(1)M 1 p,m 0 p −

ep,0δM 2,m = − p1 p0

= (Cep,m, zk,0)ek,0 + (Cep,m, zk,1)ek,0 +(Cep,m, zk,0)ek,1 (A.94) k=Xp0+1 Xk=1 p1 p0

= (Cep,m, zk,0)ek,0 + (Cep,m, zk,1)ek,0. k=Xp0+1 Xk=1

Since ep,0, p =1,...,p1 are linear independent we have

(Cep,Mp 1, zp,1)=1, p = 1(1)p0, − (Ce , z )=0, p,k = 1(1)p , m = 0(1)M 2, (A.95) p,m k,1 0 p − (Ce , z )=0, p = 1(1)p , m = 0(1)M 1, k = p + 1(1)p . p,m k,0 0 p − 0 1

2) gp,m(s) with indices p = 1(1)p0, m = Mp

ep,1 =

p1 p0

= (Cep,Mp , zk,0)ek,0 + (Cep,Mp , zk,1)ek,0 +(Cep,Mp , zk,0)ek,1 (A.96) k=Xp0+1 Xk=1 p1 p0

= (Cep,Mp , zk,0)ek,0 +(Cep,Mp , zp,0)ep,1 + (Cep,Mp , zk,1)ek,0. k=Xp0+1 Xk=1

Since ep,0, p =1,...,p1 are linear independent we have

(Cep,M , zp,1)=0, p = 1(1)p0, p (A.97) (Cep,Mp , zk,0)=0, p = 1(1)p0, k = p0 + 1(1)p1.

3) gp,m(s) with indices p = p0 + 1(1)p1, m = 0(1)Mp

ep,0δM 2,m = − p1 p0

= (Cep,m, zk,0)ek,0 + (Cep,m, zk,1)ek,0 +(Cep,m, zk,0)ek,1 (A.98) k=Xp0+1 Xk=1 p1 p0

= (Cep,m, zk,0)ek,0 + (Cep,m, zk,1)ek,0. k=Xp0+1 Xk=1 A.3. Proof of Theorem 5 119

Since ep,0, p =1,...,p1 are linear independent we have

(Cep,Mp , zp,0)=1, p = p0 + 1(1)p1, (Ce , z )=0, p,k = p + 1(1)p , m = 0(1)M 1, (A.99) p,m k,0 0 1 p − (Cep,m, zk,1)=0, p = p0 + 1(1)p1, m = 0(1)Mp, k = 1(1)p0.

Therefore, C eigenvectors z , p = p + 1(1)p are linear independent. − p,0 0 1

Representation of R for m M +1 by E(λ). m ≥ −

By analogy the following representation for the others coeﬃcients of the Laurent expansion (A.64) can be stated

for the case ν = M + i, i =1,...,M 1 − − pi

R M+i = [ek,0, zk,0] − k=Xpi−1+1 pi−1

+ [ek,0, zk,1] + [ek,1, zk,0] k=Xpi−2+1 pi−2

+ [ek,0, zk,2] + [ek,1, zk,1] + [ek,2, zk,0] (A.100) k=pi +1 X−3 ········· p1

+ [ek,0, zk,i 1] + [ek,1, zk,i 2]+ + [ek,i 1, zk,0] − − ··· − k=Xp0+1 p0

+ [ek,0, zk,i]+ + [ek,i 1, zk,1] + [ek,i, zk,0] , ··· − k=1 X where

a) ek,0, k = pi 1 + 1(1)pi are the C eigenvectors of the multiplicity (M i) for − − − the operators L;

b) zk,0, k = pi 1 + 1(1)pi are linear independent C eigenvectors and zk,m, k = − − 1(1)pi 1, m = 1(1)Mp are C associated elements from the canonical system − − for the adjoint operator L†.

And ﬁnally, denoting the elements zp,m by ǫp,m completes the proof of the Theorem 5.

Now, consider the canonical systems e (λ ) and ǫ (λ∗ ) , corresponding to dif- { p,m i } { p,m w } ferent C eigenvalues (i = w). Let us consider the following inner products − 6

λw(Cep,0, ǫl,0)=(ep,0,λw∗ C†ǫl,0)=(ep,0, L†ǫl,0)=(Lep,0, ǫl,0)= λi(Cep,0, ǫl,0) (A.101) 120 A. Appendix since λ = λ then (e , ǫ )=0. Next i 6 w p,0 l,0

λw(Cep,0, ǫl,1)=(ep,0,λ∗ C†ǫl,1)=(ep,0, L†ǫl,1 C†ǫl,0)=(ep,0, L†ǫl,1)= w − (A.102) =(Lep,0, ǫl,1)= λi(Cep,0, ǫl,1) and we obtain that (Cep,0, ǫl,1)=0. Thus, using the same method it can be proven that in the case when λ = λ the i 6 w corresponding canonical systems are biorthogonal.

