Research Article About a Partial Differential Equation-Based...

Hindawi Publishing CorporationISRN Signal ProcessingVolume 2013 Article ID 605035 18 pageshttpdxdoiorg1011552013605035

Research ArticleAbout a Partial Differential Equation-Based Interpolator forSignal Envelope Computing Existence Results and Applications

Oumar Niang123 Abdoulaye Thioune24 Eacuteric Deleacutechelle2

Mary Teuw Niane3 and Jacques Lemoine2

1 Departement Informatique et Telecommunications Ecole Polytechnique de Thies (EPT) Thies BP A10 Senegal2 Laboratoire Images Signaux et Systemes Intelligents (LISSI-EA3956) Universite Paris-Est Creteil Val-de-Marne Creteil France3 Laboratoire drsquoAnalyse Numerique et drsquoInformatique (LANI) Universite Gaston Berger (UGB) Saint-Louis BP 234 Senegal4Departement de Mathematique et Informatique Faculte des Sciences et Technique Universite Cheikh Anta Diop de DakarDakar BP 5005 Senegal

Correspondence should be addressed to Oumar Niang oniangucadsn

Received 6 November 2012 Accepted 3 December 2012

Academic Editors H Hu and S Li

Copyright copy 2013 Oumar Niang 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

This paper models and solves the mathematical problem of interpolating characteristic points of signals by a partial differentialEquation-(PDE-) based approach The existence and uniqueness results are established in an appropriate space whose regularity issimilar to cubic spline one We show how this space is suitable for the empirical mode decomposition (EMD) sifting processNumerical schemes and computing applications are also presented for signal envelopes calculation The test results show theusefulness of the new PDE interpolator in some pathological cases like input class functions that are not so regular as in the cubicsplines case Some image filtering tests strengthen the demonstration of PDE interpolator performance

1 Introduction

Interpolators are widely used in signal and processing or dataanalysis In particular for the empirical mode decomposition(EMD) algorithm [1 2] the iterative estimation of the signaltrend is based on the computing of the envelopes obtainedby the cubic spline interpolation of local extrema The splineinterpolation has been recognized as being very effective forEMD But for signals that have no local extremum the cubicspline interpolation fails We proposed a PDE-based modelwhich overcomes this limit of classical EMD implementationin the computing of the envelopes for signals that haveno local extremum [2ndash4] Recently the Spectral IntrinsicDecomposition (SID) method [5] based on the spectraldecomposition of the PDE interpolator provides a newapplication of our model This PDE interpolator contributeto the mathematical modeling of the EMD and has providedvarious applications in signal and image processing [4 6]In this paper we describe the mathematical modeling of thenew PDE interpolator by variational methodsThe resolutionof the variational problem leads to existence and uniqueness

results in appropriate spaces The paper is organized asfollows In Sections 2 and 21 recalls some PDE models insignal and image processing and in Section 22 some math-ematical preliminary notions are set out In Section 3 themathematical modeling is exposed In Sections 4 and 5 theresolution of the variational problem is dealt Subsequentlynumerical implementation and applications are presented inSection 6 At last we finish by conclusions

2 Some Preliminaries

21 PDE Models in Signal and Image Processingand EMD Principle

211 Some Diffusion Equations This part consists of a briefand nonexhaustive presentation of classical nonlinear diffu-sion filters for 1D and 2D signal processing with more focuson the 2D case In this purpose we can recall the modelfor nonlinear diffusion in image filtering proposed by Catteet al [7] This filter is a modified version of the well-know

2 ISRN Signal Processing

Perona andMalik model [8]The basic equation that governsnonlinear diffusion filtering is

119906119905(x 119905) = div (119892 (|nabla119906 (x 119905)|2) nabla119906 (x 119905)) (1)

with x = (1199091 119909

2) and where 119906(x 119905) is a filtered version of the

original image 119906(x 119905) = 1199060(x) as the initial condition with

reflecting boundary conditions In (1) 119892(sdot) is the conductivity(or diffusivity) function which is dependent (in space andtime) on the image gradient magnitude Several forms ofdiffusivity were introduced in the original paper of Peronaand Malik [8] All forms of diffusivity are chosen to be amonotonically decreasing function of the signal gradientPossible expressions for conductivity are

1198921(x 119905) = 1

1 + (|nabla119906 (x 119905)| 120573)2

1198922(x 119905) = exp (minus(|nabla119906 (x 119905)| 120573)2)

(2)

Parameter 120573 is a threshold one which influences the aniso-tropic smoothing process The nonlinear equation (1) actsas a forward parabolic equation smoothing regions whilepreserving edges Other methods based on high-order PDEare provided for image restoration like in [9ndash11] Efficientnumerical schemes were introduced in [12] based on additiveoperator splitting (AOS) scheme or based on alternatingdirection implicit (ADI) scheme See [12ndash15] for a review andextensions of these methods Unlike these methods of high-order PDE that are specially developed for denoising ourmodelwas constructed to interpolate the characteristic pointsof a signal

The major problem of nonlinear diffusion-based processis that it is generally difficult to correctly separate the highfrequency components from the low frequency ones In caseof denoising applications the objective of this process is touse the diffusivity function as a guide to retain useful dataand suppress noise

Numerous authors have proposed fourth-order PDEsfor image smoothing and denoising with the hope thatthese methods would perform better than their second-orderanalogues [16ndash18] Indeed there are good reasons to considerfourth-order equationsThefirst reason that can be retained isthe fact that fourth-order linear diffusion damps oscillationsat high frequencies (eg noise) much faster than second-order diffusion On the other hand the theory of fourth ordernonlinear PDEs is far less developed than the second orderone Also such equations often do not satisfy a maximumprinciple or comparison principle and implementation ofthe equations could thus introduce artificial singularities orother undesirable behavior In recent studies Tumblin [19]Tumblin and Turk [16] and Wei [17] proposed the followingform

119906119905(x 119905) = minus div (119892 (119898 (119906)) nablaΔ119906 (x 119905)) (3)

where 119892(sdot) = 1198921(sdot) as in (2) and 119898 is some measurement of

119906(x 119905) In [16] (3) is called a ldquolow curvature image simplifierrdquo(LCIS) and a good choice for 119898 is defined as 119898 = Δ119906 to

enforce isotropic diffusion [19] These PDE tools for digitalsignal and image processing make more reachable the 2Dextension of the 1D PDE-based method for characteristicpoints interpolation presented in this paper

212TheEmpiricalModeDecomposition Principle TheEMD[1] method decomposes iteratively a signal into amplitudemodulation-frequency modulation (AM-FM) type compo-nents called intrinsic mode functions (IMF) The underlyingprinciple of this decomposition is to locally identify in thesignal the most rapid oscillations defined as the waveforminterpolating interwoven local maxima and minima To dothis local extrema points are interpolated with a cubicspline to yield the upper and lower envelopes The meanenvelope (half sum of upper and lower envelopes) is thensubtracted from the initial signal and the same interpolationscheme is reiterated on the remainder The so-called siftingprocess stops when the mean envelope is reasonably zeroeverywhere and the resulting signal is called the first IMFThe higher-order IMFs are iteratively extracted applying thesame procedure to the initial signal after the previous IMFshave been removed

22 Some Useful Mathematical Concepts Throughout thepaperΩ denotes an open-bounded subset ofR119899 119899 = 1 or 2In the sequel we will need the following definitions andresults

Definition 1 we define one the spaces 119881 andV as follows

119881 = 119907 isin 1198673(Ω) |

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

V = 119907 isin 1198673(Ω) |

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0

V119890= 119907 isin 119867

3(Ω) such that existΓ

Ωsub Ω

open and verifying int

ΓΩ

119907119889119909 = 0 and 120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(4)

Definition 2 For a given function 119907 isin 1198671(Ω) if119863

119889(119907)(119909) and

119863119892(119907)(119909) denote respectively the right and left derivative of

119907 at 119909 the minmod derivatives of 119907 is define by

120597119907

120597119909

(119909) = minmod (119863119892(119907) (119909) 119863

119889(119907) (119909)) where

minmod (119901 119902) =

sign (119901)min (1003816100381610038161003816119901100381610038161003816100381610038161003816100381610038161199021003816100381610038161003816) if 119901119902 gt 0

0 if 119901119902 le 0

(5)

A 1198671(Ω) function being absolutely continuous admits right

and left derivatives then 1199060isin 119867

1(Ω) has obviously left

and right derivatives so that we can validate numericallycomputing of the diffusivity function 119892 defined in Section 3

ISRN Signal Processing 3

Definition 3 Let (119909119899)119899isinN be a sequence of elements in a vec-

torial normed space (119864 sdot 119864) it is said that (119909

119899)119899isinN converge

weakly [20] in 119864 and noted by 119909119899 119909 if exists an element

119909 isin 119864 such that forall119891 isin 1198641015840 lim

119899rarrinfin119891(119909

119899) = 119891(119909) where 1198641015840

denotes the set of continuous linear forms on 119864

Definition 4 (the Green formula) Let 119906 119907 isin 1198622(Ω) then

int

Ω

Δ119906119889119909 = int

120597Ω

120597119906

120597119899

119889119904

int

Ω

nabla119906nabla119907119889119909 = minusint

Ω

119906Δ119907119889119909 + int

120597Ω

120597119907

120597119899

119889119904

int

Ω

(119906Δ119907 minus 119907Δ119906) 119889119909 = int

120597Ω

119906

120597119907

120597119899

minus 119907

120597119906

120597119899

119889119904 Green formula

(6)

where 120597119906120597119899 denotes the normal derivative of 119906 on theboundary 120597Ω ofΩ

Definition 5 (the Poincare-Wirtinger inequality) LetΩ be anopen-bounded set and let 119906 isin 119867

1(Ω) then there exists a

constant 119862 gt 0 such that the norm of 119906 in 1198671(Ω) and the

norm of 119906 in 1198712(Ω) are linked by the following inequality

10038171003817100381710038171003817100381710038171003817

119906 minus

1

|Ω|

int

Ω

119906119889119904

100381710038171003817100381710038171003817100381710038171198671(Ω)

le 119862nabla1199061198712(Ω) (7)

where |Ω| denotes the length ofΩ The best constant 119862 in thePoincare-Wirtinger inequality is 1120582 for example the inverseof the first positive eigenvalue 120582 of the Laplace operator withhomogenous Neumann boundaries conditions In our caseΩ sub R and |Ω| is the diameter ofΩ

Theorem 6 (regularity theorem) Let

119906 isin 119906 isin 1198673(Ω) 119904119906119888ℎ 119905ℎ119886119905

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119906119889119909 = 0 (8)

then there exist two strictly positive reals 119862 and 119870 such thatfirstly

nabla1199062

1198712(Ω)

le 119862Δ1199062

1198712(Ω) (9)

and secondly

1199062

1198673(Ω)

le 119870Δ1199062

1198671(Ω) (10)

The main existence and uniqueness result of the solutionof our henceforth problem (29) is due to the application ofthe following Lions theorem [21]

Theorem 7 (the Lions theorem) Let119867 and 119881 be two Hilbertspaces with 119881 sub 119867 One considers a bilinear application 119886 on119881times119881 and 119861 an operator on119881 Under the following conditions

(1) exist120583 and 120588 gt 0 such that |119886(119906 119907)| ge 1205881199072119881minus 120583|119907|

2

119867

(2) exist 1198882gt 0 such that |(119907 119861119907)

119881| le 119888

2|119907|

2

119881

For 1199060= 119906(0) isin 119867 the problem

119906 isin 1198712(0 119879 119881) cap 119862 (0 119879119867) (11)

such that

forall119907 isin 119881 (

119889119906

119889119905

119861119907)

119867

+ 119886 (119906 119907) = 0 (12)

and 1199060= 119906(0) has a unique solution

Furthermore 119889119906119889119905 isin 1198712(0 119879119867

1015840)

3 Mathematical Modeling of the NewPDE-Interpolator

Recall that the PDE model aimed to contribute to themathematical modeling of the EMD as it is in [2] In a firststep to be in line with the classical EMD our goal is tomodel the upper or lower envelope as the asymptotic solutionof a PDE system whose initial condition is the input signalthat we want to interpolate local extrema (or more generallycharacteristic points) Initially this envelope is obtained bycubic spline interpolation Let Ω be the open domain of angiven finite energy signal 119906

0isin 119871

2(Ω) We construct an

operator 119860 of appropriate domain119863(119860) as follows

120597119906

120597119905

+ 119860119906 = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω

plus boundaries conditions

(13)

The need of asymptotic solution existence denoted by 119906infin

implies (120597119906120597119905)(infin 119909) = 0 for example according to (13) wehave 119860119906(infin 119909) = 0 Moreover the requirement of the sameregularity between the solution and a cubic spline leads to1205974119906120597119909

4(infin 119909) = 0 Thus in the first analysis we can choose

119860 = 1205974119906120597119909

4 To leave invariant features points during thediffusion process simply multiply in the expression 119860119906 theterm 120597

4119906120597119909

4 by function 119892 depending on spatial variable 119909and vanishing at characteristic points That gives

(119860119906) (119909) = 119892 (119909)

1205974119906

1205971199094 (14)

Another reminiscent form of long-range diffusion [22] is thefollowing

(119860119906) (119909) =

120597

120597119909

(119892 (119909)

1205973119906

1205971199093) (15)

Function 119892 can be interpreted as a diffusivity function whoserole is to control the diffusion process We take it necessarilypositive to have a direct diffusion and the existence ofsolution Other forms of operators are possible [3] we presentsome ones as follows for (14)

(119860119906) (119909) = 119892 (119909) [minus120579

1205972119906

1205971199092+ (1 minus 120579)

1205974119906

1205971199094] (16)


or for (15)

(119860119906) (119909) =

120597

120597119909

[119892 (119909) (minus120579

120597119906

120597119909

+ (1 minus 120579)

1205973119906

1205971199093)] (17)

or accomplete diffusion involving a flow

(119860119906) (119909) =

120597

120597119909

[minus120579119892 (119909)

120597119906

120597119909

+ (1 minus 120579)

120597

120597119909

(119892 (119909)

1205972119906

1205971199092)]

(18)where 0 le 120579 le 1 is the tension factor which controls theregularity of the solution

Both forms allow more freedom on the regularity ofthe solution 119906 To access the mathematical properties of thesolution we chose the second form in (17) with 120579 = 0 and thecorresponding equation (15)

At the local scale between two consecutive character-istic pointsmdashtwo local maxima for examplemdashthe diffusioninduces a smoothing phenomenon that deletes the local min-imum A simple form for 119892 to calculate the envelope is givenby a positive piecewise function lower than 1 that is constantbetween two characteristic points of 119906

0and zeroed only at

these points Characteristic points are often being definedby their values and the signs of first second and third localderivatives of119906

0We characterize the function119892 as depending

on sign(1205971199060120597119909) sign(1205972119906

0120597119909

2) and sign(1205973119906

0120597119909

3)

For the purpose of existence and regularity of the solu-tion we are led to work with the regularized version of sign

sign120599(119911) =

2

120587

arctan(120587119911120599

) (19)

where 120599 is a regularization coefficientIn the following we define 119892plusmn

119890for extrema detection

119892119891for turning points detection and 119892plusmn

mc for maximum andminimum curvature points detection

119892plusmn

119890(119909) =

1

9

[

10038161003816100381610038161003816100381610038161003816

sign(120597119906

0

120597119909

)

10038161003816100381610038161003816100381610038161003816

plusmn sign(12059721199060

1205971199092) + 1]

2

(+) for maxima and (minus) for minima

119892119891(119909) = [sign

120599(

12059721199060

1205971199092)]

