Application of Operator Theory for the Representation of Continuous and Discrete Distributed Parameter Systems

Transcription

1 Alication of Oerator Theory for the Reresentation of Continuous and Discrete Distributed Parameter Systems Der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Grades Doktor-Ingenieur vorgelegt von Vitali Dymkou Erlangen, 2006

2 Als Dissertation genehmigt von der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg Tag der Einreichung: 22. Dezember 2005 Tag der Promotion: 23. March 2006 Dekan: Prof. Dr.-Ing. Alfred Leiertz Berichterstatter: Al. Prof. Dr.-Ing. habil. Peter Steffen Al. Prof. Dr.-Ing. habil. Krzysztof Galkowski

3 Acknowledgements I would like to thank my suervisors, Al. Prof. Dr.-Ing. habil. Peter Steffen and Priv. Doz. Dr.-Ing. habil. Rudolf Rabenstein, who introduced me to the subject of engineering roblems and gave me the oortunity to work in their grou. I would like to exress them my gratitude for the excellent suervision and suort and of course for the very warm atmoshere during my work and life in Erlangen. Also, I would like to thank Al. 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 esecially Prof. Dr. Günther Greiner for their financial suort. I am deely thankful to all my colleagues from the Telecommunications Laboratory for their atience and suort over many months. I wish to thank Ursula Arnold for her hel with all my administrative questions, Wolfgang Preiss for his software suort and Manfred Lindner for his wonderful refrigerator. Esecially I would like to thank Stefan Petrausch, who was always ready to translate, exlain and answer all my rivate and scientific questions. I am deely 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 Poova for always being there for me. Finally, and most imortantly, I wish to thank my suortive family who acceted my time away from them. A secial 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.

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5 v Contents Abbreviations and Acronyms List of mathematical symbols Variables ix ix xi Introduction 2 Basic Notions from Functional Analysis 5 2. Saces Linear oerators Unbounded oerators Adjoint oerators Linear differential forms Homogeneous boundary conditions Green s formula and associated forms Linear differential oerators Adjoint homogeneous boundary conditions and the adjoint oerators Nonhomogeneous boundary conditions Chater summary Sectral Theory of Oerators 7 3. The resolvent oerator Canonical systems of the rime and the adjoint oerator Exansion of the resolvent oerator by the canonical system Sectorial oerators Definition of sectorial oerators Examles of sectorial oerators Semigrous and canonical systems Definitions

6 vi Contents Connection between the semigrou of the oerator, the resolvent and the Lalace transformation Exansion of the semigrou by the canonical system The C resolvent oerator Canonical systems of the rime and the adjoint oerator Exansion of the C resolvent oerator by the canonical systems Chater summary Mathematical Modelling of Physical Processes Ordinary differential equations Classification of ODEs State-Sace model Partial differential equations Classification of PDEs Initial-boundary-value roblems Solution of initial-boundary-value roblems in the Lalace domain Chater summary Descrition of Multidimensional Systems Multi-functional transformation (MFT) Definition and roerties Inverse MFT Alication of the MFT method to initial-boundary-value roblems with homogeneous boundary conditions Initial-boundary-value roblem with homogeneous boundary conditions Lalace transformation Satial transformation (MFT) Inverse MFT Discretization of the MFT model Satial discretization Time discretization Inverse z transformation Aroximation Alication of the MFT method to initial-boundary-value roblems with nonhomogeneous boundary conditions Initial-boundary-value roblems with nonhomogeneous boundary conditions Lalace transformation Satial transformation (MFT)

7 Contents vii Inverse MFT Discretization of the MFT model Alication of the MFT method to general vector initial-boundary-value roblems Notation General vector initial-boundary value roblem Lalace transformation Satial transformation (MFT) Inverse MFT Discretization of the MFT model Chater summary Examles of the MFT Simulations Heat flow equation Prime and adjoint oerators Eigenroblem Biorthogonality The general solution MFT Simulation Heat flow through a wall Adjoint oerator Eigenvalue roblems Associated vectors Biorthogonality The general solution MFT simulation Telegrah equation Adjoint oerator Eigenvalue roblems Associated vectors Biorthogonality The general solution Chater summary Conclusions 99 A Aendix 0 A. Proof of Theorem A.2 Proof of Examle A.3 Proof of Theorem

8 viii Contents B Titel, Inhaltsverzeichnis, Einleitung und Zusammenfassung 2 B. Titel B.2 Inhaltsverzeichnis B.3 Einleitung B.4 Zusammenfassung Bibliograhy 30

9 Abbreviations and Acronyms ix Abbreviations and Acronyms FTM IIT IBVP MD MFT ODE PDE SLT TFM functional transformation method imulse invariant transformation initial-boundary-value roblem multi dimensional multi-functional transformation ordinary differential equation artial differential equation Sturm-Liouville transformation transfer function model Mathematical symbols for all there exists subset of (or inclusion sign) belongs to / does not belongs to ( ) inverse oeration of ( ) ( ) H hermitian of ( ) ( ) T transosed of ( ) ( ) conjugate comlex of ( ) union intersection converges to <,, >, inequality signs summation sign lim limit +,, infinity Rank( ) rank of ( ) Ran( ) image or range of ( ) Ker( ) kernel of ( ) dim{ } dimension of ( ) Im{ } imaginary art of ( ) Re{ } real art of ( ) H norm in H (, ) H scalar roduct in H san( ) linear san of ( )

