An Inverse Source Problem for a One-dimensional Wave ...archive.kaust.edu.sa/.../292839/1/SharefaAsiriThesis.pdf4 ABSTRACT An Inverse Source Problem for a One-dimensional Wave Equation:

An Inverse Source Problem for a One-dimensional

Wave Equation: An Observer-Based Approach

Thesis by

Sharefa Mohammad Asiri

In Partial Fulfillment of the Requirements

For the Degree of

Masters of Science

King Abdullah University of Science and Technology, Thuwal,

Kingdom of Saudi Arabia

May, 2013

2

The thesis of Sharefa Mohammad Asiri is approved by the examination committee

Committee Chairperson: Taous-Meriem Laleg-Kirati

Committee Member: Ying Wu

Committee Member: Christian Claudel

3

Copyright ©2013


All Rights Reserved

4

ABSTRACT

An Inverse Source Problem for a One-dimensional Wave

Equation: An Observer-Based Approach


Observers are well known in the theory of dynamical systems. They are used to

estimate the states of a system from some measurements. However recently observers

have also been developed to to estimate some unknowns for systems governed by

partial differential equations.

Our aim is to design an observer to solve inverse source problem for a one-

dimensional wave equation. Firstly, the problem is discretized in both space and

time and then an adaptive observer based on partial field measurements (i.e mea-

surements taken form the solution of the wave equation) is applied to estimate both

the states and the source. We see the effectiveness of this observer in both noise-free

and noisy cases. In each case, numerical simulations are provided to illustrate the

effectiveness of this approach. Finally, we compare the performance of the observer

approach with Tikhonov regularization approach.

5

ACKNOWLEDGEMENTS

I would like to express my gratitude to all those who gave me the possibility to

complete this thesis. I sincerely would like to thank my supervisor, Prof. Taous-

Meriem Laleg-Kirati, for her support, encouragement, and advice. I also take this

opportunity to express a deep sense of gratitude to Dr. Chadia Zayane for her valuable

information and guidance. Finally, an honorable mention goes My husband, Ahmad

Ali, for his understandings, supports and great patience.

6

TABLE OF CONTENTS

Examination Committee Approval 2

Copyright 3

Abstract 4

Acknowledgements 5

List of Abbreviations 8

List of Figures 9

List of Tables 14

1 Introduction 15

2 Introduction to Inverse Problems 19

2.1 Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.1 Examples of Inverse Problems . . . . . . . . . . . . . . . . . . 20

2.2 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Functional Analysis Approach . . . . . . . . . . . . . . . . . . 24

2.2.2 Stochastic Inversion Approach . . . . . . . . . . . . . . . . . . 24

2.2.3 Regularization Approach . . . . . . . . . . . . . . . . . . . . . 25

2.3 Tikhonov regularization . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.1 Tikhonov Approach . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.2 Selecting the regularization parameter . . . . . . . . . . . . . 28

2.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Observers’ Theory 31

3.1 State-Space Representation . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Deriving the state-space representation . . . . . . . . . . . . . 32

3.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7

3.3 Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 A Tikhonov Regularization to Solve Inverse Source Problem for

Wave Equation 39

4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Inverse Problem’s Operator and its Properties . . . . . . . . . . . . . 40

4.2.1 Construct the Operator by Solving the Direct Problem [1], [2] 40

4.2.2 Operator’s Properties . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.3 Well-posedness of the Inverse Problem . . . . . . . . . . . . . 46

4.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 An Observer to Solve Inverse Source Problem for Wave Equation 54

5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.1 A State-Space Representation for the Wave Equation . . . . . 54

5.1.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3.2 Noise-Free Case . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3.3 Noise-Corrupted Case . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Comparison Between Observer and Tikhonov . . . . . . . . . . . . . 75

5.4.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Conclusion 90

References 92

Appendices 99

8

LIST OF ABBREVIATIONS

GCV Generalized Cross Validation

NCP Normalized Cumulative Periodogram

SISO Single-input, Single-output

MIMO Multiple-input, Multiple-output

ODEs Ordinary Differential Equations

PDEs Partial Differential Equations

BCs Boundary Conditions

ICs Initial Conditions

IBVPs Initial Boundary Value Problems

SNR Signal-to-Noise Ratio

FDM Finite Difference Method

CFL Courant-Fridrichs-Lewy condition

MSE Mean squared Error

9

LIST OF FIGURES

2.1 Direct problems and inverse problems. . . . . . . . . . . . . . . . . . 20

2.2 Behavior of the total error of regularization approach corresponding to

α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1 The stability region of continuous linear time-invariant systems is in the

left, and the stability region of discrete linear time-invariant systems

is in the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Observer principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 The exact source f with f = K−1uT . . . . . . . . . . . . . . . . . . . 49

4.2 Measurements with and without noise. . . . . . . . . . . . . . . . . . 50

4.3 The exact source f and the estimated source f without Tikhonov reg-

ularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 The selected regularized parameter through L-curve, GCV, and NCP. 51

4.5 The exact source f and the estimated source f after Tikhonov reg-

ularization (left) where α was chosen using Discrepency Principle of

Morozov, the error is on the right . . . . . . . . . . . . . . . . . . . . 51

4.6 The exact source f and the estimated source f after Tikhonov regu-

larization (left) where α was chosen using L-curve, the error is on the

right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.7 The exact source f and the estimated source f after Tikhonov regu-

larization (left) where α was chosen using GCV, the error is on the

right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.8 The exact source f and the estimated source f after Tikhonov reg-

ularization (left) where α was chosen using NCP, the error is on the

right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1 The exact source f(x) = 3 sin(5x). . . . . . . . . . . . . . . . . . . . . 63

5.2 The state ξ for one-dimensional wave equation where c2 = 0.9, f(x) =

3 sin(5x), and zeros boundaries and initial conditions. . . . . . . . . . 63

10

5.3 (a): the exact source f (blue) and the estimated source f (black) using

full measurements. (b): the relative error of the source estimation in %. 64

5.4 State error in the noise-free case with full measurements; (a): the state

error ξ − ξ. (b) the state relative error in %. (c): the state error, in

%, after removing the initial phase. (d): the state relative error after

removing the outliers where most of them concentrated in the initial

phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.5 The estimated source in different time steps starting form the initial

guess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6 (a): the exact source f (blue) and the estimated source f (black)

using partial measurements (50% of the state components taken from

the middle). (b): the relative error of the source estimation in %. . . 66

5.7 Zoom-in for the relative error in Figure 5.6.b . . . . . . . . . . . . . . 66

5.8 State error in the noise-free case with partial measurements (50% of

the state components taken from the middle); (a): the state error

ξ− ξ. (b) the state relative error in %. (c): the state error, in %, after

removing the initial phase. (d): the state relative error after removing

the outliers where most of them concentrated in the initial phase. . . 67

5.9 (a): the exact source f (blue) and the estimated source f (black) us-

ing observer with partial measurements (50% of the state components

taken from the end). (b): the relative error of the source estimation

in %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.10 Zoom-in for the relative error in Figure 5.9.b . . . . . . . . . . . . . . 68

5.11 State error in the noise-free case with partial measurements (75% of

the state components taken from the end); (a): the state error ξ − ξ.(b) the state relative error in %. (c): the state error, in %, after



5.12 (a): the state ξ after adding a white noise with a standard deviation

σξ = 0.0078. (b): the output z after adding a white noise with a

standard deviation σz = 0.0104. . . . . . . . . . . . . . . . . . . . . . 70


observer with full measurements. (b): the relative error of the source

estimation in %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.14 Zoom-in for the relative error in Figure 5.13.b . . . . . . . . . . . . . 71

11

5.15 State error in the noisy case with full measurements; (a): the state

error ξ − ξ. (b) the state relative error in %. (c): the state error, in

%, after removing the initial phase. (d): the state relative error after

removing the outliers where most of them concentrated in the initial

phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71



the middle). (b): the relative error of the source estimation in %. . . 72


5.18 State error in the noisy case with partial measurements (50% of the

state components taken from the middle); (a): the state error ξ − ξ.(b) the state relative error in %. (c): the state error, in %, after





the end). (b): the relative error of the source estimation in %. . . . . 74

5.20 State error in the noisy case with partial measurements (75% of the

state components taken from the end); (a): the state error ξ−ξ. (b) the

state relative error in %. (c): the state error, in %, after removing the

initial phase. (d): the state relative error after removing the outliers

where most of them concentrated in the initial phase. . . . . . . . . . 74



observer with full measurements. (b): the relative error of the source

estimation in %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


Tkhonov with full measurements. (b): is the corresponding relative

error of the source estimation in %. . . . . . . . . . . . . . . . . . . . 79

5.24 Comparison between observer and Tikhonov in noise-free case with full

measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79


observer with partial measurements in the middle. (b):the relative

error of the source estimation in %. . . . . . . . . . . . . . . . . . . . 80

12


using Tkhonov with partial measurements in the middle. (b): the

corresponding relative error of the source estimation in %. . . . . . . 80

5.27 Comparison between observer and Tikhonov in noise-free case with

partial measurements taken from the middle. . . . . . . . . . . . . . . 81


observer with partial measurements at the end. (b):the relative error

of the source estimation in %. . . . . . . . . . . . . . . . . . . . . . . 82


using Tkhonov with partial measurements in the middle. (b): the


5.30 Comparison between observer and Tikhonov in noise-free case with

partial measurements taken from the end . . . . . . . . . . . . . . . . 83


observer with full measurements. (b):the relative error of the source

estimation in %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84


using Tikhonov with full measurements in the noisy case where α was

selected manually. (b): the corresponding relative error of the source

estimation in %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.33 Comparison between observer and Tikhonov in noise-corrupted case

with full measurements. . . . . . . . . . . . . . . . . . . . . . . . . . 85


using observer with partial measurements taken from the end. (b):the

relative error of the source estimation in %. . . . . . . . . . . . . . . 86


using Tkhonov with partial measurements from the middle. (b): the


5.36 Comparison between observer and Tikhonov in the noise-corrupted

case with partial measurements taken form the middle. . . . . . . . . 86


using observer with partial measurements taken from the end. (b):

the corresponding relative error of the source estimation in %. . . . . 87


using Tkhonov with partial measurements taken from the end. (b):

the corresponding relative error of the source estimation in %. . . . . 88

13

5.39 Comparison between observer and Tikhonov in the noise-corrupted

case with partial measurements taken from the middle. . . . . . . . . 88

14

LIST OF TABLES

4.1 Values of α using the four different approaches and the total error

‖f − f‖2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1 Relative errors for noise-free case (full measurements) . . . . . . . . . 79

5.2 Relative errors for noise-free case (partial measurements form the middle) 81

5.3 Relative errors for noise-free case (partial measurements form the end) 82

5.4 Relative errors for the noisy case (full measurements) . . . . . . . . . 85

5.5 Relative errors for the noisy case (partial measurements from the middle) 87

5.6 Relative errors for noisy case (partial measurements from the end) . . 88

5.7 MSE in the noisy case (partial measurements) . . . . . . . . . . . . . 89

15

Chapter 1

Introduction

Wave equation is a crucial hyperbolic partial differential equation that arose early to

describe the motion of vibrating strings and membranes. It is a basis in many areas

such as seismic imaging and imaging steep dipping structures. In wave applications,

if the aim is to find the propagation of the wave, exactly or approximately, this is

called a direct problem. However, in most of these applications, the direct solution is

not always needed but often the wave speed, the initial state, or the source need to

be estimated. This kind of problem is called inverse problem.