2

for turning points

119892plusmn

mc (119909) =1

9

[

100381610038161003816100381610038161003816100381610038161003816

sign(12059731199060

1205971199093)

100381610038161003816100381610038161003816100381610038161003816

plusmn sign(12059721199060

1205971199092) + 1]

2

(20)

(+) for maximum curvature points and (minus) for minimumcurvature points All these functions are of the form 119892(119909) =

[ℎ(119909)]2 with ℎ(119909) = 0 at characteristic points

So we have 1198921015840(119909) = 0 if 119892(119909) = 0 This property formally

allows us to cancel119860119906 and 120597119906120597119905 of the form (15) at the pointswhere 119892 is null The factor 19 is a normalization term andverifies 0 le 119892(119909) le 1 A more general diffusivity function 119892could be envisaged

Let us consider the following differential operators119860119890and

119860119891(119890 for upper and 119891 for inflexion or curvature)

forall119906 isin 119863 (119860119890) 119860

119890119906 =

120597

120597119909

(119892plusmn

119890(119909)

1205973119906

1205971199093) (21)

or

forall119906 isin 119863 (119860119891) 119860

119891119906 =

120597

120597119909

(119892119891(119909)

1205973119906

1205971199093) (22)

If 1199060isin 119862

2(Ω) as it is the case in the classical imple-

mentation EMD there is no problem for interpolation bycubic spline method But it is indeed a restriction on theregularity of 119906

0which is due to the choice of the interpolation

technique that requires a good detection of local extremafor the envelopes calculation The basic EMD principle mustapply for input functions that are not regular as functions in119862

2(Ω) for example for signal 119906

0isin 119867

1(Ω) as it is the case

in reality [23] But numerically without even a regularizationfunction sign if the derivatives are taken in the sense ofminmod called flux limiter or slope the function 119892(119909) stillhas a meaning

31 Interpretation of the Diffusivity Action in the DiffusionProcess Between two consecutivemaxima119883max119894 and119883max119894+1 the function 119892+

119890is piecewise constant and can be written as

follows

119892 (119909) =

0 at 119883max119894 119883max119894+1

4

9

at 119883inf 119894 119883inf 119894+1 119883min119894

1 on ]119883inf 119894 119883min119894[ cup ]119883min119894 119883inf 119894+1[

1

9

sur ]119883max119894 119883inf 119894[ cup ]119883inf 119894+1 119883max119894+1[

(23)

We have a diffusion in the sense that 119906119905= 120597119906120597119905 unhook

to the maximum of the curve 1199060 The reason is that the

diffusion is more pronounced in local minimum becausethe smoothing effect tends to regularize curves Thus theupper envelope 119906(119905 cong infin sdot) of 119906

0is less oscillating than

1199060is The same behavior occurs for the mean envelope We

would then try to interpret locally the relationship 119906 ge 1199060

by evoking the maximum principle [24] But this principleis not immediately checked for equations of the type (15)The justification lies in the fact that smoothing impliesnabla119906(119905 sdot)

1198712(Ω)

le nabla11990601198712(Ω)

In summary we have just built a PDE system

(1) having the input signal 1199060to decompose as an initial

value condition

(2) such that the result of a diffusion process preservescharacteristic points of 119906

0

(3) with a regularity similar to a cubic spline

32 Formulation of the Mathematical Problem Let 119860 be oneof the above constructed interpolation operators and 119879 gt 0

be a diffusion time sufficiently long To fix ideas we solvethe problem of upper envelope calculus Indeed for the otherkinds of characteristic points the same problem arises withanalogous mathematical resolution methods


The problem posed by our mathematical modeling is tofind a solution 119906 of the system

120597119906

120597119905

+ 119860119906 = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(24)

Then the mean envelope (interpolating turning points) cal-culus is stated as follows

For given 1199060 find 119906 such that

120597119906

120597119905

+

120597

120597119909

(119892119891(119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(25)

33 Comments on theDegeneracy in Problem (24) To preventthe zero degeneracy case in the system (25) whose solution isnot directly accessible by variational methods we are led tosolve an intermediate problem with

0 lt 120572 le 119892119891120572(119909) le 120573 le 1 (26)

That is the nondegenerate problem defined hereafter

34 Formulation of the Zero Degeneracy Induced byDiffusivityFunction Thediffusivity function in (26) is very close to zeroat the characteristic points while refraining to cancel and tosolve the system (25) now just to move on to the limit when120572 tends to zero to retrieve (25) and its solution

Thus in particular we can define the function

119892119891119899(119909) = 119892

119891(119909) +

1

119899

119899 isin Nlowast (27)

which verifies the conditions of (26)The new system that we call nondegenerate problem is

fomulated as followsFind 119906

119899such that

120597119906119899

120597119905

+

120597

120597119909

(119892119899(119909)

1205973119906119899

1205971199093) = 0 in [0 119879] times Ω

119906119899(0 119909) = 119906

0in Ω


(28)

35 Formulation of the Nondegenerate Problem For 119899 suffi-ciently large 119906

119899is a close approximation of the envelope 119906

interpolating the characteristic pointsBy a density technique we can demonstrate that this

sequence (119906119899)119899isinN converges to the solution of the system (25)

By posing for the convenience of notation 0 lt 120572 le 119892119891119899

= ℎ le

120573 le 1 the non-degenerate problem is reworded as

Find 119906 such that

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(29)

Equation (29) is an operational differential equationstype The techniques for the resolution of this kind ofproblem are numerous we mainly refer to the variationalformulation to find weak solutions [25] and the method ofthe approximation of evolutionary operators The last allowsto work on a new elliptic problemThe solution is obtained bypassing to the limit of the approximate solution nstead of theequation

In the next section we solve (29) by a variationalapproach according to the resolution methods for parabolicproblems described in [7 20]

4 Existence and Uniqueness of Solutions forPDE Interpolator

41 The Guideline for the Resolution of the MathematicalProblem Now we summarize the general procedure to solvethe problem

(1) First step resolution of the problem (29) This stepincludes

(i) the variational formulation in Section 43(ii) the resolution of the variational problem in

appropriate functional space(iii) the converse showing that the solution of the

variational problem solves the departure prob-lem in Section 434

(2) Second step in Section 5 construction of the sequ-ence (119906

119899)119899isinN of solutions of problem (29) where 119906

119899

is the solution obtained with the diffusivity ℎ119899(119909) =

ℎ(119909) + (1119899) 119899 isin Nlowast see Section 5 Finally demon-strate that the sequence (119906

119899)119899isinN converges to a limit 119906

where 119906 is the solution of problem (25) in Section 5

42 The Main Results Formulation The main result which isgoing to be solves is the following

Theorem 8 For 1199060isin 119867

1(Ω) exists a unique solution 119906 isin

1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) of the variational problem

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909+int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909=0 119891119900119903 119886119899119910 119907 isin V

(30)

such that 119906(0) = 1199060and 119889119906119889119905 isin 1198712

(0 119879119867) Reversely let 119906be a solution of the variational problem (30) then (120597119906120597119905) +(120597120597119909)(ℎ(119909)(120597

3119906120597119909

3)) = 0

And the complete follows result



1(Ω) exists a unique solution

119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867

1(Ω)) (31)

verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(32)

43 Variational Formulation Let 119907 isin 1198673(Ω) and (to make

sense to integrals) multiply the equation

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (33)

of the system (29) by the test function 1205972119907120597119909

2 after weintegrate onΩ then it comes out that

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909 = 0 (34)

By using theGreen formula (seeDefinition 4) and integrationby parts we get firstly

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909

= [ℎ (119909)

1205973119906

1205971199093

1205972119907

1205971199092]

120597Ω

minus int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

(35)

and secondly

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

119906

1205972119907

1205971199092119889119909

int

Ω

119906

1205972119907

1205971199092119889119909 = int

120597Ω

119906

120597119907

120597119899

119889120590 minus int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(36)

As we are in one-dimension case the normal derivative andthe gradient on the edge in the use of the Green formula arethe same that is to say

int

120597Ω

119906

120597119907

120597119899

119889120590 = int

120597Ω

119906

120597119907

120597119909

119889120590 (37)

In the following we adopt the notation 120597119907120597119899Considering 119906 isin 1198673

(Ω) such that

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

119907 isin 119881 = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(38)

it comes out that

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0

forall119907 isin 119881

(39)

Let us consider the following bilinear forms

119886 (119906 119907) = int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

119887 (119906 119907) = int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(40)

The addition of boundaries conditions transforms our initialproblem into the system

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (on the edge of Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω


119906 (0 119909) = 1199060

in Ω

(41)

The variational formulation of our problem consequentlysubject to additional conditions that would cause a change ofspace is the following

Find 119906 isin 119881 such that 119889

119889119905

(119887 (119906 119907)) + 119886 (119906 119907) = 0


(42)

where 119886 and 119887 are like in Theorem 7This kind of problem is studied in [21] However before

continuing the resolution we explore some properties of thespace 119881

431 On the Quotient Space 119881R The space 119881 is defined by119863(119886) cap 119863(119887) where 119863(119886) and 119863(119887) are the domains of 119886 and119887 Let us consider the quotient space

119881

R= 119907 + 119888 119888 isin R such that 119907 isin 1198673

(Ω)

and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(43)

and the new spaceV defined by

V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (44)

Let 119907 isin 119881R and let 119906 be a representative of 119907 class then(120597119907120597119899)|

120597Ωis defined by

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597 (119906 + 119888)

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 with 119888 isin R (45)


Thenullity ofintΩ119907119889119909meansexist119888 isin R constant such thatint

Ω(119906+

119888)119889119909 = 0 with 119906 one representative of 119907

if 119888 = minus 1

|Ω|

int

Ω

119906119889119909 then

int

Ω

119907119889119909 = int

Ω

(119906 + 119888) 119889119909

= int

Ω

(119906 minus

1

|Ω|

int

Ω

119906119889119909)119889119909 = 0

(46)

So for all 119907 isin 119881R we have intΩ119907119889119909 = 0 Accordingly any

element of the quotient space 119881R is on one hand in1198673(Ω)

with normal derivative on the boundaries of Ω null and onthe other hand is zero average onΩThus the quotient space119881R is identified with the space V We can work later inthe space V with zero mean The final variational problemto solve is the following

find 119906isinV such that 119889

119889119905

(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V

(47)

432 Application of the Lions Theorem To apply the Lionstheorem we need to identify the applications 119886 119887 and 119861 andthe spaces119867 and119881 And finally wemust prove that the normdefined inV which derives from the scalar product given bythe bilinear application 119886 is equivalent to the norm of1198673

(Ω)Let us define 119861 as the identity and let us define 119886 and 119887 by

|119886(119906 119907)| = (radicℎ(1205973119906120597119909

3) radicℎ(120597

3119907120597119909

3))

1198712(Ω)

and 119887(119906 119907) =intΩ(120597119906120597119909)(120597119907120597119909)119889119909 obtained from (39) (40) the spaces

119881 = V and119867 are define as follows(a) The linear operator 119861 in the variational prob-

lem (11) in the Lions theorem (119889119906119889119905 119861119907)119867

denotes ascalar product on 119867 Indeed we define 119867 = 119907 isin

1198671(Ω) such that (120597119907120597119899)|

120597Ω= 0 int

Ω119907119889119909 = 0

The operator 119861 = 119868 119861119907 = 119907 is the identity whichis continuous (verify the Lions conditions Theorem 7) andthe term (119889119906119889119905 119861119907)

119867= (119889119889119905)(119887(119906 119907)) is equal to

int

Ω

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

119889119909 = int

Ω

119889

119889119905

(

120597119906

120597119909

)

120597119907

120597119909

119889119909

= (

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

)

1198712(Ω)

(48)

In this identification ((120597120597119909)(119889119906119889119905) 120597119907120597119909)1198712(Ω)

is a wellscalar product in the subspace 119867 of 1198671

(Ω) formed byfunctions with zero mean Indeed according to Poincare-Wirtinger inequality (see Definition 5) in1198671

(Ω)10038171003817100381710038171003817100381710038171003817

119906 minus

1

|Ω|

int

Ω

119906119889119909

100381710038171003817100381710038171003817100381710038171198671(Ω)

le

1

120582

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(49)

where 120582 is the first positive eigenvalue of Laplace operator inΩ But as 119906 119907 isin V then int

Ω119906119889119909 = 0 consequently there

exists 1198622gt 0 such that

1199061198671(Ω)

le 1198622

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(50)

that is to say the 1198671(Ω) norm is equivalent to the gradient

norm inV Hence one can replace the scalar product of thegradient by1198671

(Ω) scalar productHereinafter we complete the proof that the precedent

choices allow the application of the Lions theorem in ourcontext

(b) Vis an Hilbert space with equivalent norm to 1198673(Ω)

norm

Theorem 10 The spaceV resulting from (11) andV analysisdefined by

V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (51)

is an Hilbert space with the a norm equivalent to1198673(Ω) norm

Proof At first V = 119907 isin 1198673(Ω) | (120597119907120597119899)|

120597Ω= 0 int

Ω119907119889119909 =

0 is a closed vector subspace of 1198673(Ω) Note that the

condition of the nullity of the mean in V may be replacedby int

ΓΩ

119906119889119909 = 0 where ΓΩis a nonempty open subset of Ω In

this caseV becomes rather

V119890= 119907 isin 119867


Ωsub Ω verifying

int

ΓΩ

119907119889119909 = 0 and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(52)

From the fact that

0 lt 120572 le ℎ (119909) le 120573 lt 1 forall119909 isin Ω (53)

the norms built with the two bilinear applications 119886 and 119887 areequivalent to the norm constructed by the linear form notedby | |2V with 119907 isin V and given by

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(54)

Indeed | |2V is as well a norm inVThere is no difficulty toformally verify the triangular inequality If |119907|2V = 0 then120597119907120597119909

2

1198712(Ω)

= 0 and 12059731199071205971199093

2

1198712(Ω)

= 0 Consequently 119907 isnecessarily constant almost everywhere inΩ for example 119907 isequal to a constant real 119870 But as int

Ω119907119889119909 = 0 implies 119870|Ω| =

0 so119870 = 119907 = 0 We have thus built a standard scalar producton V from bilinear forms of the variational formulation Vis a Hilbert space with the scalar product associated withthe norm (54) In addition this norm is equivalent to the119867

3(Ω) norm as it shown in Lemma A2 Now we just need

to prove that the correspondence (119906 119907) isin V timesV 997891rarr 119886(119906 119907)

is continuous and coercive on the Hilbert spaceV


433 The Application 119886( sdot sdot ) Is Continuous and Coercive For119906 119907 isin V

|119886 (119906 119907)| =

1003816100381610038161003816100381610038161003816100381610038161003816

(radicℎ

1205973119906

1205971199093 radicℎ

1205973119907

1205971199093)

1198712(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

le 120573

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

sdot

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 120573|119906|V sdot |119907|V

(55)

hence the continuity of119886(sdot sdot) is onVtimesV In order to completethe application conditions of Theorem 7 we only check thefirst condition of the same theorem

For

119907 isin V minus1199072

1198671(Ω)

le minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(56)

hence

|119907|2

V minus 1199072

1198671(Ω)

le |119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(57)

nevertheless

119886 (119907 119907) =

100381710038171003817100381710038171003817100381710038171003817

radicℎ

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 120572

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= 120572[|119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]

(58)

Thus

119886 (119907 119907) ge 120572 [|119907|2

V minus 1199072

1198671(Ω)] forall119907 isin V (59)

