Research Article An Adaptive Observer-Based …downloads.hindawi.com/journals/mpe/2015/796539.pdfResearch Article An Adaptive Observer-Based Algorithm for Solving Inverse Source Problem

Research ArticleAn Adaptive Observer-Based Algorithm for Solving InverseSource Problem for the Wave Equation

Sharefa Asiri Chadia Zayane-Aissa and Taous-Meriem Laleg-Kirati

Computer Electrical and Mathematical Sciences and Engineering King Abdullah University of Science amp TechnologyPO Box 4700 Thuwal 23955-6900 Saudi Arabia

Correspondence should be addressed to Taous-Meriem Laleg-Kirati taousmeriemlalegkaustedusa

Received 12 February 2015 Revised 20 August 2015 Accepted 31 August 2015

Academic Editor Herb Kunze

Copyright copy 2015 Sharefa Asiri et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Observers are well known in control theory Originally designed to estimate the hidden states of dynamical systems given somemeasurements the observers scope has been recently extended to the estimation of some unknowns for systems governed by partialdifferential equations In this paper observers are used to solve inverse source problem for a one-dimensional wave equation Anadaptive observer is designed to estimate the state and source components for a fully discretized system The effectiveness of thealgorithm is emphasized in noise-free and noisy cases and an insight on the impact of measurementsrsquo size and location is provided

1 Introduction

In this paper we are interested in an inverse source problemfor the wave equation This problem appears frequently inmany fields especially inmodern seismology [1] One impor-tant application of this problem is to distinguish betweendifferent types of seismic events (eg earthquake implosionor explosion) [2] It is also important inmonitoring hydraulicfracturing by which fractures are created in rocks such thatentrapped hydrocarbons can be released and extracted [3]

Inverse problems are usually solved using optimizationtechniques where an appropriate cost function is minimizedHowever the ill-posedness of such problems generates insta-bility Regularization techniques are then used to restorethe stability Among the regularization techniques Tikhonovregularization [4] is probably themost used one For instanceit has been applied to the wave equation in [5 6] Other tech-niques have been also proposed to solve inverse problemsfor example in [7] a new minimization algorithm has beenproposed to solve an inverse problem for the wave equationwith unknown wave speed Most of the proposed methodsend up with an optimization step which generally turns to becomputationally heavy especially in the case of large numberof unknowns and may require an extensive storage

The objective of this paper is to present an alternativealgorithm based on observers to solve the inverse sourceproblem for the wave equation Observers are well knownin control theory for state estimation in finite dimensionaldynamical systems Presenting the distinctive feature andmain advantage of operating recursively on direct problemsobservers are gainingmore andmore interest in awide varietyof problems including partial differential equations (PDEs)systems For instance in [8] states and parameters are esti-mated using an observer depending on a discretized space fora mechanical system In [9] the initial state of a distributedparameter system has been estimated using two observersone for the forward time and the other for the backward timeA similar approach has been used in [10] using the forward-backward approach to solve inverse source problem for thewave equation An adaptive observer was applied in [11] forparameter estimation and stabilization of one-dimensionalwave equation where the boundary observation suffers froman unknown constant disturbance A similar work wasproposed in [12] with the state as unknown and the boundaryobservation suffers from an arbitrary long time delay

Dealing with PDEs either with observers or classicalinverse problems methods poses the challenge of approxi-mating infinite dimensional systems As regards observers

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 796539 8 pageshttpdxdoiorg1011552015796539

2 Mathematical Problems in Engineering

we can distinguish three approaches for studying suchsystems The first approach considers the design of theobserver in the continuous domain which requires mathe-matical analysis [13] and the application of the observer toreal application will require some adaptation The secondapproach consists in the semidiscretization of the equationin space The result of this semidiscretization can be usuallywritten in the standard state-space representation in thecontinuous domain (in time) which makes the extensionof the known methods in control theory easier The thirdapproach is the full-discretization of the PDE in space andtime In this case we can write the system in a discrete state-space representation We have chosen this latter approachsince it is more suitable for real implementation We showthat it can give good results provided that some conditionsaimed at minimizing the effect of numerical errors resultingfrom discretization are met

Another challenge related to solving inverse problems ingeneral arises when it comes to measurement constraintsIndeed from a practical point of view we usually do nothave enough measurements to estimate all the unknownsDealing with this source of ill-posedness means in observerstheory framework satisfying the equivalent property ofobservability Indeed given the PDE system together withthe measurements we can test in a prior step whether theunknown variables can be estimated fully or partially regard-less of the kind of observer to be used For instance in [8 911 12] the measurements were taken as the time derivative ofthe solution of the wave equationThis kind of measurementsgives a typical observability condition which has a positiveeffect on the stabilization but it is less readily available thanfield measurements Hence some authors sought to solveinverse problems for wave equation using observers based onpartial filedmeasurements that is measurements taken fromthe solution of the wave equation as in [14ndash16]

In this paper we consider a fully discretized version of aone-dimensional wave equation and we propose a new algo-rithm for inverse source problem based on adaptive observerfor the joint estimation of the states and the source termfrom partial measurements of the field Adaptive observersare widely used in control theory for parameter estimationin adaptive control or fault estimation in fault detection andisolation [17 18] In Section 2 problem statement is detailedThen the observer design is presented in Section 3 Finallynumerical results are presented and discussed

2 Problem Statement

Consider the one-dimensional wave equation with Dirichletboundary conditions defining in the domain Ω = (119909 119905) isin(0 119897) times (0 119879]

119906119905119905(119909 119905) minus 119888

2119906119909119909(119909 119905) = 119891 (119909)

119906 (0 119905) = 0 119906 (119897 119905) = 0

119906 (119909 0) = 1199031(119909) 119906

119905(119909 0) = 119903

2(119909)