Inverse problem is a research area that uses observed data (measurements) to

obtain knowledge about physical systems. Solving these inverse problems helps to

determine the location of oil in oil exploration applications [3], to find the shape of

a scattering object, for example in computer tomography [4], to detect tumors in

medical imaging [5], to get an image for the subsurface in marine survey acquisition,

etc. For instance, marine survey acquisition application, air guns are generally used

as a source to send sound waves into the water. The waves propagate in the water and

so can be modeled mathematically using the wave equation. These waves reflect, and

the reflected waves are received by hydrophones (sensors) located on stream where the

measurements are obtained. These receivers measure the velocity of the waves and

the time elapsed from the source to the hydrophones. Finally, these measurements

are transferred to an image of the subsurface of the earth. The first and second steps

16

of this experiment are actually the direct problem, while the final step is the inverse

problem. The same principle is used in sonography, seismology and many other

fields. However, inverse problems are usually ill-posed in the sense of Hadamard, who

proposed that the solution of any well-posed problem should satisfy three proprieties:

existence, uniqueness, and stability. If one of the proprieties is not satisfied, then the

problem is ill-posed, [6].

Inverse problems for wave equations have been studied for many decades, see [7,

8, 9, 10, 11, 12, 13, 14]. The classical way to solve these problems, or inverse problems

in general, is to minimize a suitable cost function which is solved using optimization

techniques. For instance, in [10] and [13], inverse problems for wave equation were

solved using the Tikhonov regularization method which led to optimization problems.

In [10], the optimization problem was solved using an iterative numerical algorithm

called the Pulse-Spectrum Technique. Although this method was excellent with a

two dimensional wave equation and robust with a one dimensional wave equation, it

requires many computations. This computational cost was reduced in [11] by using

the Garlerkin method to solve the appeared integral equation. However, in [13],

Tikhonov regularization was combined with the widely convergent homotopy method

in order to obtain a good initial guess for the iterative method of the optimization

problem. In [12], a new minimization algorithm was proposed to solve an inverse

problem for the wave equation where the unknown is the speed wave function inside

a bounded domain.

In all the previous work, optimization methods are required. These methods are

,in general, heavy computationally especially in the case of high order systems, or if

there are a large number of unknowns. Therefore, they require an extensive storage.

Moreover, the convergence of the optimization methods is affected by the initial guess

and the stop condition.

The objective of this thesis is to solve the inverse source problem for the wave

17

equation using an alternative method based on the concept of observers, which are

well-known in control theory. An observer is used to estimate the hidden states of a

dynamical system using only the available input and output measurements [15]. The

first observer was introduced by Luenberger in the 1960s; his observer is well known

for state estimation in linear dynamical systems [45]. Since then different types of

observers have been proposed to deal with specific applications; for instance, robust

observers for models corrupted by disturbances [49], adaptive observers for the joint

estimation of states and parameters [46], and optimal observers [52]. One advantage

of using an observer solving inverse problems is that it only requires the solution

to the direct problems which are in general well-posed and well-studied. Moreover,

observers operate recursively; thus, their implementation is straightforward with low

computational cost, especially when it comes to high order systems.

Recently researchers proposed to solve inverse problems for wave equations using

observers [16, 17, 18, 19, 20]. In [16] states and parameters are estimated using an

observer depending on a discretized space for a mechanical system. In [17], the initial

state of a distributed parameter system was estimated using two observers; one for

the forward time and the other for the backward time. Similarly in [18], but the

forward backward observer has been adapted to solve inverse source problem for the

wave equation. An adaptive observer was applied in [19] for parameter estimation

and stabilization for one-dimensional wave equation where the boundary observation

suffers from an unknown constant disturbance. A similar work was studied in [20],

however, the unknown was the state and the boundary observation suffers from an

arbitrary long time delay.

One of the difficulties in solving inverse problems consists in the lack of mea-

surements available. Indeed for physical or practical constraints, we usually do not

have enough measurements to estimate all the unknowns which makes the problem

unobservable in the sense that we can not estimate all the states or unknowns from

18

the available measurements. For this reason, the observers in the previous works

based on partial measurements. However, these measurements, in [16, 17, 19, 20],

were taken from the time derivative of the solution of wave equation. This kind of

measurements gives a typical observability condition which has a positive effect on

the stabilization, but it is less readily available than filed measurements. Hence, some

authors sought to solve inverse problems for wave equation using observers based on

partial filed measurements, i.e. measurements taken from the solution of the wave

equation, as in [21], [22], and [23]. In addition, the observer in [22] was based on a

discretized system, in both space and time, which can be considered as a different

methodology comparing with the previous works. This methodology improved the

convergence properties of observers based on partial field measurements.

In this thesis, we use an observer to solve an inverse problem for the one-dimensional

wave equation where the source is unknown. The problem is first discretized in both

space and time and then an adaptive observer based on partial measurements of the

field is applied to estimate both the states and the source of the discrete dynamical

system. Moreover, we test the method in two cases: noise-free case noise-corrupted

case.

This thesis is organized as follows: Chapter 2 provides the reader with an in-

troduction to inverse problems field and regularization methods. Chapter 3 is on

observers’ theory. In Chapter 4, inverse source problem for a one-dimensional wave

equation using Tikhonov regularization is studied, and the same problem but using

observer is studied in Chapter 5. In addition, a comparison between observer and an

original Tikhonov approach is presented also in Chapter 5. Finally, The conclusion

is drawn in Chapter 6.

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Chapter 2

Introduction to Inverse Problems

Inverse problem field has appeared since the first half of the 20th century. It is a

research area that uses the observed data (measurements) to obtain some information

about a physical system. In other words, it is a determination of some unknowns from

measurements and other known information.

Any problem can be either a direct problem or an inverse problem. In the di-

rect problems, we try to find the solution which describes phenomena, for example,

the propagation of heat or waves where the model parameter, the initial state, and

boundary properties are known. However, model parameters such as speed, density,

and conductivity are often unknowns, and we need to estimate them; this is an inverse

problem.

Inverse problems arise in many fields. They arise in geophysical field such as

seismic imaging [24], in image processing like medical imaging [25], in physical sciences

as in deconvolution problems for ground based telescopes [26], and in many other

areas. This reflects the importance of inverse problem.

In this chapter, general concepts on inverse problems with some examples are in-

troduced. Then, the definition of a well-posed versus ill-posed problem is presented.

In the third section, three classical approaches to overcome the ill-posedness of a

problem are presented, which are functional analysis, stochastic inversion, and regu-

larization. Finally, the last section focuses on the Tikhonov regularization and how

20

to choose the regularization parameter.

2.1 Inverse Problem

Consider the following mathematical model:

K(x) = y, (2.1)

where y denotes the data (measurements), x denotes the unknown that can be some

parameters or the input, and K an operator which represents the relation between the

output and the unknowns (see Figure. (2.1)). The problem is called direct (forward)

problem if x is known and y to be determined, and it can be solved directly from

(2.1). If the data y is measured, and the unknown x is to be estimated, this is called

inverse problem. In this case, the problem consists in inverting the operator K, which

is not easy in general.

Figure 2.1: Direct problems and inverse problems.

2.1.1 Examples of Inverse Problems

Example 1. Linear equation:

Consider the linear equation y(ϑ) = aϑ + b. First, the problem can be written

in the form Kx = y where K =

(ϑ 1

)and x =

a

b

. If the constants a

21

and b are known, then it is easy to solve the direct problem to obtain y for any ϑ.

However, if y is given, and the problem is to find the constants a and b that satisfy

the linear equation; this is an inverse problem. Solving this inverse problem is fitting

straight line to the data while solving the direct problem is evaluating a polynomial

of first order; accordingly, it is clear through this simple example that solving inverse

problem is not easy as its corresponding direct problem.

Example 2. Integral of First Kind:

Frequently, inverse problems can be written as an integral of first kind such as

inverse wave equation (see Chapter 5), inverse heat equation [27]. It describes a linear

relation between the data and the unknown [28] . If the operator K is an integral of

first kind, then it can be written as:

K(x(t)) =

∫ 1

0

k(t, s)x(s)ds = y(t), 0 ≤ t ≤ 1 (2.2)

where k is a kernel which is known, x is unknown, and y denotes the data (measure-

ments).

Example 3. Wave Equation:

Consider the following one-dimensional wave equation

∂2u(x, t)

∂t2− c2∂

2u(x, t)

∂x2= Q(x, t),

u(0, t) = g1(t), u(l, t) = g2(t);

u(x, 0) = r1(x), ut(x, 0) = r2(x);

0 ≤ x ≤ l, t ≥ 0;

(2.3)

where u(x, t) is the displacement, c is the wave speed, and Q(x, t) is the source

function. g1(t) and g2(t) are the boundary conditions. r1(x) and r2(x) are the initial

position and the initial velocity, respectively, and they represent the initial conditions.

22

The direct problem is to find the solution u(x, t) such that the wave speed c,

the source Q(x, t), the boundary conditions, and the initial conditions are known.

If one of the parameters is unknown, such as c, g1, g2, r1, r2, or Q, and we would

like to determine it using available measurements, then this problem is called inverse

problem.

Based on these unknowns, the inverse problems can be classified as follows: if the

problem is required to estimate the wave speed or generally any model parameter,

then it is called coefficients inverse problem or inverse media problem. If the source

is the unknown then the problem is inverse source problem. The inverse problem is

called retrospective if the initial conditions are unknowns, and it is called boundary

problem if the boundary conditions are knowns. These are not all the classes; there

are some mixed cases e.g. the unknowns are the initial and boundary conditions; for

more details in the classification of inverse problems see [29].

As seen through these examples, some questions on the existence of the inverse of

K raise, this leads to the definition of well-posedness.

2.2 Well-posedness

The definition of a well-posed problem has been given in 1902 by Hadamard. In the

sense of Hadamard, a mathematical problem is well posed if and only if the following

three conditions are satisfied [6]:

1. Existence: the solution of the problem exists.

2. Uniqueness: the problem has at most one solution.

3. Stability: the solution depends continuously on the data which can be related

to the stability when dealing with numerical solution.

Next definition gives a mathematical description for the three conditions.

23

Definition 1. Let X and Y be normed spaces, K : X → Y a (linear or non-linear)

mapping. The equation Kx = y is called well-posed if the following holds:

1. Existence: For every y ∈ Y there is (at least one) x ∈ X such that Kx = y.

2. Uniqueness: For every y ∈ Y there is at most one x ∈ X such that Kx = y.

3. Stability: The solution x depends continuously on y; that is, for every sequence

(xn) ⊂ X with Kxn → Kx (n→∞), it follows that xn → x (n→∞).

A problem that loses one of the previous conditions is called ill-posed problem.

In fact, inverse problems are usually ill-posed. If the solution does not exist, then it

can be solved by extending the solution’s space; and if it is not unique, then adding

additional information or some constraints can solve the uniqueness issue. However,

the stability is a crucial condition and it is mostly violated.

The existence and the uniqueness conditions can be explained simply through

Example 2; in case k(t, s) = 1, (2.2) will be

∫ 1

0

x(s)ds = y(t). (2.4)

Calculating the left hand side of (2.4) gives a constant because it is independent on

t. If y(t) is not constant, then (2.4) has no solution. Thus, the existence condition is

not satisfied.

Now, if we assume that the solution exists, the solution is not unique. Because

it can be found an infinite number of solutions x(s) such that the integral over [0, 1]

gives the same constant and thus satisfy (2.4) exactly.

The stability issue can be more clear if we choose x(s) to be x(s) = sin(ηs). Then

by taking the infinite limit for (2.2) and using Riemann-Lebesgue lemma [30] (see

24

Theorem 7 in Appendix A), one can get:

∫ 1

0

k(t, s) sin(ηs)ds→ 0 as η →∞. (2.5)

From (2.5), it is clear that a very small change in the data y leads to a huge change

in the solution x; thus, the problem is not stable [31].

Generally, inverse problems are solved by minimizing the error between predicted

data and observed data (measurements) i.e. we seek to minimize the following cost

function:

J(x) = ‖Kx− y‖2p, (2.6)

where p ≥ 1.

To restore the numerical stability of an inverse problem, one can distinguish be-

tween three approaches: functional analysis, stochastic inversion, and regularization.

A description for each approach is provided in the next section.

2.2.1 Functional Analysis Approach

Here, the ill-posedness is solved by changing the space of the variables and their

topologies. This change is under physical considerations [32].