We have found thus (120588 120583) = (120572 120572) such that

119886 (119907 119907) ge 120588|119907|2

V minus 1205831199072

1198671(Ω) forall119907 isin V (60)

All the conditions of Theorem 7 are acquired now the mainexistence result for solution of system (41) can be enunciatedas follows



1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) of the problem

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0

119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905

119906 (0) = 1199060

119889119906

119889119905

isin 1198712(0 119879119867)

(61)

434 Equivalence with the Initial Problem Reversely let 119906 bea solution of the variational problem

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0 forall119907 isin V (62)

As we must consider a test function

119907 isin 1198673(Ω) such that 120597

2119907

1205971199092still in 119863 (Ω)

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(63)

Let us take a function test 120601 isin 119863(Ω) and119870 isin R such that

int

Ω

(120601 + 119870) 119889119909 = 0 (64)

We pose the following Neumann problem

1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(65)

Its variational formulation gives a unique solution in119867 =

119908 isin 1198671(Ω) | int

Ω119908119889119909 = 0 and the regularity of 120601 isin 1198672

(Ω)

implies the regularity of the solution which is in 119907 isin 1198673(Ω)

consequently 119907 isin VReturning to (62) with 119907 solution of the problem (65) the

first term can be written as

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

(int

Ω

119906

1205972119907

1205971199092119889119909 minus int

120597Ω

119906

120597119907

120597119899

119889120590)

as 1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

and it comes out that

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906 (120601 + 119870) 119889119909

=

119889

119889119905

(int

Ω

119906120601119889119909 + 119870int

Ω

119906119889119909)

(66)

However intΩ119906119889119909 = 0 subsequently

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906120601119889119909 = int

Ω

119889119906

119889119905

120601119889119909 (67)

The second term of (62) is written as follows

int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

ℎ (119909)

1205973119906

1205971199093

120597120601

120597119909

119889119909 (68)

which gives in the distribution sense

int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909 (69)

from which

int

Ω

119889119906

119889119905

120601119889119909+int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)

(70)


We deduce that

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (71)

We verify now that (12059731199061205971199093)|

120597Ω= 0 Returning to (70) with

120601 = 1205972119907120597119909

2 and according to the Green formula we have

int

Ω

[

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)]

1205972119907

1205971199092119889119909

minus int

120597Ω

ℎ (119909)

1205973119906

1205971199093

120597

120597119899

(

120597119907

120597119909

) sdot 119899119889120590 = 0

forall119907 isin V

(72)

But as (120597120597119899)(120597119907120597119909) sdot 119899 = (120597120597119909)(120597119907120597119909) = 12059721199071205971199092 it givesaccording to (71) int

120597Ωℎ(119909)(120597

3119906120597119909

3)(120597

2119907120597119909

2)119889120590 = 0 forall119907 isin

V and consequently ℎ(119909)(12059731199061205971199093) = 0 on boundaries ofΩ

or as ℎ(119909) = 0 12059731199061205971199093= 0 on boundaries ofΩ

In conclusion 119906 is solution of the initial equation (41)then we enunciate the following theorem



1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(73)

435 Some Remarks on the Modeling Space The nullitycondition of the average for functions inV has been decisivein the construction of the norm on this space This spaceis prima facie clearly not natural for EMD because of entryfunction119906

0that is not necessarily zeromean But at the cost of

working with an initial condition 1199070isin 119867

1(Ω) such that 119907

0=

1199060minus (1|Ω|) int

Ω1199060119889119909 we can consider119867 as1198671

(Ω) functionswith null local mean

The Nullity of the Mean Envelope A Characteristic of V andEMD Algorithm This condition far from being superfluouswas even expected because the space V is the space of themean envelopes which must be zero mean in EMD stiftingprocess for mode extraction If it were to calculate the upperor lower envelopes it should be noted that the conditionof the mean nullity raised a query In EMD algorithm thisnullity condition is affected on the mean envelope and not onupper or lower envelopes This question does not arise in themodel giving the envelope interpolating inflection points ofa signal The elements ofV

119890are not zero mean on Ω but on

a non-empty subset ΓΩofΩ

The nullity of the mean of the signal on a ΓΩ

is notimmediately visible andwarranty to decompose any function

We show in Lemma A1 that each relevant function to bedecomposed by EMD is an integrable function possessing atleast three extrema belonging to V

119890 The modeling space of

the extrema interpolation isV119890defined in (52)

Let 119891 be a function admitting at least three extrema wecan consider that 119891 passes at least once a zero at a point 119909

119911

in its definition domain Otherwise we just work with thetranslatory (as our model is invariant by translation of theinput function) vector which is equals to the half amplitudeof 119891

5 Convergence of the Sequence (119906119899)119899isinN of

Solutions of (41) to a Solution of theCalled Degenerate Problem

In the following sections we demonstrate that the sequence(119906

119899)119899isinN of solutions of the non degenerate problem is

bounded Next we prove that there exists a subsequence of(119906

119899)119899isinN which converges weakly to an element 119906 isin V and

finally that this element is solution of the degenerate initialproblem

51 Some Estimations From (28)we have in the distributionssense (multiplying by 1205972119906

119899120597119909

2)

int

Ω

120597119906119899

120597119905

1205972119906119899

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092119889119909 = 0 (74)

which is equivalent to

⟨

120597119906119899

120597119905

1205972119906119899

1205971199092⟩

1198811015840119881

+⟨

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092⟩

1198811015840119881

=0

(75)

After integration by parts it comes out that

1

2

int

Ω

120597

120597119905

(

10038161003816100381610038161003816100381610038161003816

120597119906119899

120597119909

10038161003816100381610038161003816100381610038161003816

2

)119889119909 + int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 = 0 (76)

then by integrating with respect to the variable 119905 isin]0 119879[ itcomes out that

1

2

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

minus

1

2

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904 = 0

(77)

which implies that

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

=

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(78)


But as ℎ119899(119909) ge 120572 we have

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

int

Ω

(

1205973119906119899

1205971199093)

2

119889119909 119889119904

(79)

or

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

100381710038171003817100381710038171003817100381710038171003817

1205973119906119899

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904

(80)

from which10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904

(81)

Thus on one hand10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(82)

But10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 12058221003817100381710038171003817119906119899

1003817100381710038171003817

2

1198671(Ω) (83)

where 120582 comes from inequality (49) which implies that

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le

1

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(84)

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(85)

Then the sequence (119906119899)119899isinN is bounded in119871infin

(0 1198791198671(Ω)) and

in 1198712(0 119879119867

1(Ω)) And on the other hand

2120572int

119879

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904 le

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(86)

which gives

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(87)

But as10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω) (88)

therefore

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω)119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(89)

Thus

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le 119879

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(90)

which implies according to (84) that

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(91)

We finally deduced the second estimate

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V)

=1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V119890)

le (

119879

1205822+

1

2120572

)

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(92)

Then the sequence (119906119899)119899isinN is bounded in 1198712

(0 119879V)

52 Weak Convergence of the Subsequence of (119906119899)119899isinN The

sequence (119906119899)119899isinN is bounded in the space 1198712

(0 119879V) Thenthere exists a subsequence (119906

119899119896)119896isinN

that converge weakly to 119906in 1198712

(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881

1015840)

Leting 119907 isin V be a test function we have the following

Firstly As 119906119899119896rarr 119906weakly in 1198712

(0 119879 119881) therefore 119906119899119896rarr 119906

weakly in 1198712(0 119879119867

1(Ω))

Thus in the distributions sense

int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 (93)

by integrating on [0 119905] we have

int

119905

0

int

Ω

119889119906119899119896

119889119904

1205972119907

1205971199092119889119909 119889119904 = minusint

119905

0

119889

119889119904

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 119889119904

= minusint

119905

0

⟨

119889119906119899119896

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

= minusint

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909

(94)

By integrating by parts again the last equation we have

int

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909 = int

Ω

120597119906119899119896(119905)

120597119909

120597119907

120597119909

119889119909

minus int

Ω

120597119906119899119896(0)

120597119909

120597119907

120597119909

119889119909

(95)


This expression converges to

int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when

119899119896tends to infinity to

int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)

Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712

(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909

And because 1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093

converges to 1205973119906

1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)

converges to zero when 119899119896tends to infinity

On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816

int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)

which converges to zero in 1198712(0 119879 119871

2(Ω)) according to weak

convergence (see Definition 3) in 1198673(Ω) of 1205973119906

119899119896120597119909

3 to

1205973119906120597119909

3 Thus we have shown that in the distributions senseof the one hand

int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)

and on the other hand

int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)

Consequently for each 119907 isin 1198712(0 119879 119881

1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)

converges in the distributions sense to

int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899

119896tends to infinity to ⟨(119889119906119889119905) 119907⟩

1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩

1198811015840119881 It remains to show that the

limit 119906 is a solution of the degenerate problem ButintΩ(119889119906

119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0

which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +

intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909

3))119907119889119909 = 0 In other words the

weak limit of the sequence (119906119899119896)119896isinN

is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871

2(0 119879V) cap 119862([0 119879]119867

1(Ω)) Better using the

compact inclusion of V (which behaves like 1198673(Ω) because

of the equivalence of the norms) in 1198672(Ω) we conclude

that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672

(Ω) to thesolution of the initial degenerate problem The space 1198672

(Ω)

is moreover the space of strong solutions of the initial system(25)


6 Numerical Implementationand Applications

Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals

In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by

(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896

119894the approximate value of the solution

119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the

discrete time 119905119896and is denoted by 119880119896 An approximation of

the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905

119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905

61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows

119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)

where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively

to the spatial coordinate 119909119894

To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892

119894119895defined by

119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)

Recall that 119892 is strictly positive which allows its inversion inthe previous system

We define also the numerical derivatives of 119907 by

119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)

forward difference on the 119909119894-direction

119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)

backward difference on the 119909119894-direction

1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)

central difference on the 119909119894-direction

119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)

and finally we pose 119908119894119895119906(119894 119895) = 119863

+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-

tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of

1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)

We present the first natural explicit scheme

611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1

minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880

119896 Let us poseD = 119868minusΔ119905sum2

119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)

If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732

However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD

612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows

119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)


This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0

613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder

It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following

119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)

Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time

62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886

31199093+

11988621199092+ 119886

1119909 + 119886

0 The nullity of its derivatives of orders 3 and

1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886

2119909 + 119886

1= 0 on

the boundary Hence 1198862= 119886

1= 0 which finally gives 119906(119909) =

1198860 We have taken as boundary conditions that outside the

domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere

63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time

631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function

Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope

In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different

15 20 25 30 35 40 45145514601465147014751480148514901495

Mean envelopeUpper envelopeLower envelope

Signal s

Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines

tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero

In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0

In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904

1

and 1199041 When it comes to interpolate the classical local

extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly

632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling

In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =

1199041+ 119904

2 The convergence is slower if the number of sampling

points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904

1of the signal 119904 To capture

the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling

633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations

(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000

200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06

(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000

200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06

(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000

Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000

200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005

200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05

200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1

Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE


200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06

(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000

200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06

(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000

200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06

(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000

Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution

corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough

7 Conclusion

In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867

1(Ω) The PDE interpolator is not

based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal

Appendix

LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema

and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909

119888isin Ω

such that the local mean at 119909119888is null for example

119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)

Proof Let 119909119888and 120598 gt 0 such that 119891(119909

119888) = 0 119891(119909

119888minus 120598) lt

0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the

local mean is continuous and 119872loc120575[119891] not null implies

that it keeps a constant sign on Ω Suppose that forall120575 119909 gt

0 119872loc120575[119891](119909) gt 0 In particular the positive sequence

(119872loc1119899[119891](119909

119888minus 120598))

119899isinNlowastconverges to 119891(119909

119888minus 120598) ge 0 This is

absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1

is performed

From this lemma any eligible function for EMD is inV119890

Lemma A2 There exist two constants 119862119881 119862

119867gt 0 such that

forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)

Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907

2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max

PDE interpolator min-max

PDE interpolator courbure

119904

Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes

80 100 120 140 160 180

256 points-40 iterations

minus1

minus05

0

05

1

(a) Envelope after 40 iterations

80 100 120 140 160 180


minus1

minus05

0

05

1

(b) Envelope after 400 iterations

Figure 6 The envelope of the signal test 119904 = 1199041+ 119904

2sampling with 256 points

and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)

we have immediately |119907|2V le 1199072

1198673(Ω)

posing 119862119867= 1

Step 2 (first part of (A2)) Firstly

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)

and by integration by parts

int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)

As 119907 isin V we have

int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)

Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage

and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)

More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger

theorem [20] it comes out that 11990721198671(Ω)

le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)

The inegality (A2) is established and the lemmarsquos proof iscompleted

References

[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998


[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005

[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007

[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010

[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012

[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012

[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992

[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990

[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002

[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003

[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003

[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998

[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997

[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955

[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000

[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999

[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999

[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000

[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson

Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes

aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany

1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode

decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004

[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981

[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972

[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002

[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996

[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996

[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003

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DistributedSensor Networks



Perona andMalik model [8]The basic equation that governsnonlinear diffusion filtering is

119906119905(x 119905) = div (119892 (|nabla119906 (x 119905)|2) nabla119906 (x 119905)) (1)

with x = (1199091 119909

2) and where 119906(x 119905) is a filtered version of the

original image 119906(x 119905) = 1199060(x) as the initial condition with

reflecting boundary conditions In (1) 119892(sdot) is the conductivity(or diffusivity) function which is dependent (in space andtime) on the image gradient magnitude Several forms ofdiffusivity were introduced in the original paper of Peronaand Malik [8] All forms of diffusivity are chosen to be amonotonically decreasing function of the signal gradientPossible expressions for conductivity are

1198921(x 119905) = 1

1 + (|nabla119906 (x 119905)| 120573)2

1198922(x 119905) = exp (minus(|nabla119906 (x 119905)| 120573)2)

(2)

Parameter 120573 is a threshold one which influences the aniso-tropic smoothing process The nonlinear equation (1) actsas a forward parabolic equation smoothing regions whilepreserving edges Other methods based on high-order PDEare provided for image restoration like in [9ndash11] Efficientnumerical schemes were introduced in [12] based on additiveoperator splitting (AOS) scheme or based on alternatingdirection implicit (ADI) scheme See [12ndash15] for a review andextensions of these methods Unlike these methods of high-order PDE that are specially developed for denoising ourmodelwas constructed to interpolate the characteristic pointsof a signal

The major problem of nonlinear diffusion-based processis that it is generally difficult to correctly separate the highfrequency components from the low frequency ones In caseof denoising applications the objective of this process is touse the diffusivity function as a guide to retain useful dataand suppress noise

Numerous authors have proposed fourth-order PDEsfor image smoothing and denoising with the hope thatthese methods would perform better than their second-orderanalogues [16ndash18] Indeed there are good reasons to considerfourth-order equationsThefirst reason that can be retained isthe fact that fourth-order linear diffusion damps oscillationsat high frequencies (eg noise) much faster than second-order diffusion On the other hand the theory of fourth ordernonlinear PDEs is far less developed than the second orderone Also such equations often do not satisfy a maximumprinciple or comparison principle and implementation ofthe equations could thus introduce artificial singularities orother undesirable behavior In recent studies Tumblin [19]Tumblin and Turk [16] and Wei [17] proposed the followingform