Remark 19 The equalities (A.81),(A.95),(A.97),(A.99) and similar equalities which can be obtained from (A.100) state the biorthogonality of the two canonical systems with respect to weighting operator C (see Remark (9))

(Cep,m(λi), ǫl,n(λw∗ )) = δi,wδp,lδm,Mp n. (A.103) − 121 B Titel, Inhaltsverzeichnis, Einleitung und Zusammenfassung

The following german translations of the title (Section B.1), the table of contents (Sec- tion B.2), the introduction (Section B.3), and the summary (Section B.4) are a manda- tory requirement for the dissertation at the Faculty of Engineering of the University of Erlangen-Nuremberg.

B.1 Titel

Anwendung der Operator-Theorie zur Darstellung kontinuierlicher und diskreter Systeme mit verteilten Parametern.

B.2 Inhaltsverzeichnis

Abkurzungen¨ ix

Liste der mathematischen Symbole ix

Variablen xi

1 Einfuhrung¨ 1

2 Grundlegende Begriffe der Funktionalanalysis 5 2.1 Räume...... 5 2.2 LineareOperatoren...... 8 2.2.1 UnbeschränkteOperatoren...... 8 2.2.2 AdjungierteOperatoren ...... 9 2.2.3 LineareDifferentialformen...... 10 2.2.3.1 Homogene Randbedingungen ...... 10 2.2.3.2 Greensche Formel und assoziierte Formen ...... 11 2.2.4 LineareDifferentialoperatoren...... 12 2.2.4.1 Adjungierte homogene Randbedingungen und adjungierte Operatoren ...... 13 122 B.Titel,Inhaltsverzeichnis,EinleitungundZusammenfassung

2.2.4.2 Inhomogene Randbedingungen ...... 15 2.3 Kapitelzusammenfassung ...... 16

3 Spektraltheorie von Operatoren 17 3.1 DieResolvente ...... 17 3.1.1 Kanonische Systeme von Operatoren und deren Adjungierten. . . . 20 3.1.2 Entwicklung der Resolvente durch das kanonische System...... 24 3.1.3 SektorielleOperatoren ...... 25 3.1.3.1 Deﬁnition von sektoriellen Operatoren ...... 26 3.1.3.2 Beispiele zu sektoriellen Operatoren ...... 26 3.1.4 Halbgruppen und kanonische Systeme ...... 28 3.1.4.1 Deﬁnitionen...... 28 3.1.4.2 Zusammenhang zwischen der Halbgruppe, der Resolvente undderLaplaceTransformation ...... 29 3.1.5 Erweiterung der Halbgruppe durch das kanonische System ..... 30 3.2 Die C Resolvente ...... 32 − 3.2.1 Kanonische Systeme des primalen und adjungierten Operators . . . 37 3.2.2 Entwicklung des C Resolvente durch die kanonischen Systeme . . . 39 − 3.3 Kapitelzusammenfassung ...... 40

4 Mathematische Modellierung Physikalischer Prozesse 43 4.1 Gewöhnliche Differentialgleichungen (GDGen) ...... 43 4.1.1 KlassifikationvonGDGen ...... 43 4.1.2 Zustandsraummodell ...... 44 4.2 Partielle Differentialgleichungen (PDGen) ...... 46 4.2.1 KlassifikationvonPDGen ...... 46 4.2.2 Anfangs-Randwert-Probleme...... 48 4.2.3 Lösung von Anfangs-Randwert-Problemen im Laplace Bereich . .. 49 4.3 Kapitelzusammenfassung ...... 51

5 Beschreibung von Multidimensionalen Systemen 53 5.1 Multifunktional Transformation (MFT) ...... 54 5.1.1 Deﬁnition und Eigenschaften ...... 55 5.1.2 InverseMFT ...... 56 5.2 Anwendung der MFT Methode auf Anfangs-Randwert-Problemen mit ho- mogenenRandbedingungen ...... 57 5.2.1 Anfangs-Randwert-Probleme mit homogenen Randbedingungen . . 57 5.2.2 LaplaceTransformation ...... 58 5.2.3 OrtlicheTransformation(MFT)¨ ...... 59 5.2.4 InverseMFT ...... 60 B.2. Inhaltsverzeichnis 123