(1)

where 119909 is the space coordinate 119905 is the time coordinate1199031(119909) and 119903

2(119909) are the initial conditions in L2[0 119897] 119891(119909) isin

L2[0 119897] is the source function which is assumed for simplic-ity to be independent of time and 119888 is the velocity which isknownThe notations 119906

119886and 119906

119886119886refer to the first and second

derivatives of 119906 with respect to 119886 respectivelyOur inverse problem falls in the estimation of the source

119891(119909) in (1) using an adaptive observer with partial measure-ments of the field 119906 available We first propose to rewrite (1)in a system of first order PDEs by introducing two auxiliaryvariables V(119909 119905) = 119906(119909 119905) and 119908(119909 119905) = 119906

119905(119909 119905) and let

120585 (119909 119905) = [V (119909 119905) 119908 (119909 119905)]tr (2)

where tr refers to transposeThen (1) can bewritten as follows

120597120585 (119909 119905)

120597119905= A120585 (119909 119905) + 119865

V (0 119905) = 0

V (119897 119905) = 0

V (119909 0) = 1199031(119909)

V119905(119909 0) = 119903

2(119909)

119911 =H120585 (119909 119905)

(3)

where the operatorA is given byA = ( 0 119868

1198882(12059721205971199092) 0) 119865 = ( 0119891 )

119911 is the output and H is the observation operator such thatH = [H

00] where H

0is a restriction operator on the

measured domainDiscretizing system (3) using implicit Euler scheme in

time and central finite difference discretization for the spacegives the following discrete state-space representation

120585119895+1= 119866120585119895+ 119861119891119895+ 119887

119911119895= 119867120585119895

119891119895+1= 119891119895

119895 = 1 2 119873119905

(4)

where

119866 = (

Δ119905119864 + 119868 Δ119905119868

119864 119868)

119864 =1198882Δ119905

(Δ119909)2(

minus2 1

1 minus2 d

d d 1

1 minus2

)

119861 = ((Δ119905)2119868

Δ119905119868

) 119867 = (119867119898

0)

(5)

and 119867119898= (

0 sdotsdotsdot 0

119868119898

0 sdotsdotsdot 0

) 119868119898is the identity matrix of dimension

119898 where119898 refers to the number of measurements and 119887 is aterm that includes the boundary conditions such that

Mathematical Problems in Engineering 3

119887 = (1198882(Δ119905)2

(Δ119909)2V119895101times(119873119909minus2)

1198882(Δ119905)2

(Δ119909)2V119895119873119909

1198882(Δ119905)

(Δ119909)2V119895101times(119873119909minus2)

1198882(Δ119905)

(Δ119909)2V119895119873119909

)

tr

(6)

This system is linear multiple-input multiple-output discretetime invariant If119873

119909refers to the space grid size and119898 refers

to the number of measurements then the state matrix 119866 is ofdimension 2119873

119909times2119873119909 the observer matrix119867 is of dimension

119898 times 2119873119909 and the input matrix 119861 is of dimension 2119873

119909times 119873119909

The numerical scheme (4) is consistent In addition it isstable if and only if 119888Δ119905 le Δ119909 (the CFL condition) Thus if119888Δ119905 le Δ119909 scheme (4) converges as Δ119905 rarr 0 Δ119909 rarr 0 to (3)and therefore to (1)

3 Observer Design

We propose to use an adaptive observer for the joint estima-tion of the states V and 119908 and the source 119891 This observerhas been proposed in [17] and it has been developed forjoint estimation of the state and the parameters Howeverwe propose to generalize the idea behind this observer toestimate the input considering each spatial sample of theinput as an independent parameter The adaptive observer isgiven by the following system of equations

119895= 119867120585119895

Υ119895+1= (119866 minus 119871119867)Υ

119895+ 119861

119891119895+1= 119891119895+ 120590Υ119895119879

119867119879(119911119895minus 119895)

120585119895+1= 119866120585119895+ 119861119891119895+ 119887 + 119871 (119911

119895minus 119895)

+ Υ119895+1(119891119895+1minus 119891119895)

(7)

where 119871 is the observer gain matrix of dimension 2119873119909times 119898

120585119895 and 119891119895 are the state and source estimates respectively Υ119895is a matrix sequence obtained by linearly filtering 119861 and 120590 isa scalar gain satisfying the following assumption as in [17]

Assumption 1 The scalar gain 120590 satisfies the following

(1) radic120590119867Υ1198952le 1

(2) (1120581)sum119895+120581minus1119894=119895120590Υ119894119879

119867119879119867Υ119894ge 120573119868 for some constant 120573 gt

0 integer 120581 gt 0 and all 119895

Remark 2 In the proposed method no particular formfor the matrices 119866 119861 and 119867 is required However allthese matrices are assumed bounded In our problem waveequation with constant velocity these matrices are actuallyconstant and therefore always bounded

Remark 3 From Remark 2 any consistent and stable numer-ical method can be used to discretize system (3) provided thatit ends up with bounded matrices (119866 119861 and119867)

Under Assumption 1 Algorithm (7) converges exponen-tially fast when 119895 tends to infinity in noise-free case and

the estimation errors remain bounded in the noisy case aslong as the noises are bounded Moreover the estimationerrors converge in the mean to zero if the noises have zeromeans see Theorems 1 and 2 with their proofs in [17]

4 Numerical Simulations

To test the performance of the observer we generated a setof synthetic data using the following parameters Δ119909 = 001119897 = 2 Δ119905 = 001 and 119879 = 100 Thus 119873