2.2.2 Stochastic Inversion Approach

In this approach, all the variables are considered as random variables in order to

take into account the uncertainties, and the solution is a probability distribution for

the unknowns. Bayesian approach is one of stochastic inversion approaches. In this

approach priory information on the solution is expressed as prior distribution, then

it is combined with the data to obtain posterior distribution through Bayes’ rule.

Ultimately, the solution is the maximizer of this a posterior distribution (MAP), [31]

[33].

25

2.2.3 Regularization Approach

The idea of regularization methods is to define a regularized solution that depends on

the data and takes into account available prior information about the exact solution.

In order to obtain better understanding regularization approaches, which is part of

the contributions of this thesis, some definitions and theorems in regularization are

presented [34]. Definitions and theorems on operator’s properties such as linearity,

boundedness, compactness, and self adjointness can be found in Appendix A.

Definition 2. Let K : X −→ Y be a compact and one-to-one operator between two

Hilbert spaces X and Y such that K(x) = y, x ∈ X and y ∈ Y . A regularization

strategy can be defined as a family of operators Rα(y) : Y −→ X, that depend on a

parameter α > 0 such that

limα→0

Rα(K(x)) = x, (2.7)

i. e. the operators RαK converge pointwise to the identity; then Rα is a regularized

operator for K(x) = y.

Theorem 1. Let Rα : Y −→ X be a regularization operator where dim(X) =∞ then

there exists a sequence (αi) with ‖Rαi‖ → ∞ as i→∞.

The defined notation for a regularization strategy in Definition 2 is based on

unperturbed data; that is the regularizer Rαy converges to the exact solution x for

y = Kx. However, if perturbed data yδ was considered such that ‖y − yδ‖ ≤ δ, then

a regularized solution can be defined as xδα = Rαyδ. Thus, the error in the solution

can be formulated as

‖xδα − x‖ = ‖Rαyδ − x‖

= ‖Rαyδ −Rαy +Rαy − x‖

≤ ‖Rα‖‖yδ − y‖+ ‖RαKx− x‖

26

thus,

‖xδα − x‖ ≤ δ‖Rα‖+ ‖RαKx− x‖. (2.8)

It appears form (2.8) that the total error between the exact solution x and the regu-

larized solution xδα occurs due to two sources of errors. The first error is the error due

to uncertainty in the measurements δ‖Rα‖; this error goes to infinity when α goes to

zero (by Theorem 1). The second one is the regularization error ‖RαKx− x‖, and it

goes to zero when α goes to zero (by Definition 2). Figure (2.2) illustrates the effect

of α on the two types of errors.

Figure 2.2: Behavior of the total error of regularization approach corresponding to α.

The choice of the regularization operator defines the regularization strategy. Dif-

ferent regularization techniques aim to construct this operator; for example, Tikhonov

regularization, Landweber iteration, total variation, and so on. Tikhonov regular-

ization (1977) is the most widely used technique for regularizing discrete ill-posed

problems [31].

27

2.3 Tikhonov regularization

2.3.1 Tikhonov Approach

In a simple description, it is a least square problem with a penalization term that

includes priory information multiplied by a regularization parameter α > 0. Thus,

Tikhonov functional of the system Kx = y can be written as

Jα(x) =1

2‖Kx− y‖22 +

α

2‖x‖22 (2.9)

One can minimize (2.9) as follows:

∂Jα∂x

= 0,

⇒ K∗(Kx− y) + αx = 0,

⇒ (K∗K + αI)x = K∗y.

Thus, the regularized solution can be written as:

xα = Rαy, (2.10)

such that Rα = (K∗K + αI)−1K∗; where K∗ is the adjoint operator of K and I is

the identity operator.

Consequently, the regularization parameter α has to be chosen dependently on δ

such that the right hand side of (2.8) is minimum as possible. Different methods exist

to find this parameter such as Discrepancy Principle of Morozov, L-curve, Generalized

Cross Validation (GCV), and Normalized cumulative Periodogram (NCP) analysis,

[35]. All these methods seek to find the best trade-off between these two errors. In

the next section, a short description for these methods is shown.

28

2.3.2 Selecting the regularization parameter

Discrepancy Principle of Morozov, [36] [37]

Definition 3. In Morozov’s Discrepancy Principle, α = α(δ, yδ) is chosen and xδα

such that for 1 < µ1 ≤ µ2

µ1δ ≤ ‖K(xδα)− yδ‖ ≤ µ2δ holds.

In this principle, the measurements error (δ) is assumed to be known, which is

not often the case. Thus, small α can be chosen to gain the accuracy. This method

is simple to apply and good for theoretical study, but it is risky when dealing with

real data because δ is unknown in general.

L-curve, [38] [39] [40]

In this method, the regularization parameter α is chosen such that the regulariza-

tion and perturbation errors are balanced. No guarantee that good results will be

obtained using this method, but in general it is a good heuristic approach.

Generalized Cross Validation, [41]

Generalized Cross Validation, GCV, method is derived from a classical statistical

technique called cross validation. In cross validation, we leave out the ith element if

the data, yi, and then compute the regularized solution xδα(i)

such that xδα(i)

= R(i)α y(i)

where (i) indicates that yi was left out. Then, yi is estimated such that yi = Kixδα(i)

.

The aim is to select α that minimizes the estimated errors for all i. Finally, after

some technical steps, the following formula of generalized cross validation is obtained:

arg minα

1

m

m∑i=1

(Kixδα − yi

1− trace(Rα)/m

)2

. (2.11)

29

where m refers to the number of measurements. GCV is considered as a robust

method for finding the regularization parameter.

Normalized cumulative Periodogram (NCP) analysis, [41], [31]

NCP method is based on the Fourier transform of the residual vector. Let r =

y −Kxδα. Then after taking the discrete Fourier transformation, one can get:

ζ = F(r) = (ζ1, ζ2, · · · , ζq+1)T , (2.12)

where q =n

2, and n refers to the dimension of x. After that, we define the Peri-

odogram P vector with the coefficient

pj =|ζ2|+ |ζ3|+ · · ·+ |ζj+1||ζ2|+ |ζ3|+ · · ·+ |ζq+1|

, j = 1, · · · , q. (2.13)

Finally, we search for regularization parameter α such that the coefficients of P lie

(approximately) on a right line. One of the advantages of NCP is that it is not

expensive computationally. Also, it is good when the noise is white noise.

As we see, inverse problems lead to optimization problems. Optimization tech-

niques are heavy computationally especially if the number of parameters is high. Also

for large systems, they require extensive storage. Moreover, they need good initial

guess and clever stop condition to obtain good results; which are generally not easy

tasks. For more details on solving inverse problems see [34], [41], [42], and [32].

2.4 Chapter Summary

From the previous discussion, it appears that solving inverse problems is not easy

at least by comparing them with their corresponding direct problems. Moreover,

they are in general ill-posed. We highlighted three standard approaches to overcome

30

ill-posedness, which are functional analysis, stochastic inversion, and regularization.

In the regularization approach, Tikhonov regularization for constructing the regular-

ized operator was explained. Then different methods for choosing the regularization

parameter have been presented. These methods aim to restore the stability while

minimizing the error between the regularized and the exact solutions.

There is an alternative approach derived form control theory for solving inverse

problems [43]. It is an observer based-approach. Observers operate recursively which

implies their implementation low computational cost. The concepts of observers are

highlighted in the next chapter.

31

Chapter 3

Observers’ Theory

We recall basic definitions of the observers. In the first section, we introduce the

definition of the state space representation, how it is derived, and how can the stability

be studied through this representation. Then, the observability property, which is an

essential condition for applying observer, is presented. Finally, the observer method

is defined and presented.

3.1 State-Space Representation

State-space form is a mathematical representation that can describe the dynamics

in physical systems such as biological systems, mechanical systems, and economic

systems. Linear continuous-time state-space systems can be written in the following

state-space representation [44]:

ξ = A(t)ξ(t) +B(t)ν(t),

z(t) = C(t)ξ(t) +D(t)ν(t),(3.1)

where ξ(t) ∈ Rn is the state vector, z(t) ∈ Rm is the output vector, ν(t) ∈ Rr is the

input vector , A is the state matrix of dimension n × n, B is the input matrix of

dimension n×r, C is the output matrix with dimension m×n, and D is transmission

(feedthrough) matrix from input to output with dimension m × r. Moreover, in the

32

system (3.1) the first equation is called the state equation while the second equation

is called the output equation. Also, (3.1) has the following solution [44]:

ξ(t) = Φ(t, t0)ξ(t0) +

∫ t

t0

Φ(t, τ)B(τ)ν(τ)dτ, (3.2)

where Φ(t, τ) = U(t)U(t)−1, and U(t) is the solution of U(t) = A(t)U(t). If the

system is invariant then Φ(t, t0) can be defined as:

Φ(t, t0) = eA(t−t0). (3.3)

Similar to system (3.1), linear discrete-time system can be put on the state-space

form: ξ(k + 1) = A(k)ξ(k) +B(k)ν(k),

z(k) = C(k)ξ(k) +D(k)ν(k),(3.4)

where k refers to the time step.

3.1.1 Deriving the state-space representation

We define the state variables using the input-output differential or difference equations

to obtain the state-space representation. The general idea is to move from an nth

order of the differential equation to n fist order differential equations. To illustrate

the procedure, we consider the following linear ordinary differential equation which

represents a single input, single output (SISO) system

dnz(t)

dtn+ an−1

dn−1z(t)

dtn−1+ · · ·+ a1

dz(t)

dt+ a0z(t) = ν(t). (3.5)

33

Now the state variables are defined as the following:

ξ1 = z,

ξ2 = dzdt,

ξ3 = d2zdt2,

...

ξn = dzn−1

dtn−1 ,

(3.6)

It is worth mentioning that the used method in (3.6) to define the sate variable is not

unique. System (3.6) has n state equations, by differentiating (3.6), one can get the

following n− 1 first order differential equations:

ξ1 = ξ2,

ξ2 = ξ3,

...

ξn−1 = ξn,

ξn = a0ξ1 − a1ξ2 − · · · − an−1ξn + ν(t).

(3.7)

Therefore, (3.7) can be written as

ξ =

0 1 0 0 · · · 0

0 0 1 0 · · · 0

0 0 0 1 · · · 0

...

0 0 0 0 · · · 1

−a0 −a1 −a2 −a3 . . . −an−1

ξ +

0

0

0

...

0

1

ν(t) = Aξ +Bν(t). (3.8)

34

And the output can be

z(t) = ξ1(t) =

[1 0 0 · · · 0

]ξ(t) = Cξ(t). (3.9)

Similarly for the discrete-time systems where the difference can be written as:

z(k + n) + an−1z(k + n− 1) + · · ·+ a1z(k + 1) + a0z(k) = ν(k). (3.10)

Thus, the state variables can be written as

ξ1(k) = z(k),

ξ2(k) = z(k + 1),

ξ3(k) = z(k + 2),

...

ξn(k) = z(k + n− 1),

(3.11)

Thus, the state-variables are:

ξ1(k + 1) = ξ2(k),

ξ2(k + 1) = ξ3(k),

...

ξn−1(k + 1) = ξn(k),

ξn(k + 1) = −a0ξ1(k)− a1ξ2(k)− · · · − an−1ξn(k) + ν(k).

(3.12)

Thus, we get exactly the same presentation form in (3.8).

It appears from the previous study of both continuous and discrete systems have

the same structures of the matrices A,B, and C, but the coefficients ai are different.

If the coefficients are functions of time, the system (continuous or discrete) is called

time-varying system; otherwise, it called invariant-system.

35

One of the most important properties of a system is stability. Stability of a system

studies the behavior of the state vector relative to an equilibrium state. There are

many definitions for stability such as uniform stability, asymptotic stability, expo-

nential stability, and bounded input, bounded output stability [44]. However, if the

systems were in time-invariant case and were written in the state-space representation,

then the stability can be studied easily through the eigenvalues of the state matrix.