10 x List of mathematical symbols inf( ) infimum of ( ) (greatest lower bound) su( ) suremum of ( ) (least uer bound) min( ) minimum of ( ) max( ) maximum of ( ) det( ) ( ) determinant of ( ) Cn k or n k binomial coefficient π circle constant j imaginary unit L{} Lalace transformation T {} multi-functional transformation Z{} z transformation δ m,n Kronecker symbol N set of natural numbers Z set of integers R set of real numbers C set of comlex numbers C n (or R n ) set of comlex (or real) n tules {a n } sequence (a, b) oen 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 u to order n Ω subset of R n Ω boundary of Ω Ω closure of Ω L (Ω) Lebesgue sace on Ω W k (Ω) Sobolev sace on Ω D t or t first-order temoral derivative D x or x first-order satial derivative ẏ first-order temoral derivative of y y first-order satial derivative of y X, Y Banach saces H, H Hilbert saces I identity oerator L, A, C linear oerators L adjoint of linear oerator L rank(l) rank of the oerator L ρ(l) resolvent set of the oerator L

11 Variables xi σ(l) R(s, L) sectrum of the oerator L resolvent of the oerator L emty set Variables λ eigenvalue x continuous sace variable x vector of continuous satial coordinates z comlex frequency variable of the z transformation s comlex frequency variable of the Lalace transformation t continuous time variable i, m,, indices µ, ν

12 xii Variables

13 Introduction The develoment of comuter technologies, the availability of high-seed rocessors and various rogramming languages allow nowadays the researchers in different areas of science to investigate and design numerous algorithms to solve hysical henomena on the comuter. However, to construct high recision models of a real rocess one has to begin with its mathematical descrition and analysis in order to obtain secific characteristics of the considered roblem. This hels to design very efficient numerical methods which can be imlemented directly on the comuter. The ast decades, in articular, have seen a continually growing interest in the alication of functional analysis and esecially general oerator theory to engineering roblems. This develoment is clearly related to the wide variety of alications of both ractical and theoretical interests. Many hysical and information rocesses in various fields ossess identical mathematical structures, that can be described in a common oerator form. Such a generalization allows to construct general algorithms to solve a wide class of roblems. It is, however, not sufficient to analyze a given roblem on a ure theoretical basis. The ractical alication may ose additional constraints like real-time erformance of the system, low-delay requirements or restrictions on comuting ower or memory. Therefore, it is necessary to develo a discrete model for the solution of the roblem which is suitable for comuter imlementation. The general stes in such a develoment are illustrated in Figure.. The resented thesis is oriented towards the develoment of algorithms based on sectral oerator theory and their alication for solving ractical roblems arising in the discrete simulation of continuous systems governed by artial differential equations with unbounded and non-self-adjoint oerators, in general. We deal with the theory of wellosed roblems arising in systems simulation. The need to consider the case of unbounded oerators has been stimulated by numerous engineering alications where system models of hysical rocesses yield this class of oerators. In site of the large oularity of these rocesses, relatively few results of the rigorous mathematical theory have been alied to ractice. The main roblem is that a number of results obtained in the classical case of bounded and self-adjoint oerators is no longer valid or is incomlete in the case of unboundedness and non-self-adjointness, in general. Moreover, as is well known, there is no general sectral theory for unbounded and non-self-adjoint oerators in a Hilbert sace. Sectral theory treats only some secial classes of such oerators (see [GK69]).

14 2. Introduction Physical Problem Mathematical Descrition Set of PDEs Discrete Model Discretization and Aroximation Transformations in the Continuous Domain Simulation Figure.: The general rocedure between the formulation of the hysical roblems and their comuter realization. Here the hysical roblem is a distributed arameter system, described by a set of artial differential equations (PDEs). The most elementary case of finite dimensional saces and matrices as oerators, which we assume to be known to the reader, is well established (see [Gan59]). It has a set of owerful tools which also might be useful to have in the more general case of infinite dimensional saces and general oerators. Of articular interest is the roblem of a canonical form for arbitrary linear transformations in a Hilbert sace 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 secial basis, which in the most simle case is formed by the set of linear indeendent 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 oerators in a general Banach sace can be treated essentially in the same way. M.V. Keldysh was the first who generalized the notion of the Jordan chain of vectors [Gan59] to a wide class of non-self-adjoint bounded oerators (see [Kel5],[Kel7]). 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 [Fur0b],[Fur0a] resectively. For further references see [GK69]. In this work, methods of sectral analysis are alied to the class of unbounded and non-self-adjoint oerators with comact resolvent. It summarizes, extends and generalizes recent research, erformed by the author in [DRS04a],[DRS04b],[DRS05],[DRS]. The basic rinciles are based on the method of Keldysh. The intention is to obtain a sectral decomosition of the solution of initial-boundary-value roblems, which is adated to the structure of the satial differential oerator which leads to its reresentation with resect to its canonical system and its adjoint. It is a consequence of the satial searation that the resulting structure is well adated to the simulation of the comlete system with