119909= 201 and 119873

119905=

10001 The velocity is chosen to be 1198882 = 09 and the sourceis equal to 119891(119909) = 3 sin(5119909) The matrix sequence Υ119895 and thescalar gain 120590 are chosen such that Assumption 1 is satisfiedThe algorithm was implemented in Matlab and the tests wererun for two main cases noise-free and noisy datasets Inthe noise-corrupted case zero mean white Gaussian randomnoises were added to the states and to the measurementswith standard deviations 120590

120585= 0007816 and 120590

119911= 001044

respectively The gain matrix is selected to have fast andaccurate convergence of the observer We took advantage ofthe particular structure of 119866 to design the gain 119871 Indeedthe matrix 119866 is sparse so we selected 119871 to be also a sparsematrix The number of unknown entries is then reduced andwe identified them such that the eigenvalues of (119866 minus 119871119867) areinside the unit circle In general standard pole placement canbe used to select the gain matrix 119871

Figure 1 shows the error in the estimated state andFigure 2 presents the exact and the estimated source bothfigures exhibit noise-free and noisy cases with respect to fulland partial measurements For the partial measurementswe supposed that the field is available on half of the spacedomain only Tables 1 and 2 show the minimum squareerror (MSE) in the estimated source in noise-free and noisycases respectively Both tables show the error in case of fullmeasurements partial measurements taken from themiddleand partial measurements taken from the end

In the noise-free case the adaptive observer used in thispaper provides a good estimate of the unknown source forthe wave equation both when the field is available on thewhole space domain and when it is available only on halfthe domain However we noticed in the second case a smallerror at the end of the interval In the noisy case also thereconstruction is good but can be improved by a good choiceof the gain 119871

5 Comparison between Observer-Based andTikhonov-Based Approaches

To assess the observer performance in the source estima-tion a comparison to optimization-based method has beenperformed The initial problem has been first written in


(d)(c)

(b)(a)

2minus1

1

1

2

3

0

0

0

0

10

20

50

100

tx

Full measurements

times10minus3

times10minus4

04

06

02

02

0

01

minus01

minus02

0

0

0

50

100

xt

1

2

Noise-free case

Partial measurements

04

02

0

2

1

0 0

xt

0

01

50

100

minus02

minus02

minus01

minus04

Full measurementsNoise-corrupted case

0

2

1

0 0

0

02

50

tx

100

02

04

minus04

minus02

minus02


Figure 1 The state error (120585 minus 120585) (a) and (b) present the noise-free case with respect to full measurements and partial measurementsrespectively (c) and (d) show the noise-corrupted case with respect to full measurements and partial measurements respectively In thepartial measurements cases 50 of the state components are taken from the end

Table 1 Source estimation errors in the noise-free case

Measurements MSE = radic( 1119873119909

)

119873119909

sum

119894=1

(119891119894minus 119891119894)

Full 18168 times 10minus14

Partial (middle) 03354

Partial (end) 02096

a suitable inputoutput formulation allowing us to havecomparable problems Then due to the ill-posedness of theproblem a Tikhonov regularization has been used whensolving the optimization problem The two methods theobserver-based method and the optimization-based method

Table 2 Source estimation errors in the noisy case


)

119873119909

sum

119894=1

(119891119894minus 119891119894)

Full 02865


Partial (end) 03213

have been used for the estimation of the source for the sameset of measurements and the same level of noise

To formulate the initial problem we propose to derive thestate and output at time 119896 + 119901 119901 isin N from the state at time 119896and the input sequence using the state-space matrices 119866 119861and119867Thus by repeating substitution from (4) and for some


Full measurementsNoise-free case

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f


(d)(c)

(b)(a)

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

Figure 2 The exact source 119891 (blue) and the estimated source 119891 (black) (a) and (b) present the noise-free case with respect to fullmeasurements and partial measurements respectively (c) and (d) show the noise-corrupted case with respect to full measurements andpartial measurements respectively In the partial measurements cases 50 of the state components are taken from the end

119901 isin N we obtain a new state-space representation where thetransmission matrix is given by a Hankel matrix as follows

120585119895+119901= 119866119901120585119895+C119901119891119895

119901+ b

119911119895

119901= O120585119895+ 120591119891119895

119901

(8)

where

119891119895

119901= [119891119895119891119895+1sdot sdot sdot 119891

119895+119901minus1]tr

119911119895

119901= [119911119895119911119895+1sdot sdot sdot 119911119895+119901minus1]tr

C119901= [119866119901minus1119861 sdot sdot sdot 119866119861 119861]

O = [119867 119867119866 sdot sdot sdot 119867119866119901minus1]tr

120591 =

[[[[[[[[[[

[

0119898times119873119909

0 0 sdot sdot sdot 0

119867119861 0119898times119873119909

d d

119867119866119861 119867119861 0119898times119873119909

d 0

d d d 0

119867119866119901minus2119861 sdot sdot sdot 119867119866119861 119867119861 0

119898times119873119909

]]]]]]]]]]

]

(9)

and b = 1119901otimes 119887 (otimes is the Kronecker product)


Thus from the second equation in (8) new set of meas-urements can be defined as

119895

119901= 120591119891119895

119901 (10)

where 119895119901= 119911119895

119901minusO120585119895

The aim is to estimate the source 119891 at time step 119895 byminimizing the following cost function where Tikhonovregularization is used

119869120572(119891119895

119901) =1

2

10038171003817100381710038171003817120591119891119895

119901minus 11989510038171003817100381710038171003817

2

2+120572

2

10038171003817100381710038171003817119891119895

119901

10038171003817100381710038171003817

2

2 (11)

where 120572 is the regularization parameter There are well-studied approaches for selecting this parameter such as L-curve GCV and NCP [19]

