An Introduction to Identification Problems Via Functional Analysis 3110364115, 9783110364118

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 3110364115, 9783110364118

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INVERSE AND ILL-POSED PROBLEMS SERIES

An Introduction to Identification Problems via Functional Analysis

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Also available in the Inverse and Ill-Posed Problems Series: Coefficient Inverse Problems for Parabolic Type Equations and Their Application P.G. Danilaev Inverse Problems for Kinetic and other Evolution Equations Yu.E Anikonov Inverse Problems ofWave Processes A.S. Blagoveshchenskii Uniqueness Problems for Degenerating Equations and Nonclassical Problems S.P. Shishatskii, A. Asanov and ER. Atamanov Uniqueness Questions in Reconstruction of Multidimensional Tomography-Type Projection Data V.P Golubyatnikov Monte Carlo Method for Solving Inverse Problems of Radiation Transfer V.S.Antyufeev Introduction to the Theory of Inverse Problems A.L Bukhgeim Identification Problems ofWave Phenomena - Theory and Numerics S.I. Kabanikhin and A. Lorenzi Inverse Problems of Electromagnetic Geophysical Fields P.S. Martyshko Composite Type Equations and Inverse Problems A.I. Kozhanov Inverse Problems ofVibrational Spectroscopy A.G.Yagola, I.V. Kochikov, GM. Kuramshina andYuA. Pentin Elements of the Theory of Inverse Problems A.M. Denisov Volterra Equations and Inverse Problems A.L Bughgeim Small Parameter Method in Multidimensional Inverse Problems A.S. Barashkov Regularization, Uniqueness and Existence ofVolterra Equations of the First Kind A. Asanov Methods for Solution of Nonlinear Operator Equations V.P.Tanana Inverse and Ill-Posed Sources Problems Yu.EAnikonov, B.A. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P.Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A.Asanov and ER.Atamanov Formulas in Inverse and Ill-Posed Problems Yu.E. Anikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.EAnikonov Ill-Posed Problems with A Priori Information V.V.Vasin and A.LAgeev Integral Geometry ofTensor Fields V.A. Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin Unauthenticated Download Date | 6/15/17 8:09 AM

INVERSE AND ILL-POSED PROBLEMS SERIES

An Introduction to Identification Problems via Functional Analysis

A. Lorenzi

my BPm UTRECHT · BOSTON · KÖLN · TOKYO

2001 Unauthenticated Download Date | 6/15/17 8:09 AM

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© V S P B V 2001 First p u b l i s h e d in 2 0 0 1 ISBN 90-6764-349-1

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To my wife Adriana and my son Luca

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Preface The present book was originated by two courses dedicated to the Theory of Inverse Problems that I held in the academic years 1996-97 and 1997-1998 at the Department of Mathematics "F. Enriques" of the Universitä degli Studi di Milano. The first year the course was delivered to the students of the Doctorate in Computational Mathematics and Operative Research (MA. C. R. O.), while in the second year the course was dedicated to fourthyear students and Ph.D. students in Mathematics and/or MA.C.R.O. The text is essentially devoted to that aspect of the Theory of Inverse Problems which is usually related to as the Identification of Parameters (numbers, vectors, matrices, functions) appearing in differential (or integrodifferential) equations. When the latter describe some physical, chemical, geological, . . . phenomena such parameters are either quite unknown or only partially known (more often they are only qualitatively known). However, their knowledge is in most cases essential, since they describe, e.g., the intrinsic properties of the material or the substance under consideration. As a consequence, such characteristics, usually invariant, are the actual aim of investigation, in contrast to the determination of the so-called "state variable" (i. e. of the solution to the differential equation). The latter is not often of interest in itself, as it varies accordingly with the initial and/or boundary conditions or with the exterior forces, currents, concentrations and so on. The presentation of the topics contained in this book is essentially mathematical, but addressed to people who are not excessively familiar with modern Functional Analysis. The prerequisites are, in fact, restricted to the first 10 sections of Chapter 5 of the famous book by W. Rudin (1991). The main aim of the first two chapters consists in showing how it is possible to solve some identification problems related to linear differential equations when having initially at our disposal only simple tools from the Theory of Matrices and Ordinary Differential Equations. Therefore the problems illustrated in Chapters 1 and 2 essentially have an introductory nature (in some cases we restrict ourselves to setting the problem, in others the problem Brought to you by | New York University Authenticated Download Date | 6/25/17 6:19 AM