15 . Introduction 3 distributed arameters, i.e. the initial-boundary-value roblem. The desired discrete simulation can be finally done by standard aroximation and simulation techniques. To this end, one needs to counteract the non-regularity of the system oerators by exloiting roerties of the bases in suitable functional saces formed by the eigenfunctions and associated functions of certain oerators. There are two reresentations of the solution in the case of bounded oerators, a series exansion and an integral formula with a comlex contour integral. However, in the case of unbounded oerators, the contour integral admits a generalization to sectorial oerators. If an oerator is sectorial, the solution of the corresonding initial-boundary-value roblem is given by an holomorhic semigrou. Sectorial oerators and holomorhic semigrous are basic tools in the theory of abstract arabolic roblems, and in the solution of systems of artial differential equations of arabolic tye. They will be introduced and investigated in the following chaters. The main goals of this work are: (i) to resent the state of the art of the theory of unbounded and non-self-adjoint oerators in the context of holomorhic semigrous, (ii) to exand the solution of initial-boundary-value roblems with resect to the canonical systems of the sectorial oerators, (iii) to design functional transformations for a generalized frequency domain reresentation, and for the subsequent discretization and (iv) to aly the resented methods to selected differential equations of technical interest. These four goals are reflected by Chater 2 through Chater 6. Chater 2 contains the basic concets of functional analysis such as saces and oerators. This section is imortant in allowing us to be consistent with a rigorous mathematical formulation of the roblem. In the sense that for each roblem one needs to define the corresonding sace in which the roblem is considered and the corresonding oerators which reflect the hysical behavior or relations. Moreover, the roblem can ossess some additional constraints such as boundary or initial conditions. Chater 3 rovides a summary of sectral theory of unbounded oerators with comact resolvent. This detailed overview is included there because it is not easily accessible in the engineering literature and also some fundamental results are resented only in the Russian literature. The resentation is centered around general linear oerators in Hilbert saces and their canonical systems, i.e. their eigenvalues, eigenvectors and associated vectors. An imortant subclass form the so-called sectorial oerators. They describe the evolution of dynamical systems through an aroriate holomorhic semigrou. This section closes with new results in the reresentation of a certain holomorhic semigrou by the canonical system of the underlying sectorial oerator and the exansion of the generalized resolvent oerator by its canonical system. Chater 4 introduces initial-boundary-value roblems which are defined by a sectorial differential oerator with comact resolvent. It is shown how the methods comiled in Chater 3 allow to reresent the solution of initial-boundary-value roblems by the

16 4. Introduction canonical systems of the corresonding satial differential oerators. In short, Chater 4 resents a reresentation of the general solution of a wide class of roblems frequently encountered in hysics and engineering. Chater 5 resents the main results of this contribution. It shifts the focus from mathematics to multidimensional systems theory. This shift is subtle, but imortant. Rather than emhasizing the solution of an initial-boundary-value roblem, i.e. a function of time and sace, we consider the initial-boundary-value roblem as the descrition of a multidimensional system. The reresentation of all functions under consideration by the canonical system of its satial differential oerator 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 temoral and the generalized satial frequency domain by an infinite set of one-dimensional systems. In turn, each one-dimensional system has a very simle structure. Its details are determined by the structure of the canonical system. The corresonding blocks are given in terms of the temoral and generalized satial frequency variables. The uniform nature of these blocks makes it easy to aly standard discretization methods well-known for one-dimensional systems. This section concludes with a discrete-time, discrete-sace aroximation of multidimensional systems under consideration. The new results resented in this section may be summarized as follows: The structure of initial-boundary-value roblems is maed to the block structure of discrete-time, discrete-sace multidimensional systems. These results rovide a mathematically rigorous link between the theory of multidimensional systems (e.g. [Bos82],[DM84]) and their discrete comuter imlementation. The resented rocedure is exlained by examles of hysical roblems in Chater 6.

17 5 2 Basic Notions from Functional Analysis In this chater we describe the aroriate context in which one can define and analyze the sectral roerties of unbounded linear oerators, articularly those which are closed and non-self-adjoint. The recise descrition of the oerators will be the main focus of attention throughout this chater. The reader is suosed to be familiar with the fundamental tools of alied functional analysis which will be used systematically throughout the text, otherwise we refer to [Bal76],[Yos80],[Gol66],[Has00]. 2. Saces Before one can start to study the behaviour of oerators one has to choose an aroriate sace on which they act. It turns out that some roerties of the oerator can change deending uon the saces on which it acts. Definition A set X is called a linear sace (or vector sace) over a field R (C) if the following conditions are satisfied:. 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 multilication 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. x = x;

18 6 2. Basic Notions from Functional Analysis 9. α(x + y) = αx + αy; 0. (α + β)x = αx + βx; Definition 2 A linear sace X is called a normed linear sace, if for every x X, there is associated a real number x X, the norm of the element x, such that:. x X 0 and x X = 0 iff x = 0; 2. αx X = α x X, α R(C); 3. x + y X x X + y X (triangle inequality); Definition 3 A sequence {x n } in a normed linear sace X converges to the element x X if x n x X 0, n. Definition 4 A sequence {x n } in a normed linear sace X is said to be a Cauchy sequence if ε > 0 N n > N, N : x n+ x n X < ε. Definition 5 If every Cauchy sequence is convergent, a normed linear sace is said to be comlete. A Banach sace is a comlete normed linear sace. Definition 6 A subset S of a normed linear sace X is dense in X if the closure of S is the entire sace X ( S = X). Definition 7 A subset S of a normed linear sace X is called comact if every (infinite) sequence in X has a convergent subsequence. Definition 8 An inner roduct (or scalar roduct) on a linear sace X defined over the field R (C) is a ma (, ) : X X R(C) such that. (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 sace X with an inner roduct (, ) is called an inner roduct sace. We can consider any inner roduct sace as a normed linear sace (X, X ) by defining the norm as X = (x, x). Definition 9 A Hilbert sace is an inner roduct sace which is comlete as a normed sace under the induced norm.