For the numerical simulations it is important to note thatthe size of the Hankel matrix depends on the space step Δ119909time step Δ119905 and the final time 119879 To have reasonable sizewhich allows computation using Matlab the values of theseparameters have been chosen as follows Δ119909 = 01 Δ119905 =005 and 119879 = 2 respectively This decrease in the final time119879 will affect the estimation errors convergence as discussedin Section 3 especially in the noisy case Consequently weconsider a small modification on the observerrsquos structure (7)in order to increase the robustness of the algorithm Thismodification has been inspired by sliding mode observersand consists in adding tanh to the correction term asdescribed in the following

119895= 119867120585119895

Υ119895+1= (119866 minus 119871119867)Υ

119895+ 119861

119891119895+1

= 119891119895

+ ΣΥ119895119879

119867119879[(119911119895minus 119895) + 1205741tanh (120574

2(119911119895minus 119895))]

120585119895+1

= 119866120585119895+ 119861119891119895+ 119887

+ 119871 [(119911119895minus 119895) + 1205743tanh (120574

4(119911119895minus 119895))]

+ Υ119895+1(119891119895+1minus 119891119895)

(12)

where 1205741 1205742 1205743 and 120574

4are scalers

The results for source estimation using observer-basedand Tikhonov-based methods with full and partial measure-ments in noise-free and noisy cases are depicted in Figure 3The corresponding MSE are presented in Tables 3 and 4 fornoise-free and noise-corrupted cases respectively

Under the described conditions the observer approachgives comparable results in some cases better results than theoptimization-based methods

6 Discussion

Wehave studied the effect of number ofmeasurements on theconvergence of the proposed observer Obviously increasing

Table 3 MSE for the source estimation using observer andTikhonov methods in the noise-free case

Measurements Observer TikhonovFull 12212 times 10

minus599112 times 10

minus13

Partial 05033 15539

Table 4 MSE for the source estimation using observer andTikhonov methods in the noisy case

Measurements Observer TikhonovFull 02060 02355Partial 06805 15466

number of measurements means increasing information onthe state thus insuring the observability condition for all thestates However for some applications only few measure-ments can be available and the idea is to study the effect ofthis number on the convergence of the observer

The analysis of the error of estimation of the sourcewith respect to the number of measurements shows thatnumerical issues may happen when we reduce the number ofmeasurements below a threshold These numerical problemscome in fact from the ill-conditioning of the observabilitymatrix119882

119882 = (119867 119867119866 1198671198662sdot sdot sdot 119867119866

119899minus1)tr (13)

The decay of the condition number of the observabilitymatrix 119882 as a function of the number of measurements isillustrated in Figure 4 It is well known in control theorythat the rank of 119882 gives the number of observable statesIt is known also that a high condition number for theobservability matrix leads to nearly unobservable states [20]

It is also important to study the effect of discretizationon the performance of the method which was not includedin this paper as the objective was to assess the possibility ofusing this method and to discuss its performance in presenceof noise The scheme that we used in the paper works wellbut the step discretization may affect the performance andespecially in case of few measurements More investigationson this question are required

The objective of this paper was to propose a newmethodas an alternative to the standard optimization methods inorder to solve inverse source problems for the wave equationOf course this problem has several important applicationsin different fields ranging from geophysics to medical fieldand especially when few measurements are available Thefirst results obtained on simulations are promising andthe observer approach seems to be suitable for real onlineestimation problems thanks to its recursive structure How-ever we still have to investigate more the approach beforewe can claim its performance on real application One ofthe points to assess carefully is the number of availablemeasurements Through this work we studied the effect ofmeasurements on the performance and from the comparisonto the optimization-based methods in Section 5 it is clearthat the adaptive observer gives interesting results which ispromising for real applications The second important point


(d)(c)

(b)(a)

Full measurements

Noise-free case

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs

Full measurements

Noise-corrupted case

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f

fTik

obs


0 05 1 15 2minus4

minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs

Figure 3 The exact source 119891 (solid blue line) and the estimated one 119891 using observer solution (red) and modified-Tikhonov (green) (a)and (b) present the noise-free case with respect to full measurements and partial measurements respectively (c) and (d) show the noise-corrupted case with respect to full measurements and partial measurements respectively In the partial measurements cases 50 of the statecomponents are taken from the end

is the effect of noise Even if we succeed to obtain goodresults in noisy cases we believe that some improvement canbe suggested leading to some modifications of the observerstructure aiming at improving the robustness properties

7 Conclusion

In this paper an adaptive observer for the joint estimationof the source and the states in the wave equation has been

designed Numerical simulations for the source and statesestimation using observer have been presented and theyhave proven the capability of observer to estimate both thesource and the states in noise-free and noisy cases A com-parison between observer algorithm and an optimization-based method has been performedThis comparison consid-ered also the different cases of noise (noise-free and noise-corrupted) with full and partial measurements The resultsshow the outperformance of the observer-based approach


150 160 170 180 190 2000

1000

2000

3000

4000

5000

6000

7000

8000

Number of measurements

Con

ditio

n nu

mbe

r of W

Figure 4 Number of measurements versus the condition numberof the observability matrix119882

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Kim Q Liu and J Tromp ldquoAdjoint centroid-moment tensorinversionsrdquoGeophysical Journal International vol 186 no 1 pp264ndash278 2011

[2] B Sjogreen andN A Petersson ldquoSource estimation by full waveform inversionrdquo Journal of Scientific Computing vol 59 no 1pp 247ndash276 2014

[3] G E King ldquoHydraulic fracturing 101 what every representa-tive environmentalist regulator reporter investor universityresearcher neighbor and engineer should know about estimat-ing frac risk and improving frac performance in unconventionalgas and oil wellsrdquo in Proceedings of the SPEHydraulic FracturingTechnology Conference Society of Petroleum Engineers pp 651ndash730 February 2012