iv

A. Lorenzi.

An Introduction to identification Problems

is solved only under simplified, though sufficiently meaningful, hypotheses). Basically we can say that the keystone to solving such problems is the Spectral Theory in finite-dimensional spaces. In turn, this gives a motivation to introduce the Functional Analysis for Linear Bounded Operators as well as the corresponding Spectral Theory. Once we have introduced the basic concepts and have developed a convenient theory for the questions under examination, we again analyze the previous finite-dimensional problems as well as their infinite-dimensional variants from the point of view of the theory of Linear Bounded Operators in abstract Banach spaces. Only subsequently the abstract results are applied to actual cases, after the formal analogies have played their role of psychological persuasion. Consequently, speaking about, or considering, identification problems related to particular classes of partial differential or integrodifferential equations will no longer be as hard as climbing sixth-grade routes. Finally, the last conceptual leap consists in a limited (and guided) introduction to the theory of Analytic Semigroups of Linear Bounded Operators related to linear closed sectorial operators. This approach is made possible by the fact that any such analytic semigroup admits a simple representation by a Dunford integral. Consequently, for its treatment we will only need a good introduction to the Theory of the Riemann Integral in Banach spaces and the basic properties of Banach-valued holomorphic functions. Using the Theory of Analytic Semigroups, we can solve a number of identification problems for parabolic equations related to space dimension 1. However strange it may seem, in the first chapter, which has an introductory character, we have privileged the analysis of the existence and uniqueness of the solutions to identification problems rather than that of their continuous dependence on the data. Apart from the fact that this is more natural for people approaching Inverse Problems when provided with a mathematical culture, we note that for the majority of problems explicitly dealt with in this book, with the exception of those in Chapter 2, the existence and uniqueness of the solution implies almost everywhere its continuous dependence on the data in the same spaces where the existence has been proved. In fact, the basic tool used to solve the identification problems discussed in this book is the Principle of the Contraction Mapping after Banach-Caccioppoli. Such a principle, as is well-known, provides, in addition to existence and uniqueness, also the continuous dependence on the data. However, Chapter 10 is explicitly devoted to a general enough treatment of the continuous dependence on the data. Brought to you by | New York University Authenticated Download Date | 6/25/17 6:19 AM

Preface

ν

On the other hand, in Chapter 2, where we exhibit some simple severely ill-posed identification problems, the question of the continuous dependence is immediately dealt with. Summing up, we can say that the spirit pervading the entire book consists in the attempt of transferring to the reader the capability of determining a suitable system of fixed-point equations, rather than committing everything to the study of resolution techniques, even though these have been sufficiently outlined. I would also like to point out that part of the material contained in Chapter 1 is an elaboration of some problems dealt with in the fine book by A.M. Denisov Introduction to the Theory of Inverse Problems (2000). Finally, I want to thank my students of the 9-th cycle of the Doctorate MA. C. R. 0 . Elisabetta Cordero, Francesca Lunardini, Dario Malchiodi, Simona Perotto and Claudia Salani to have actively attended my course and to have contributed with patience, but with personality and enthusiasm, to a large part of the first version of this book. Moreover, I want to thank Jonathan Hacon, student of the 13-th cycle of the Doctorate in Mathematics for his help and criticism in preparing the final version of the book. I hope that the extensive discussions with my students may have given an original character to the material that was initially only sketched.