5.2.5 Diskretisierung des MFT-Modells ...... 61 5.2.5.1 OrtlicheDiskretisierung¨ ...... 62 5.2.5.2 Zeitliche Diskretisierung ...... 62 5.2.6 Inverse z Transformation ...... 63 − 5.2.7 Approximation ...... 64 5.3 Anwendung der MFT Methode auf Anfangs-Randwert-Probleme mit nichthomogenen Randbedingungen ...... 64 5.3.1 Anfangs-Randwert-Probleme mit nichthomogenen Randbedingungen 64 5.3.2 LaplaceTransformation ...... 66 5.3.3 OrtlicheTransformation(MFT)¨ ...... 66 5.3.4 InverseMFT ...... 69 5.3.5 Diskretisierung des MFT-Modells ...... 69 5.4 Anwendung der MFT Methode auf allgemeine vektorielle Anfangs- Randwert-Probleme...... 70 5.4.1 Notation...... 70 5.4.2 Allgemeine vektorielle Anfangs-Randwert-Probleme ...... 70 5.4.3 LaplaceTransformation ...... 71 5.4.4 OrtlicheTransformation(MFT)¨ ...... 72 5.4.5 InverseMFT ...... 74 5.4.6 Diskretisierung des MFT-Modells ...... 74 5.5 Kapitelzusammenfassung ...... 74

6 Beispiele der MFT Simulationen 77 6.1 Wärmeleitungsgleichung ...... 77 6.1.1 PrimaleundadjungierteOperatoren ...... 78 6.1.2 Eigenwertproblem...... 78 6.1.3 Biorthogonalität ...... 79 6.1.4 Die allgemeine Lösung ...... 79 6.1.5 MFT-Simulation ...... 80 6.2 WärmeleitungdurcheineWand ...... 81 6.2.1 AdjungierterOperator ...... 82 6.2.2 Eigenwertprobleme ...... 82 6.2.3 AssoziierteVektoren ...... 85 6.2.4 Biorthogonalität ...... 87 6.2.5 Die allgemeine Lösung ...... 88 6.2.6 MFT-Simulation ...... 89 6.3 Telegrafen-Gleichung ...... 89 6.3.1 AdjungierterOperator ...... 91 6.3.2 Eigenwertprobleme ...... 92 124 B.Titel,Inhaltsverzeichnis,EinleitungundZusammenfassung

6.3.3 AssoziierteVektoren ...... 95 6.3.4 Biorthogonalit¨at ...... 97 6.3.5 Die allgemeine L¨osung ...... 98 6.4 Kapitelzusammenfassung ...... 98

7 Zusammenfassung 99

A Anhang 101 A.1 BeweisdesTheorems2...... 101 A.2 Beweis des Beispiels 5 ...... 109 A.3 BeweisdesTheorems5...... 112

B Titel, Inhaltsverzeichnis, Einleitung und Zusammenfassung 121 B.1 Titel ...... 121 B.2 Inhaltsverzeichnis ...... 121 B.3 Einleitung ...... 124 B.4 Zusammenfassung...... 128