[4] C W Groetsch The Theory of Tikhonov Regularization forFredholm Equations of the First Kind vol 105 Pitman BostonMass USA 1984

[5] Y M Chen and J Q Liu ldquoA numerical algorithm for solvinginverse problems of two-dimensional wave equationsrdquo Journalof Computational Physics vol 50 no 2 pp 193ndash208 1983

[6] F Hong-Sun and H Bo ldquoA regularization homotopy methodfor the inverse problem of 2-d wave equation and well logconstraint inversionrdquo Chinese Journal of Geophysics vol 48 no6 pp 1509ndash1517 2005

[7] L Oksanen ldquoSolving an inverse problem for the wave equationby using aminimization algorithm and time-reversedmeasure-mentsrdquo Inverse Problems and Imaging vol 5 no 3 pp 731ndash7442011

[8] PMoireau D Chapelle and P Le Tallec ldquoJoint state and param-eter estimation for distributed mechanical systemsrdquo ComputerMethods in Applied Mechanics and Engineering vol 197 no 6pp 659ndash677 2008

[9] K Ramdani M Tucsnak and G Weiss ldquoRecovering andinitial state of an infinite-dimensional system using observersrdquoAutomatica vol 46 no 10 pp 1616ndash1625 2010

[10] M Chapouly and M Mirrahimi ldquoDistributed source identi-fication for wave equations an observer-based approachrdquo inProceedings of the 19th International Symposium on Mathemat-ical Theory of Networks and Systems pp 389ndash394 BudapestHungary 2010

[11] W Guo and B-Z Guo ldquoParameter estimation and stabilisationfor a one-dimensional wave equation with boundary outputconstant disturbance and non-collocated controlrdquo InternationalJournal of Control vol 84 no 2 pp 381ndash395 2011

[12] B-Z Guo C-Z Xu and H Hammouri ldquoOutput feedbackstabilization of a one-dimensional wave equation with anarbitrary time delay in boundary observationrdquo ESAIM ControlOptimisation and Calculus of Variations vol 18 no 1 pp 22ndash352012

[13] M Tucsnak and GWeissObservation and Control for OperatorSemigroups Springer 2009

[14] J Chauvin ldquoObserver design for a class of wave equation drivenby an unknown periodic inputrdquo in Proceedings of the IEEEInternational Conference on Control Applications (CCA rsquo12) pp922ndash926 October 2012

[15] D Chapelle N Cındea and P Moireau ldquoImproving conver-gence in numerical analysis using observersmdashthe wave-likeequation caserdquo Mathematical Models and Methods in AppliedSciences vol 22 no 12 Article ID 1250040 35 pages 2012

[16] D Chapelle N Cındea M De Buhan and P MoireauldquoExponential convergence of an observer based on partial fieldmeasurements for the wave equationrdquo Mathematical Problemsin Engineering vol 2012 Article ID 581053 12 pages 2012

[17] A Guyader and Q Zhang ldquoAdaptive observer for discretetime linear time varying systemsrdquo in Proceedings of the 13thSymposium on System Identification (SYSID rsquo03) pp 1743ndash1748Rotterdam The Netherlands 2003

[18] R J Patton and J Chen ldquoObserver-based fault detection andisolation robustness and applicationsrdquo Control EngineeringPractice vol 5 no 5 pp 671ndash682 1997

[19] T Correia A Gibson M Schweiger and J Hebden ldquoSelectionof regularization parameter for optical topographyrdquo Journal ofBiomedical Optics vol 14 no 3 Article ID 034044 2009

[20] Z Chen ldquoLocal observability and its application to multiplemeasurement estimationrdquo IEEE Transactions on Industrial Elec-tronics vol 38 no 6 pp 491ndash496 1991

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


we can distinguish three approaches for studying suchsystems The first approach considers the design of theobserver in the continuous domain which requires mathe-matical analysis [13] and the application of the observer toreal application will require some adaptation The secondapproach consists in the semidiscretization of the equationin space The result of this semidiscretization can be usuallywritten in the standard state-space representation in thecontinuous domain (in time) which makes the extensionof the known methods in control theory easier The thirdapproach is the full-discretization of the PDE in space andtime In this case we can write the system in a discrete state-space representation We have chosen this latter approachsince it is more suitable for real implementation We showthat it can give good results provided that some conditionsaimed at minimizing the effect of numerical errors resultingfrom discretization are met

Another challenge related to solving inverse problems ingeneral arises when it comes to measurement constraintsIndeed from a practical point of view we usually do nothave enough measurements to estimate all the unknownsDealing with this source of ill-posedness means in observerstheory framework satisfying the equivalent property ofobservability Indeed given the PDE system together withthe measurements we can test in a prior step whether theunknown variables can be estimated fully or partially regard-less of the kind of observer to be used For instance in [8 911 12] the measurements were taken as the time derivative ofthe solution of the wave equationThis kind of measurementsgives a typical observability condition which has a positiveeffect on the stabilization but it is less readily available thanfield measurements Hence some authors sought to solveinverse problems for wave equation using observers based onpartial filedmeasurements that is measurements taken fromthe solution of the wave equation as in [14ndash16]

In this paper we consider a fully discretized version of aone-dimensional wave equation and we propose a new algo-rithm for inverse source problem based on adaptive observerfor the joint estimation of the states and the source termfrom partial measurements of the field Adaptive observersare widely used in control theory for parameter estimationin adaptive control or fault estimation in fault detection andisolation [17 18] In Section 2 problem statement is detailedThen the observer design is presented in Section 3 Finallynumerical results are presented and discussed