Milan, February 2000

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Contents

Introduction

1

Chapter 1. Parameter Identification in Linear Ordinary Differential Equations

5

1.1. Introduction

5

1.2. Examples of inverse problems

8

1.3. Appendix. Exponential of a square matrix

29

Chapter 2. Identification Problems in Hilbert Spaces

33

2.1. An identification problem related to a first-order differential equation

33

2.2. Analysis of the dependence on the data of the solution ({ΐ/η}>λ) to problem (2.1.5), (2.1.6)

36

2.3. The minimization method

39

2.4. A further identification problem

44

2.5. A generalization to the vector case

47

2.6. An identification problem for a second-order differential equation

50

Chapter 3. Proper Riemann Integrals for Banach-valued Functions. Curvilinear Integrals and Banach-valued Holomorphic Functions

57

3.1. Proper Riemann Integrals: basic properties

57

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viii

A. Lorenzi.

An Introduction to Identification Problems

3.2. Curvilinear Integrals and Banach-valued Holomorphic Functions

63

Chapter 4. Riemann-Stieltjes Integrals of Banach-valued Functions

67

4.1. Riemann-Stieltjes integrals over compact intervals

67

Chapter 5. Improper Riemann Integrals for Banach Space-valued functions

83

5.1. Improper integrals

83

Chapter 6. Banach Algebras and Spectral Analysis for Linear Bounded Operators

91

6.1. Introduction

91

6.2. Basic properties of the spectrum of an element

93

6.3. Integration of Banach algebra-valued functions

97

6.4. Banach algebra-valued holomorphic functions and the spectral theorem

99

Chapter 7. Identifying Parameters in First-Order Partial Differential Equations

103

7.1. An identification problem relative to a first-order linear partial differential equation

103

7.2. An identification problem relative to a non-linear first-order partial differential equation

107

Chapter 8. Identification Problems relative to Linear Bounded Operators I

113

8.1. An identification problem relative to a first-order differential equation

113

8.1.1. The singular case

117

8.1.2. The supersingular case

118

8.1.3. Continuous dependence on the data

118

8.2. An identification problem relative to a second-order differential equation

121 Unauthenticated Download Date | 6/15/17 8:11 AM

Contents

ix

8.3. A particular case

124

8.4. An integro-differential identification problem

127

8.5. A onedimensional integro-differential identification problem

134

Chapter 9. Identification Problems Relative to Linear Bounded Operators II

139

9.1. An abstract control problem

139

9.2. A concrete example

141

Chapter 10. Analysis of the Continuous Dependence on the Data

149

10.1. Construction of an abstract model for the analysis of the continuous dependence on the data

149

10.2. Continuous dependence on the data of the solution to the identification problem (8.4.1)

152

10.3. Appendix. Gronwall's generalized inequality

158

Chapter 11. Linear Closed operators and Analytic Semigroups of Linear Bounded Operators

161

11.1. Linear closed operators

161

11.2. Resolvent set and spectrum of a linear operator

165

11.3. Sectorial operators

168

Chapter 12. Cauchy Problems for Linear Abstract Differential Equations Relative to Sectorial Operators and Applications

181

12.1. Abstract differential equations and analytic semigroups. Applications to Cauchy problems

181

12.2. Applications

195

Chapter 13. Identification Problems for Linear Abstract Differential Equations Relative to Sectorial Operators and Applications 13.1. An abstract identification problem

209 209

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χ

Α. Lorenzi.