19 2.. Saces 7 The saces taken into consideration are the usual saces of comlex valued functions defined on R M or on an oen set Ω of R M. The following two classes of Banach saces are widely used in functional analysis and the theory of differential equations. Definition 0 [Ada75] The Lebesgue sace L (Ω), < +, is the set of all comlexvalued functions y(x) = y(x, x 2,...,x M ) defined in Ω and such that ( y L(Ω) = where the integral is taken in the Lebesgue-sense. Ω ) y(x) dx <, (2.) Definition [Ada75] Let k be a ositive integer, and < +. The Sobolev sace W k (Ω) is the set of all comlex-valued functions y(x) = y(x, x 2,..., x M ) defined on Ω and such that ( ) y W k (Ω) = D a y(x) dx <, (2.2) Ω a k where a = (α,...,α M ) is a vector of nonnegative integers, a = α + + α M and D a = α x α... α M x α M is the weak or distributional artial derivative of order a. M The following inequalities will be useful in the next chaters. Let < < + and let the conjugate exonent of be defined by Hölder s inequality fgdx Ω holds for any functions f L (Ω) and g L (Ω). Minkowski s inequality Ω + =. (2.3) fg dx f L(Ω) g L (Ω) (2.4) f + g L(Ω) f L(Ω) + g L(Ω) (2.5) holds for any f, g L (Ω). Minkowski s integral inequality dx 2 f(x,x 2 )dx Ω 2 Ω dx f(x,x 2 ) dx 2 Ω Ω 2 (2.6) holds for any integrable function f(x,x 2 ) defined on the set Ω Ω 2 (x Ω R M, x 2 Ω 2 R M 2 ) and such that integral in the right art of (2.6) is finite.

20 8 2. Basic Notions from Functional Analysis 2.2 Linear oerators Let X,Y be Banach saces over the comlex field C and A be a ma acting from X to Y. Denote by D(A) the set of elements from X where the maing 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 2 The maing A : X Y is a linear oerator if i) the set D(A) is linear subsace of X; ii) the maing A is linear on D(A), i. e. A(α x + α 2 x 2 ) = α Ax + α 2 Ax 2, x, x D(A), α, α 2 C. (2.7) Note that D(A) is a linear subsace of X, while Ran(A) is a linear subsace of Y. The case D(A) X and/or Ran(A) Y may occur Unbounded oerators One of the basic instruments in functional analysis is a bounded linear oerator. The need to consider the case of unbounded oerators has been stimulated by numerous engineering alications where system models of the hysical rocess yield this class of oerators. Moreover, the most differential oerators are unbounded when considered as acting on any usual Banach or Hilbert saces. Definition 3 The linear oerator A : X Y is bounded if i) D(A) = X; ii) there exists a constant c > 0 such that Ax Y c x X for all x X. The norm A of the linear bounded oerator A is defined as Ax Y A = su, (2.8) x =0 x X Remark The infimum of all such constants c is equal to the norm of the oerator A, inf(c) = A. Thus, the linear oerator A can be unbounded if the condition i) or/and the condition ii) of Definition 3 are violated. It is known that in the case where X and Y are normed linear saces the boundedness of the linear oerator A is equivalent to its continuity. It is said that the linear oerator A : X Y is continuous at the oint x 0 D(A) if x n x 0 X 0 leads to Ax n Ax 0 Y 0. The linear oerator A is continuous if it is continuous at an arbitrary oint x 0 D(A).

21 2.2. Linear oerators 9 Examle [KF8] Unbounded oerator Denote by C([0, 2π]) the Banach sace of all continuous comlex valued functions defined on the comact interval [0, 2π] R with the norm x C = max x(t) ( x C = x ). 0 t 2π Define the differential oerator A : C([0, 2π]) C([0, 2π]) by the formula Af = f, with D(A) = C ([0, 2π]) C([0, 2π]), where C ([0, 2π]) is the set of all continuously differentiable functions on [0, 2π]. It is obvious that C ([0, 2π]) is a dense subsace in C([0, 2π]). This oerator is linear since d dt (α dx x + α 2 x 2 ) = α dt + α dx 2 2 dt. This oerator is not bounded. To show this consider, for examle, the sequences x n (t) = sin nt, x n C, dxn dt C = max n cosnt = n. Hence su dxn 0 t 2π dt C = +, which roves n the unboundedness of the oerator A. Examle 2 [KF8] Bounded oerator In contrast to the case above, consider the differential oerator A acting from the Banach sace C ([0, 2π]) with the norm x C = max x(t) + max 0 t 2π 0 t 2π x (t) to the Banach sace C([0, 2π]) by the same formula: Af = f. This oerator is linear and bounded since Ax C = x C = max 0 t 2π x (t) max x(t) + max 0 t 2π 0 t 2π x (t) = x C, x C ([0, 2π]). Remark 2 The given examles show that the oerator given by the same formula can be bounded or unbounded with resect to different norms associated with the underlying Banach saces Adjoint oerators Now we consider Hilbert saces H and H 2 over the field C with inner roducts denoted as (, ) H and (, ) H2, resectively. L is assumed to be a linear oerator acting from H into H 2 with the domain D(L) H. Definition 4 Let {f n } n= be a sequence of elements f n D(L). If from f n f, n and Lf n g, n it follows that f D(L) and Lf = g then the oerator L is called a closed oerator. Definition 5 A bounded oerator is called a comact oerator if it mas each bounded subset of H into a comact subset of H 2.