2 Problem Statement

Consider the one-dimensional wave equation with Dirichletboundary conditions defining in the domain Ω = (119909 119905) isin(0 119897) times (0 119879]

119906119905119905(119909 119905) minus 119888

2119906119909119909(119909 119905) = 119891 (119909)

119906 (0 119905) = 0 119906 (119897 119905) = 0

119906 (119909 0) = 1199031(119909) 119906

119905(119909 0) = 119903

2(119909)

(1)

where 119909 is the space coordinate 119905 is the time coordinate1199031(119909) and 119903

2(119909) are the initial conditions in L2[0 119897] 119891(119909) isin

L2[0 119897] is the source function which is assumed for simplic-ity to be independent of time and 119888 is the velocity which isknownThe notations 119906

119886and 119906

119886119886refer to the first and second

derivatives of 119906 with respect to 119886 respectivelyOur inverse problem falls in the estimation of the source

119891(119909) in (1) using an adaptive observer with partial measure-ments of the field 119906 available We first propose to rewrite (1)in a system of first order PDEs by introducing two auxiliaryvariables V(119909 119905) = 119906(119909 119905) and 119908(119909 119905) = 119906

119905(119909 119905) and let

120585 (119909 119905) = [V (119909 119905) 119908 (119909 119905)]tr (2)

where tr refers to transposeThen (1) can bewritten as follows

120597120585 (119909 119905)

120597119905= A120585 (119909 119905) + 119865

V (0 119905) = 0

V (119897 119905) = 0

V (119909 0) = 1199031(119909)

V119905(119909 0) = 119903

2(119909)

119911 =H120585 (119909 119905)

(3)

where the operatorA is given byA = ( 0 119868

1198882(12059721205971199092) 0) 119865 = ( 0119891 )

119911 is the output and H is the observation operator such thatH = [H

00] where H

0is a restriction operator on the

measured domainDiscretizing system (3) using implicit Euler scheme in

time and central finite difference discretization for the spacegives the following discrete state-space representation

120585119895+1= 119866120585119895+ 119861119891119895+ 119887

119911119895= 119867120585119895

119891119895+1= 119891119895

119895 = 1 2 119873119905

(4)

where

119866 = (

Δ119905119864 + 119868 Δ119905119868

119864 119868)

119864 =1198882Δ119905

(Δ119909)2(

minus2 1

1 minus2 d

d d 1

1 minus2

)

119861 = ((Δ119905)2119868

Δ119905119868

) 119867 = (119867119898

0)

(5)

and 119867119898= (

0 sdotsdotsdot 0

119868119898

0 sdotsdotsdot 0

) 119868119898is the identity matrix of dimension

119898 where119898 refers to the number of measurements and 119887 is aterm that includes the boundary conditions such that


119887 = (1198882(Δ119905)2

(Δ119909)2V119895101times(119873119909minus2)

1198882(Δ119905)2

(Δ119909)2V119895119873119909

1198882(Δ119905)

(Δ119909)2V119895101times(119873119909minus2)

1198882(Δ119905)

(Δ119909)2V119895119873119909

)

tr

(6)






119909times 119873119909


3 Observer Design


119895= 119867120585119895

Υ119895+1= (119866 minus 119871119867)Υ

119895+ 119861

119891119895+1= 119891119895+ 120590Υ119895119879

119867119879(119911119895minus 119895)

120585119895+1= 119866120585119895+ 119861119891119895+ 119887 + 119871 (119911

119895minus 119895)

+ Υ119895+1(119891119895+1minus 119891119895)

(7)




(1) radic120590119867Υ1198952le 1

(2) (1120581)sum119895+120581minus1119894=119895120590Υ119894119879









119909= 201 and 119873

119905=


120585= 0007816 and 120590

119911= 001044







(d)(c)

(b)(a)

2minus1

1

1

2

3

0

0

0

0

10

20

50

100

tx

Full measurements

times10minus3

times10minus4

04

06

02

02

0

01

minus01

minus02

0

0

0

50

100

xt

1

2

Noise-free case


04

02

0

2

1

0 0

xt

0

01

50

100

minus02

minus02

minus01

minus04


0

2

1

0 0

0

02

50

tx

100

02

04

minus04

minus02

minus02





)

119873119909

sum

119894=1

(119891119894minus 119891119894)



Partial (end) 02096




)

119873119909

sum

119894=1

(119891119894minus 119891119894)

Full 02865


Partial (end) 03213





0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f


(d)(c)

(b)(a)

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f



120585119895+119901= 119866119901120585119895+C119901119891119895

119901+ b

119911119895

119901= O120585119895+ 120591119891119895

119901

(8)

where

119891119895

119901= [119891119895119891119895+1sdot sdot sdot 119891

119895+119901minus1]tr

119911119895




120591 =

[[[[[[[[[[

[

0119898times119873119909


119867119861 0119898times119873119909

d d

119867119866119861 119867119861 0119898times119873119909

d 0

d d d 0


119898times119873119909

]]]]]]]]]]

]

(9)




119895

119901= 120591119891119895

119901 (10)

where 119895119901= 119911119895

119901minusO120585119895


119869120572(119891119895

119901) =1

2

10038171003817100381710038171003817120591119891119895

119901minus 11989510038171003817100381710038171003817

2

2+120572

2

10038171003817100381710038171003817119891119895

119901

10038171003817100381710038171003817

2

2 (11)



119895= 119867120585119895

Υ119895+1= (119866 minus 119871119867)Υ

119895+ 119861

119891119895+1

= 119891119895

+ ΣΥ119895119879

119867119879[(119911119895minus 119895) + 1205741tanh (120574

2(119911119895minus 119895))]