The continuous time-invariant systems is stable if eigenvalues of A are located at the

left half plane while discrete time-invariant systems is stable if eigenvalues of A are

located inside the unit circle (see Figure 3.1). If the system is not stable then it should

be stabilized; otherwise, human losses, financial losses and other losses may occur.

The ability to stabilize a system depends on some conditions; the most prominent of

these conditions are controllability and its dual notation observability.

Figure 3.1: The stability region of continuous linear time-invariant systems is in theleft, and the stability region of discrete linear time-invariant systems is in the right.

In general, only few measurements (system output) are available, and we often

need to know the states for control purposes for example. So the hidden states have

to be estimated under some conditions using for instance an observer. Next section

presents what the observability of a system is.

36

3.2 Observability

Observability is a structural property of a system, and it means the ability to recon-

struct the state vector using system outputs. In other words; it means the possibility

to determine the behavior of the state using some measurements . The next definition

defines the observability in linear system [44].

Definition 4. A linear system is observable at t0 ∈ T if it is possible to determine

ξ(t0) from the output z[t0,t1] where t1 is finite time in T . If this condition is satisfied

for all t0 and ξ(t0), then the system is completely observable.

For linear time-invariant systems, we have the following theorem [44].

Theorem 2. A linear time-invariant state-space system is completely observable if

and only if the observability matrix W has full rank, i.e. rank(W ) = n; where

W =

C

CA

CA2

...

CAn−1

, (3.13)

and n is the dimension of the state matrix.

3.3 Observer

An observer is a dynamical system used to estimate the state or part of the state of an

observable dynamical system using the available input and output measurements (see

Figure 3.2). Its concept was defined by Luenberger many decades ago [45], [15]. The

Luenberger observer is well known for states estimation in linear dynamical systems.

37

Many other kinds of observer have been proposed to deal with specific and more real-

istic situations. We can classify them into adaptive observer for the joint estimation

of states and parameters [46] [47] [48], robust observers against perturbations such as

sliding mode observers [49] [50] [51], and optimal observers such as Kalman filter [52]

[53].

Figure 3.2: Observer principle

This thesis is focused on discrete linear time invariant (LTI) systems which has

the form ξ(k + 1) = Aξ(k) +Bν(k),

z(k) = Cξ(k),(3.14)

where the matrix D set to be a null matrix as the case in many physical systems. To

explain the basic idea behind the observer, we propose, as an example, the following

observer for (3.14):

ξ(k + 1) = Aξ(k) +Bν(k) + L(z(k)− z(k)),

z(k) = Cξ(k),(3.15)

where L is the observer gain matrix which will be determined to insure the convergence

of the error of estimation to zero.

If the observer error is defined as e(k) = ξ(k)− ξ(k), then the dynamics of the error

38

of (3.15) can be written as

e(k + 1) = (A− LC)e(k). (3.16)

To insure the convergence of the error to zero, the matrix (A − LC) must be

Hurwitz, which means that the eigenvalues of this matrix must be inside the unit

circle. Therefore, the observer gain matrix L should be chosen appropriately to obtain

stable system. In other words, L is chosen such that the dynamics of the observer

is much faster than the system itself; in this case the error converges asymptotically

exponentially to zero.

Observer gain matrix can be obtained by pole placement [44]. This method

consists in choosing the matrix L such that the system is still stable, i.e., the eigen-

values of the matrix (A− LC) has a magnitude strictly less than one for this discrete

system. To get this L, we first fix the appropriate eigenvalues of (A− LC), say

{λ1, λ2, · · · , λn}; then we solve the problem of determining the coefficients of the

matrix L such that

det (λI − (A− LC)) = (λ− λ1)(λ− λ2) · · · (λ− λn). (3.17)

3.4 Chapter Summary

This chapter has introduced the concept of observer. It has been discussed how a

differential equation can be written into state-space representation which is a stan-

dard formulation for dynamical systems. Also, it was clarified how the stability of a

system can be studied through this representation. Some definitions and theorems in

observability terminology were presented. Finally, the idea of observer was shown in

the last section.

39

Chapter 4

A Tikhonov Regularization to

Solve Inverse Source Problem for

Wave Equation

This chapter presents an inverse source problem for the wave equation. We start by

analyzing the solution to the direct problem that allows us to define an operator relat-

ing the unknown source and the measurements which are the position at some points.

Then some properties for this operator are proved. Finally, Tikhonov regularization

is applied to solve instability problems where its regularization parameter is chosen

through different approaches: Discrepency Principle of Morozov, L-curve, Generalized

Cross Validation (GCV), and Normalized cumulative Periodogram (NCP).

4.1 Problem Statement

Consider the following one-dimensional wave equation with Dirichlet boundary con-

ditions utt(x, t)− c2uxx(x, t) = f(x),

u(0, t) = 0, u(l, t) = 0,

u(x, 0) = r1(x), ut(x, 0) = r2(x),

(4.1)

40

where x is the space coordinate defined in [0, l], t is the time coordinate defined in

[0, T ], ux denotes the derivative of u with respect to x and ut the derivative with

respect to t, r1(x) and r2(x) are the initial conditions in L2[0, l], and f(x) ∈ L2[0, l]

is the source function which is assumed, for simplicity, to be independent on time.

The direct problem and examples of inverse problems for the wave equation were

described in Chapter 2, Example 3. In this chapter we focus on the inverse source

problem. First, We propose to determine the operator linking the unknowns to the

measurements.

4.2 Inverse Problem’s Operator and its Properties

We try in the next section to find the operator of this inverse problem through the

analytic solution of the direct problem of (4.1).

4.2.1 Construct the Operator by Solving the Direct Problem

[1], [2]

(4.1) can not be solved directly by separation of variables method. Because separation

of variables method requires both the PDE and the boundary conditions, BCs, to

be homogeneous. Although, some transformations can be applied first on initial

boundary value problems, IBVPs, then separation of variables method can be used.

However, in some cases this transformation may not solve the inhomogeneity issue.

For a problem such as (4.1) where the inhomogeneity is in the PDE and the BCs

are zeros, eigenfunctions expansion can be used. If the boundary conditions are not

zeros, we need to zero out them using some transformation functions.

Proposition 1. Using the eigenfunction expansion method, the solution of the direct

41

problem of system (4.1) is:

u(x, t) =

∫ l

0

r1(x)∂

∂tG(x, t, x, 0)dx+

∫ l

0

r2(x)G(x, t, x, 0)dx+

∫ l

0

∫ t

0

f(x)G(x, t, x, t)dtdx,

(4.2)

where

G(x, t, x, t) =∞∑k=1

2

ckπsin

kπ

lx sin

ckπ

l(t− t) sin

kπ

lx is the Green function. (4.3)

However, if the source f(x) needs to be estimated where the wave speed and the

initial conditions are known, the initial conditions are zeros for simplicity, and we

have some measurements for u at time t = T , then this is an inverse source problem,

and it has an operator K such that

(Kf)(x) =

∫ l

0

H(x, x) f(x)dx = g(x). (4.4)

where K is an integral of first kind oprator with kernel H(x, x) such that

H(x, x) =∞∑k=1

2

ckπ

(1− cos

ckπT

l

)sin

kπ

lx sin

kπ

lx. (4.5)

Proof. In the eigenfunction expansion method, we look for a solution of the form:

u(x, t) =∞∑k=1

ak(t)φk(x), (4.6)

where φk(x) are the eigenfunctions. They can be found after solving the homogenous

version of (4.1) which is:

utt(x, t)− c2uxx(x, t) = 0,

u(0, t) = 0, u(l, t) = 0,

u(x, 0) = 0, ut(x, 0) = 0.

(4.7)

42

By applying separation of variables method on the homogenous problem (4.7) where

the boundary conditions are Dirichlet boundary conditions, one can get:

φk(x) = sinkπ

lx; k = 1, 2, · · · (4.8)

Thus, the solution of (4.1) will be in the form:

u(x, t) =∞∑k=1

ak(t) sinkπ

lx. (4.9)

Differentiate (4.9) with respect to t and with respect to x then substitute in

(4.1), one can get:

∞∑k=1

[d2ak(t)

dt2+ (

kπ

l)2c2ak(t)

]sin

kπ

lx = f(x). (4.10)

Let qk(t) be the kth Fourier coefficient of f decomposition i.e.

d2ak(t)

dt2+ c2(

kπ

l)2ak(t) = qk(t), (4.11)

then qk(t) can be expressed as:

qk(t) =2

l

∫ l

0

f(x) sinkπ

lx dx. (4.12)

Moreover, (4.11) is just an inhomogeneous second order ODE, and its solution is

in the form

ak(t) = akh + akp , (4.13)

where akh is the homogeneous solution and anp is the particular solution.

The homogeneous solution: first: the characteristic equation of the homoge-

neous version of (4.11) is r2 + c2(kπl

)2 = 0; thus, its solution is r = ±ckπli, where

43

i =√−1. Therefore, the homogeneous solution can be written as:

akh(t) = c1 cosckπ

lt︸︷︷︸

a1

+c2 sinckπ

lt︸︷︷︸

a2

. (4.14)

Second, the particular solution: it can be obtained using variation of param-

eter method:

akp = u1a1 + u2a2, (4.15)

where

a1 = cosckπ

lt; a2 = sin

ckπ

lt;

u1 = − 1

w

∫ t

0

qk(t)a2 dt; u2 =1

w

∫ t

0

qk(t)a1 dt;

; and w is the Wronskian such that w =

∣∣∣∣∣∣∣a1 a2da1dt

da2dt

∣∣∣∣∣∣∣ =ckπ

l.

Thus,

akp =l

ckπ

∫ t

0

qk(t) sinckπ

l(t− t) dt. (4.16)

By substituting from (4.14) and (4.16) into (4.13), one can get:

ak(t) = c1 coskπ

lt+ c2 sin

kπ

lt+

l

ckπ

∫ t

0

qk(t) sinckπ

l(t− t) dt. (4.17)

Now, Initial conditions are used to find the constant c1 and c2, and they give

c1 =2

l

∫ l

0

r1(x) sinkπ

lx dx.

c2 =2

ckπ

∫ l

0

r2(x) sinkπ

lx dx.

(4.18)

Finally, we can plug (4.17) with its known values c1 and c2 into (4.9), so we get:

u(x, t) =

∫ l

0

r1(x)∂

∂tG(x, t, x, 0)dx+

∫ l

0

r2(x)G(x, t, x, 0)dx+

∫ l

0

∫ t

0

f(x)G(x, t, x, t)dtdx,

44

where

G(x, t, x, t) =∞∑k=1

2

ckπsin

kπ

lx sin

ckπ

l(t− t) sin

kπ

lx is the Green function.

Now, let us write the problem in the form (2.1) using (4.2.1). First, for simplicity

let , r1(x) = r2(x) = 0. So,

u(x, T ) =

∫ l

0

f(x)

∫ T

0

G(x, T, x, t)dt dx,

=

∫ l

0

H(x, x) f(x)dx,

= g(x),

where

H(x, x) =∞∑k=1

2

(ckπ)2

(1− cos

ckπT

l

)sin

kπ

lx sin

kπ

lx.

Thus,

(Kf)(x) =

∫ l

0

H(x, x) f(x)dx = g(x).

where K is the operator we are looking for, and it is an integral of first kind with

kernel H(x, x).

4.2.2 Operator’s Properties

Proposition 2. Let f, f1 and f2 ∈ L2[0, l], and β1, β2 be scalars; thus, the operator

K is linear, bounded, continuous, and self adjoint.

Proof. •Linearity:

K(β1f1 + β2f2) =

∫ L

0

H(x, x)[β1f1(x) + β2f2(x)]dx

= β1

∫ l

0

H(x, x)f1(x)dx+ β2

∫ l

0

H(x, x)f2(x)dx

45

= β1(Kf1)(x) + β2(Kf2)(x).

Thus, using Definition 5 in Appendix A, K is linear.