119906119905(x 119905) = minus div (119892 (119898 (119906)) nablaΔ119906 (x 119905)) (3)

where 119892(sdot) = 1198921(sdot) as in (2) and 119898 is some measurement of

119906(x 119905) In [16] (3) is called a ldquolow curvature image simplifierrdquo(LCIS) and a good choice for 119898 is defined as 119898 = Δ119906 to

enforce isotropic diffusion [19] These PDE tools for digitalsignal and image processing make more reachable the 2Dextension of the 1D PDE-based method for characteristicpoints interpolation presented in this paper

212TheEmpiricalModeDecomposition Principle TheEMD[1] method decomposes iteratively a signal into amplitudemodulation-frequency modulation (AM-FM) type compo-nents called intrinsic mode functions (IMF) The underlyingprinciple of this decomposition is to locally identify in thesignal the most rapid oscillations defined as the waveforminterpolating interwoven local maxima and minima To dothis local extrema points are interpolated with a cubicspline to yield the upper and lower envelopes The meanenvelope (half sum of upper and lower envelopes) is thensubtracted from the initial signal and the same interpolationscheme is reiterated on the remainder The so-called siftingprocess stops when the mean envelope is reasonably zeroeverywhere and the resulting signal is called the first IMFThe higher-order IMFs are iteratively extracted applying thesame procedure to the initial signal after the previous IMFshave been removed

22 Some Useful Mathematical Concepts Throughout thepaperΩ denotes an open-bounded subset ofR119899 119899 = 1 or 2In the sequel we will need the following definitions andresults

Definition 1 we define one the spaces 119881 andV as follows

119881 = 119907 isin 1198673(Ω) |

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

V = 119907 isin 1198673(Ω) |

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0

V119890= 119907 isin 119867


Ωsub Ω

open and verifying int

ΓΩ

119907119889119909 = 0 and 120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(4)

Definition 2 For a given function 119907 isin 1198671(Ω) if119863

119889(119907)(119909) and

119863119892(119907)(119909) denote respectively the right and left derivative of

119907 at 119909 the minmod derivatives of 119907 is define by

120597119907

120597119909

(119909) = minmod (119863119892(119907) (119909) 119863

119889(119907) (119909)) where

minmod (119901 119902) =

sign (119901)min (1003816100381610038161003816119901100381610038161003816100381610038161003816100381610038161199021003816100381610038161003816) if 119901119902 gt 0

0 if 119901119902 le 0

(5)

A 1198671(Ω) function being absolutely continuous admits right

and left derivatives then 1199060isin 119867

1(Ω) has obviously left

and right derivatives so that we can validate numericallycomputing of the diffusivity function 119892 defined in Section 3







119899rarrinfin119891(119909

119899) = 119891(119909) where 1198641015840



int

Ω

Δ119906119889119909 = int

120597Ω

120597119906

120597119899

119889119904

int

Ω


Ω

119906Δ119907119889119909 + int

120597Ω

120597119907

120597119899

119889119904

int

Ω

(119906Δ119907 minus 119907Δ119906) 119889119909 = int

120597Ω

119906

120597119907

120597119899

minus 119907

120597119906

120597119899


(6)






10038171003817100381710038171003817100381710038171003817

119906 minus

1

|Ω|

int

Ω

119906119889119904

100381710038171003817100381710038171003817100381710038171198671(Ω)

le 119862nabla1199061198712(Ω) (7)



119906 isin 119906 isin 1198673(Ω) 119904119906119888ℎ 119905ℎ119886119905

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119906119889119909 = 0 (8)


nabla1199062

1198712(Ω)

le 119862Δ1199062

1198712(Ω) (9)

and secondly

1199062

1198673(Ω)

le 119870Δ1199062

1198671(Ω) (10)




2

119867


119881| le 119888

2|119907|

2

119881


119906 isin 1198712(0 119879 119881) cap 119862 (0 119879119867) (11)

such that

forall119907 isin 119881 (

119889119906

119889119905

119861119907)

119867

+ 119886 (119906 119907) = 0 (12)



1015840)



0isin 119871



120597119906

120597119905

+ 119860119906 = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(13)




119860 = 1205974119906120597119909


4119906120597119909


(119860119906) (119909) = 119892 (119909)

1205974119906

1205971199094 (14)


(119860119906) (119909) =

120597

120597119909

(119892 (119909)

1205973119906

1205971199093) (15)


(119860119906) (119909) = 119892 (119909) [minus120579

1205972119906

1205971199092+ (1 minus 120579)

1205974119906

1205971199094] (16)


or for (15)

(119860119906) (119909) =

120597

120597119909

[119892 (119909) (minus120579

120597119906

120597119909

+ (1 minus 120579)

1205973119906

1205971199093)] (17)


(119860119906) (119909) =

120597

120597119909

[minus120579119892 (119909)

120597119906

120597119909

+ (1 minus 120579)

120597

120597119909

(119892 (119909)

1205972119906

1205971199092)]




0and zeroed only at



on sign(1205971199060120597119909) sign(1205972119906

0120597119909

2) and sign(1205973119906

0120597119909

3)


sign120599(119911) =

2

120587

arctan(120587119911120599

) (19)





119892plusmn

119890(119909) =

1

9

[

10038161003816100381610038161003816100381610038161003816

sign(120597119906

0

120597119909

)

10038161003816100381610038161003816100381610038161003816

plusmn sign(12059721199060

1205971199092) + 1]

2


119892119891(119909) = [sign

120599(

12059721199060

1205971199092)]

2

for turning points

119892plusmn

mc (119909) =1

9

[

100381610038161003816100381610038161003816100381610038161003816

sign(12059731199060

1205971199093)

100381610038161003816100381610038161003816100381610038161003816

plusmn sign(12059721199060

1205971199092) + 1]

2

(20)







forall119906 isin 119863 (119860119890) 119860

119890119906 =

120597

120597119909

(119892plusmn

119890(119909)

1205973119906

1205971199093) (21)

or

forall119906 isin 119863 (119860119891) 119860

119891119906 =

120597

120597119909

(119892119891(119909)

1205973119906

1205971199093) (22)

If 1199060isin 119862






0isin 119867





follows

119892 (119909) =

0 at 119883max119894 119883max119894+1

4

9

at 119883inf 119894 119883inf 119894+1 119883min119894


1

9


(23)








1198712(Ω)

le nabla11990601198712(Ω)



value condition


0






120597119906

120597119905

+ 119860119906 = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(24)



120597119906

120597119905

+

120597

120597119909

(119892119891(119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(25)


0 lt 120572 le 119892119891120572(119909) le 120573 le 1 (26)




119892119891119899(119909) = 119892

119891(119909) +

1

119899




119899such that

120597119906119899

120597119905

+

120597

120597119909

(119892119899(119909)

1205973119906119899

1205971199093) = 0 in [0 119879] times Ω

119906119899(0 119909) = 119906

0in Ω


(28)






= ℎ le



120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(29)











119899








1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909+int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909=0 119891119900119903 119886119899119910 119907 isin V

(30)



3119906120597119909

3)) = 0





119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867

1(Ω)) (31)

verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(32)



120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (33)



int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909 = 0 (34)


int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909

= [ℎ (119909)

1205973119906

1205971199093

1205972119907

1205971199092]

120597Ω

minus int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

(35)

and secondly

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

119906

1205972119907

1205971199092119889119909

int

Ω

119906

1205972119907

1205971199092119889119909 = int

120597Ω

119906

120597119907

120597119899

119889120590 minus int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(36)


int

120597Ω

119906

120597119907

120597119899

119889120590 = int

120597Ω

119906

120597119907

120597119909

119889120590 (37)


(Ω) such that

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

119907 isin 119881 = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(38)

it comes out that

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0


(39)


119886 (119906 119907) = int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

119887 (119906 119907) = int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(40)


120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω


120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω


119906 (0 119909) = 1199060

in Ω

(41)



119889119905

(119887 (119906 119907)) + 119886 (119906 119907) = 0


(42)




119881


(Ω)

and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(43)


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (44)



120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597 (119906 + 119888)

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 with 119888 isin R (45)



Ω(119906+


if 119888 = minus 1

|Ω|

int

Ω

119906119889119909 then

int

Ω

119907119889119909 = int

Ω

(119906 + 119888) 119889119909

= int

Ω

(119906 minus

1

|Ω|

int

Ω

119906119889119909)119889119909 = 0

(46)





119889119905

(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V

(47)



|119886(119906 119907)| = (radicℎ(1205973119906120597119909

3) radicℎ(120597

3119907120597119909

3))

1198712(Ω)





1198671(Ω) such that (120597119907120597119899)|

120597Ω= 0 int

Ω119907119889119909 = 0


119867= (119889119889119905)(119887(119906 119907)) is equal to

int

Ω

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

119889119909 = int

Ω

119889

119889119905

(

120597119906

120597119909

)

120597119907

120597119909

119889119909

= (

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

)

1198712(Ω)

(48)




(Ω)10038171003817100381710038171003817100381710038171003817

119906 minus

1

|Ω|

int

Ω

119906119889119909

100381710038171003817100381710038171003817100381710038171198671(Ω)

le

1

120582

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(49)




1199061198671(Ω)

le 1198622

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(50)






norm


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (51)



120597Ω= 0 int

Ω119907119889119909 =



ΓΩ



V119890= 119907 isin 119867


Ωsub Ω verifying

int

ΓΩ

119907119889119909 = 0 and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(52)

From the fact that



|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(54)


2

1198712(Ω)

= 0 and 12059731199071205971199093

2

1198712(Ω)


Ω119907119889119909 = 0 implies 119870|Ω| =







|119886 (119906 119907)| =

1003816100381610038161003816100381610038161003816100381610038161003816

(radicℎ

1205973119906

1205971199093 radicℎ

1205973119907

1205971199093)

1198712(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

le 120573

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

sdot

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 120573|119906|V sdot |119907|V

(55)


For

119907 isin V minus1199072

1198671(Ω)

le minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(56)

hence

|119907|2

V minus 1199072

1198671(Ω)

le |119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(57)

nevertheless

119886 (119907 119907) =

100381710038171003817100381710038171003817100381710038171003817

radicℎ

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 120572

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= 120572[|119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]

(58)

Thus

119886 (119907 119907) ge 120572 [|119907|2

V minus 1199072

1198671(Ω)] forall119907 isin V (59)


119886 (119907 119907) ge 120588|119907|2

V minus 1205831199072

1198671(Ω) forall119907 isin V (60)




1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0

119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905

119906 (0) = 1199060

119889119906

119889119905

isin 1198712(0 119879119867)

(61)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0 forall119907 isin V (62)


119907 isin 1198673(Ω) such that 120597

2119907

1205971199092still in 119863 (Ω)

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(63)


int

Ω

(120601 + 119870) 119889119909 = 0 (64)


1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(65)


119908 isin 1198671(Ω) | int


(Ω)




119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

(int

Ω

119906

1205972119907

1205971199092119889119909 minus int

120597Ω

119906

120597119907

120597119899

119889120590)

as 1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906 (120601 + 119870) 119889119909

=

119889

119889119905

(int

Ω

119906120601119889119909 + 119870int

Ω

119906119889119909)

(66)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906120601119889119909 = int

Ω

119889119906

119889119905

120601119889119909 (67)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

ℎ (119909)

1205973119906

1205971199093

120597120601

120597119909

119889119909 (68)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909 (69)

from which

int

Ω

119889119906

119889119905

120601119889119909+int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)

(70)


We deduce that

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (71)



120601 = 1205972119907120597119909


int

Ω

[

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)]

1205972119907

1205971199092119889119909

minus int

120597Ω

ℎ (119909)

1205973119906

1205971199093

120597

120597119899

(

120597119907

120597119909

) sdot 119899119889120590 = 0

forall119907 isin V

(72)


120597Ωℎ(119909)(120597

3119906120597119909

3)(120597

2119907120597119909

2)119889120590 = 0 forall119907 isin






1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(73)





0=

1199060minus (1|Ω|) int












119911










119899120597119909

2)

int

Ω

120597119906119899

120597119905

1205972119906119899

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092119889119909 = 0 (74)


⟨

120597119906119899

120597119905

1205972119906119899

1205971199092⟩

1198811015840119881

+⟨

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092⟩

1198811015840119881

=0

(75)


1

2

int

Ω

120597

120597119905

(

10038161003816100381610038161003816100381610038161003816

120597119906119899

120597119909

10038161003816100381610038161003816100381610038161003816

2

)119889119909 + int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 = 0 (76)


1

2

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

minus

1

2

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904 = 0

(77)

which implies that

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

=

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(78)


But as ℎ119899(119909) ge 120572 we have

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

int

Ω

(

1205973119906119899

1205971199093)

2

119889119909 119889119904

(79)

or

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

100381710038171003817100381710038171003817100381710038171003817

1205973119906119899

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904

(80)

from which10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904

(81)

Thus on one hand10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(82)

But10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 12058221003817100381710038171003817119906119899

1003817100381710038171003817

2

1198671(Ω) (83)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le

1

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(84)

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(85)


(0 1198791198671(Ω)) and

in 1198712(0 119879119867


2120572int

119879

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904 le

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(86)

which gives

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(87)

But as10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω) (88)

therefore

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω)119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(89)

Thus

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le 119879

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(90)


int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(91)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V)

=1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V119890)

le (

119879

1205822+

1

2120572

)

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(92)


(0 119879V)




119899119896)119896isinN


(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881

1015840)



(0 119879 119881) therefore 119906119899119896rarr 119906

weakly in 1198712(0 119879119867

1(Ω))


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 (93)


int

119905

0

int

Ω

119889119906119899119896

119889119904

1205972119907

1205971199092119889119909 119889119904 = minusint

119905

0

119889

119889119904

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 119889119904

= minusint

119905

0

⟨

119889119906119899119896

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

= minusint

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909

(94)


int

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909 = int

Ω

120597119906119899119896(119905)

120597119909

120597119907

120597119909

119889119909

minus int

Ω

120597119906119899119896(0)

120597119909

120597119907

120597119909

119889119909

(95)



int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when


int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)


(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909


1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093


1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)



int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)




119899119896120597119909

3 to

1205973119906120597119909


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)


int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)


1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)


int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899


1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩



119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0


intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909




2(0 119879V) cap 119862([0 119879]119867






(Ω)






(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















119899rarrinfin119891(119909

119899) = 119891(119909) where 1198641015840



int

Ω

Δ119906119889119909 = int

120597Ω

120597119906

120597119899

119889119904

int

Ω


Ω

119906Δ119907119889119909 + int

120597Ω

120597119907

120597119899

119889119904

int

Ω

(119906Δ119907 minus 119907Δ119906) 119889119909 = int

120597Ω

119906

120597119907

120597119899

minus 119907

120597119906

120597119899


(6)






10038171003817100381710038171003817100381710038171003817

119906 minus

1

|Ω|

int

Ω

119906119889119904

100381710038171003817100381710038171003817100381710038171198671(Ω)

le 119862nabla1199061198712(Ω) (7)



119906 isin 119906 isin 1198673(Ω) 119904119906119888ℎ 119905ℎ119886119905

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119906119889119909 = 0 (8)


nabla1199062

1198712(Ω)

le 119862Δ1199062

1198712(Ω) (9)

and secondly

1199062

1198673(Ω)

le 119870Δ1199062

1198671(Ω) (10)




2

119867


119881| le 119888

2|119907|

2

119881


119906 isin 1198712(0 119879 119881) cap 119862 (0 119879119867) (11)

such that

forall119907 isin 119881 (

119889119906

119889119905

119861119907)

119867

+ 119886 (119906 119907) = 0 (12)



1015840)



0isin 119871



120597119906

120597119905

+ 119860119906 = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(13)




119860 = 1205974119906120597119909


4119906120597119909


(119860119906) (119909) = 119892 (119909)