An Introduction to Identißcation Problems

13.2. An application to a concrete case

212

13.3. An identification problem relative to an abstract non-autonomous first-order differential equation

215

13.4. Analysis of a Cauchy problem with coefficients depending on time only

220

13.5. Analysis of a Cauchy problem with non-homogeneous boundary value conditions on a bounded interval

222

13.6. Solving the identification problem (13.3.1)-(13.3.4)

224

Bibliography

233

List of Symbols

237

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Introduction The first complication we meet when dealing with Inverse Problems is giving a clear definition of what an inverse problem is. Unfortunately such a definition is (or should be) strictly related to that of a direct problem. In turn this cannot be defined in an authoritative way, since those problems that are now called direct have usually been introduced before the inverse ones. Consequently, there has been an historical ordering in naming such problems rather than a theoretical one. However, basing ourselves on our experience concerning differential or integral equations from Applied Sciences, we could call direct a problem of the following type: we are given three (topological, metric, normed ...) spaces Χ, Υ, Z, an element y G Y and a family of (nonlinear and continuous) operators A(z) : X -»· Y depending on a parameter ζ G Z. Under the assumption that the parameter ζ is known we are required to determine χ Ε X solving the equation A(z)(x) = y. According to this, we might call inverse to the previous one the problem of determining both χ Ε X and ζ Ε Ζ when the parameter ζ itself is unknown and some additional information is available. Problems of this type are more properly named Identification Problems. More generally, but more abstractly, we might consider as a direct problem the following: we are given two (topological, metric, normed ...) spaces X, Y, an element y G Y and a (nonlinear and continuous) operator A : X -» Y. We are required to compute y = In other words, we have to compute a mapping at some given point. This may seem a trivial task, but it is usually not so: it is enough to think of the computation of a definite integral when we are given the integrand and the domain of integration, which defines our operator A. In this more general case we are naturally led to dealing with two inverse problems. The first is what we have previously called a direct problem and consists in determining χ in correspondence with a given pair ( A , y ) , while Brought to you by | New York University Authenticated Download Date | 6/25/17 6:20 AM

2

A. Lorenzi.

An Introduction to Identißcation

Problems

the latter is just what we have previously called an inverse problem and consists in determining the pair (A, x) when some additional information is provided. In this book, however, we will restrict ourselves to identifying one or more parameters in the equation modelling the phenomenon we are trying to describe, i. e. we will be dealing with identification problems only. We add that in direct problems the existence, uniqueness and continuous dependence of the solution on the data usually occur. In contrast to this, in inverse problems none of these properties can be taken for granted. It is this peculiarity that makes inverse problems challenging and interesting. We stress the fact that in most identification problems small changes in y G Y result in large changes in (χ, ζ) G X x Z. This occurrence seemingly prevents us from making use of approximation of the data and apparently destroys any reasonable application of Numerical Analysis. Likewise, when dealing with an identification problem, existence and uniqueness of the solution may fail. This occurrence is not so strange to us as the instability we have just now recalled. Indeed, it may simply depend on bad knowledge of the domain of the (nonlinear) operator we are trying to invert. For instance, the operator under consideration maps the pair (x,y) in a "small" subset Yo of Y (e. g. a subset of "smooth" elements), while our data y (e. g. a "rough" physical measurement) belongs to Υ \ Υό· Notice that, in practice, we may not have a good characterization of YQ or, which is worse, the physical measurement may not reasonably yield any element in loin spite of these nontrivial complications, the theory of inverse problems has succeeded in several cases to overcome the theoretical difficulties so that to restore the uniqueness and continuous dependence of solution on the data as well as (sometimes) the existence itself. We conclude this introduction by noticing that around the turn of the century, the French mathematician Hadamard first stated that the existence and uniqueness of the solution and its continuous dependence upon the data characterize any well-posed problem. Unfortunately, he also expressed the view that mathematical models from Applied Sciences should always lead to well-posed problems. In a sense this sounded as a death penalty for ill-posed problems, i. e. for most of inverse problems. However, Courant and Hilbert, although formally supporting Hadamard's position, recognized that "properly posed problems are by far not the only ones which appropriately reflect real phenomena." Yet, the invasion of Applied Sciences and Technology in our life has shown that a lot of important questions in those areas are modelled by matheBrought to you by | New York University Authenticated Download Date | 6/25/17 6:20 AM

Introduction

3

matical problems that turn out to be ill-posed in Hadamard's sense. Consequently, Inverse Problems have become a new and fascinating field in todays mathematical research. This book is devoted to showing the deep links of this subject with Pure Mathematics and Modern Functional Analysis. In order to draw the attention of young people I have tried to keep the expository level of many chapters to an introductory one.

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Chapter 1. Parameter Identification in Linear Ordinary Differential Equations

1.1.

INTRODUCTION

It is well-known that the Cauchy problem ' ip'(t) = A(t)0,

V?G{1,... ,n}.