Literaturverzeichnis 130

B.3 Einleitung

Die Entwicklung der Computertechnologien, die Verfugbarkeit¨ von Hochgeschwindigkeits- Prozessoren und unterschiedliche Programmiersprachen ermöglichen es den Wissenschaft- lern unterschiedlicher Gebiete zahlreiche Algorithmen zu erforschen und zu entwickeln um physikalische Phänomene am Computer zu simulieren. Um aber Hochpräzisions-Modelle eines wirklichen Prozesses zu konstruieren, muss man mit seiner mathematischen Be- schreibung und Analyse beginnen um spezifische Charakteristiken des zu beschreibenden Problems zu erhalten. Dieser Vorgang ist sehr hilfreich um effiziente numerische Methoden zu entwickeln, die direkt am Computer implementiert werden können. In den letzen Jahrzehnten wuchs das Interesse kontinuierlich an der Anwendung der Funktional-Analysis und speziell der allgemeinen Operatortheorie auf technische Proble- me. Diese Entwicklung ist deutlich verknupft¨ mit der großen Nachfrage an Anwendungen und ist gleichermaßen von praktischem als auch theoretischem Interesse. Viele physikalische und Informations-Prozesse unterschiedlicher Gebiete besitzen identische mathematische Strukturen, die in einem gemeinsamen Operator-Kalkul¨ beschrieben werden können. Eine solche Abstraktion ermöglicht es einen allgemeinen Algorithmus zu konstruieren, um zahlreiche Probleme zu lösen. Es ist jedoch nicht ausreichend ein gegebenes Problem anhand seiner reinen theore- tischen Basis zu analysieren. Die praktische Anwendung kann zusätzliche Einschränkun- B.3. Einleitung 125 gen aufwerfen wie z.B. die Echtzeit-Lösung des Systems, die geringe laufzeiten oder Be- schränkungen der Computerleistung oder der Speicherkapazität. Deshalb ist es nötig ein differenziertes Model der Problemlösung zu entwickeln, die der Computer-Implementation angepasst ist. Die allgemeine Vorgehensweise einer solchen Entwicklung ist in der Tabelle B.1 zusammengefasst.

Mathematische Physikalisches Problem Beschreibung Satz von PDEs

Discretisierung und Ubertragung in den Diskretes Modell Approximation Kontinuierlichen Bereich

Simulation

Abbildung B.1: Das allgemeine Vorgehen zwischen Formulierung des physikalischen Problems und ihre Computerrealisierung. Hier ist das physikalische Problem ein verteil- tes Parameter System, beschrieben durch einen Satz partieller Diﬀerenzialgleichungen (PDGen).

Die vorliegende Doktorarbeit orientiert sich an der Entwicklung von Algorithmen, die auf der spektralen Operator-Theorie und ihrer Anwendung basiert, um praktische Pro- bleme zu lösen, die im Allgemeinen bei der diskreten Simulation von kontinuierlichen Systemen mit partiellen Differenzialgleichungen von unbeschränkten und nicht-selbst- adjungierten Operatoren, entstehen. Wir behandeln die Theorie von ”wohl definierten” Problemen, die bei der Simulation von Systemen entstehen. Die Notwendigkeit den Fall von unbeschränkten Operatoren zu berucksichtigen,¨ wurde durch zahlreiche technische Anwendungen angeregt, wo die Systemmodelle der physikalischen Prozesse diese Klasse der Operatoren hervorrufen. Trotz der weiten Verbreitung dieser Prozesse wurden bis jetzt relativ wenig Ergebnisse dieser strengen mathematischen Theorie in der Praxis angewandt. Im Allgemeinen ist das Hauptproblem, dass ein Teil der Ergebnisse, die im klassischen Fall von beschränkten und selbst-adjungierten Operatoren erhalten wird, ungultig¨ oder unzureichend wird im Falle von unbeschränkten und nicht-selbst-adjungierten Operatoren. Daruberhinaus¨ gibt es, wie man weiß, keine allgemeine spektrale Theorie fur¨ unbeschränk- te und nicht-selbst-adjungierte Operatoren im Hilbert Raum. Spektral-Theorie behandelt nur einige spezielle Klassen solcher Operatoren (siehe [GK69]). Der einfachste Fall von endlichdimensionalen Räumen und Matrizen als Operatoren, von dem wir ausgehen, dass er dem Leser bekannt ist, ist weit verbreitet (siehe [Gan59]). 126 B.Titel,Inhaltsverzeichnis,EinleitungundZusammenfassung