1205974119906

1205971199094 (14)


(119860119906) (119909) =

120597

120597119909

(119892 (119909)

1205973119906

1205971199093) (15)


(119860119906) (119909) = 119892 (119909) [minus120579

1205972119906

1205971199092+ (1 minus 120579)

1205974119906

1205971199094] (16)


or for (15)

(119860119906) (119909) =

120597

120597119909

[119892 (119909) (minus120579

120597119906

120597119909

+ (1 minus 120579)

1205973119906

1205971199093)] (17)


(119860119906) (119909) =

120597

120597119909

[minus120579119892 (119909)

120597119906

120597119909

+ (1 minus 120579)

120597

120597119909

(119892 (119909)

1205972119906

1205971199092)]




0and zeroed only at



on sign(1205971199060120597119909) sign(1205972119906

0120597119909

2) and sign(1205973119906

0120597119909

3)


sign120599(119911) =

2

120587

arctan(120587119911120599

) (19)





119892plusmn

119890(119909) =

1

9

[

10038161003816100381610038161003816100381610038161003816

sign(120597119906

0

120597119909

)

10038161003816100381610038161003816100381610038161003816

plusmn sign(12059721199060

1205971199092) + 1]

2


119892119891(119909) = [sign

120599(

12059721199060

1205971199092)]

2

for turning points

119892plusmn

mc (119909) =1

9

[

100381610038161003816100381610038161003816100381610038161003816

sign(12059731199060

1205971199093)

100381610038161003816100381610038161003816100381610038161003816

plusmn sign(12059721199060

1205971199092) + 1]

2

(20)







forall119906 isin 119863 (119860119890) 119860

119890119906 =

120597

120597119909

(119892plusmn

119890(119909)

1205973119906

1205971199093) (21)

or

forall119906 isin 119863 (119860119891) 119860

119891119906 =

120597

120597119909

(119892119891(119909)

1205973119906

1205971199093) (22)

If 1199060isin 119862






0isin 119867





follows

119892 (119909) =

0 at 119883max119894 119883max119894+1

4

9

at 119883inf 119894 119883inf 119894+1 119883min119894


1

9


(23)








1198712(Ω)

le nabla11990601198712(Ω)



value condition


0






120597119906

120597119905

+ 119860119906 = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(24)



120597119906

120597119905

+

120597

120597119909

(119892119891(119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(25)


0 lt 120572 le 119892119891120572(119909) le 120573 le 1 (26)




119892119891119899(119909) = 119892

119891(119909) +

1

119899




119899such that

120597119906119899

120597119905

+

120597

120597119909

(119892119899(119909)

1205973119906119899

1205971199093) = 0 in [0 119879] times Ω

119906119899(0 119909) = 119906

0in Ω


(28)






= ℎ le



120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(29)











119899








1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909+int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909=0 119891119900119903 119886119899119910 119907 isin V

(30)



3119906120597119909

3)) = 0





119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867

1(Ω)) (31)

verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(32)



120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (33)



int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909 = 0 (34)


int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909

= [ℎ (119909)

1205973119906

1205971199093

1205972119907

1205971199092]

120597Ω

minus int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

(35)

and secondly

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

119906

1205972119907

1205971199092119889119909

int

Ω

119906

1205972119907

1205971199092119889119909 = int

120597Ω

119906

120597119907

120597119899

119889120590 minus int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(36)


int

120597Ω

119906

120597119907

120597119899

119889120590 = int

120597Ω

119906

120597119907

120597119909

119889120590 (37)


(Ω) such that

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

119907 isin 119881 = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(38)

it comes out that

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0


(39)


119886 (119906 119907) = int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

119887 (119906 119907) = int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(40)


120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω


120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω


119906 (0 119909) = 1199060

in Ω

(41)



119889119905

(119887 (119906 119907)) + 119886 (119906 119907) = 0


(42)




119881


(Ω)

and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(43)


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (44)



120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597 (119906 + 119888)

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 with 119888 isin R (45)



Ω(119906+


if 119888 = minus 1

|Ω|

int

Ω

119906119889119909 then

int

Ω

119907119889119909 = int

Ω

(119906 + 119888) 119889119909

= int

Ω

(119906 minus

1

|Ω|

int

Ω

119906119889119909)119889119909 = 0

(46)





119889119905

(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V

(47)



|119886(119906 119907)| = (radicℎ(1205973119906120597119909

3) radicℎ(120597

3119907120597119909

3))

1198712(Ω)





1198671(Ω) such that (120597119907120597119899)|

120597Ω= 0 int

Ω119907119889119909 = 0


119867= (119889119889119905)(119887(119906 119907)) is equal to

int

Ω

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

119889119909 = int

Ω

119889

119889119905

(

120597119906

120597119909

)

120597119907

120597119909

119889119909

= (

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

)

1198712(Ω)

(48)




(Ω)10038171003817100381710038171003817100381710038171003817

119906 minus

1

|Ω|

int

Ω

119906119889119909

100381710038171003817100381710038171003817100381710038171198671(Ω)

le

1

120582

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(49)




1199061198671(Ω)

le 1198622

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(50)






norm


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (51)



120597Ω= 0 int

Ω119907119889119909 =



ΓΩ



V119890= 119907 isin 119867


Ωsub Ω verifying

int

ΓΩ

119907119889119909 = 0 and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(52)

From the fact that



|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(54)


2

1198712(Ω)

= 0 and 12059731199071205971199093

2

1198712(Ω)


Ω119907119889119909 = 0 implies 119870|Ω| =







|119886 (119906 119907)| =

1003816100381610038161003816100381610038161003816100381610038161003816

(radicℎ

1205973119906

1205971199093 radicℎ

1205973119907

1205971199093)

1198712(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

le 120573

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

sdot

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 120573|119906|V sdot |119907|V

(55)


For

119907 isin V minus1199072

1198671(Ω)

le minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(56)

hence

|119907|2

V minus 1199072

1198671(Ω)

le |119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(57)

nevertheless

119886 (119907 119907) =

100381710038171003817100381710038171003817100381710038171003817

radicℎ

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 120572

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= 120572[|119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]

(58)

Thus

119886 (119907 119907) ge 120572 [|119907|2

V minus 1199072

1198671(Ω)] forall119907 isin V (59)


119886 (119907 119907) ge 120588|119907|2

V minus 1205831199072

1198671(Ω) forall119907 isin V (60)




1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0

119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905

119906 (0) = 1199060

119889119906

119889119905

isin 1198712(0 119879119867)

(61)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0 forall119907 isin V (62)


119907 isin 1198673(Ω) such that 120597

2119907

1205971199092still in 119863 (Ω)

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(63)


int

Ω

(120601 + 119870) 119889119909 = 0 (64)


1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(65)


119908 isin 1198671(Ω) | int


(Ω)




119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

(int

Ω

119906

1205972119907

1205971199092119889119909 minus int

120597Ω

119906

120597119907

120597119899

119889120590)

as 1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906 (120601 + 119870) 119889119909

=

119889

119889119905

(int

Ω

119906120601119889119909 + 119870int

Ω

119906119889119909)

(66)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906120601119889119909 = int

Ω

119889119906

119889119905

120601119889119909 (67)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

ℎ (119909)

1205973119906

1205971199093

120597120601

120597119909

119889119909 (68)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909 (69)

from which

int

Ω

119889119906

119889119905

120601119889119909+int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)

(70)


We deduce that

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (71)



120601 = 1205972119907120597119909


int

Ω

[

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)]

1205972119907

1205971199092119889119909

minus int

120597Ω

ℎ (119909)

1205973119906

1205971199093

120597

120597119899

(

120597119907

120597119909

) sdot 119899119889120590 = 0

forall119907 isin V

(72)


120597Ωℎ(119909)(120597

3119906120597119909

3)(120597

2119907120597119909

2)119889120590 = 0 forall119907 isin






1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(73)





0=

1199060minus (1|Ω|) int












119911










119899120597119909

2)

int

Ω

120597119906119899

120597119905

1205972119906119899

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092119889119909 = 0 (74)


⟨

120597119906119899

120597119905

1205972119906119899

1205971199092⟩

1198811015840119881

+⟨

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092⟩

1198811015840119881

=0

(75)


1

2

int

Ω

120597

120597119905

(

10038161003816100381610038161003816100381610038161003816

120597119906119899

120597119909

10038161003816100381610038161003816100381610038161003816

2

)119889119909 + int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 = 0 (76)


1

2

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

minus

1

2

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904 = 0

(77)

which implies that

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

=

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(78)


But as ℎ119899(119909) ge 120572 we have

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

int

Ω

(

1205973119906119899

1205971199093)

2

119889119909 119889119904

(79)

or

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

100381710038171003817100381710038171003817100381710038171003817

1205973119906119899

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904

(80)

from which10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904

(81)

Thus on one hand10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(82)

But10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 12058221003817100381710038171003817119906119899

1003817100381710038171003817

2

1198671(Ω) (83)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le

1

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(84)

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(85)


(0 1198791198671(Ω)) and

in 1198712(0 119879119867


2120572int

119879

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904 le

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(86)

which gives

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(87)

But as10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω) (88)

therefore

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω)119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(89)

Thus

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le 119879

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(90)


int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(91)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V)

=1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V119890)

le (

119879

1205822+

1

2120572

)

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(92)


(0 119879V)




119899119896)119896isinN


(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881

1015840)



(0 119879 119881) therefore 119906119899119896rarr 119906

weakly in 1198712(0 119879119867

1(Ω))


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 (93)


int

119905

0

int

Ω

119889119906119899119896

119889119904

1205972119907

1205971199092119889119909 119889119904 = minusint

119905

0

119889

119889119904

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 119889119904

= minusint

119905

0

⟨

119889119906119899119896

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

= minusint

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909

(94)


int

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909 = int

Ω

120597119906119899119896(119905)

120597119909

120597119907

120597119909

119889119909

minus int

Ω

120597119906119899119896(0)

120597119909

120597119907

120597119909

119889119909

(95)



int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when


int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)


(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909


1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093


1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)



int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)




119899119896120597119909

3 to

1205973119906120597119909


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)


int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)


1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)


int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899


1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩



119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0


intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909




2(0 119879V) cap 119862([0 119879]119867






(Ω)






(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










or for (15)

(119860119906) (119909) =

120597

120597119909

[119892 (119909) (minus120579

120597119906

120597119909

+ (1 minus 120579)

1205973119906

1205971199093)] (17)


(119860119906) (119909) =

120597

120597119909

[minus120579119892 (119909)

120597119906

120597119909

+ (1 minus 120579)

120597

120597119909

(119892 (119909)

1205972119906

1205971199092)]




0and zeroed only at



on sign(1205971199060120597119909) sign(1205972119906

0120597119909

2) and sign(1205973119906

0120597119909

3)


sign120599(119911) =

2

120587

arctan(120587119911120599

) (19)





119892plusmn

119890(119909) =

1

9

[

10038161003816100381610038161003816100381610038161003816

sign(120597119906

0

120597119909

)

10038161003816100381610038161003816100381610038161003816

plusmn sign(12059721199060

1205971199092) + 1]

2


119892119891(119909) = [sign

120599(

12059721199060

1205971199092)]

2

for turning points

119892plusmn

mc (119909) =1

9

[

100381610038161003816100381610038161003816100381610038161003816

sign(12059731199060

1205971199093)

100381610038161003816100381610038161003816100381610038161003816

plusmn sign(12059721199060

1205971199092) + 1]

2

(20)







forall119906 isin 119863 (119860119890) 119860

119890119906 =

120597

120597119909

(119892plusmn

119890(119909)

1205973119906

1205971199093) (21)

or

forall119906 isin 119863 (119860119891) 119860

119891119906 =

120597

120597119909

(119892119891(119909)

1205973119906

1205971199093) (22)

If 1199060isin 119862






0isin 119867





follows

119892 (119909) =

0 at 119883max119894 119883max119894+1

4

9

at 119883inf 119894 119883inf 119894+1 119883min119894


1

9


(23)








1198712(Ω)

le nabla11990601198712(Ω)



value condition


0






120597119906

120597119905

+ 119860119906 = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(24)



120597119906

120597119905

+

120597

120597119909

(119892119891(119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(25)


0 lt 120572 le 119892119891120572(119909) le 120573 le 1 (26)




119892119891119899(119909) = 119892

119891(119909) +

1

119899




119899such that

120597119906119899

120597119905

+

120597

120597119909

(119892119899(119909)

1205973119906119899

1205971199093) = 0 in [0 119879] times Ω

119906119899(0 119909) = 119906

0in Ω


(28)






= ℎ le



120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(29)











119899








1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909+int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909=0 119891119900119903 119886119899119910 119907 isin V

(30)



3119906120597119909

3)) = 0





119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867

1(Ω)) (31)

verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(32)



120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (33)



int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909 = 0 (34)


int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909

= [ℎ (119909)

1205973119906

1205971199093

1205972119907

1205971199092]

120597Ω

minus int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

(35)

and secondly

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

119906

1205972119907

1205971199092119889119909

int

Ω

119906

1205972119907

1205971199092119889119909 = int

120597Ω

119906

120597119907

120597119899

119889120590 minus int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(36)


int

120597Ω

119906

120597119907

120597119899

119889120590 = int

120597Ω

119906

120597119907

120597119909

119889120590 (37)


(Ω) such that

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

119907 isin 119881 = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(38)

it comes out that

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0


(39)


119886 (119906 119907) = int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

119887 (119906 119907) = int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(40)


120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω


120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω


119906 (0 119909) = 1199060

in Ω

(41)



119889119905

(119887 (119906 119907)) + 119886 (119906 119907) = 0


(42)




119881


(Ω)

and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(43)


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (44)



120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597 (119906 + 119888)

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 with 119888 isin R (45)



Ω(119906+


if 119888 = minus 1

|Ω|

int

Ω

119906119889119909 then

int

Ω

119907119889119909 = int

Ω

(119906 + 119888) 119889119909

= int

Ω

(119906 minus

1

|Ω|

int

Ω

119906119889119909)119889119909 = 0

(46)





119889119905

(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V

(47)



|119886(119906 119907)| = (radicℎ(1205973119906120597119909

3) radicℎ(120597

3119907120597119909

3))

1198712(Ω)





1198671(Ω) such that (120597119907120597119899)|

120597Ω= 0 int

Ω119907119889119909 = 0


119867= (119889119889119905)(119887(119906 119907)) is equal to

int

Ω

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

119889119909 = int

Ω

119889

119889119905

(

120597119906

120597119909

)

120597119907

120597119909

119889119909

= (

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

)

1198712(Ω)

(48)




(Ω)10038171003817100381710038171003817100381710038171003817

119906 minus

1

|Ω|

int

Ω

119906119889119909

100381710038171003817100381710038171003817100381710038171198671(Ω)

le

1

120582

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(49)




1199061198671(Ω)

le 1198622

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(50)






norm


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (51)



120597Ω= 0 int

Ω119907119889119909 =



ΓΩ



V119890= 119907 isin 119867


Ωsub Ω verifying

int

ΓΩ

119907119889119909 = 0 and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(52)

From the fact that



|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(54)


2

1198712(Ω)

= 0 and 12059731199071205971199093

2

1198712(Ω)


Ω119907119889119909 = 0 implies 119870|Ω| =







|119886 (119906 119907)| =

1003816100381610038161003816100381610038161003816100381610038161003816

(radicℎ

1205973119906

1205971199093 radicℎ

1205973119907

1205971199093)

1198712(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

le 120573

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

sdot

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 120573|119906|V sdot |119907|V