The next problem we consider differs from those dealt with so far in that in equation

Vn G N.

2/OjTi

Such a system is uniquely solvable (in L 2 ((0,T))) if we make the following assumptions: Brought to you by | New York University Authenticated Download Date | 6/25/17 5:54 AM

Chapter 2. Identification Problems i) {gn} is α orthonormal

in Hilbert

complete system in

35

Spaces

L2((0,T));

+00

ii) Σ Ι 1 θ δ ( ^ ΐ Λ η ) | 2 < + 0 ° · n=l Under such hypotheses the solution to problem (2.1.7) is given, for any t e [0,T], by +0O

λ(ί) = £

log ( ^ k ( i ) ,

+00 yn{t) = exp [ V ] log (^^)

(2.1.8)

/ qm(s)qn(s)

ds

2 / 0 ,η •

The convergence of the series in (2.1.8) is understood in the sense of the L 2 ((0,T))- or C([0,T])-metrics according as {log(^)}e/ c/0,71

2

or j l o g ( — ) } 6 I1 VO ,71

(ii\\qn\\cw,Tl)i,n} are the two sequences appearing in the problems ' v'n(t)-ß{t)qn{t)vn(t)

= 0,

«n(0) = «o>n, k

η e Ν.

νη(Τ) = υι,η,

As space Ω of the admissible data we can choose the following set, where δ, Ν e Rf: Ω = \{yo,n}, {yi,n} 6 I2 :2/0,η Φ 0 Vn G N, 5 < ^ < L i/o, η

n\. >

By such a choice we can show the continuous dependence of (y, λ) on the data with respect to the metrics of C([0,T]) χ L 2 ((0,T)) (or, better, of W^l{0,T) χ L 2 ((0,T))) and 12 χ I2. On the contrary, starting from the estimate +00, n=l

where we take advantage of the hypothesis of equiboundedness of the function sequence {qn}, the space of admissible data changes into Ω = i(yo,n}, K n } G 11 : yo,n Φ ο Vn G N, «5 < ^ < n\, VO,n > Brought to you by | New York University Authenticated Download Date | 6/25/17 5:54 AM

37

Chapter 2. Identißcation Problems in Hilbert Spaces where δ, Ν 6 1 + .

For the sake of simplicity let us analyse the case of the Z 2 ((0, T))-metrics. Observe that from the bound |log(l+i)| i)) = Lmin(|y |,N) 2 (ll/i - wi|2 + |wo - i/o|2) 0 is a metrics on R+ χ R+. Finally, let us consider the difference y — v, where y = {yn} and ν — denote the first components of the solutions to the problems (2.1.5), (2.1.6) corresponding to the data (yo,n, yi,n) and {νοίΤΙ,νιΓ)), ΙΗΙί,^ο,Γ))),

6 N,

9η(β)[λ(β) - ß(s)] ds < ||λ - m||l2((0,t))i

Vn 6 N,

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Chapter 2. Identification Problems in Hilbert Spaces

39

and from equations (2.2.1) and (2.2.3) we easily deduce the following estimates, which also prove the continuity of the map {yn} (yo,n, yi,n)'·

2.3.

THE MINIMIZATION METHOD

Let us now reconsider problem (2.1.1)-(2.1.3). We know that it admits infinitely many solutions. Among these we look for that (or those) which minimize the L 2 ([0,T]) norm of λ. We are going to discuss three distinct cases. First case. Assume we are given only an integral of the function Λ solution to the problem (2.1.1)—(2.1.3):

Let us introduce the closed vector subspace VQ of L2((0, Τ)) and the constant function λο defined by:

As Vo + λο is a closed and convex set of the Hilbert space L 2 ((0,T)), a well-known theorem of Functional Analysis (cf. Rudin, 1966) guarantees the existence of a unique function λ G VQ + λο such that: Ι|λ||ί,2((0,Γ)) =

min ||Ä||L2((0jr)) = pev min ||p + A0||z,2((o,r))AeVo+Ao 0

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40

A. Lorenzi.

An Introduction to Identißcation Problems

In order to explicitly determine the function λ we observe that, after setting ρ = λ — λο G Vo, the previous equation becomes ΙΙλο + p\\h((0,T)) =

lip + λοΙΙ^2((ο,Γ))·

To determine ρ 6 Vo we restrict the functional rT

A(p)= [

Jo

[X0(s) + p(s)]2ds

to the line ρ = p + tu, (r G I ) . As a consequence, the function rT A{p + Tu)= f [\0{s) + p{s) + rv{s)]2 ds Jo

must have an (absolute) minimum in r = 0. Therefore we get: rT

DTA(p + τν)

=2 [ [λο(β) + p ( e ) M a ) d s = 0, τ=ο Jo 10

W £ V0.

(2.3.1)

As Vo is a closed subspace of L 2 ( ( 0 , T ) ) , Vo is a Hilbert space itself. Moreover, since λο = cost = μ/Τ, from (2.3.1) it follows rT

,,

rT

0 = [ [\0(s) + p(s)]v{s)ds = % [ 1 Jo Jo = [ p(s)is(s) ds, Jo Ό

rT

v(s)ds+ [ p{s)u(s)ds Jo

Vi^eVb.

Hence ρ = 0: this implies ufcAM-MV-l

rT .,2 f j d ^ f .

Consequently, AmmÄJ|Ä||i2((o,T))

=^

and X(t) = X0(t) = ^

We[0,Tj.

fT Remark 2.3.1. Observe that 1 Ε Vo: indeed (l,v)L2 = / i^(s)ds = 0 for any u £ Vq.

Jo

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41

Chapter 2. identification Problems in Hilbert Spaces

Second case. Suppose now we are given two integrals of the unknown function λ: rii / λ(β) ds = μι, Jo

ri2 / λ(β) ds = μ2, Jo

(0 < Τι 0,

we get _ μ ι Γ | - μ2Τΐ _ ~ τ,τ2{τ2-τ{) •

a

μ

2{μ2Τ,-μιΤ2) ΓιΤ 2 (Τ 2 -Τι)

As a consequence, λο admits the representation Ao(s) = ao + ßos for any se[o,T). Let us now introduce the Hilbert space 2 Vb = ( i / 6 l ( 0 , T 2 ) : J

1

v(s) da = J

2

i/(s)ds = o}.

Observe that λο ^ Vq. Reasoning as in the previous case we derive the implication min||Äo + p||L2((o,r))

A!{p)v = 0,

Vi/ e Vb,

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42

A. Lorenzi.

An Introduction to Identißcation

Problems

where we have set A(p)=

[T\\o(s)+p(s)]2ds.

Jo

We see that the vector space V0X = £ ( α ι , ο 2 ) , which is orthogonal to Vo in L 2 (0,T 2 ), is generated by the two functions a2(s) = 1,

«1 (s) = Χ[0>Γι](β)>

Vs G

[°>T2]>

χ denoting the characteristic function of the interval [Ο,Τχ]. Indeed rTi / a2{s)i/(s)ds Jo

rTi =

i/(s)ds = 0,

Vi/ G Vo,

/•T2 rT\ / ai(s)i/(s)ds = / u(s) ds = 0, Jo Jo

Vi/ G V0.

Jo

Observe that a 2 G C([0,T 2 ];K), while αχ £ C([0,T2];M). We now project λο onto Vo and VQ1: λ0 = λ! + λ2

where

λ2(β) = 7 + ίχ[0ΙΏ](Β)

VsG[0,T2].

Hence, Ai(s) = Ao(s) — 7 — ίχ[ 0 r i ](s) for any s G [0,T2]. Imposing the condition λχ G Vo, we find that (7, δ) solves the following linear system: rTi Tij + Τ\δ = [ 1 Jo Τ2η + Τλδ = f Jo

λo{s)ds:=μ1,

2

X0{s)ds:=ß2.