•Boundedness:

‖(Kf)(x)‖2L2 = ‖∫ l

0

H(x, x)f(x)dx‖2L2

=

∫ l

0

∣∣∣∣∫ l

0

H(x, x)f(x)dx

∣∣∣∣2 dx≤∫ l

0

f 2(x)

(∫ l

0

H2(x, x)dx

)dx.

≤∫ l

0

λ

∫ l

0

f 2(x)dx dx,

= l λ‖f‖2L2

= κ2‖f‖2L2 , where κ2 = λl

⇒ ‖(Kf)(x)‖ ≤ κ‖f‖L2 .

Therefore, K is a bounded operator, see Definition 6 in Appendix A . In addition,

the operator K is continuous, from the Theorem 3 in Appendix A.

• Self adjointness

(Kf1, f2) =

(∫ l

0

H(x, x)f1(x)dx, f2

)

=

∫ l

0

[∫ l

0

H(x, x)f1(x)dx

]f2(x)dx

=

∫ l

0

[∫ l

0

H(x, x)f1(x)f2(x)dx

]dx

46

By using Fubini’s Theorem (see Theorem 4):

=

∫ l

0

∫ l

0

H(x, x)f1(x)f2(x)dxdx

=

∫ l

0

[∫ l

0

H(x, x)f2(x)dx

]f1(x)dx

= (f1, K∗f2)

⇒ K∗f2 =

∫ l

0

H(x, x)f2(x)dx.

However,

H(x, x) = H(x, x);

thus, K is self adjoint, see Definition 8 in Appendix A.

4.2.3 Well-posedness of the Inverse Problem

Our inverse problem is well-posed if the three conditions: existence, uniqueness, and

stability are satisfied. However, stability condition is violated mostly; therefore, we

can check the well-posednees by examine the stability condition first.

Stability

First, we define a noise δω as δω = βω sin(ωx) such that

limω→∞

βω = 0.

Now, let the data f(x) = f(x) + δω; thus,

‖f − f‖2L2 =

∫ l

0

β2ω sin2(ωx)dx

47

=

∫ l

0

β2ω

2(1− cos(2ωx)dx

=β2ω

2

[x− 1

2ωsin(2ωx)

]l0

=β2ω

2

l − 1

2ωsin(2ωl)︸︷︷︸

→ 0 as ω →∞

−0 +1

2ωsin(0)︸︷︷︸= 0

Thus,

‖f − f‖2L2 −→1

2β2ωl as ω −→∞. (4.19)

Now the solution,

‖Kf −Kf‖2L2 = ‖K(f − f)‖2L2

=

∫ l

0

∣∣∣∣∫ l

0

H(x, x)βω sin(ωx)dx

∣∣∣∣2 dx.But H(x, x) is continuous function; thus, it is a Riemann integrable function (see

Theorem 5 in Appendix A). And by using Riemann- Lebesgue theorem [30] (see

Theorem 7 in Appendix A), one can get:

limω→∞

∣∣∣∣∫ l

0

H(x, x)βω sin(ωx)dx

∣∣∣∣ = 0

Thus,

‖Kf −Kf‖2L2 −→ 0 as ω −→∞. (4.20)

(4.19) and (4.20), the error in the data tends to zero while the error in the solution

tends to a constant. Thus, the solution does not depend continuously on the data.

Therefore, the problem is not stable, so it is ill-posed.

In another way, because Kf = g is linear and K is an integral of first kind which

is compact [34], the problem is ill-posed using Theorem 1.17 in [34]. The general

idea of this theorem is that any linear equation on the form Kx = y with compact

48

operator K is ill-posed.

To solve the instability issue here, we have to regularize the problem. We propose

to use a Tikhonov regularization (see section 2.3). Therefore, we seek to minimize

the Tikhonov functional Jα(f) where

Jα(f) = ‖Kf − u|T‖22 + α‖f‖22. (4.21)

Moreover, the regularization parameter α should be chosen such that the total

error ‖f δα− f‖ is minimum as possible. We apply four different approaches to choose

α as present in the next section.

4.3 Numerical Simulations

Discretization

Equation (4.4) can be discretized using the midpoint quadrature rule [41] which gives

the following:Nx∑j=1

ρjH(xi, xj)f(xj) = g(xi), i = 1, 2, · · · , Nx (4.22)

where xj are the abscissas of the quadrature rule, ρj are the corresponding weights,

and Nx is the space grid size. In the midpoint rule, xj =j − 1

2

Nx

, and ρj = ρ =l − 0

Nx

=

l

Nx

for all j = 1, 2, · · · , Nx. Thus, (4.22) can be written as Kf = g where

K = ρ

H(x1, x1) H(x1, x2) . . . H(x1, xNx)

H(x2, x1) H(x2, x2) . . . H(x2, xNx)

...

H(xNx , x1) H(xNx , x2) . . . H(xNx , xk)

;

49

f =

f(x1)

f(x2)

...

f(xNx)

; and g =

g(x1)

g(x2)

...

g(xNx)

.

Estimation Results

For numerical simulation purpose, a Matlab code has been written, and the param-

eters were set as follows: the space step ∆x = 0.01, l = 2, T = 100, the velocity is

chosen to be c2 = 0.9, and the source is f(x) = 3 sin(5x). Figure 4.1 compares the

source f with the one obtained by inverting K.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−3

−2

−1

0

1

2

3

x

sou

rce

f

f

Figure 4.1: The exact source f with f = K−1uT .

However, in practice the measurements are usually combined with a noise; there-

fore, we added to the measurements a gaussian white noise δ with standard deviation

σ = 0.03580; where signal-to-noise ratio SNR = 20db, and Nx refers to the space

grid size, see Figure 4.2. The relation between the standard deviation σ and the

signal-to-noise ratios (SNR) is:

σ =

√‖A‖2F

n m 10SNR10

(4.23)

50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x

u(x

,T)

u |Tu|T

Figure 4.2: Measurements with and without noise.

Table 4.1: Values of α using the four different approaches and the total error ‖f− f‖2Method α ‖f − f‖2 MSE

Discrepency Principle of Morozov 0.0132 2.1014 0.14822L-curve 0.0127 3.2196 0.31201GCV 0.0082 5.8385 0.44729NPC 0.0223 2.1958 0.25327

for any n× m matrix A.

Therefore, the estimated source f , will be totally affected by this noise as illus-

trated in Figure (4.3).

Tikhonov regularization method was applied to regularize the estimated source.

Moreover, the regularization parameter α was chosen using Discrepency Principle of

Morozov, L-curve, GCV, and NCP (see Figure (4.4)).

Figure 4.5, Figure 4.6, Figure 4.7, and Figure 4.8 shows the regularized source fαδ

versus the exact source f(x) = 3 sin(5x).

The values of α through these different approaches, the absolute error between f

and f , and the mean squared error (MSE) are shown in Table 4.1.

Remark: It can be seen from the relative errors figures that there are few points

have a high relative errors; This happen due to dividing by f in the relative error

51

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1500

−1000

−500

0

500

1000

1500

x

source

f

f

Figure 4.3: The exact source f and the estimated source f without Tikhonov regu-larization

100

100

102

104

106

108

1010

1012

1014

0.0416510.0014704

5.1906e−051.8324e−066.4686e−082.2835e−098.0612e−11

2.8457e−12

1.0046e−13

3.5464e−15

residual norm || A x − b ||2

solu

tion n

orm

|| x || 2

L−curve, Tikh. corner at 0.012735

10−15

10−10

10−5

100

10−6

10−5

10−4

10−3

λ

G(λ

)

GCV function, minimum at λ = 0.0082123

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Selected NCPs. Most white for λ = 0.022329

Figure 4.4: The selected regularized parameter through L-curve, GCV, and NCP.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

4

x

Sour

ce

f

f

0 0.5 1 1.5 2−1200

−1000

−800

−600

−400

−200

0

200

x

Rel

ativ

e er

ror

in %

Figure 4.5: The exact source f and the estimated source f after Tikhonov regular-ization (left) where α was chosen using Discrepency Principle of Morozov, the erroris on the right

52

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

x

Sour

ce

f

f

0 0.5 1 1.5 2−100

0

100

200

300

400

500

x

Rek

ativ

e er

ror

in %

Figure 4.6: The exact source f and the estimated source f after Tikhonov regular-ization (left) where α was chosen using L-curve, the error is on the right

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

x

Sour

ce

f

f

0 0.5 1 1.5 2−400

−200

0

200

400

600

800

1000

1200

1400

x

Rel

ativ

e er

ror

in %

Figure 4.7: The exact source f and the estimated source f after Tikhonov regular-ization (left) where α was chosen using GCV, the error is on the right

53

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

x

Sour

ce

f

f

0 0.5 1 1.5 2−100

−50

0

50

100

150

200

250

300

x

Rel

ativ

e er

ror

in %

Figure 4.8: The exact source f and the estimated source f after Tikhonov regular-ization (left) where α was chosen using NCP, the error is on the right

where f at these points is zero. Thus, Table 4.1 presents the absolute errors and the

Mean squared Error (MSE).

Previous figures and table illustrates that there are different approaches for se-

lecting the regularization parameter α which give different regularized solutions. One

can notice the ability of Tikhonov method to regularize the solution if α is chosen

appropriately.

4.4 Chapter Summary

In this chapter we focused on solving inverse source problem for a one-dimensional

wave equation using Tikhonov regularization method. The operator of this inverse

problem was obtained through the solution of the direct problem. Tikhonov reg-

ularization method was applied using different regularization parameters which are

obtained using Discrepency principle, L-curve, GCV, and NCP. In the next chapter,

the same inverse problem is solved using an observer-based approach.

54

Chapter 5

An Observer to Solve Inverse

Source Problem for Wave Equation

In this chapter, we propose to apply an observer on a one-dimensional wave equation

to estimate the source. For this purpose, we propose to write the one dimensional wave

equation (4.1) in a state space representation firstly. Then, the system is discretized

in both space and time. Finally, the observer design is presented. Using this method,

the state and the source are estimated. We propose then to compare the results

obtained with observer to an original Tikhonov approach.

5.1 Problem Statement

5.1.1 A State-Space Representation for the Wave Equation

Consider IBVP of one-dimensional wave equation as in (4.1):

utt(x, t)− c2uxx(x, t) = f(x),

u(0, t) = 0, u(l, t) = 0,

u(x, 0) = r1(x), ut(x, 0) = r2(x),

(5.1)

Out aim is to solve the inverse source problem of (5.1) using an adaptive observer

55

with partial measurements of the solution u available. We first propose to rewrite

(5.1) in an appropriate form by introducing two auxiliary variables v(x, t) = u(x, t)

and w(x, t) = ut(x, t) and let

ξ(x, t) =

[v(x, t), w(x, t)

]T. (5.2)

Therefore, the (5.1) can be written as follows,

∂ξ(x, t)

∂t= Aξ(x, t) + F,

v(0, t) = 0, v(l, t) = 0,

v(x, 0) = r1(x), vt(x, 0) = r2(x),

z = Hξ(x, t),

(5.3)

where the operator A is given by A =

0 I

c2 ∂2

∂x20

, F =

0

f

, z is the output,

and H is the observation operator such that H = [H 0] where H is a restriction

operator on the measured domain.

5.1.2 Discretization

There are three known methods for discretization: finite difference method, FDM;

finite element method, FEM, and finite volume method, FVM. For simplicity and

validation, we propose to apply FDM to discretize system (5.3).

Discretizing system (5.3) using implicit Euler scheme in time and the central finite

56

difference discretization for the space gives:

vj+1i − vji

∆t= wj+1

i ,

wj+1i − wji

∆t=

c2

(∆x)2(vji−1 − 2vji + vji+1) + f ji ,

vj1 = 0, vjNx= 0,

v1i = r1(xi), v2i = ∆t r2(xi) + v1i ,

i = 1, 2, · · · , Nx, j = 1, 2, · · · , Nt,

(5.4)

where ∆x refers to the space step, ∆t refers to the time step, Nx is the space grid

size, and Nt is the time grid size. Simplifying the first two parts in (5.4) leads to:

vj+1i =

c2k2

(∆x)2(vji−1 − 2vji + vji+1) + ∆t2f ji + kwji + vji ,

wj+1i =

c2k

(∆x)2(vji−1 − 2vji + vj+1

i ) + ∆tf ji + wji ,

vj1 = 0, vjNx= 0,

v1i = r1(xi), v2i = ∆t r2(xi) + v1i ,

i = 1, 2, · · · , Nx, j = 1, 2, · · · , Nt.