120585119895+1

= 119866120585119895+ 119861119891119895+ 119887

+ 119871 [(119911119895minus 119895) + 1205743tanh (120574

4(119911119895minus 119895))]

+ Υ119895+1(119891119895+1minus 119891119895)

(12)

where 1205741 1205742 1205743 and 120574

4are scalers



6 Discussion





minus13

Partial 05033 15539





119882 = (119867 119867119866 1198671198662sdot sdot sdot 119867119866

119899minus1)tr (13)





(d)(c)

(b)(a)

Full measurements

Noise-free case

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs

Full measurements


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f

fTik

obs


0 05 1 15 2minus4

minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs



7 Conclusion




150 160 170 180 190 2000

1000

2000

3000

4000

5000

6000

7000

8000


Con

ditio

n nu

mbe

r of W




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119887 = (1198882(Δ119905)2

(Δ119909)2V119895101times(119873119909minus2)

1198882(Δ119905)2

(Δ119909)2V119895119873119909

1198882(Δ119905)

(Δ119909)2V119895101times(119873119909minus2)

1198882(Δ119905)

(Δ119909)2V119895119873119909

)

tr

(6)






119909times 119873119909


3 Observer Design


119895= 119867120585119895

Υ119895+1= (119866 minus 119871119867)Υ

119895+ 119861

119891119895+1= 119891119895+ 120590Υ119895119879

119867119879(119911119895minus 119895)

120585119895+1= 119866120585119895+ 119861119891119895+ 119887 + 119871 (119911

119895minus 119895)

+ Υ119895+1(119891119895+1minus 119891119895)

(7)




(1) radic120590119867Υ1198952le 1

(2) (1120581)sum119895+120581minus1119894=119895120590Υ119894119879









119909= 201 and 119873

119905=


120585= 0007816 and 120590

119911= 001044







(d)(c)

(b)(a)

2minus1

1

1

2

3

0

0

0

0

10

20

50

100

tx

Full measurements

times10minus3

times10minus4

04

06

02

02

0

01

minus01

minus02

0

0

0

50

100

xt

1

2

Noise-free case


04

02

0

2

1

0 0

xt

0

01

50

100

minus02

minus02

minus01

minus04


0

2

1

0 0

0

02

50

tx

100

02

04

minus04

minus02

minus02





)

119873119909

sum

119894=1

(119891119894minus 119891119894)



Partial (end) 02096




)

119873119909

sum

119894=1

(119891119894minus 119891119894)

Full 02865


Partial (end) 03213





0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f


(d)(c)

(b)(a)

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f



120585119895+119901= 119866119901120585119895+C119901119891119895

119901+ b

119911119895

119901= O120585119895+ 120591119891119895

119901

(8)

where

119891119895

119901= [119891119895119891119895+1sdot sdot sdot 119891

119895+119901minus1]tr

119911119895




120591 =

[[[[[[[[[[

[

0119898times119873119909


119867119861 0119898times119873119909

d d

119867119866119861 119867119861 0119898times119873119909

d 0

d d d 0


119898times119873119909

]]]]]]]]]]

]

(9)




119895

119901= 120591119891119895

119901 (10)

where 119895119901= 119911119895

119901minusO120585119895


119869120572(119891119895

119901) =1

2

10038171003817100381710038171003817120591119891119895

119901minus 11989510038171003817100381710038171003817

2

2+120572

2

10038171003817100381710038171003817119891119895

119901

10038171003817100381710038171003817

2

2 (11)



119895= 119867120585119895

Υ119895+1= (119866 minus 119871119867)Υ

119895+ 119861

119891119895+1

= 119891119895

+ ΣΥ119895119879

119867119879[(119911119895minus 119895) + 1205741tanh (120574

2(119911119895minus 119895))]

120585119895+1

= 119866120585119895+ 119861119891119895+ 119887

+ 119871 [(119911119895minus 119895) + 1205743tanh (120574

4(119911119895minus 119895))]

+ Υ119895+1(119891119895+1minus 119891119895)

(12)

where 1205741 1205742 1205743 and 120574

4are scalers



6 Discussion





minus13

Partial 05033 15539





119882 = (119867 119867119866 1198671198662sdot sdot sdot 119867119866

119899minus1)tr (13)





(d)(c)

(b)(a)

Full measurements

Noise-free case

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs

Full measurements


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f

fTik

obs


0 05 1 15 2minus4

minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs



7 Conclusion




150 160 170 180 190 2000

1000

2000

3000

4000

5000

6000

7000

8000


Con

ditio

n nu

mbe

r of W




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(d)(c)

(b)(a)

2minus1

1

1

2

3

0

0

0

0

10

20

50

100

tx

Full measurements

times10minus3

times10minus4

04

06

02

02

0

01

minus01

minus02

0

0

0

50

100

xt

1

2

Noise-free case


04

02

0

2

1

0 0

xt

0

01

50

100

minus02

minus02

minus01

minus04


0

2

1

0 0

0

02

50

tx

100

02

04

minus04

minus02

minus02





)

119873119909

sum

119894=1

(119891119894minus 119891119894)



Partial (end) 02096




)

119873119909

sum

119894=1

(119891119894minus 119891119894)

Full 02865


Partial (end) 03213





0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f


(d)(c)

(b)(a)

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f



120585119895+119901= 119866119901120585119895+C119901119891119895

119901+ b

119911119895

119901= O120585119895+ 120591119891119895

119901

(8)

where

119891119895

119901= [119891119895119891119895+1sdot sdot sdot 119891

119895+119901minus1]tr

119911119895




120591 =

[[[[[[[[[[

[

0119898times119873119909


119867119861 0119898times119873119909

d d

119867119866119861 119867119861 0119898times119873119909

d 0

d d d 0


119898times119873119909

]]]]]]]]]]