(55)


For

119907 isin V minus1199072

1198671(Ω)

le minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(56)

hence

|119907|2

V minus 1199072

1198671(Ω)

le |119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(57)

nevertheless

119886 (119907 119907) =

100381710038171003817100381710038171003817100381710038171003817

radicℎ

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 120572

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= 120572[|119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]

(58)

Thus

119886 (119907 119907) ge 120572 [|119907|2

V minus 1199072

1198671(Ω)] forall119907 isin V (59)


119886 (119907 119907) ge 120588|119907|2

V minus 1205831199072

1198671(Ω) forall119907 isin V (60)




1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0

119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905

119906 (0) = 1199060

119889119906

119889119905

isin 1198712(0 119879119867)

(61)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0 forall119907 isin V (62)


119907 isin 1198673(Ω) such that 120597

2119907

1205971199092still in 119863 (Ω)

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(63)


int

Ω

(120601 + 119870) 119889119909 = 0 (64)


1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(65)


119908 isin 1198671(Ω) | int


(Ω)




119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

(int

Ω

119906

1205972119907

1205971199092119889119909 minus int

120597Ω

119906

120597119907

120597119899

119889120590)

as 1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906 (120601 + 119870) 119889119909

=

119889

119889119905

(int

Ω

119906120601119889119909 + 119870int

Ω

119906119889119909)

(66)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906120601119889119909 = int

Ω

119889119906

119889119905

120601119889119909 (67)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

ℎ (119909)

1205973119906

1205971199093

120597120601

120597119909

119889119909 (68)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909 (69)

from which

int

Ω

119889119906

119889119905

120601119889119909+int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)

(70)


We deduce that

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (71)



120601 = 1205972119907120597119909


int

Ω

[

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)]

1205972119907

1205971199092119889119909

minus int

120597Ω

ℎ (119909)

1205973119906

1205971199093

120597

120597119899

(

120597119907

120597119909

) sdot 119899119889120590 = 0

forall119907 isin V

(72)


120597Ωℎ(119909)(120597

3119906120597119909

3)(120597

2119907120597119909

2)119889120590 = 0 forall119907 isin






1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(73)





0=

1199060minus (1|Ω|) int












119911










119899120597119909

2)

int

Ω

120597119906119899

120597119905

1205972119906119899

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092119889119909 = 0 (74)


⟨

120597119906119899

120597119905

1205972119906119899

1205971199092⟩

1198811015840119881

+⟨

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092⟩

1198811015840119881

=0

(75)


1

2

int

Ω

120597

120597119905

(

10038161003816100381610038161003816100381610038161003816

120597119906119899

120597119909

10038161003816100381610038161003816100381610038161003816

2

)119889119909 + int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 = 0 (76)


1

2

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

minus

1

2

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904 = 0

(77)

which implies that

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

=

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(78)


But as ℎ119899(119909) ge 120572 we have

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

int

Ω

(

1205973119906119899

1205971199093)

2

119889119909 119889119904

(79)

or

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

100381710038171003817100381710038171003817100381710038171003817

1205973119906119899

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904

(80)

from which10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904

(81)

Thus on one hand10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(82)

But10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 12058221003817100381710038171003817119906119899

1003817100381710038171003817

2

1198671(Ω) (83)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le

1

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(84)

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(85)


(0 1198791198671(Ω)) and

in 1198712(0 119879119867


2120572int

119879

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904 le

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(86)

which gives

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(87)

But as10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω) (88)

therefore

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω)119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(89)

Thus

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le 119879

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(90)


int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(91)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V)

=1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V119890)

le (

119879

1205822+

1

2120572

)

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(92)


(0 119879V)




119899119896)119896isinN


(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881

1015840)



(0 119879 119881) therefore 119906119899119896rarr 119906

weakly in 1198712(0 119879119867

1(Ω))


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 (93)


int

119905

0

int

Ω

119889119906119899119896

119889119904

1205972119907

1205971199092119889119909 119889119904 = minusint

119905

0

119889

119889119904

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 119889119904

= minusint

119905

0

⟨

119889119906119899119896

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

= minusint

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909

(94)


int

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909 = int

Ω

120597119906119899119896(119905)

120597119909

120597119907

120597119909

119889119909

minus int

Ω

120597119906119899119896(0)

120597119909

120597119907

120597119909

119889119909

(95)



int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when


int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)


(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909


1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093


1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)



int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)




119899119896120597119909

3 to

1205973119906120597119909


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)


int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)


1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)


int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899


1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩



119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0


intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909




2(0 119879V) cap 119862([0 119879]119867






(Ω)






(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120597119906

120597119905

+ 119860119906 = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(24)



120597119906

120597119905

+

120597

120597119909

(119892119891(119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(25)


0 lt 120572 le 119892119891120572(119909) le 120573 le 1 (26)




119892119891119899(119909) = 119892

119891(119909) +

1

119899




119899such that

120597119906119899

120597119905

+

120597

120597119909

(119892119899(119909)

1205973119906119899

1205971199093) = 0 in [0 119879] times Ω

119906119899(0 119909) = 119906

0in Ω


(28)






= ℎ le



120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

119906 (0 119909) = 1199060

in Ω


(29)











119899








1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909+int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909=0 119891119900119903 119886119899119910 119907 isin V

(30)



3119906120597119909

3)) = 0





119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867

1(Ω)) (31)

verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(32)



120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (33)



int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909 = 0 (34)


int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909

= [ℎ (119909)

1205973119906

1205971199093

1205972119907

1205971199092]

120597Ω

minus int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

(35)

and secondly

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

119906

1205972119907

1205971199092119889119909

int

Ω

119906

1205972119907

1205971199092119889119909 = int

120597Ω

119906

120597119907

120597119899

119889120590 minus int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(36)


int

120597Ω

119906

120597119907

120597119899

119889120590 = int

120597Ω

119906

120597119907

120597119909

119889120590 (37)


(Ω) such that

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

119907 isin 119881 = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(38)

it comes out that

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0


(39)


119886 (119906 119907) = int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

119887 (119906 119907) = int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(40)


120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω


120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω


119906 (0 119909) = 1199060

in Ω

(41)



119889119905

(119887 (119906 119907)) + 119886 (119906 119907) = 0


(42)




119881


(Ω)

and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(43)


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (44)



120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597 (119906 + 119888)

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 with 119888 isin R (45)



Ω(119906+


if 119888 = minus 1

|Ω|

int

Ω

119906119889119909 then

int

Ω

119907119889119909 = int

Ω

(119906 + 119888) 119889119909

= int

Ω

(119906 minus

1

|Ω|

int

Ω

119906119889119909)119889119909 = 0

(46)





119889119905

(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V

(47)



|119886(119906 119907)| = (radicℎ(1205973119906120597119909

3) radicℎ(120597

3119907120597119909

3))

1198712(Ω)





1198671(Ω) such that (120597119907120597119899)|

120597Ω= 0 int

Ω119907119889119909 = 0


119867= (119889119889119905)(119887(119906 119907)) is equal to

int

Ω

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

119889119909 = int

Ω

119889

119889119905

(

120597119906

120597119909

)

120597119907

120597119909

119889119909

= (

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

)

1198712(Ω)

(48)




(Ω)10038171003817100381710038171003817100381710038171003817

119906 minus

1

|Ω|

int

Ω

119906119889119909

100381710038171003817100381710038171003817100381710038171198671(Ω)

le

1

120582

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(49)




1199061198671(Ω)

le 1198622

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(50)






norm


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (51)



120597Ω= 0 int

Ω119907119889119909 =



ΓΩ



V119890= 119907 isin 119867


Ωsub Ω verifying

int

ΓΩ

119907119889119909 = 0 and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(52)

From the fact that



|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(54)


2

1198712(Ω)

= 0 and 12059731199071205971199093

2

1198712(Ω)


Ω119907119889119909 = 0 implies 119870|Ω| =







|119886 (119906 119907)| =

1003816100381610038161003816100381610038161003816100381610038161003816

(radicℎ

1205973119906

1205971199093 radicℎ

1205973119907

1205971199093)

1198712(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

le 120573

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

sdot

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 120573|119906|V sdot |119907|V

(55)


For

119907 isin V minus1199072

1198671(Ω)

le minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(56)

hence

|119907|2

V minus 1199072

1198671(Ω)

le |119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(57)

nevertheless

119886 (119907 119907) =

100381710038171003817100381710038171003817100381710038171003817

radicℎ

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 120572

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= 120572[|119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]

(58)

Thus

119886 (119907 119907) ge 120572 [|119907|2

V minus 1199072

1198671(Ω)] forall119907 isin V (59)


119886 (119907 119907) ge 120588|119907|2

V minus 1205831199072

1198671(Ω) forall119907 isin V (60)




1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0

119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905

119906 (0) = 1199060

119889119906

119889119905

isin 1198712(0 119879119867)

(61)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0 forall119907 isin V (62)


119907 isin 1198673(Ω) such that 120597

2119907

1205971199092still in 119863 (Ω)

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(63)


int

Ω

(120601 + 119870) 119889119909 = 0 (64)


1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(65)


119908 isin 1198671(Ω) | int


(Ω)




119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

(int

Ω

119906

1205972119907

1205971199092119889119909 minus int

120597Ω

119906

120597119907

120597119899

119889120590)

as 1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906 (120601 + 119870) 119889119909

=

119889

119889119905

(int

Ω

119906120601119889119909 + 119870int

Ω

119906119889119909)

(66)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906120601119889119909 = int

Ω

119889119906

119889119905

120601119889119909 (67)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

ℎ (119909)

1205973119906

1205971199093

120597120601

120597119909

119889119909 (68)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909 (69)

from which

int

Ω

119889119906

119889119905

120601119889119909+int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)

(70)


We deduce that

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (71)



120601 = 1205972119907120597119909


int

Ω

[

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)]

1205972119907

1205971199092119889119909

minus int

120597Ω

ℎ (119909)

1205973119906

1205971199093

120597

120597119899

(

120597119907

120597119909

) sdot 119899119889120590 = 0

forall119907 isin V

(72)


120597Ωℎ(119909)(120597

3119906120597119909

3)(120597

2119907120597119909

2)119889120590 = 0 forall119907 isin






1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(73)





0=

1199060minus (1|Ω|) int












119911










119899120597119909

2)

int

Ω

120597119906119899

120597119905

1205972119906119899

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092119889119909 = 0 (74)


⟨

120597119906119899

120597119905

1205972119906119899

1205971199092⟩

1198811015840119881

+⟨

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092⟩

1198811015840119881

=0

(75)


1

2

int

Ω

120597

120597119905

(

10038161003816100381610038161003816100381610038161003816

120597119906119899

120597119909

10038161003816100381610038161003816100381610038161003816

2

)119889119909 + int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 = 0 (76)


1

2

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

minus

1

2

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904 = 0

(77)

which implies that

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

=

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(78)


But as ℎ119899(119909) ge 120572 we have

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

int

Ω

(

1205973119906119899

1205971199093)

2

119889119909 119889119904

(79)

or

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

100381710038171003817100381710038171003817100381710038171003817

1205973119906119899

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904

(80)

from which10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904

(81)

Thus on one hand10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(82)

But10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 12058221003817100381710038171003817119906119899

1003817100381710038171003817

2

1198671(Ω) (83)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le

1

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(84)

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(85)


(0 1198791198671(Ω)) and

in 1198712(0 119879119867


2120572int

119879

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904 le

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(86)

which gives

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(87)

But as10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω) (88)

therefore

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω)119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(89)

Thus

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le 119879

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(90)


int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(91)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V)

=1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V119890)

le (

119879

1205822+

1

2120572

)

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(92)


(0 119879V)




119899119896)119896isinN


(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881

1015840)



(0 119879 119881) therefore 119906119899119896rarr 119906

weakly in 1198712(0 119879119867

1(Ω))


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 (93)


int

119905

0

int

Ω

119889119906119899119896

119889119904

1205972119907

1205971199092119889119909 119889119904 = minusint

119905

0

119889

119889119904

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 119889119904

= minusint

119905

0

⟨

119889119906119899119896

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

= minusint

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909

(94)


int

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909 = int

Ω

120597119906119899119896(119905)

120597119909

120597119907

120597119909

119889119909

minus int

Ω

120597119906119899119896(0)

120597119909

120597119907

120597119909

119889119909

(95)



int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when


int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)


(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909


1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093


1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)



int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)




119899119896120597119909

3 to

1205973119906120597119909


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)


int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)


1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)


int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899


1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩



119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0


intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909




2(0 119879V) cap 119862([0 119879]119867






(Ω)






(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867

1(Ω)) (31)

verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(32)



120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (33)



int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909 = 0 (34)


int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)

1205972119907

1205971199092119889119909

= [ℎ (119909)

1205973119906

1205971199093

1205972119907

1205971199092]

120597Ω

minus int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

(35)

and secondly

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

119906

1205972119907

1205971199092119889119909

int

Ω

119906

1205972119907

1205971199092119889119909 = int

120597Ω

119906

120597119907

120597119899

119889120590 minus int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(36)


int

120597Ω

119906

120597119907

120597119899

119889120590 = int

120597Ω

119906

120597119907

120597119909

119889120590 (37)


(Ω) such that

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

119907 isin 119881 = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(38)

it comes out that

119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0


(39)


119886 (119906 119907) = int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909

119887 (119906 119907) = int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909

(40)


120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 in [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω


120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω


119906 (0 119909) = 1199060

in Ω

(41)



119889119905

(119887 (119906 119907)) + 119886 (119906 119907) = 0


(42)




119881


(Ω)

and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(43)


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (44)



120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597 (119906 + 119888)

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

=

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 with 119888 isin R (45)



Ω(119906+


if 119888 = minus 1

|Ω|

int

Ω

119906119889119909 then

int

Ω

119907119889119909 = int

Ω

(119906 + 119888) 119889119909

= int

Ω

(119906 minus

1

|Ω|

int

Ω

119906119889119909)119889119909 = 0

(46)





119889119905

(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V

(47)



|119886(119906 119907)| = (radicℎ(1205973119906120597119909

3) radicℎ(120597

3119907120597119909

3))

1198712(Ω)





1198671(Ω) such that (120597119907120597119899)|

120597Ω= 0 int

Ω119907119889119909 = 0


119867= (119889119889119905)(119887(119906 119907)) is equal to

int

Ω

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

119889119909 = int

Ω

119889

119889119905

(

120597119906

120597119909

)

120597119907

120597119909

119889119909

= (

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

)

1198712(Ω)

(48)




(Ω)10038171003817100381710038171003817100381710038171003817

119906 minus

1

|Ω|

int

Ω

119906119889119909

100381710038171003817100381710038171003817100381710038171198671(Ω)

le

1

120582

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(49)




1199061198671(Ω)

le 1198622

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(50)






norm


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (51)



120597Ω= 0 int

Ω119907119889119909 =



ΓΩ



V119890= 119907 isin 119867


Ωsub Ω verifying

int

ΓΩ

119907119889119909 = 0 and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(52)

From the fact that



|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(54)


2

1198712(Ω)

= 0 and 12059731199071205971199093

2

1198712(Ω)


Ω119907119889119909 = 0 implies 119870|Ω| =







|119886 (119906 119907)| =

1003816100381610038161003816100381610038161003816100381610038161003816

(radicℎ

1205973119906

1205971199093 radicℎ

1205973119907

1205971199093)

1198712(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

le 120573

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

sdot

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 120573|119906|V sdot |119907|V

(55)