22 0 2. Basic Notions from Functional Analysis Definition 6 Let the oerator L have a dense domain D(L) in H. The adjoint oerator L is that oerator with domain D(L ) containing all those elements g H 2 for which there exists an element h H such that the following equality holds: In this case, by definition, L g = h. Equivalently we can write: f D(L) (Lf, g) H2 = (f, h) H. (2.9) f D(L), g D(L ) (Lf, g) H2 = (f, L g) H. (2.0) Lemma [Yos80] Let the oerator L have a dense domain D(L) in H. Assume that the oerator L exists and its domain D(L ) is dense in H 2. Then (L ) exists and satisfies (L ) = (L ). (2.) In the following we assume that the Hilbert saces H i are identical, i.e H = H 2 = H. The scalar roduct and its corresonding induced norm will now be simly denoted by (, ) and, resectively Linear differential forms In this section we introduce the basic concet of linear differential forms and their adjoints. We will consider the set C (n) ([a, b]) of continuously differentiable functions u to the order n over finite interval [a, b]. An arbitrary linear differential form l alied to an arbitrary element y C (n) ([a, b]) is defined as n l(y)(x) := n (x)y (n) (x) + n (x)y (n ) (x) (x)y(x) = ν (x)y (ν) (x), (2.2) where the functions ν (x), ν = 0,, n are continuously differentiable u to the order ν over the given interval [a, b] Homogeneous boundary conditions Denote by y a, y a,, y a (n ) and y b, y b,, y(n ) b the values of the function y(x) and its derivatives at the oints x = a and x = b, resectively. Let m linear indeendent forms be given by ν=0 U µ (y) :=α µ 0 y a + α µ y a + + αµ n y(n ) a + +β µ 0 y b + β µ y b + + β µ n y (n ) b, α µ i, βµ i C. (2.3)

23 2.2. Linear oerators For an arbitrary number m 2n of such kind of forms we say that the equalities U µ (y) = 0, µ =,...,m (2.4) reresent homogeneous boundary conditions. If m = 2n, it is obvious, that from (2.4) follow the equalities y a = 0, y a = 0,, y (n ) a = Green s formula and associated forms Let us introduce the following vectors with n comonents y a = y a y () a y (2) a y (n ) a, y b = y b y () b y (2) b y (n ) b, z a = z a z () a z (2) a z (n ) a, z b = z b z () b z (2) b z (n ) b. (2.5) We consider the integral b a ν (x)y (ν) (x)z(x) dx for ν = 0,, n. Reeated integration by arts shows that for any y, z C (n) ([a, b]) and ν (x) C (ν) ([a, b]) the equality b [ ν (x)y (ν) (x)z(x) dx = ν (x)z (x)y (ν ) (x) ( ν (x)z (x)) y (ν 2) (x) + a ] x=b b + ( ) ν ( ν (x)z (x)) (ν ) y(x) + ( ) ν y(x) ( ν (x)z(x) ) (ν) dx x=a a (2.6) holds. The sum of all equalities (2.6) over the indices ν = 0,, n gives us that the differential form l(y) satisfies the following identity Here (l(y), z) = P(η, ζ) + (y, l (z)). (2.7) n l (z) = ( ) n ( n z)(n) + ( ) n ( n z)(n ) z = ( ) ν ( ν z)(ν) (2.8) ν=0 reresents an associated (formally adjoint) differential form to l(y) and [ n ν ] x=b P(η, ζ) = ( ) ν i ( ν z ) (ν i) y (i ) ν= i= x=a (2.9) is a bilinear form of the variables η = (y a,y b ) and ζ = (z a,z b ). The relation (2.7) is called the Green s formula.

24 2 2. Basic Notions from Functional Analysis If we introduce the matrix Q as follows, Q = () ( ) n (n ) n ( ) n 2 n ( ) n n 2 () ( ) n 2 (n 2) n ( ) n 2 n 0 3 () ( ) n 3 (n 3) n n 2 () n + (2) n n n n () n n n (2.20) the bilinear form P(η, ζ) can be written in comact form as P(η, ζ) = y T b Q(b)z b y T a Q(a)z a. (2.2) Linear differential oerators Define the differential oerator L : C (n) ([a, b]) C([a, b]) as follows: i) the domain D(L) of the oerator is D(L) = { y C (n) ([a, b]) : U µ (y) = 0, µ =,...,m } ; (2.22a) ii) the oerator is given by the formula Ly = l(y), y D(L). (2.22b) Assigning different boundary conditions (2.4) to a fixed differential form l will generate different differential oerators L with different domains, in general. Remark 3 The recise secification of the domain of the oerator is very imortant since it turns out that the same linear differential form (2.2) with different boundary conditions (2.4) roduces different differential oerators L, in general. For these reasons when using the term differential oerator we shall understand that we have already chosen the boundary conditions if we are thinking in more alied terms, or that we have already chosen the recise domain of definition of the oerator if we are thinking in abstract terms. It is obvious that D(L) is a linear subsace 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 oerator with domain D(l) = C (n) ([a, b]). Also, we always have the inclusion D(L) D(l).

25 2.2. Linear oerators Adjoint homogeneous boundary conditions and the adjoint oerators We consider m linear indeendent forms U,, U m of the variables y a, y a,, y(n ) a, y b, y b,, y(n ) given by α0 α αn β0 β βn α0 2 α 2 αn 2 β0 2 β 2 β 2 n α0 m α m αn m β0 m β m βn m b m 2n y a y () a y (n ) a y b y () b y (n ) b 2n = (2.23) Using block-matrices A,B and the vector U of aroriate dimensions we rewrite (2.23) in more comact form as [ A B ] [ y a y b ] = U U 2 U m m [ U ]. (2.24) If m < 2n we can add 2n m arbitrary linear indeendent forms U m+,, U 2n in such a way that the resulting forms U,, U 2n are linear indeendent as well. Then we can write. α0 α αn β0 β βn α0 2 α 2 αn 2 β0 2 β 2 βn 2 α 0 m α m αn m β0 m β m βn m α0 2n α 2n αn 2n β0 2n β 2n βn 2n or equivalently 2n 2n y a y () a y (n ) a y b y () b y (n ) b = 2n U U 2 U 2n, 2n (2.25)