Es hat eine Reihe von starken Aussagen, die in einem allgemeineren Fall von unendlichen dimensionalen Räumen und allgemeinen Operatoren nutzlich¨ sein könnten. Von speziellem Interesse ist hierbei das Problem der kanonischen Form bei beliebigen Transformationen im Hilbert Raum als Verallgemeinerung der Jordan Form fur¨ Matrizen (einige Wissen- schaftliche Arbeiten beschreiben es als die kanonische Jordan Form fur¨ Matrizen oder die normale Jordan Form). Die Jordan Struktur fur¨ Matrizen wird durch die spezielle Basis erreicht, welche im einfachsten Fall von einer Reihe von linear unabhängigen Eigenvek- toren und im allgemeinen Fall der assoziierten Vektoren (auch bekannt als allgemeine Eigenfunktionen) die dann zusammen die Basis bilden. Der Fall der beschränkten Operatoren im allgemeinen Banach-Raum kann genauso behandelt werden. M.V. Keldysh war der erste, der den Begriff der Jordan Vektorenkette [Gan59] zu einer breiten Klasse von nicht-selbst-adjungierten beschränkten Operatoren generalisiert hat (siehe [Kel51],[Kel71]). Aus diesem Grund nennt man sie die Keldysh- Kette. Weitere Ergebnisse wurden in den Arbeiten von M.G. Krein [Kre59], I. Gohberg [IGK90], V.B. Lidskii [Lid59], A.S. Markus [Mar88], M.A. Naimark [Nai62], A.V. Fursikov [Fur01b],[Fur01a] erreicht. Fur¨ nähere Informationen siehe [GK69]. In dieser Arbeit werden die Methoden der Spektral-Analyse auf die Klasse der un- beschränkten und nicht-selbst-adjungierten Operatoren mit kompakter Resolvente angewandt. Es fasst die Forschungsergebnisse des Autors [DRS04a],[DRS04b],[DRS05],[DRS] zusammen, erweitert und generalisiert sie. Die Basisprinzipien sind auf der Methode von Keldysh basiert. Die Absicht besteht darin eine spektrale Zerlegung der Lösung vo Anfangs-Randwert-Problemen zu gewinnen, die der Struktur von räumlichen Differential- Operatoren angepaßt sind, die dann zu ihrer Darstellung in Bezug auf ihr kanonisches System und des adjungierten Systems. Es ist eine Konsequenz aus der Raumtrennung, das die resultierende Struktur der Simulation des kompletten Systems mit den verteilten Pa- rametern gut angepasst ist, z.B. das Anfangs-Randwert-Problem. Die gewunschte¨ diskrete Simulation kann durch Standard-Approximationen und Standard-Simulationstechnik erreicht werden. Generell muss man der Nicht-Regularität der Systemoperatoren entgegenwirken, durch Ausnutzen der Eigenschaften der Basis in einem passenden Funktional-Raum, definiert durch die Eigen- und assoziierten Funktionen eines bestimmten Operators. Es gibt zwei Darstellungen der Lösung im Falle der beschränkten Operatoren, eine Reihen-Entwicklung und eine Integralformel mit einem komplexen Konturintegral. Im Falle der unbeschränk- ten Operatoren solltem jedoch Konturintegrals eine Generalisation der Sektor-Operatoren haben. Ist der Operator sektoriell dann ist die Lösung des korrespondierenden Anfangs- Randwertes durch eine holomorphische Halbgruppe gegeben. Sektorielle Operatoren und holomorphe Halbgruppen sind die Basis der abstrakten parabolischen Theorie und fur¨ die Lösung der partiellen Differentialgleichungen vom parabolischen Typ. Diese werden im folgenden Kapitel näher erläutert und untersucht. Die Hauptziele dieser Arbeit sind: (i) den B.3. Einleitung 127 gegenwärtigen Stand der Forschung der Theorie von unbeschränkten und nicht-selbst- adjungierten Operatoren im Zusammenhang mit holomorphischen Halbgruppen darzustellen, (ii) die Lösung von Anfangs-Randwert-Problemen in Zusammenhang mit den kanonischen Systemen von sektoriellen Operatoren zu erweitern, (iii) Funktiona- Trans- formationen fur¨ eine generalisierte Frequenzbereichsdarstellung und fur¨ eine nachfolgende Diskretisierung zu entwickeln und (iv) die dargestellten Methoden fur¨ ausgewählte Dif- ferenzialgleichungen fur¨ technische Interessen anzuwenden. Diese vier Ziele werden durch die Kapitel 2-6 widergespiegelt. Kapitel 2 enthält Grundlagen der Funktionalanalysis wie Räume und Operatoren. Die- ses Kapitel ist wichtig, weil es uns die strenge mathematische Formulierung des Problems erlaubt. In diesem Sinne muss man fur¨ jedes Problem den korrespondierenden Raum sowie die entsprechenden Operatoren definieren, die das physikalische Verhalten oder die physikalische Beziehung beschreiben. Außerdem wird das Problem, i.a. zusätzliche Ein- schränkungen wie Rand- oder Anfangsbedinungen besitzen. Kapitel 3 liefert eine Zusammenfassung der Spektraltheorie unbeschränkter Opera- toren mit kompakter Resolvente. Diese ausfuhrliche¨ Darstellung ist deshalb erfolgt weil sie zum einen in der Technikliteratur nicht leicht zugänglich ist und zum anderen einige fundamentale Arbeiten nur in der russischen Literatur zur Verfugung¨ stehen. Die vorliegende Arbeit konzentriert sich auf generelle lineare Operatoren in Hilbert-Räumen und ihr kanonisches System wie z.B. ihre Eigenwerte und assoziierte Vektoren. Eine wichtige Un- terklasse bildet die Klasse der sektoriellen Operatoren. Sie beschreiben die Evolution von dynamischen Systemen durch eine passende holomorph Halbgruppe. Das Kapitel schließt mit neuen Ergebnissen in der Darstellung bestimmter holomorph Halbgruppen durch das kanonische System des zugrundeliegenden sektoriellen Operators sowie die Erweiterung des verallgemeinerten Resolventen-Operators durch sein kanonisches System. Kapitel 4 fuhrt¨ in die Anfangs-Randwert-Problematik ein, die durch den sektoriellen Differentialoperator mit kompakter Resolventen definiert ist. Gezeigt wird, wie die Metho- den, die in Kapitel 3 zusammengestellt sind, eine Darstellung der Lösung des Anfangs- Randwerte-Problems durch das kanonische System der korrespondierenden räumlichen Differentialoperators ermöglichen. Kurz gesagt, Kapitel 4 präsentiert eine Darstellung der allgemeinen Lösung einer breiten Klasse von Problemen, denen man in der Physik und der Technik häufig begegnet. Kapitel 5 stellt das Hauptergebnis dieser Arbeit vor. Es verlagert den Schwerpunkt von der Mathematik zur multidimensionalen Systemtheorie. Dieser Wechsel ist subtil aber durchaus wichtig. Anstatt die Lösung eines Anfangs-Randwert-Problem direckt zu er- mitteln, z.B. als Funktion von Zeit und Raum, betrachten wir das Anfangs-Randwert- Problem als eine Beschreibung eines multidimensionalen Systems. Die Darstellung al- ler Funktionen unter Berucksichtigung¨ des kanonischen Systems und dessen räumli- chen Differentialoperators wird dann durch die Einfuhrung¨ einer ziemlich allgemeinen 128 B.Titel,Inhaltsverzeichnis,EinleitungundZusammenfassung