Whence we obtain = 7

Ά(μ2-μχ) Τχ(Τ2 — Γ ι ) '

=

Τ2μι - ΤΧμ2 Τ\{Τ2 — Τχ)

Therefore the equation 0 = [ '(λχ + λ 2 + p)vds = [ '(λχ + p)vds, Jo Jo

VI/ G V&,

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43

Chapter 2. Identification Problems in Hilbert Spaces

is equivalent to λ χ + Ρ = 0, since λ χ + Ρ € Vq. Hence, we deduce that Ρ is given by p(s) = - λ ι ( β ) = - α ο -ßos-

7 - ίχ[0ιΓι](β)>

Vs e

[°' Γ 2]·

Consequently, ρ does not belong to C([0, T2]; M)! Third case. We assume that η integrals fTj / A(s)ds = ^·, Jo

(j = 1,... ,n, 0 0 for a. e. χ G (0,T n ).

A F U R T H E R IDENTIFICATION P R O B L E M

Let us consider the sequence of differential problems 'y'n(t)-X(t)yn(t) !/n(0) = 0 , k Vn(T)

= qn(t),

Vi G [0,T], η G N,

(2.4.1)

= J/Ι,Π) Brought to you by | New York University Authenticated Download Date | 6/25/17 5:54 AM

Chapter 2. Identification Problems in Hilbert Spaces

45

where λ G L 2 ((0,T)) is an unknown function. The solution to the Cauchy problem in (2.4.1) with initial data at t — 0 is given by yn(t) = J rt e x p ( ^ λ(σ) dσ) 9 η (β) de,

Vi G [Ο,Τ],

(

A

0



Imposing the condition t = Τ we derive infinitely many equations where A(s) = e x p ( ^ λ(σ)άσ), Vs e [0,T]. y ι,ι = J rT A(s)qn{s)ds , rT If {?n} C C 1 ([0,T]) is a complete orthonormal system in L 2 ((0,T)) and Ση=ι Vi,η < +°°> then Λ is uniquely determined by the formula +00 Λ 5

( ) = Σ

y^nQn(s),

(convergence in

L2((0,T))).

n=l

Let, in addition, the coefficients yi)ri enjoy the properties iii) A (s) > 0 for all s G [Ο,Τ]; iv) Λ G C^O.T] and A'(s) = Ση=ι V i M * ) for all s G [Ο,Τ], Then from the equation e x p ( j λ(σ) da) = A(s),

Vs G [Ο,Τ],

we immediately deduce the following relationships for any s G [Ο,Τ]: fT

da = log A{s)

=

=

Remark 2.4.1. Suppose now that the sequences {qn} and {yi, n } enjoy the following properties: v) lkn||c([o,r]) < M 0 ; vi

) IKIIc([o,r]) < M m a for some α G M+;

vii) {(l + n Q )yi, n } G/ 1 ; Brought to you by | New York University Authenticated Download Date | 6/25/17 5:54 AM

A. Lorenzi.

46

An Introduction

to Identification

Problems

+00

viii)

> ο for any s G [0,T].

n=1

Then λ belongs to C^fO, T]) and satisfies the estimate llAllcwi) ^

o\\{yi,n}h

M

+M1\\{nayhn}\\lu

In practice, instead of the sequence {yi,n}> it is more convenient to assign a positive function Λο G C 1 ([0,T]) and to denote with {yi, n } the sequence of its Fourier coefficients. Let us now consider the sequence of differential problems ' y'n{t) - X(t)yn(t) = qn{t),

Vi G [0, Γ],

< Vn{0) = yo,n, k

(2.4.2)

Vn(T) = yi,n

that generalize problems (2.4.1) since the initial data at t = 0 now may not vanish. Observe that the solution to the Cauchy problem (2.4.2) is given by y „ ( t ) = e x p ( j f λ(σ) da^yo,n + J

e x p [ j λ(σ) da)g n (a) ds, ViG[0,T].

Imposing the additional condition at t = Τ we derive the equations yi,n = A(0)yo,n + [ Α(β)ς„(β) ds, Jο

η G N,

(2.4.3)

where we have set Λ ( s ) = F \{σ)άσ, Js

VsG[0,T].

Let us now assume that {