26 4 2. Basic Notions from Functional Analysis [ ] [ ] [ ] A B ya U =, (2.26) A 2 B 2 y b U 2 where A 2 and B 2 are block-matrices of aroriate dimensions and U 2 is a vector of length 2n m. In this case, by inverting the matrix in 2.26, each y a and y b can be reresented as a linear combination of these forms Thus [ ya y b ] = [ C D C 2 D 2 ] [ U y a = C U + D U 2, U 2 ]. (2.27) (2.28a) y b = C 2 U + D 2 U 2. (2.28b) Substitution of these reresentations into P(η, ζ) yields P(η, ζ) = (C 2 U + D 2 U 2 ) T Q(b)z b (C U + D U 2 ) T Q(a)z a = U T [CT 2 Q(b)z b C T Q(a)z a] + U T 2 [DT 2 Q(b)z b D T Q(a)z a] (2.29) = U T V 2 + U T 2 V, where V 2 = [V 2n, V 2n,...,V 2n m ] T = C T 2 Q(b)z b C T Q(a)z a, V = [V 2n m, V 2n m 2,..., V ] T = D T 2 Q(b)z b D T Q(a)z a, (2.30a) (2.30b) and V i, i =,...,2n are linear indeendent forms of the variables z a and z b, too. Now we rewrite the bilinear form (2.29) in terms of U i and V i P(η, ζ) = U V 2n + + U m V 2n m+ + U m+ V 2n m + + U 2n V. (2.3) Substitution of the rime boundary conditions U = 0, U 2 = 0,, U m = 0 yields We say that the exressions P(η, ζ) = U m+ V 2n m + + U 2n V. (2.32) V = 0, V 2 = 0,, V 2n m = 0 (2.33) are the adjoint boundary conditions associated with the rime boundary conditions: U = 0, U 2 = 0,, U m = 0. The urose of the secial artitioning into two sets U i and V i

27 2.2. Linear oerators 5 is that under conditions (2.4) and (2.33) the exression 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 V,, V 2n m. Since the adjoint oerator L to the oerator L can be defined only in some Hilbert sace (see Definition 6), let C (n) ([a, b]) H, where H is the Hilbert sace with the scalar roduct (y, z) = b a y(x)z (x)dx (for examle we can set H = L 2 ([a, b])). Therefore, the adjoint linear differential form l (z) and the adjoint boundary conditions V,, V 2n m generate the adjoint oerator L defined as: i ) the domain D(L ) is D(L ) = {z C (n) ([a, b]) : V µ (z) = 0, µ =,...,2n m}; (2.34a) ii ) the oerator acts by the formula L z = l (z), z D(L ). (2.34b) Nonhomogeneous boundary conditions In the case of nonhomogeneous boundary conditions m U µ (y) = φ µ, φ µ φ µ > 0 (2.35) the differential oerator L is defined by the following conditions i) the domain D(L) of the oerator is µ= D(L) = { y C (n) ([a, b]) : U µ (y) = φ µ, µ =,..., m } ; (2.36a) ii) the oerator is given by the formula Ly = l(y), y D(L). (2.36b) When no confusion may arise, we introduce some new notations. In articular, denote by l the differential form of the oerator under consideration, by L 0 the oerator with homogeneous boundary conditions and by L φ the oerator with nonhomogeneous boundary conditions. Thus and L 0 : D(L 0 ) = {y C (n) ([a, b]) : U µ (y) = 0, µ =,...,m}, (2.37) L φ : D(L φ ) = {y C (n) ([a, b]) : U µ (y) = φ µ, µ =,..., m}. (2.38) Remark 4 Note that due to the nonhomogeneous boundary condition the oerator L φ is no longer linear.

28 6 2. Basic Notions from Functional Analysis 2.3 Chater summary This chater described briefly the basic notions and facts from functional analysis and accomanied by the aroriate author s interretation. More exactly, the main focus was on linear oerators in Hilbert saces. It has been shown that to define an oerator in rigorous mathematical terms, in addition to its formal definition it is necessary to define the sace where oerator acts and to secify its domain. The corresonding sace contains those elements which satisfy the natural constraints such as continuity, differentiability or integrability. In turn, the domain of the oerator is a subsace of this abstract or general sace, and consists of such a set of elements which satisfy the desired roerties or more secific restrictions generated in many cases by the nature of hysical rocesses. Also, since we have defined the Hilbert sace, it is ossible to consider some relations between the elements such as scalar roduct and norm. And of course, as we will see in the next chaters, the notion of the adjoint oerator is a very owerful tool in the investigation of the rime oerator. In this chater we have restricted our detailed consideration to the case of ordinary differential oerators of order n which act on the subsace C (n) ([a, b]) of scalar functions of the Hilbert sace L 2 ([a, b]). But in a similar way the case of saces with functions f : R C M and ordinary differential oerators on it can be treated (see [Naj68]). Moreover, the case of artial differential oerators can be also considered analogously. However, since artial differential oerators include more comlicated boundary conditions (due to the multidimensional argument), the generalization to a such comact resentation described above involves the corresonding generalization of the D Green s formula (2.7) to its MD analogy (see [Tay], [Sho77]). Also, the strong requirements to have a continuous artial derivative for functions in a usual (classical) sense (C (n) ([a, b])), can be substantially weakened by introducing the corresonding Sobolev saces and weak or distributional artial derivatives. For further details about Sobolev saces for use in some alication we refer to [Ada75], [Eva02]. On the one hand, the main attemt has been made in this chater to resent all necessary basic material of oerator theory in sufficient generality, in order to be able to maniulate in abstract terms. On the other hand, it was intended to cover in detail the rocedure of defining the oerators in secial cases, in order to understand the alied asects of the resented theory.