For

119907 isin V minus1199072

1198671(Ω)

le minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(56)

hence

|119907|2

V minus 1199072

1198671(Ω)

le |119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(57)

nevertheless

119886 (119907 119907) =

100381710038171003817100381710038171003817100381710038171003817

radicℎ

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 120572

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= 120572[|119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]

(58)

Thus

119886 (119907 119907) ge 120572 [|119907|2

V minus 1199072

1198671(Ω)] forall119907 isin V (59)


119886 (119907 119907) ge 120588|119907|2

V minus 1205831199072

1198671(Ω) forall119907 isin V (60)




1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0

119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905

119906 (0) = 1199060

119889119906

119889119905

isin 1198712(0 119879119867)

(61)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0 forall119907 isin V (62)


119907 isin 1198673(Ω) such that 120597

2119907

1205971199092still in 119863 (Ω)

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(63)


int

Ω

(120601 + 119870) 119889119909 = 0 (64)


1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(65)


119908 isin 1198671(Ω) | int


(Ω)




119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

(int

Ω

119906

1205972119907

1205971199092119889119909 minus int

120597Ω

119906

120597119907

120597119899

119889120590)

as 1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906 (120601 + 119870) 119889119909

=

119889

119889119905

(int

Ω

119906120601119889119909 + 119870int

Ω

119906119889119909)

(66)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906120601119889119909 = int

Ω

119889119906

119889119905

120601119889119909 (67)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

ℎ (119909)

1205973119906

1205971199093

120597120601

120597119909

119889119909 (68)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909 (69)

from which

int

Ω

119889119906

119889119905

120601119889119909+int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)

(70)


We deduce that

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (71)



120601 = 1205972119907120597119909


int

Ω

[

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)]

1205972119907

1205971199092119889119909

minus int

120597Ω

ℎ (119909)

1205973119906

1205971199093

120597

120597119899

(

120597119907

120597119909

) sdot 119899119889120590 = 0

forall119907 isin V

(72)


120597Ωℎ(119909)(120597

3119906120597119909

3)(120597

2119907120597119909

2)119889120590 = 0 forall119907 isin






1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(73)





0=

1199060minus (1|Ω|) int












119911










119899120597119909

2)

int

Ω

120597119906119899

120597119905

1205972119906119899

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092119889119909 = 0 (74)


⟨

120597119906119899

120597119905

1205972119906119899

1205971199092⟩

1198811015840119881

+⟨

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092⟩

1198811015840119881

=0

(75)


1

2

int

Ω

120597

120597119905

(

10038161003816100381610038161003816100381610038161003816

120597119906119899

120597119909

10038161003816100381610038161003816100381610038161003816

2

)119889119909 + int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 = 0 (76)


1

2

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

minus

1

2

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904 = 0

(77)

which implies that

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

=

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(78)


But as ℎ119899(119909) ge 120572 we have

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

int

Ω

(

1205973119906119899

1205971199093)

2

119889119909 119889119904

(79)

or

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

100381710038171003817100381710038171003817100381710038171003817

1205973119906119899

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904

(80)

from which10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904

(81)

Thus on one hand10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(82)

But10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 12058221003817100381710038171003817119906119899

1003817100381710038171003817

2

1198671(Ω) (83)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le

1

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(84)

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(85)


(0 1198791198671(Ω)) and

in 1198712(0 119879119867


2120572int

119879

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904 le

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(86)

which gives

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(87)

But as10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω) (88)

therefore

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω)119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(89)

Thus

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le 119879

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(90)


int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(91)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V)

=1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V119890)

le (

119879

1205822+

1

2120572

)

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(92)


(0 119879V)




119899119896)119896isinN


(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881

1015840)



(0 119879 119881) therefore 119906119899119896rarr 119906

weakly in 1198712(0 119879119867

1(Ω))


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 (93)


int

119905

0

int

Ω

119889119906119899119896

119889119904

1205972119907

1205971199092119889119909 119889119904 = minusint

119905

0

119889

119889119904

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 119889119904

= minusint

119905

0

⟨

119889119906119899119896

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

= minusint

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909

(94)


int

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909 = int

Ω

120597119906119899119896(119905)

120597119909

120597119907

120597119909

119889119909

minus int

Ω

120597119906119899119896(0)

120597119909

120597119907

120597119909

119889119909

(95)



int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when


int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)


(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909


1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093


1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)



int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)




119899119896120597119909

3 to

1205973119906120597119909


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)


int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)


1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)


int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899


1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩



119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0


intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909




2(0 119879V) cap 119862([0 119879]119867






(Ω)






(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Ω(119906+


if 119888 = minus 1

|Ω|

int

Ω

119906119889119909 then

int

Ω

119907119889119909 = int

Ω

(119906 + 119888) 119889119909

= int

Ω

(119906 minus

1

|Ω|

int

Ω

119906119889119909)119889119909 = 0

(46)





119889119905

(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V

(47)



|119886(119906 119907)| = (radicℎ(1205973119906120597119909

3) radicℎ(120597

3119907120597119909

3))

1198712(Ω)





1198671(Ω) such that (120597119907120597119899)|

120597Ω= 0 int

Ω119907119889119909 = 0


119867= (119889119889119905)(119887(119906 119907)) is equal to

int

Ω

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

119889119909 = int

Ω

119889

119889119905

(

120597119906

120597119909

)

120597119907

120597119909

119889119909

= (

120597

120597119909

(

119889119906

119889119905

)

120597119907

120597119909

)

1198712(Ω)

(48)




(Ω)10038171003817100381710038171003817100381710038171003817

119906 minus

1

|Ω|

int

Ω

119906119889119909

100381710038171003817100381710038171003817100381710038171198671(Ω)

le

1

120582

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(49)




1199061198671(Ω)

le 1198622

10038171003817100381710038171003817100381710038171003817

120597119906

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

(50)






norm


V = 119907 isin 1198673(Ω) |

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 int

Ω

119907119889119909 = 0 (51)



120597Ω= 0 int

Ω119907119889119909 =



ΓΩ



V119890= 119907 isin 119867


Ωsub Ω verifying

int

ΓΩ

119907119889119909 = 0 and 120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(52)

From the fact that



|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(54)


2

1198712(Ω)

= 0 and 12059731199071205971199093

2

1198712(Ω)


Ω119907119889119909 = 0 implies 119870|Ω| =







|119886 (119906 119907)| =

1003816100381610038161003816100381610038161003816100381610038161003816

(radicℎ

1205973119906

1205971199093 radicℎ

1205973119907

1205971199093)

1198712(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

le 120573

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

sdot

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 120573|119906|V sdot |119907|V

(55)


For

119907 isin V minus1199072

1198671(Ω)

le minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(56)

hence

|119907|2

V minus 1199072

1198671(Ω)

le |119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(57)

nevertheless

119886 (119907 119907) =

100381710038171003817100381710038171003817100381710038171003817

radicℎ

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 120572

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= 120572[|119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]

(58)

Thus

119886 (119907 119907) ge 120572 [|119907|2

V minus 1199072

1198671(Ω)] forall119907 isin V (59)


119886 (119907 119907) ge 120588|119907|2

V minus 1205831199072

1198671(Ω) forall119907 isin V (60)




1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0

119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905

119906 (0) = 1199060

119889119906

119889119905

isin 1198712(0 119879119867)

(61)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0 forall119907 isin V (62)


119907 isin 1198673(Ω) such that 120597

2119907

1205971199092still in 119863 (Ω)

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(63)


int

Ω

(120601 + 119870) 119889119909 = 0 (64)


1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(65)


119908 isin 1198671(Ω) | int


(Ω)




119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

(int

Ω

119906

1205972119907

1205971199092119889119909 minus int

120597Ω

119906

120597119907

120597119899

119889120590)

as 1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906 (120601 + 119870) 119889119909

=

119889

119889119905

(int

Ω

119906120601119889119909 + 119870int

Ω

119906119889119909)

(66)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906120601119889119909 = int

Ω

119889119906

119889119905

120601119889119909 (67)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

ℎ (119909)

1205973119906

1205971199093

120597120601

120597119909

119889119909 (68)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909 (69)

from which

int

Ω

119889119906

119889119905

120601119889119909+int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)

(70)


We deduce that

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (71)



120601 = 1205972119907120597119909


int

Ω

[

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)]

1205972119907

1205971199092119889119909

minus int

120597Ω

ℎ (119909)

1205973119906

1205971199093

120597

120597119899

(

120597119907

120597119909

) sdot 119899119889120590 = 0

forall119907 isin V

(72)


120597Ωℎ(119909)(120597

3119906120597119909

3)(120597

2119907120597119909

2)119889120590 = 0 forall119907 isin






1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(73)





0=

1199060minus (1|Ω|) int












119911










119899120597119909

2)

int

Ω

120597119906119899

120597119905

1205972119906119899

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092119889119909 = 0 (74)


⟨

120597119906119899

120597119905

1205972119906119899

1205971199092⟩

1198811015840119881

+⟨

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092⟩

1198811015840119881

=0

(75)


1

2

int

Ω

120597

120597119905

(

10038161003816100381610038161003816100381610038161003816

120597119906119899

120597119909

10038161003816100381610038161003816100381610038161003816

2

)119889119909 + int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 = 0 (76)


1

2

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

minus

1

2

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904 = 0

(77)

which implies that

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

=

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(78)


But as ℎ119899(119909) ge 120572 we have

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

int

Ω

(

1205973119906119899

1205971199093)

2

119889119909 119889119904

(79)

or

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

100381710038171003817100381710038171003817100381710038171003817

1205973119906119899

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904

(80)

from which10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904

(81)

Thus on one hand10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(82)

But10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 12058221003817100381710038171003817119906119899

1003817100381710038171003817

2

1198671(Ω) (83)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le

1

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(84)

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(85)


(0 1198791198671(Ω)) and

in 1198712(0 119879119867


2120572int

119879

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904 le

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(86)

which gives

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(87)

But as10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω) (88)

therefore

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω)119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(89)

Thus

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le 119879

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(90)


int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(91)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V)

=1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V119890)

le (

119879

1205822+

1

2120572

)

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(92)


(0 119879V)




119899119896)119896isinN


(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881

1015840)



(0 119879 119881) therefore 119906119899119896rarr 119906

weakly in 1198712(0 119879119867

1(Ω))


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 (93)


int

119905

0

int

Ω

119889119906119899119896

119889119904

1205972119907

1205971199092119889119909 119889119904 = minusint

119905

0

119889

119889119904

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 119889119904

= minusint

119905

0

⟨

119889119906119899119896

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

= minusint

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909

(94)


int

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909 = int

Ω

120597119906119899119896(119905)

120597119909

120597119907

120597119909

119889119909

minus int

Ω

120597119906119899119896(0)

120597119909

120597119907

120597119909

119889119909

(95)



int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when


int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)


(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909


1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093


1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)



int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)




119899119896120597119909

3 to

1205973119906120597119909


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)


int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)


1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)


int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899


1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩



119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0


intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909




2(0 119879V) cap 119862([0 119879]119867






(Ω)






(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











|119886 (119906 119907)| =

1003816100381610038161003816100381610038161003816100381610038161003816

(radicℎ

1205973119906

1205971199093 radicℎ

1205973119907

1205971199093)

1198712(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

le 120573

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

sdot

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

le 120573|119906|V sdot |119907|V

(55)


For

119907 isin V minus1199072

1198671(Ω)

le minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(56)

hence

|119907|2

V minus 1199072

1198671(Ω)

le |119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(57)

nevertheless

119886 (119907 119907) =

100381710038171003817100381710038171003817100381710038171003817

radicℎ

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 120572

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= 120572[|119907|2

V minus

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]

(58)

Thus

119886 (119907 119907) ge 120572 [|119907|2

V minus 1199072

1198671(Ω)] forall119907 isin V (59)


119886 (119907 119907) ge 120588|119907|2

V minus 1205831199072

1198671(Ω) forall119907 isin V (60)




1198712(0 119879V) cap 119862([0 119879]119867


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0

119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905

119906 (0) = 1199060

119889119906

119889119905

isin 1198712(0 119879119867)

(61)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 + int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = 0 forall119907 isin V (62)


119907 isin 1198673(Ω) such that 120597

2119907

1205971199092still in 119863 (Ω)

120597119907

120597120584

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(63)


int

Ω

(120601 + 119870) 119889119909 = 0 (64)


1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0

(65)


119908 isin 1198671(Ω) | int


(Ω)




119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

(int

Ω

119906

1205972119907

1205971199092119889119909 minus int

120597Ω

119906

120597119907

120597119899

119889120590)

as 1205972119907

1205971199092= 120601 + 119870

120597119907

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906 (120601 + 119870) 119889119909

=

119889

119889119905

(int

Ω

119906120601119889119909 + 119870int

Ω

119906119889119909)

(66)


119889

119889119905

int

Ω

120597119906

120597119909

120597119907

120597119909

119889119909 =

119889

119889119905

int

Ω

119906120601119889119909 = int

Ω

119889119906

119889119905

120601119889119909 (67)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

ℎ (119909)

1205973119906

1205971199093

120597120601

120597119909

119889119909 (68)


int

Ω

ℎ (119909)

1205973119906

1205971199093

1205973119907

1205971199093119889119909 = int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909 (69)

from which

int

Ω

119889119906

119889119905

120601119889119909+int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)

(70)


We deduce that

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (71)



120601 = 1205972119907120597119909


int

Ω

[

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)]

1205972119907

1205971199092119889119909

minus int

120597Ω

ℎ (119909)

1205973119906

1205971199093

120597

120597119899

(

120597119907

120597119909

) sdot 119899119889120590 = 0

forall119907 isin V

(72)


120597Ωℎ(119909)(120597

3119906120597119909

3)(120597

2119907120597119909

2)119889120590 = 0 forall119907 isin






1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(73)





0=

1199060minus (1|Ω|) int












119911










119899120597119909

2)

int

Ω

120597119906119899

120597119905

1205972119906119899

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092119889119909 = 0 (74)


⟨

120597119906119899

120597119905

1205972119906119899

1205971199092⟩

1198811015840119881

+⟨

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092⟩

1198811015840119881

=0

(75)


1

2

int

Ω

120597

120597119905

(

10038161003816100381610038161003816100381610038161003816

120597119906119899

120597119909

10038161003816100381610038161003816100381610038161003816

2

)119889119909 + int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 = 0 (76)


1

2

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

minus

1

2

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904 = 0

(77)

which implies that

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

=

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(78)


But as ℎ119899(119909) ge 120572 we have

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

int

Ω

(

1205973119906119899

1205971199093)

2

119889119909 119889119904

(79)

or

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

100381710038171003817100381710038171003817100381710038171003817

1205973119906119899

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904

(80)

from which10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904

(81)

Thus on one hand10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(82)

But10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 12058221003817100381710038171003817119906119899

1003817100381710038171003817

2

1198671(Ω) (83)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le

1

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(84)

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(85)


(0 1198791198671(Ω)) and

in 1198712(0 119879119867


2120572int

119879

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904 le

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(86)

which gives

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(87)

But as10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω) (88)

therefore

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω)119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(89)

Thus

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le 119879

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(90)


int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(91)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V)

=1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V119890)

le (

119879

1205822+

1

2120572

)

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(92)


(0 119879V)




119899119896)119896isinN


(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881

1015840)



(0 119879 119881) therefore 119906119899119896rarr 119906

weakly in 1198712(0 119879119867

1(Ω))


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 (93)


int

119905

0

int

Ω

119889119906119899119896

119889119904

1205972119907

1205971199092119889119909 119889119904 = minusint

119905

0

119889

119889119904

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 119889119904

= minusint

119905

0

⟨

119889119906119899119896

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

= minusint

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909

(94)


int

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909 = int

Ω

120597119906119899119896(119905)