]

(9)




119895

119901= 120591119891119895

119901 (10)

where 119895119901= 119911119895

119901minusO120585119895


119869120572(119891119895

119901) =1

2

10038171003817100381710038171003817120591119891119895

119901minus 11989510038171003817100381710038171003817

2

2+120572

2

10038171003817100381710038171003817119891119895

119901

10038171003817100381710038171003817

2

2 (11)



119895= 119867120585119895

Υ119895+1= (119866 minus 119871119867)Υ

119895+ 119861

119891119895+1

= 119891119895

+ ΣΥ119895119879

119867119879[(119911119895minus 119895) + 1205741tanh (120574

2(119911119895minus 119895))]

120585119895+1

= 119866120585119895+ 119861119891119895+ 119887

+ 119871 [(119911119895minus 119895) + 1205743tanh (120574

4(119911119895minus 119895))]

+ Υ119895+1(119891119895+1minus 119891119895)

(12)

where 1205741 1205742 1205743 and 120574

4are scalers



6 Discussion





minus13

Partial 05033 15539





119882 = (119867 119867119866 1198671198662sdot sdot sdot 119867119866

119899minus1)tr (13)





(d)(c)

(b)(a)

Full measurements

Noise-free case

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs

Full measurements


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f

fTik

obs


0 05 1 15 2minus4

minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs



7 Conclusion




150 160 170 180 190 2000

1000

2000

3000

4000

5000

6000

7000

8000


Con

ditio

n nu

mbe

r of W




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f


(d)(c)

(b)(a)

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f



120585119895+119901= 119866119901120585119895+C119901119891119895

119901+ b

119911119895

119901= O120585119895+ 120591119891119895

119901

(8)

where

119891119895

119901= [119891119895119891119895+1sdot sdot sdot 119891

119895+119901minus1]tr

119911119895




120591 =

[[[[[[[[[[

[

0119898times119873119909


119867119861 0119898times119873119909

d d

119867119866119861 119867119861 0119898times119873119909

d 0

d d d 0


119898times119873119909

]]]]]]]]]]

]

(9)




119895

119901= 120591119891119895

119901 (10)

where 119895119901= 119911119895

119901minusO120585119895


119869120572(119891119895

119901) =1

2

10038171003817100381710038171003817120591119891119895

119901minus 11989510038171003817100381710038171003817

2

2+120572

2

10038171003817100381710038171003817119891119895

119901

10038171003817100381710038171003817

2

2 (11)



119895= 119867120585119895

Υ119895+1= (119866 minus 119871119867)Υ

119895+ 119861

119891119895+1

= 119891119895

+ ΣΥ119895119879

119867119879[(119911119895minus 119895) + 1205741tanh (120574

2(119911119895minus 119895))]

120585119895+1

= 119866120585119895+ 119861119891119895+ 119887

+ 119871 [(119911119895minus 119895) + 1205743tanh (120574

4(119911119895minus 119895))]

+ Υ119895+1(119891119895+1minus 119891119895)

(12)

where 1205741 1205742 1205743 and 120574

4are scalers



6 Discussion





minus13

Partial 05033 15539





119882 = (119867 119867119866 1198671198662sdot sdot sdot 119867119866

119899minus1)tr (13)





(d)(c)

(b)(a)

Full measurements

Noise-free case

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs

Full measurements


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f

fTik

obs


0 05 1 15 2minus4

minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs



7 Conclusion




150 160 170 180 190 2000

1000

2000

3000

4000

5000

6000

7000

8000


Con

ditio

n nu

mbe

r of W




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119895

119901= 120591119891119895

119901 (10)

where 119895119901= 119911119895

119901minusO120585119895


119869120572(119891119895

119901) =1

2

10038171003817100381710038171003817120591119891119895

119901minus 11989510038171003817100381710038171003817

2

2+120572

2

10038171003817100381710038171003817119891119895

119901

10038171003817100381710038171003817

2

2 (11)



119895= 119867120585119895

Υ119895+1= (119866 minus 119871119867)Υ

119895+ 119861

119891119895+1

= 119891119895

+ ΣΥ119895119879

119867119879[(119911119895minus 119895) + 1205741tanh (120574

2(119911119895minus 119895))]

120585119895+1

= 119866120585119895+ 119861119891119895+ 119887

+ 119871 [(119911119895minus 119895) + 1205743tanh (120574

4(119911119895minus 119895))]

+ Υ119895+1(119891119895+1minus 119891119895)

(12)

where 1205741 1205742 1205743 and 120574

4are scalers



6 Discussion





minus13

Partial 05033 15539





119882 = (119867 119867119866 1198671198662sdot sdot sdot 119867119866

119899minus1)tr (13)





(d)(c)

(b)(a)

Full measurements

Noise-free case

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs

Full measurements


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f

fTik

obs


0 05 1 15 2minus4

minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs



7 Conclusion




150 160 170 180 190 2000

1000

2000

3000

4000

5000

6000

7000

8000


Con

ditio

n nu

mbe

r of W




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(d)(c)

(b)(a)

Full measurements

Noise-free case

0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs

Full measurements


0 05 1 15 2minus3

minus2

minus1

0

1

2

3

4

x

f

f

fTik

obs


0 05 1 15 2minus4

minus3

minus2

minus1

0

1

2

3

x

f

f

fTik

obs



7 Conclusion




150 160 170 180 190 2000

1000

2000

3000

4000

5000

6000

7000

8000


Con

ditio

n nu

mbe

r of W




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










150 160 170 180 190 2000

1000

2000

3000

4000

5000

6000

7000

8000


Con

ditio

n nu

mbe

r of W




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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