29 7 3 Sectral Theory of Oerators In the theory of linear and time-invariant systems and network theory one of the fundamental concets is the system function or transfer function. It hels to describe the system, its behaviour and its transfer roerties from the inut to the outut (see [GRS0]). Most alications lead to models with meromorhic transfer functions. This means such functions are comletely determined by all their oles and zeroes u to a multilicative constant. We remark that the number of oles and zeroes might be infinite. In any case, the class of rational functions is included in the more general class of meromorhic functions. Since rescribed oles and zeroes are easy to imlement in realizing structures, frequency-domain models are often used. In rincile, there is a similar situation in the general oerator theory, as will become obvious in this chater. The alication of sectral methods in the resence of general oerators rovides similar advantages as the introduction of transfer function above. In this chater we will resuose a closed linear oerator L : H H, the domain D(L) of which is dense in the Hilbert sace H. In addition we will consider the oerator L s = (si L) : H H, s C, where I is the identity oerator. The investigation of the oerator L s is called the sectral theory for the oerator L. It includes the characterization of the distribution of the values of s for which L s has an inverse and the roerties of this inverse when it exists. Similar to the former case, the general theory of the inverse of L s leads to sectral decomosition based on a certain set of oles. As we shall see later, the inverse of the oerator L s lays a similar role in obtaining the solution of initial-boundary-value roblems as the system function in common systems. 3. The resolvent oerator Definition 7 If s 0 is such that the image Ran(L s0 ) is dense in H and L s0 has a bounded inverse (s 0 I L), we say that s 0 is a regular oint of L and we denote this inverse (s 0 I L) by R(s 0, L) and call it the resolvent oerator of L at s 0. Definition 8 The set of all regular oints of the oerator L is called the resolvent set, and is denoted by ρ(l). The set C\ρ(L) is called the sectrum of the oerator L and is denoted by σ(l).

30 8 3. Sectral Theory of Oerators In the following we assume that neither the sectrum nor the resolvent set is emty. Thus, for any s ρ(l) the resolvent oerator is defined as R(s, L) = (si L), s ρ(l). (3.) Let L : H H be the adjoint oerator of L. Since the oerator L is assumed to be 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 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.). Definition 9 The comlex number λ σ(l) is called an eigenvalue of the oerator L, if Ker(λI L) {0}. (3.5) Any vector e = e(λ) Ker(λI L) \ {0}, is called an eigenvector of the oerator L corresonding to the eigenvalue λ. It will turn out for all relevant cases that the eigensaces Ker(λI L) have finite dimension, i.e. P = P(λ) = dim { Ker(λI L) } <. Hence, we can assume that a basis B 0 of this eigensace is given by the set B 0 = B 0 (λ) = { e,0, e 2,0,, e P,0 } (3.6) of the linear indeendent eigenvectors e,0 corresonding to the eigenvalue λ. For later uroses we have introduced the subscrit 0. Using the simle roerty (3.4) of the resolvent oerator of L we are already able to obtain a useful characterization of the sectrum of L. Lemma 2 [Bal76] Let L be a closed oerator with dense domain in H. If there exists s 0 ρ(l) such that the resolvent oerator R(s 0, L) is comact, then R(s, L) is comact in any oint of the resolvent set ρ(l); moreover, the sectrum σ(l) consists of a discrete set of oints and hence is denumerable.

31 3.. The resolvent oerator 9 The imortant statement of this lemma is the fact that it guaranties that the sectrum of L consists of a countable set of oints. Consequently, all those values λ are isolated oints. They are the oles of the resolvent oerator R(s, L). It can therefore be reresented in a certain neighborhood of such a singular oint λ by the following Laurent series exansion R(s, L) = + ν= (s λ) ν R ν, (3.7) where the coefficients R ν are oerators and can be reresented by the formula ( ) R ν = (s λ) ν R(s, L)ds (3.8) 2πj s λ =r for sufficiently small r > 0. Here j denotes the imaginary unit. The next two lemmas contain some roerties of the oerators R ν in the series exansion (3.7). Lemma 3 [Yos80] a) the oerators R ν in (3.7) commute with each other and the oerator L; b) the oerator R is a rojection oerator, i.e. R 2 = R ; c) the following formula is valid R ν + (λi L)R ν = δ 0,ν I, ν Z. (3.9) Here δ µ,ν is the standard Kronecker symbol defined as {, µ = ν δ µ,ν = 0, µ ν. (3.0) Lemma 4 [Fur0a][Fur0b] Let L be a closed oerator with dense domain and comact resolvent oerator 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 owers of (s λ) i.e. R(s, L) = + ν= M (s λ) ν R ν, (3.) R ν = 0, ν < M; (3.2)