Funktional-Transformation, die sog. multifunktionale Transformation (MFT) formuliert. Es beschreibt das multidimensionale System im zeitlichen und im verallgemeinerten räum- lichen Frequenzbereich mit einer Reihe unendlich vieler eindimensionaler Systeme. Je- des dieser eindimensionalen Systeme hat wiederum eine sehr einfache Struktur. Seine Merkmale sind von der Struktur des kanonischen Systems bestimmt. Die korrespondierenden Blöcke sind in Bezug auf die zeitlichen und generalisierten räumlichen Fre- quenzvariablen gegeben. Die gleichförmige Natur dieser Blöcke macht es leicht, geläufi- ge Standard-Diskretisations-Methoden fur¨ eindimensionale Systeme anzuwenden. Dieses Kapitel schließt mit Beschreibung einer diskreten zeitlichen und diskreten räumlichen Approximation der multidimensionalen Systeme. Die neuen Ergebnisse, die in diesem Kapitel vorgestellt werden können folgendermaßen zusammengefasst werden: die Struk- tur von Anfangs-Randwert-Problemen bildet die Blockstruktur der diskreten zeitlichen und diskreten räumlichen multidimensionalen Systeme ab. Diese Ergebnisse stellen eine mathematisch strenge Verbindung zwischen der multidimensionalen Systemtheorie (siehe [Bos82],[DM84]) und ihrer diskreten Computerimplementierung dar. Das vorgestellte Vorgehen wird durch Beispiele physikalischer Probleme in Kapitel 6 verdeutlich.