(5.5)

From (5.5), the state-space system can be written as:

ξj+1 = Gξj +Bf j + b, (a)

zj = Hξj, (b)

f j+1 = f j, (c)

j = 1, 2, · · · , Nt;

(5.6)

such that

G =

∆tE + I ∆tI

E I

;

57

E =c2∆t

(∆x)2

−2 1

1 −2. . .

. . . . . . 1

1 −2

;B =

(∆t)2I

∆tI

;

, and b is a term that includes the boundary conditions such that

b =

[c2(∆t)2

(∆h)2vj1 0 · · · 0

c2(∆t)2

(∆h)2vjNx

c2(∆t)

(∆h)2vj1 0 · · · 0

c2(∆t)

(∆h)2vjNx

]T.

The observation operator H has the form H = [Hm 0] where Hm =

0 · · · 0

... Im...

0 · · · 0

and m refers to the number of measurements. Im is the identity matrix of dimension

m.

This system is linear multiple-input, multiple-output (MIMO) discrete time-invariant.

If Nx refers to the space grid size, and m refers to the number of measurements; thus,

the state matrix G is of dimension 2Nx× 2Nx, the observer matrix H is of dimension

m× 2Nx , and the input matrix B is of dimension 2Nx ×Nx.

For the observer to be applied, it is important to check first the observability of

this system which can be done using the observability matrix as in Theorem 2 . Let

W be the observability matrix, so it has the form:

W =

H

HG

HG2

...

HG2Nx−1

, (5.7)

58

The system is observable if rank(W ) = dimension(G) = 2Nx . This depends on the

number of measurements (outputs) that are taken into account. This number affects

the convergence, and we study the convergence error with respect to this number.

In this thesis, both cases, full measurements and partial measurements, are pre-

sented in chapter (5). After satisfying the observability condition, the observer can

be designed.

5.2 Observer Design

As recalled in Chapter 3, a state observer is a system that provides an estimate of its

internal state, given measurements of the input and the output of the real system. We

propose to use an adaptive observer for the joint estimation of the states v and w and

the source f . This observer has been proposed in [47], and it has been developed for

joint estimation of the state and the parameters for a class of systems. However, we

propose to generalize the idea behind this observer to estimate the input considering

each spatial sample of the input as an independent parameter. The adaptive observer

is given by the following system of equations,

zj = Hξj, (a)

Υj+1 = (G− LH)Υj +B, (b)

f j+1 = f j + ΣΥjTHT (zj − zj), (c)

ξj+1 = Gξj +Bf j + b+ L(zj − zj) + Υj+1(f j+1 − f j), (d)

(5.8)

where L is the observer gain matrix of dimension 2Nx×m, ξj and f j are the state

and source estimates respectively, Υj is a matrix sequence obtained by linearly filter-

ing B, and Σ a bounded diagonal positive definite matrix satisfying the assumptions

in Assumption 1 as in [47]:

59

Assumption 1. The diagonal positive definite matrix Σ satisfy:

1. ‖HΥjΣ12‖2 ≤ 1.

2.1

κ

∑j+κ−1i=j ΣΥiTHTHΥi ≥ βI for some constant β > 0, integer κ > 0, and all

j.

Proposition 3. Observer (5.8) is a global exponential adaptive observer for discrete

finite dimensional systems, i.e. the state estimation error ξj − ξj and the source

estimation error f j−f j converge to zero exponentially fast as j tends to infinity [47].

Proof. Let ejξ = ξj − ξj be the state error and ejf = f j − f j be the source error, thus

ej+1ξ = Gξj +Bf j + b+ L(Hξj −Hξj) + Υj+1(f j+1 − f j)−Gξj −Bf j − b

= Gejξ +Bejf − LHejξ + Υj+1(ej+1

f − ejf ) since f j+1 = f j from (5.6.c).

Therefore,

ej+1ξ = (G− LH)ejξ +Bejf + Υj+1(ej+1

f − ejf ). (5.9)

The key step of the proof is to define linear combined error sequence; let:

ηj = ejξ −Υjejf . (5.10)

Now compute the dynamic of ηj:

ηj+1 = ej+1ξ −Υj+1ej+!

f

= (G− LH)ejξ +Bejf + Υj+1(ej+1f − ejf )−Υj+1ej+1

f

= (G− LH)ejξ − (G− LH)Υjejf by using (5.8b)

= (G− LH)(ejξ −Υjejf ).

Thus,

ηj+1 = (G− LH)ηj. (5.11)

60

Since the eigenvalues of G − LH are inside the unit circle, the sequence ηj tends to

zero exponentially fast. Now, the error dynamics of the source is:

ej+1f = f j+1 − f j+1

= f j + ΣΥjTHT (Hξj −Hξj)− f j+1.

But f j+1 = f j from (5.6 c); thus,

ej+1f = ejf − ΣΥjTHTHejξ.

By substituting from (5.10),

ej+1f = [I − ΣΥjTHTHΥj]ejf − ΣΥjTHTHηj. (5.12)

First, Let us study the first part, (i.e)

ej+1f = [I − Σ(HΥj)THΥj]ejf (5.13)

Because the two conditions in Assumptions 1 are satisfied and by using Lemma

2 in Appendix A, (5.13) is exponential stable. In addition, because Σ, H, and Υ

are bounded, and the sequence ηj tends to zero fast, the second term in (5.12),

ΣΥjTHTHηj, goes to zero exponentially fast; and therefore, (5.12) goes to zero ex-

ponentially fast (by using Lemma 1 in Appendix A). Ultimately, the state error

ejξ = ηj + Υejf converges also to zero exponentially fast.

To achieve the convergence and the stability of this observer some parameters

such ∆x,∆t, and c should be tuned precisely [54].

61

5.3 Numerical Simulations

5.3.1 Preliminary

First of all, we try to explain all the conditions that appeared during the numerical

simulations work starting from the discretization. Our system was based on discrete

version of wave equation. During the work in the simulation, we found that the results

are related to five connected conditions: Courant-Fridrichs-Lewy condition (CFL),

number of measurements, observability condition, condition number of observability

matrix, and observer gain matrix. Now, we give a short description for each condition.

1. CFL condition:

CFL condition is one of the necessary condition to guarantee the stability of

the chosen scheme, especially for hyperbolic PDEs (e.g wave equation). In this

condition the time and the space steps are chosen appropriately such that

∆x

∆tc ≤ S,

where c is the wave speed and S is a scheme dependent constant.

2. Observability condition:

We can estimate only the observable states with the observers, so the observabil-

ity condition should be satisfied. This condition is affected by the discretization

scheme; moreover, it is affected by ∆t, ∆x, and the wave speed c.

3. Number of measurements:

Obviously, increasing number of measurements means increasing information

about the state; thus, insuring the observability condition for all the states.

However, for some applications, only few measurements can be available and

the idea is to study the effect of this number on the convergence of the ob-

62

server in order to find the minimum number of measurements that insures the

reconstruction of the source.

4. Condition number of observability matrix:

While the rank of the observability matrix indicates whether the system is

theoretically observable or not, its condition number measures the degree of

observability. In other words, it gives information on the potential complexity

of tuning the observer parameter. Indeed, a high condition number means that

the system is nearly unobservable, which make it difficult to choose the adequate

observer gain [55]. Moreover, as the case in metrics, the condition number of

the observability matrix affects by the size of this matrix and its kind [56]. For

distributed system such our system, the condition number is generally high.

This condition is affected by the discretization scheme and the chosen parameter

∆t, ∆x, and c. In our work, We put 104 as an upper bound for the condition

number of the observability matrix. In other words, If the condition number

exceeds 104, we try to tuning ∆t, ∆x, c, or number of measurements until we

achieve a result.

5. Observer gain matrix:

Once the full rank condition of the observability matrix is satisfied, and its

condition number is acceptable, we need to find the observer gain matrix L

such that the system (5.6) is remains stable. Finding this L is not an easy task

for such distributed system. In general, pole placement method is used to find

the observer gain matrix (see section3.3). However in our case, pole placement

often fails due to the size of the state matrix and the restricted number of

measurements. Thus, we construct L such that it has a similar structure to the

structure of the state matrix G which is sparse matrix meanwhile the eigenvalues

of (G− LH) are inside the unit circle. In this way, the degree of freedom in

63

choosing the coefficients of L is reduced.

For numerical simulation purpose, we used Matlab (2012) to write a code for the

simulation, and the parameters were set as follows. For the space, ∆x = 0.01 and

l = 2. For the time, ∆t = 0.01 and T = 100. Thus, Nx = 201 and Nt = 10001. The

velocity is chosen to be c2 = 0.9, and the source is f(x) = 3 sin(5x). The matrix Σ is

chosen to improve the estimation accuracy meanwhile satisfies the two conditions in

Assumption 1. The initial guess for the estimated source is f(x) = 0, and the initial

state is choosen to be ξ = 0.

The exact source f(x) = 3 sin(5x), and the state ξ of (5.6) are presented in Figure

5.1 and Figure 5.2

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f(x)

Figure 5.1: The exact source f(x) =3 sin(5x).

Figure 5.2: The state ξ for one-dimensionalwave equation where c2 = 0.9, f(x) =3 sin(5x), and zeros boundaries and initialconditions.

We discuss in this section two cases. First case, noise-free case. Second case, when

there are some noises in the state and the measurements. Next part gives the results

of the simulation in case of noise-free.

64

5.3.2 Noise-Free Case

Full measurements

In case of full measurements, the observation operator has the form H =

[I 0

].

Figure 5.3.a shows the efficiency of this observer to estimate the source, and Figure

5.3.b displays the relative error where the maximum relative error is 6.14× 10−11%,

and the mean squared error, (MSE), is 1.8168× 10−14.

Remark: It can be seen from Figure 5.3.b that there are a few points have relative

errors more than the others; actually, this happen due to the dividing by f in the

relative error where f is zero.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

sour

ce

f

f

0 0.5 1 1.5 2−4

−2

0

2

4

6

8x 10

−11

x

Rel

ativ

e er

ror

in %

a b

Figure 5.3: (a): the exact source f (blue) and the estimated source f (black) usingfull measurements. (b): the relative error of the source estimation in %.

Moreover, the efficiency of this observer appears in the state estimation; it esti-

mated the state with relative error 0.044%, see Figure 5.4.

65

a b

c d

Figure 5.4: State error in the noise-free case with full measurements; (a): the stateerror ξ− ξ. (b) the state relative error in %. (c): the state error, in %, after removingthe initial phase. (d): the state relative error after removing the outliers where mostof them concentrated in the initial phase.

Figure 5.5 displays the estimated source starting form the initial guess f j=1 then,

f j=20, and finally the final estimated source f j=Nt .

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f(x

)

f j = 1

f j = 2 0

f j = N k

Figure 5.5: The estimated source in different time steps starting form the initial guess.

66

Partial measurements

In the partial measurements case, Hm = H[m0,mf ] where [m0,mf ] is the observation

interval (see section 5.1.2). Figure 5.6 and Figure 5.8 display the estimated source

and the state error, respectively; where only 50% of the state components taken from

the middle. The MSE in the source estimation is 0.3354.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−1000

−500

0

500

1000

1500

x

Rel

ativ

e er

ror

in %

a b

Figure 5.6: (a): the exact source f (blue) and the estimated source f (black) usingpartial measurements (50% of the state components taken from the middle). (b): therelative error of the source estimation in %.