32 20 3. Sectral Theory of Oerators e) the oerator R can be characterized by the condition Ran(R ) = Ker([λI L] M ); (3.3a) the oerators R ν, ν > 0 have finite dimensional images dim{ran(r ν )} <, ν > 0 (3.3b) and satisfy the relations R ν = ( ) ν (λi L) ν R, ν =,...,M. (3.3c) It is obvious that the Laurent series exansion of the adjoint resolvent oerator R(s, L ) in the neighborhood of λ σ(l ) is R(s, L ) = + ν= M (s λ ) ν R ν. (3.4) In articular, it has the same order M of the ole λ and the corresonding oerators R ν are adjoint to the oerators R ν. Thus, by analogy with Lemma 3 and Lemma 4 we have R ν + (λ I L )R ν = δ 0,νI, ν Z, dim{ran(r ν)} = dim{ran(r ν )} <, ν > 0, Ran(R ) = Ker([λ I L ] M ), (3.5) R ν =( ) ν (λ I L ) ν R, ν =,...,M. 3.. Canonical systems of the rime and the adjoint oerator An imortant art of matrix theory is the transformation which leads to Jordan form of a matrix and the corresonding Jordan chains of vectors (the sets of eigenvectors and generalized eigenvectors) (see [Gan59]). There is a similar situation in the general sectral theory of oerators. M.V. Keldysh was the first who generalized the notions of the Jordan chains of vectors to a more general classes of oerators (see [Kel5],[Kel7]). He has considered the canonical systems for the oerator encil of the form L(λ) = A 0 +λba + +λ n B n A n +λ n B n in the Hilbert sace H, where A 0, A,...,A n are arbitrary comact oerators, B is a self-adjoint oerator and Bf 0, if f 0. Here we reformulate the definitions introduced by Keldysh in a context which is suitable for our urose. We consider a closed linear oerator L : H H, the domain D(L) of which is dense in the Hilbert sace H.

33 3.. The resolvent oerator 2 Definition 20 For a fixed eigenvalue λ and a fixed eigenvector e,0, where P(λ) (P(λ) is the dimension of the eigenvector sace belonging to the eigenvalue λ, see (3.6)), we say that the vector e,m 0 is an associated vector (or generalized eigenvector) of order m of the eigenvector e,0, if it satisfies the following equations (λi L)e,0 = 0, e,0 + (λi L)e, = 0,, e,m + (λi L)e,m = 0. (3.6) The maximum ossible number m of associated vectors is denoted by M = M (λ) and (M + ) is called the multilicity of the eigenvector e,0. The set E = E (λ) = { e,0, e,,, e,m } (3.7) is called the comlete chain (or Keldysh chain) of associated vectors that belongs to the eigenvector e,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 multilicities M + are less or equal to the order M of the corresonding ole in the Laurent series exansion (3.7) for the considered eigenvalue λ, M + M. Definition 2 For each fixed eigenvalue λ we consider the union E(λ) = E (λ) (3.8) P(λ) = over the set of linear indeendent eigenvectors e,0, =,..., P(λ). E(λ) is called a canonical system of the oerator L corresonding to the fixed eigenvalue λ. P(λ) The number N(λ) = M + + M M P(λ) + = P(λ) + M is called the multilicity of the considered eigenvalue λ. In the following we assume without loss of generality that the sequence of eigenvectors is ordered with resect to their multilicities (M ) = M M 2 M P(λ) 0. To illustrate the notion used in (3.6)-(3.8), the canonical system in (3.8) can be arranged as =

34 22 3. Sectral Theory of Oerators E(λ) = { E (λ), E 2 (λ),, E (λ),, E P(λ) (λ) } e,0 e 2,0 e,0 e P(λ),0 = e, e 2, e, e P(λ), e,m e 2,m e,m e P(λ),m e P(λ),MP(λ) e,m e 2,M2 e,m. (3.9) The first row contains the set of P(λ) eigenvectors, and the th column shows the comlete chain of associated vectors corresonding to the eigenvector e,0 as defined in (3.6). In addition to the canonical system E(λ) with the elements e,m in (3.8) we define another canonical system E consisting of elements ǫ,m, that corresonds to the eigenvalue λ of the adjoint oerator L. This collection of eigenvectors together with the corresonding set of associated vectors satisfies the following equations (λ I L )ǫ,0 = 0, ǫ,0 + (λ I L )ǫ, = 0,, ǫ,m + (λ I L )ǫ,m = 0,, ǫ,m + (λ I L )ǫ,m = 0, for each =,, P(λ ). By analogy with (3.6)-(3.8), we introduce the comlete chains E = E (λ ) = { ǫ,0, ǫ,,, ǫ,m } (3.20) (3.2) and the canonical system of the oerator L for a fixed eigenvalue λ as E (λ ) = P(λ ) = In addition, we will use the following notations : E = E (λ ). (3.22) E(λ i ) (3.23) λ i σ(l)

35 3.. The resolvent oerator 23 is called the canonical system of the oerator L; E = E (λ λ i ) (3.24) i σ(l ) is called the canonical system of the adjoint oerator 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 After roer normalization of ǫ,m for each eigenvalue the introduced canonical systems E and E are biorthogonal (e,m (λ i ), ǫ l,n (λ w )) = δ i,wδ,l δ m,m n. (3.25) Proof. see Remark 8 in Aendix. To illustrate the biorthogonality relations of the canonical systems we assume that it has for examle only 4 (M = 3) elements in the chain E (λ). In Figure 3. solid lines mean nonorthogonal relations, and dotted lines show orthogonality. e,0 ǫ,3 e, ǫ,2 e,2 ǫ, e,3 ǫ,0 Figure 3.: Biorthogonality relations of a canonical systems Definition 22 A system A = {a i } i Z, a i H is called minimal if and only if none of the elements a i A is contained in the closed linear san of the remaining elements a i / san(a\{a i }). (3.26) 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 indeendent. Remark 6 Note, that for an arbitrary linear indeendent system it does not necessary follow the existence of the corresonding biorthogonal system. For examle, the sequence {a i = x i } i=0 is linear indeendent system in L 2 ([0, ]) and has no biorthogonal system.