B.4 Zusammenfassung

Diese Arbeit präsentiert eine Ubersicht¨ uber¨ die moderne Operator-Theorie und ihre An- wendungen. Diese Theorie wurde punktuell erweitert und fur¨ spezielle Anforderungen der Systemtheorie weiterentwickelt. Fur¨ die Realisierung dieser Aussagen wurden Algorith- men angegeben und optimiert. Der Text beabsichtig, die Klasse nicht selbstadjungierter und nicht beschränkter linearer Operatoren so ausfuhrlich¨ darzustellen, wie es fur¨ die praktischen Anwendungen erforderlich ist. Besondere Aufmerksamkeit wird den sektoriellen Operatoren zuteil. Diese haben sehr gunstige¨ Eigenschaften, trotzdem ist der Nachweis dieser Sektor-Eigenschaft i.a. äßerst kompliziert. Ein möglicher Ansatz zur Lösung dieses speziellen Problems findet sich im Anhang A.2. Obwohl diese Klasse von Operatoren be- reits sehr einschränkend ist, ist sie noch hinreichend allgemein, um ein breites Spektrum von Anwendungen zu behandeln (siehe Kapitel 5-6). In der vorliegenden Arbeit werden einige neue Ergebnisse beschrieben, die die Darstel- lung zeit- und raumabhängiger Vorgänge betreffen, wenn sie als Lösung relativ allgemei- ner Anfangs-Randwert-Probleme erzeugt werden können. Insbesondere wurde die Lösung dieses Problems durch strikte Verwendung des kanonischen Systems des zugrunde lie- genden Operators und des zugehörigen adjungierten Operators dargestellt (siehe Kapitel 3). Die derart gewonnene Lösung fuhrt¨ auf naturliche¨ Weise zur Einfuhrung¨ der Mul- tifunktionaltransformation (MFT) bzgl. der Ortsvariablen. Zusammen mit der Laplace- Transformation hinsichtlich der Zeitvariablen wird dabei das ursprungliche¨ transzendente B.4. Zusammenfassung 129

Problem in einen Satz separierter algebraischer Probleme uberf¨ uhrt.¨ Ihre Lösung fuhrt¨ auf Blockstrukturen, die praktisch die Struktur des kanonischen Systems wiederspiegeln (siehe Kapitel 5). Der Hauptvorteil dieser gemischten Zeit-Raum-Frequenzbereichsdarstellung ist, daß eine sehr weite Klasse von Problemen durch dieses einfache Konzept dargestellt und gelöst werden kann. Das klassische Konzept der Systemtheorie, das auf eine Beschreibung mit Ubertragungsfunktionen¨ fuhrt,¨ wird hier auf naturliche¨ Art erweitert. Dies ermöglicht konsequenter Weise die Herleitung passender diskreter Modelle und ihre effiziente Reali- sierung auf dem Computer. Das Ergebnis dieser Arbeit ist eine bestechende, leistungsfähige und vereinheitlich- te Theorie. Der Anwender dieser Ergebnisse muß selbst noch die folgenden Schritte durchfuhren,¨ um eine Implementierung am Rechner zu ermöglichen:

Formulierung des Problems im Operator-Kalkul;¨ • Uberprufung,¨ ob ein sektorieller Operator vorliegt; • Bestimmung des primalen und des adjungierten Operators; • L¨osung der zugeh¨origen Eigenwertprobleme und Bestimmung der entsprechenden • kanonischen Systeme;

Festlegung der Multifunktionaltransformation (MFT); • Durchfuhrung¨ der MFT fur¨ alle in Frage kommenden Signale; • Geeignete Diskretisierung und Approximation (Uberf¨ uhrung¨ in ein ﬁnites Modell) • des kontinuierlichen (inﬁniten) Systems;

Nach Durchfuhrung¨ dieser Schritte kann das diskrete Modell z.B. als Programm reali- siert werden. Es ist bemerkenswert, daß insgesamt gleiche Prinzipien verwendet werden, wie es bei der Untersuchung einfacher elektrischer Netzwerke passiert. Die Anwendung der dargestellten Methode auf reale Probleme wurde im Kapitel 6 durch Beispiele demon- striert. 130 B.Titel,Inhaltsverzeichnis,EinleitungundZusammenfassung 131

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Name: Vitali M. Dymkou

Birth: December 20, 1979 Minsk, Belarus

Family status: Unmarried

Nationality: Belarus

School Education: Sept. 1986 – June 1995 Secondary School, Minsk, Belarus Sept. 1995 – July 1997 Lyceum of Byelorussian State University, Minsk, Belarus

University Sept. 1997 – July. 2002 Mechanics and Mathematics Faculty at the Education: Moscow State University, Moscow, Russia July 2002 Reception of the Dipl.–Math. degree

Professional Live: since Nov. 2002 Scientiﬁc assistant at the Telecommunications Laboratory, University of Erlangen/Nuremberg, Germany