0 0.1 0.2 0.3 0.4 0.5−10

−8

−6

−4

−2

0

2

4

6

8

10

x

Rel

ativ

e er

ror

in %

Figure 5.7: Zoom-in for the relative error in Figure 5.6.b

As appears in Figure 5.6.b, some points have huge errors. This also happens due

67

to division by f in the computation on their relative errors where f is zero. The

actual error can be seen in Figure 5.7.

a b

c d

Figure 5.8: State error in the noise-free case with partial measurements (50% of thestate components taken from the middle); (a): the state error ξ − ξ. (b) the staterelative error in %. (c): the state error, in %, after removing the initial phase. (d):the state relative error after removing the outliers where most of them concentratedin the initial phase.

However in practice, the measurements are more accessible on the boundary. For

that purpose, we took some measurements form the end, 75% of the state compo-

nents. Figure 5.9 and Figure 5.11 display the estimated source and the state error,

respectively. In addition, MSE for the source estimation is equal 0.2096. Figure 5.10

presents the relative error regardless the points that have huge relative errors due to

deviding by zero. In both cases, partial measurements in the middle or at the end,

68

observer displays good performance.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

4

x

f

f

0 0.5 1 1.5 2−100

−50

0

50

100

150

200

x

Rel

ativ

e er

ror

in %

a b

Figure 5.9: (a): the exact source f (blue) and the estimated source f (black) usingobserver with partial measurements (50% of the state components taken from theend). (b): the relative error of the source estimation in %.

0 0.1 0.2 0.3 0.4 0.5−10

−8

−6

−4

−2

0

2

4

6

8

10

x

Rel

ativ

e er

ror

in %


69

a b

c d

Figure 5.11: State error in the noise-free case with partial measurements (75% ofthe state components taken from the end); (a): the state error ξ − ξ. (b) the staterelative error in %. (c): the state error, in %, after removing the initial phase. (d):the state relative error after removing the outliers where most of them concentratedin the initial phase.

5.3.3 Noise-Corrupted Case

White Gaussian random noises with zero means were added to the states and to

the measurements with standard deviations σξ = 0.007816 (SNRξ = 30db) and

σz = 0.01044 (SNRz = 20db), respectively. The relation between σ and SNR can

bee seen in (4.23). The effect of the noise on the state and the measurements can be

seen in Figure 5.12

70

a b

Figure 5.12: (a): the state ξ after adding a white noise with a standard deviationσξ = 0.0078. (b): the output z after adding a white noise with a standard deviationσz = 0.0104.

Full measurements

The estimated source in the noisy case using full measurements can be seen in Figure

5.13.a, its corresponding relative error is in Figure 5.13.b, and MSE is equal 0.28655.

The state error in this case is shown in Figure 5.15

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

4

x

f

f

0 0.5 1 1.5 2−600

−500

−400

−300

−200

−100

0

100

200

x

Rel

ativ

e er

ror

in %

a b

Figure 5.13: (a): the exact source f (blue) and the estimated source f (black) usingobserver with full measurements. (b): the relative error of the source estimation in%.

71

0 0.1 0.2 0.3 0.4 0.5−20

−15

−10

−5

0

5

10

15

20

x

Rel

ativ

e er

ror

in %


a b

c d

Figure 5.15: State error in the noisy case with full measurements; (a): the state errorξ − ξ. (b) the state relative error in %. (c): the state error, in %, after removing theinitial phase. (d): the state relative error after removing the outliers where most ofthem concentrated in the initial phase.

72

Partial measurements

In partial measurements noisy case, the estimated source, its corresponding relative

error, and the error in the state are displayed in Figure 5.16.a, Figure 5.16.b and

Figure 5.18, respectively; where the measurements taken form the middle. MSE for

source estimation is equal 0.4014.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

4

x

f

f

0 0.5 1 1.5 2−1000

−500

0

500

1000

x

Rel

ativ

e er

ror

in %

a b

Figure 5.16: (a): the exact source f (blue) and the estimated source f (black) usingpartial measurements (50% of the state components taken from the middle). (b): therelative error of the source estimation in %.

0 0.1 0.2 0.3 0.4 0.5−10

−8

−6

−4

−2

0

2

4

6

8

10

x

Rel

ativ

e er

ror

in %


73

a b

c d

Figure 5.18: State error in the noisy case with partial measurements (50% of thestate components taken from the middle); (a): the state error ξ − ξ. (b) the staterelative error in %. (c): the state error, in %, after removing the initial phase. (d):the state relative error after removing the outliers where most of them concentratedin the initial phase.

For the practical case where the measurements taken from the end, Figure 5.19.a

and Figure 5.19.b display the estimated source and it’s corresponding relative error,

respectively; where MSE is equal 0.3213 while the error in the state is shown in Figure

5.20.

74

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−500

−400

−300

−200

−100

0

100

200

300

400

500

x

Rela

tive

erro

r in

%

a b

Figure 5.19: (a): the exact source f (blue) and the estimated source f (black) usingpartial measurements (50% of the state components taken from the end). (b): therelative error of the source estimation in %.

a b

c d

Figure 5.20: State error in the noisy case with partial measurements (75% of thestate components taken from the end); (a): the state error ξ− ξ. (b) the state relativeerror in %. (c): the state error, in %, after removing the initial phase. (d): the staterelative error after removing the outliers where most of them concentrated in theinitial phase.

75

0 0.1 0.2 0.3 0.4 0.5−10

−8

−6

−4

−2

0

2

4

6

8

10

x

Rel

ativ

e er

ror

in %


5.4 Comparison Between Observer and Tikhonov

To asses the observer performance in the source estimation, we need to compare

its performance with a standard method such as Tikhonov regularization. However,

Tikhonov does not give a recursive (sequential) estimation for the source f . Therefore,

we propose a new Tikhonov approach based on the use of Hankel matrix in order to

estimate the unknown recursively. The general idea is to derive the state and output

at time k + p from the state at time k and the input sequence, using the state space

matrices where p is a constant, p ≥ 0; thus, we get a new state-space representation

where the transmission matrix is a Hankel matrix [57].

As was presented before, the linear time-invariant discrete system can be written

as:

ξ(k + 1) = Aξ(k) +Bu(k),

z(k) = Cξ(k) +D(k)u(k).(5.14)

Thus, By repeating substitution, one can get for some p ≥ 0,

76

ξ(k + p) = Apξ(k) + Cpup(k),

zp(k) = Oξ(k) + τup(k),(5.15)

where

up(k) =

u(k)

u(k + 1)

...

u(k + p− 1)

; (5.16)

zp(k) =

z(k)

z(k + 1)

...

z(k + p− 1)

; (5.17)

Cp =

[Ap−1B · · · AB B

](5.18)

O =

C

CA

...

CAp−1

; (5.19)

and

τ =

D 0 0 · · · 0

CB D. . . . . .

...

CAB CB D. . . 0

.... . . . . . . . . 0

CAp−2B · · · CAB CB D

. (5.20)

77

In light of that, the system (5.6) can be written as:

ξj+p = Gpξj + Cpf jp + b,

zj = Oξj + τf jp

(5.21)

where b = 1p⊗

b, and⊗

is the Kronecker product. Thus, from the second equations

in (5.6) and (5.21), a new measurements can be defined as

zj = τf j, (5.22)

where zj = zj −Oξj=1.

The aim is to estimate the source f by minimizing the following cost function

where a Tikhonov regularization is used:

Jα(f) =1

2‖τf j − zj‖22 +

α

2‖f j‖22 (5.23)

By differentiation (5.23) and equate it with zero, one can get

(τ ∗τ + αI)f j = τ ∗zj (5.24)

Thus,

f j = Rατ∗zj (5.25)

where the Tikhonov operator Rα = (τ ∗τ + αI)−1τ ∗.

5.4.1 Numerical Simulations

If the space step ∆x, time step ∆t and the final time T are used as in section 5.3,

the size of the Hankel matrix τ will be very huge, and we can not deal with it using

Matlab. Therefore, we replace the space step, time step and the final time to be

78

∆x = 0.1, ∆t = 0, 05, and T = 2, respectively. In this section, we discuss the

two cases: noise-free case and noise-corrupted case as were studied in section 5.3. In

addition, in each case we discuss the cases of full measurements, partial measurements

in the middle and partial measurements at the end. For both partial measurements

in the middle and partial measurements at the end, 50% of the state components are

taken. In the simulations of the original Tikhonov regularization, we applied L-curve,

GCV, and NCP approaches for selecting the regularization parameter α, then we

present the best result.

The Noise-Free Case

Full measuremets

Figure 5.22 and Figure 5.23 represent the estimated source using observer and Tikhonov,

respectivly. From Figure 5.24, it appears that both observer and Tikhonov approaches

presented a good performance to estimate the source whatever the used approach for

selecting the regularization parameter α. Relative errors for this case can be seen in

Table 5.1.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−2

0

2

4

6

8

10

12

14x 10

−3

x

Rel

ativ

e er

ror

in %

a b

Figure 5.22: (a): the exact source f (blue) and the estimated source f (black) usingobserver with full measurements. (b): the relative error of the source estimation in%.

79

Table 5.1: Relative errors for noise-free case (full measurements)

method‖f − f‖‖f‖

× 100 max(f − ff× 100)

Observer 0.000577662% 0.0131826%Tikhonov with L-curve 4.68842705436554e− 11% 1.32058070134541e− 10%Tikhonov with GCV 4.68842705436554e− 11% 1.32058070134541e− 10%Tikhonov with NCP 9.37414665016344e− 11% 2.50190556173606e− 11%

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2x 10

−10

xR

elat

ive

erro

r in

%

a b

Figure 5.23: (a): the exact source f (blue) and the estimated source f (black)using Tkhonov with full measurements. (b): is the corresponding relative error ofthe source estimation in %.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f T i k .

f o b s

Figure 5.24: Comparison between observer and Tikhonov in noise-free case with fullmeasurements

80

Partial measurements taken from the middle

Figure 5.25 and Figure 5.26 present the estimated source using observer and Tikhonov,

respectively; where 50% of the state components taken form the middle. From the

two figures and Figure 5.27, it appears that observer approach gives better estimation,

see also Table 5.2.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−200

−100

0

100

200

300

400

xR

elat

ive

erro

r in

%

a b

Figure 5.25: (a): the exact source f (blue) and the estimated source f (black) usingobserver with partial measurements in the middle. (b):the relative error of the sourceestimation in %.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−100

−50

0

50

100

x

Rel

ativ

e er

ror

in %

a b

Figure 5.26: (a): the exact source f (blue) and the estimated source f (black) usingTkhonov with partial measurements in the middle. (b): the corresponding relativeerror of the source estimation in %.

81

Table 5.2: Relative errors for noise-free case (partial measurements form the middle)


× 100

Observer 28.3223%Tikhonov with L-curve 75.8694%Tikhonov with GCV 75.8694%Tikhonov with NCP 75.8694%

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f T i k .

f o b s

Figure 5.27: Comparison between observer and Tikhonov in noise-free case withpartial measurements taken from the middle.

Partial measurements taken from the end

As mentioned before, the measurements are more accessible on the boundary. Thus,

we took partial measurements form the end of the domain, 50% of the state compo-

nents. The results for this case can be seen in Figure 5.28, Figure 5.29, and Figure

5.30. It is clear that Tikhonov is completely unable to recover the interval where no

measurements are available while observer has somewhat a good estimation in this

interval. Thus, also the in case of partial measurements from the end, the estimated

source using observer is better than the estimation using Tikhonov.

82

Table 5.3: Relative errors for noise-free case (partial measurements form the end)


× 100


0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−100

0

100

200

300

400

xR

elat

ive

erro

r in

%

a b

Figure 5.28: (a): the exact source f (blue) and the estimated source f (black) usingobserver with partial measurements at the end. (b):the relative error of the sourceestimation in %.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−20

0

20

40

60

80

100

x

Rel

ativ

e er

ror

in %

a: L-curve b

Figure 5.29: (a): the exact source f (blue) and the estimated source f (black) usingTkhonov with partial measurements in the middle. (b): the corresponding relativeerror of the source estimation in %.