120597119909

120597119907

120597119909

119889119909

minus int

Ω

120597119906119899119896(0)

120597119909

120597119907

120597119909

119889119909

(95)



int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when


int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)


(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909


1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093


1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)



int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)




119899119896120597119909

3 to

1205973119906120597119909


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)


int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)


1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)


int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899


1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩



119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0


intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909




2(0 119879V) cap 119862([0 119879]119867






(Ω)






(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










We deduce that

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 (71)



120601 = 1205972119907120597119909


int

Ω

[

119889119906

119889119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)]

1205972119907

1205971199092119889119909

minus int

120597Ω

ℎ (119909)

1205973119906

1205971199093

120597

120597119899

(

120597119907

120597119909

) sdot 119899119889120590 = 0

forall119907 isin V

(72)


120597Ωℎ(119909)(120597

3119906120597119909

3)(120597

2119907120597119909

2)119889120590 = 0 forall119907 isin






1198712(0 119879V) cap 119862([0 119879]119867

1(Ω)) verifying

120597119906

120597119905

+

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093) = 0 119894119899 [0 119879] times Ω

1205973119906

1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω

= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)

119906 (0 119909) = 1199060

119894119899 Ω

(73)





0=

1199060minus (1|Ω|) int












119911










119899120597119909

2)

int

Ω

120597119906119899

120597119905

1205972119906119899

1205971199092119889119909 + int

Ω

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092119889119909 = 0 (74)


⟨

120597119906119899

120597119905

1205972119906119899

1205971199092⟩

1198811015840119881

+⟨

120597

120597119909

(ℎ119899(119909)

1205973119906119899

1205971199093)

1205972119906119899

1205971199092⟩

1198811015840119881

=0

(75)


1

2

int

Ω

120597

120597119905

(

10038161003816100381610038161003816100381610038161003816

120597119906119899

120597119909

10038161003816100381610038161003816100381610038161003816

2

)119889119909 + int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 = 0 (76)


1

2

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

minus

1

2

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904 = 0

(77)

which implies that

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

=

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(78)


But as ℎ119899(119909) ge 120572 we have

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

int

Ω

(

1205973119906119899

1205971199093)

2

119889119909 119889119904

(79)

or

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

100381710038171003817100381710038171003817100381710038171003817

1205973119906119899

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904

(80)

from which10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904

(81)

Thus on one hand10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(82)

But10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 12058221003817100381710038171003817119906119899

1003817100381710038171003817

2

1198671(Ω) (83)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le

1

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(84)

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(85)


(0 1198791198671(Ω)) and

in 1198712(0 119879119867


2120572int

119879

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904 le

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(86)

which gives

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(87)

But as10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω) (88)

therefore

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω)119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(89)

Thus

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le 119879

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(90)


int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(91)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V)

=1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V119890)

le (

119879

1205822+

1

2120572

)

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(92)


(0 119879V)




119899119896)119896isinN


(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881

1015840)



(0 119879 119881) therefore 119906119899119896rarr 119906

weakly in 1198712(0 119879119867

1(Ω))


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 (93)


int

119905

0

int

Ω

119889119906119899119896

119889119904

1205972119907

1205971199092119889119909 119889119904 = minusint

119905

0

119889

119889119904

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 119889119904

= minusint

119905

0

⟨

119889119906119899119896

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

= minusint

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909

(94)


int

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909 = int

Ω

120597119906119899119896(119905)

120597119909

120597119907

120597119909

119889119909

minus int

Ω

120597119906119899119896(0)

120597119909

120597119907

120597119909

119889119909

(95)



int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when


int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)


(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909


1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093


1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)



int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)




119899119896120597119909

3 to

1205973119906120597119909


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)


int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)


1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)


int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899


1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩



119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0


intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909




2(0 119879V) cap 119862([0 119879]119867






(Ω)






(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










But as ℎ119899(119909) ge 120572 we have

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

int

Ω

(

1205973119906119899

1205971199093)

2

119889119909 119889119904

(79)

or

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2int

119905

0

int

Ω

ℎ119899(119909) (

1205973119906119899

1205971199093)

2

119889119909 119889119904

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

100381710038171003817100381710038171003817100381710038171003817

1205973119906119899

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904

(80)

from which10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+ 2120572int

119905

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904

(81)

Thus on one hand10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge

10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(82)

But10038171003817100381710038171003817100381710038171003817

120597119906119899(119905 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

ge 12058221003817100381710038171003817119906119899

1003817100381710038171003817

2

1198671(Ω) (83)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

le

1

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(84)

1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879119867

1(Ω))

le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(85)


(0 1198791198671(Ω)) and

in 1198712(0 119879119867


2120572int

119879

0

[1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

Vminus

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

]119889119904 le

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(86)

which gives

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(87)

But as10038171003817100381710038171003817100381710038171003817

120597119906119899(119904 sdot)

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω) (88)

therefore

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le int

119879

0

1003817100381710038171003817119906119899(119904 sdot)

1003817100381710038171003817

2

1198671(Ω)119889119904 +

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(89)

Thus

int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le 119879

1003817100381710038171003817119906119899

1003817100381710038171003817

2

119871infin(0119879119867

1(Ω))

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(90)


int

119879

0

1003816100381610038161003816

1003817100381710038171003817119906119899

1003817100381710038171003817

1003816100381610038161003816

2

V119889119904 le

119879

1205822

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2120572

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(91)


1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V)

=1003817100381710038171003817119906119899

1003817100381710038171003817

2

1198712(0119879V119890)

le (

119879

1205822+

1

2120572

)

10038171003817100381710038171003817100381710038171003817

1205971199060

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(92)


(0 119879V)




119899119896)119896isinN


(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881

1015840)



(0 119879 119881) therefore 119906119899119896rarr 119906

weakly in 1198712(0 119879119867

1(Ω))


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 =

119889

119889119905

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 (93)


int

119905

0

int

Ω

119889119906119899119896

119889119904

1205972119907

1205971199092119889119909 119889119904 = minusint

119905

0

119889

119889119904

int

Ω

120597119906119899119896

120597119909

120597119907

120597119909

119889119909 119889119904

= minusint

119905

0

⟨

119889119906119899119896

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

= minusint

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909

(94)


int

Ω

(119906119899119896(119905) minus 119906

119899119896(0))

1205972119907

1205971199092119889119909 = int

Ω

120597119906119899119896(119905)

120597119909

120597119907

120597119909

119889119909

minus int

Ω

120597119906119899119896(0)

120597119909

120597119907

120597119909

119889119909

(95)



int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when


int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)


(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909


1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093


1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)



int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)




119899119896120597119909

3 to

1205973119906120597119909


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)


int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)


1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)


int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899


1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩



119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0


intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909




2(0 119879V) cap 119862([0 119879]119867






(Ω)






(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











int

Ω

120597119906 (119905)

120597119909

120597119907

120597119909

119889119909 minus int

Ω

1205971199060

120597119909

120597119907

120597119909

119889119909 = int

Ω

[

120597119906 (119905)

120597119909

minus

1205971199060

120597119909

]

120597119907

120597119909

119889119909

= int

Ω

[119906 (119905) minus 1199060]

1205972119907

1205971199092119889119909

= int

119905

0

⟨

119889119906

119889119904

1205972119907

1205971199092⟩

1198811015840119881

119889119904

(96)

Therefore int119905

0intΩ(119889119906

119899119896119889119904)(120597

2119907120597119909

2)119889119909119889119904 converges when


int

119905

0

int

Ω

119889119906

119889119904

1205972119907

1205971199092119889119909 119889119904 (97)


(0 119879V1015840)

Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)

we have 120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093minus ℎ (119909)

1205973119906

1205971199093)

=

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) +

1

119899119896

1205974119906119899119896

1205971199094

However intΩ

1205974119906119899119896

1205971199094119907119889119909 = int

120597Ω

1205973119906

1205971199093119907 sdot 119899119889120590

minus int

Ω

120597119907

120597119909

1205973119906119899119896

1205971199093119889119909


1205971199093

100381610038161003816100381610038161003816100381610038161003816120597Ω

= 0

1205973119906119899119896

1205971199093


1205971199093

in 1198712(0 119879 119871

2(Ω))

(98)

then

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205974119906119899119896

1205971199094119907119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

=

1

119899119896

1003816100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973119906119899119896

1205971199093

120597119907

120597119909

119889119909

1003816100381610038161003816100381610038161003816100381610038161003816

le

1

119899119896

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

1003817100381710038171003817100381710038171003817100381710038171003817

1205973119906119899119896

1205971199093

10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(99)



int

Ω

120597

120597119909

(ℎ (119909)

1205973

1205971199093(119906

119899119896minus 119906)) 119907119889119909

100381610038161003816100381610038161003816100381610038161003816

le 120573

100381610038161003816100381610038161003816100381610038161003816

int

Ω

1205973

1205971199093(119906

119899119896minus 119906)

120597119907

120597119909

119889119909

100381610038161003816100381610038161003816100381610038161003816

(100)




119899119896120597119909

3 to

1205973119906120597119909


int

Ω

119889119906119899119896

119889119905

1205972119907

1205971199092119889119909 (101)

converges to

int

Ω

119889119906

119889119905

1205972119907

1205971199092119889119909 (102)


int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (103)

converges to

int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (104)


1015840)

int

Ω

119889119906119899119896

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ119899119896

1205973119906119899119896

1205971199093)119907119889119909 (105)


int

Ω

119889119906

119889119905

119907119889119909 + int

Ω

120597

120597119909

(ℎ (119909)

1205973119906

1205971199093)119907119889119909 (106)

that is to say ⟨(119889119906119899119896119889119905) 119907⟩

1198811015840119881+ ⟨(120597120597119909)(ℎ

119899119896(119909)(120597

3119906119899119896120597119909

3))

119907⟩1198811015840119881convergeswhen 119899


1198811015840119881+

⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909

3)) 119907⟩



119899119896119889119905)119907119889119909 + int

Ω(120597120597119909)(ℎ

119899119896(120597

3119906119899119896120597119909

3))119907119889119909 = 0


intΩ(120597120597119909)(ℎ(119909)(120597

3119906120597119909




2(0 119879V) cap 119862([0 119879]119867






(Ω)






(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













(119905119896 119909

119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)

We note by 119880119896





119896=

119896Δ119905 is then (119880119896+1minus 119880

119896)Δ119905


119906119905= minus

2

sum

119894119895=1

1205971

119909119894(119892

1198941198951205971

1199091198941205972

119909119895119906) (108)





119892119894+12119895

= [

119892minus1

119894119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894minus12119895

= [

119892minus1

119894minus1119895+ 119892

minus1

119894+1119895

2

]

minus1

119892119894119895+12

= [

119892minus1

119894119895+ 119892

minus1

119894119895+1

2

]

minus1

119892119894119895minus12

= [

119892minus1

119894119895+ 119892

minus1

119894119895minus1

2

]

minus1

(109)



119863+

119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)


119863minus

119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)


1198630

119894119907 (119894 119895) = 119907 (119894 +

1

2

119895) minus 119907 (119894 minus

1

2

119895)


119863+

119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)


119863minus

119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)


(110)


+

119894119863

minus

119895119863

+

119895119906(119894 119895) For posi-


1198921198941198951205971

1199091198941205972

119909119895119906 in (108) is given by

119871119894119895= 119892

119894+12119895119863

+

119894119908

119894119895minus 119892

119894minus12119895119863

minus

119894119908

119894119895

+ 119892119894119895+12

119863+

119895119908

119895119894minus 119892

119894119895minus12119863

minus

119895119908

119895119894

(111)



minus 119880119896)Δ119905 = minussum

2

119894119895=1119871

119894119895119880

119896 or119880

119896+1= (119868minusΔ119905sum

2

119894119895=1119871

119894119895)119880


119894119895=1119871

119894119895

Thus

119880119896+1

= D119880119896 with 119880

0= 119906

0 (112)




119880119896+1

=

1

2

2

sum

119901=1

(119868 + 2Δ119905119871119901119901)

minus1

(119868 minus Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (113)





119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119880119896+1

= (

2

prod

119901=1

(119868 minus Δ119905119871119901119901))

minus1

(119868 + Δ119905

2

sum

119894=1

sum

119895 = 119894

119871119894119895)119880

119896 (114)



31199093+

11988621199092+ 119886

1119909 + 119886



2119909 + 119886

1= 0 on









15 20 25 30 35 40 45145514601465147014751480148514901495


Signal s





1





1199041+ 119904







200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06

4000 iterations


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06



200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

minus06

minus04

minus02

0

02

04

06




200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










200 250 300 350 400 450 500

4000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

40000 iterations

minus06

minus04

minus02

0

02

04

06


200 250 300 350 400 450 500

400000 iterations

minus06

minus04

minus02

0

02

04

06




7 Conclusion




Appendix



119888isin Ω


119872loc120575[119891] (119909

119888) =

1

2120575

int

119909119888+120575

119909119888minus120575

119891 (119904) 119889119904 = 0 (A1)


119888) = 0 119891(119909

119888minus 120598) lt





(119872loc1119899[119891](119909

119888minus 120598))




is performed




forall119907 isin V

119862119881119907

2

1198673(Ω)

le |119907|2

V le 119862119867119907

2

1198673(Ω) (A2)


2

1198673(Ω)

= sum3

119904=0120597

119904119907120597119909

1199042

1198712

(Ω)

that is to say

1199072

1198673(Ω)

= 1199072

1198712(Ω)

+

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A3)


100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










100 200 300 400 500 600 700 800 900 1000

0

2

0

1

280 300 320 340 360 380 400 420 440 460 480

Env infEnv sup

minus2

minus1

0

1

minus1

0

1

minus1

Spline min-max



119904


80 100 120 140 160 180


minus1

minus05

0

05

1


80 100 120 140 160 180


minus1

minus05

0

05

1




and secondly

|119907|2

V =

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A4)


1198673(Ω)

posing 119862119867= 1


100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

= int

Ω

(

1205972119907

1205971199092)

2

119889119909 (A5)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = int

120597Ω

120597119907

120597119899

1205972119907

1205971199092119889120590 minus int

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A6)


int

Ω

(

1205972119907

1205971199092)

2

119889119909 = minusint

Ω

120597119907

120597119909

1205973119907

1205971199093119889119909 (A7)


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










350 400 450 500 550 600 650 700


minus1

minus05

0

05

1


350 400 450 500 550 600 650 700


minus1

minus05

0

05

1




(a) (b) (c)



1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

100381710038171003817100381710038171003817100381710038171198712(Ω)

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

1003817100381710038171003817100381710038171003817100381710038171198712(Ω)

(A8)

then we deduce100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A9)



le 119862Ω120597119907120597119909

2

1198712(Ω)

Therefore

1199072

1198673(Ω)

= 1199072

1198671(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205972119907

1205971199092

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

le 119862Ω

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

1

2

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

(A10)

Thus

1199072

1198673(Ω)

le max(12

119862Ω

3

2

)(

10038171003817100381710038171003817100381710038171003817

120597119907

120597119909

10038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

+

100381710038171003817100381710038171003817100381710038171003817

1205973119907

1205971199093

100381710038171003817100381710038171003817100381710038171003817

2

1198712(Ω)

)

(A11)

If we pose

119862119881=

1

(max ((12) + 119862Ω 32))

(A12)

we have

119862119881119907

2

1198673(Ω)

le |119907|2

V (A13)


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article About a Partial Differential Equation-Based...

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