83

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f T i k .

f o b s

Figure 5.30: Comparison between observer and Tikhonov in noise-free case withpartial measurements taken from the end

The Noise-Corrupted Case

In the simulations of this noisy case, we found that observer need to be more robust;

therefore, we add minor modifications to the observer (5.8) to obtain an adaptive

observer but with a sliding mode-like term as follows:

zj = Hξj,

Υj+1 = (G− LH)Υj +B,

f j+1 = f j + ΣΥjTHT [(zj − zj) + γ1 tanh(γ2(zj − zj))] ,

ξj+1 = Gξj +Bf j + b+ L [(zj − zj) + γ3 tanh(γ4(zj − zj))] + Υj+1(f j+1 − f j),

(5.26)

where γ1, γ2, γ3, and γ4 are scalers.

In addition, L-curve, GCV, and NCP sometimes in the noisy case fail to investigate

a good regularization parameter. To treat that, we tried to find the regularization

parameter manually.

84

Full measurements

Figure 5.31 shows the estimation source using observer which is a good estimation in

this noisy case. The estimated source using Tikhonov with the selected α is shows in

Figure 5.32. Figure 5.33 compares the estimated source using the two approaches, and

it illustrates that observer has better estimation than Tikhonov, see also Table 5.4.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

4

x

f

f

0 0.5 1 1.5 2−10

0

10

20

30

40

xR

elat

ive

erro

r in

%

a b

Figure 5.31: (a): the exact source f (blue) and the estimated source f (black) usingobserver with full measurements. (b):the relative error of the source estimation in %.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−50

0

50

100

150

200

250

300

x

Rel

ativ

e er

ror

in %

a: selected α b

Figure 5.32: (a): the exact source f (blue) and the estimated source f (black) usingTikhonov with full measurements in the noisy case where α was selected manually.(b): the corresponding relative error of the source estimation in %.

85

Table 5.4: Relative errors for the noisy case (full measurements)


× 100


Tikhonov with selected α 11.14185%

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

4

x

f

f T i k .

f o b s .

Figure 5.33: Comparison between observer and Tikhonov in noise-corrupted casewith full measurements.

Partial measurements in the middle

The estimated source using observer and Tikhonov can be seen in Figure 5.34 and

Figure 5.35, respectively, and the relative errors can be seen in Table 5.5. The compar-

ison between the two methods is displayed in Figure 5.36. These figures illustrate that

both observer and Tikhonov have approximately close results with a little excellence

in the observer estimation.

86

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−100

−50

0

50

100

150

200

250

x

Rel

ativ

e er

ror

in %

a b

Figure 5.34: (a): the exact source f (blue) and the estimated source f (black) usingobserver with partial measurements taken from the end. (b):the relative error of thesource estimation in %.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−100

−50

0

50

100

150

x

Rel

ativ

e er

ror

in %

a b

Figure 5.35: (a): the exact source f (blue) and the estimated source f (black) usingTkhonov with partial measurements from the middle. (b): the corresponding relativeerror of the source estimation in %.

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

x

f

f T i k .

f o b s .

Figure 5.36: Comparison between observer and Tikhonov in the noise-corrupted casewith partial measurements taken form the middle.

87

Table 5.5: Relative errors for the noisy case (partial measurements from the middle)


× 100

Observer 42.92428%Tikhonov with L-curve 100%Tikhonov with GCV 32.58996%Tikhonov with NCP 32.34354%

Tikhonov with selected α 49.04133%

Partial measurements at the end

By seeing Figure 5.37, Figure 5.38, Figure 5.39, and Table 5.6, one can conclude that

observer gave better estimation than Tikhonov. Table 5.7 presents the MSE in the

noise-corrupted case with partial measurements.

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−200

−100

0

100

200

300

400

500

x

Rel

ativ

e er

ror

in %

a b

Figure 5.37: (a): the exact source f (blue) and the estimated source f (black) usingobserver with partial measurements taken from the end. (b): the correspondingrelative error of the source estimation in %.

88

Table 5.6: Relative errors for noisy case (partial measurements from the end)

Method‖f − f‖‖f‖

× 100

Observer 32.18883%Tikhonov with L-curve 100%Tikhonov with GCV 50.34026%Tikhonov with NCP 48.83239%

Tikhonov with selected α 50.29478

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

f

f

0 0.5 1 1.5 2−50

0

50

100

150

200

250

300

350

x

Rel

ativ

e er

ror

in %

a b

Figure 5.38: (a): the exact source f (blue) and the estimated source f (black) usingTkhonov with partial measurements taken from the end. (b): the correspondingrelative error of the source estimation in %.

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

x

f

f T i k .

f o b s .

Figure 5.39: Comparison between observer and Tikhonov in the noise-corrupted casewith partial measurements taken from the middle.

The numerical simulations in this section are proven that in general observer

approach gave better estimation than Tikhonove regularization method, and in the

worst cases, observer gives the same performance as Tikhonov.

89

Table 5.7: MSE in the noisy case (partial measurements)method Middle End

Observer 0.907404289806715 0.680460568436796Tikhonov with L-curve 2.11396522810324 2.11396522810324Tikhonov with GCV 0.796284023328078 1.58249288930699Tikhonov with NCP 0.777861070191947 1.54960272106339

Tikhonov with selected α 1.03663795787186 1.54659139143788

5.5 Chapter Summary

In this chapter, we presented how observers can be applied on PDE through the one-

dimensional wave equation. Then, numerical simulations for estimating the source

and the state of our one-dimensional wave equation system are displayed. These

numerical results were presented in absence and presence of noise. They proved the

efficiency of observer-based approach to estimate unknowns in a distributed system.

To emphasis this efficiency, a comparison between observer method and the opti-

mization of cost functions with a Tikhonov regularization was built. This comparison

considered also the two main cases: noise-free case and noisy case. In both cases,

the full measurements and the partial measurements cases were studied. The results

proved generally that observer has better result than Tkihnonv approach. More-

over, the Tikhonov regularization that we introduced is approximately ineffective to

estimate the source in the interval where no measurements are available.

90

Chapter 6

Conclusion

Observers for solving problems governed by partial differential equations are being

actively used. In this thesis, we have demonstrated the performance of the observer

to estimate the states and the unknown source for a one-dimensional wave equation.

Moreover, the study covered the two cases: noise-free case and noise-corrupted case.

We investigated that the states and source estimation errors tend to zero exponentially

fast. The effectiveness of the observer-based approach is strongly confirmed through

a comparison between observer and Tikhonov regularization approaches.

In Chapter 2, the general idea of inverse problems was introduced with some ex-

amples. We highlighted the fact that inverse problems are generally ill-posed due to

the non continuity between the data and the unknown which leads to an instability.

We saw that to solve an inverse problem we usually use optimization techniques.

However to overcome the ill-posedness of the problem, we usually require some regu-

larization. We focused more on Tikhonov regularization which is widely used. Four

different approaches for selecting the regularization parameter were presented.

In Chapter 3, we recalled some basic definitions on observer theory starting from

state-space representation, passing by the observability condition, and ending with

the concept of observers.

In Chapter 4, we stated our problem then the direct problem was solved to to find

an operator relating the unknown to the measurements. After that, the ill-posedness

91

of the problem was proved. Then we proposed to solve the problem using a mini-

mization approach with tikhonov regularization method. Moreover, the regularized

parameter was chosen through the four different methods: Discrepancy Principle of

Morozov, L-curve, GCV, and NCP.

In Chapter 5, first, we rewrote the one dimensional wave equation in a state-space

representation. Then we wrote down the problem in a discretized version. After that,

an adaptive observer for the joint estimation of the source and the states was designed.

Numerical simulations for the source and states estimation using observer were pre-

sented, and they have proven the capability of observer to estimate both the source

and the states. Finally, to asses the observer in the source estimation, we compared

its performance with a standard approach which is Tikhonov method considering dif-

ferent number of measurements full and partial and considering the noise-free and

noisy cases. The estimation results confirmed the superiority of observer to estimate

the unknown source.

Finally, we point out that future work can extend to the following research topics:

• Study the discretization effect on the convergence of the adaptive observer that

has been used in this thesis.

• Analyze some numerical issues encountered during some simulations with partial

measurements such as the ill-conditioning of the observability matrix.

• Study the inverse source problem for a two-dimensional wave equation using

observer-based approach.

• Study inverse problem for wave equation where the speed of the wave is the

unknown.

92

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APPENDICES

A Definitions and Theorems

Definition 5. Let K : X → Y be an operator between two vector spaces X and Y ,

then K is said to linear if the following two condition satisfy for all x ∈ X, y ∈ Y ,

and any scalar γ:

1. K(x+ y) = K(x) +K(y)

2. K(γx) = γK(x)

Or equivalently, K is linear if

K(γx+ βy) = γK(x) + βK(y), (A.1)

for any scalers γ and β.

Definition 6. A linear operator K is called bounded if there exist constant κ > 0

such that

‖Ax‖ ≤ κ‖x‖, ∀x ∈ X. (A.2)

Theorem 3. The following assertions are equivalent

• K is bounded;

100

• K is continuous at x = 0, i.e., xj → 0⇒ AXj → 0;

• K is continuous for every x ∈ X.

Definition 7. Let K : X → Y be a linear operator between two Banach spaces X

and Y , then K is compact if for any bounded set S ⊆ X, L(S) is a relatively compact

subset of Y .

Definition 8. Let K : X → Y be an operator between two Hilbert spaces X and Y ,

then there exists one and only one linear bounded operator K∗ : X → Y with the

property

(Kx, y) = (x,K∗y), ∀x ∈ X, y ∈ Y. (A.3)

K∗ is called adjoint operator to K. If X = Y , then it is called self-adjoint.

Theorem 4. Fubini: Suppose X and Y are complete measure spaces. Suppose f(x,y)

is X × Y measurable. If

∫X×Y|f(x, y)| d(x, y) <∞,

where the integral is taken with respect to a product measure on the space over X×Y

, then

∫X

(∫Y

f(x, y) dy

)dx =

∫Y

(∫X

f(x, y) dx

)dy =

∫X×Y

f(x, y) d(x, y), (A.4)

Corollary 1. If f(x, y) = g(x)h(y) for some functions g and h, then

∫X

g(x) dx

∫Y

h(y) dy =

∫X×Y

f(x, y) d(x, y), (A.5)

Theorem 5. Lef be a continuous real valued function on a closed interval [a, b], then

f is Riemann integrable on this closed interval.

101

Theorem 6. Lef be Riemann integrable function on [a, b], then

limλ→±∞

∫ b

a

f(t) cos(λt)dt = 0, (A.6)

limλ→±∞

∫ b

a

f(t) sin(λt)dt = 0, (A.7)

limλ→±∞

∫ b

a

f(t)eiλtdt = 0. (A.8)

Lemma 1. If a system ηj+1 = F jηj is exponentially stable, then

1. for any bounded sequence rj, the sequence zj define by zj+1 = F jzj + rj is

bounded;

2. if rj converging to zero exponentially fast, then the sequence zj converges also

to zero exponentially fast [47].

Lemma 2. Let φj be a matrix sequence such that ‖φj‖ ≤ 1 for all j ≥ 0. If there

exist a real constant β > 0 and an integer κ > 0 such that for all j ≥ 0 the following

inequality holds

1

κ

j+κ−1∑i=j

φiTφi ≥ βI (A.9)

then the system

zj+1 = (I − φTj φj)zj (A.10)

is exponentially stable.

102

B Published and Under

Preparation Papers

•Published Paper

Sharefa Asiri, Taousmeriem Laleg-Kirati, and Chadia Zayane, “Inverse source prob-

lem for a one- dimensional wave equation using observers”, in 11th International

Conference on Mathematical and Numerical Aspects of Waves, Tunisia, 2013.

•Under Preparation Paper

”Comparison Between Tikhonov Regularization Approach and Observer-Based Ap-

proach ”

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