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Research ArticleDissipativity Analysis of Descriptor Systems Using ImageSpace Characterization

Liang Qiao12 Qingling Zhang12 and Wanquan Liu3

1 Institute of Systems Science Northeastern University Shenyang 110819 China2 State Key Laboratory of Synthetical Automation for Process Industries Northeastern University China3Department of Computing Curtin University Perth WA 6102 Australia

Correspondence should be addressed to Qingling Zhang qlzhangmailneueducn

Received 20 March 2014 Revised 14 June 2014 Accepted 14 June 2014 Published 15 July 2014

Academic Editor Masoud Hajarian

Copyright copy 2014 Liang Qiao 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

In this paper we analyze the dissipativity for descriptor systems with impulsive behavior based on image space analysis First a newimage space is used to characterize state responses for descriptor systems Based on such characterization and an integral propertyof delta function a new necessary and sufficient condition for the dissipativity of descriptor systems is derived using the linearmatrix inequality (LMI) approach Also some of the earlier related results on dissipativity for linear systems are investigated in theframework proposed in this paper Finally two examples are given to show the validity of the derived results

1 Introduction

Since the notion of dissipative dynamical system was pro-posed by Willems in [1] it has played an important rolein system analysis and synthesis of various control systemssuch as in circuits power systems and mechanical systems[1 2] In fact this concept is an extension of passivity (orpositive realness) and 119867

infinperformance and is heavily used

in stability analysis of nonlinear systems [3 4] As we knowthe concept of dissipativity in control systems generalizesseveral important results including the passivity theorembounded real lemma Kalman-Yakubovich lemma and thecircle criterion [4 5]

As to the importance of dissipativity dissipative controlfor normal systems has attracted much attention [6ndash10]With mild assumptions the strict quadratic dissipativity ofcontinuousdiscrete normal systems is equivalent to their119867

infin

performance problem Based on this observation the exis-tence conditions of feedback controllers to achieve asymp-totic stability and strict dissipativity are derived respectivelyfor continuousdiscrete normal systems in [6 7] In [8] thestability and strict dissipativity conditions for normal delaysystems are investigated In [9] the dissipativity conditionsfor symmetric normal systems are given In [10] the optimalcontroller design is addressed for a class of normal systems

which are dissipative with respect to a quadratic ldquopowerfunctionrdquo

There has been a growing research interest in descrip-tor systems in control community since such systems canprovide accurate representations for interconnected large-scale systems economic systems electrical networks powersystems and mechanical systems [11 12] Many importantresults for descriptor systems have been obtained in the lasttwo decades such as controllability and observability [13]1198672119867infincontrol [14ndash16] positive realness [17 18] and passive

control [19] Dissipative control for descriptor systems hasalso attracted particular interest for its various applicationsin the literature [20ndash26] In [20 23] some necessary andsufficient conditions are given respectively to make con-tinuousdiscrete descriptor systems admissible and strictlydissipative In [21] matrix inequalities are used to charac-terize necessary and sufficient conditions for dissipativity ofdescriptor systems Based on results in [21] design of outputfeedback controllers to achieve dissipativity of the closed-loop system is investigated in [22] The dissipativity analysisfor Takagi-Sugeno (T-S) fuzzy descriptor systems with time-delay is studied in [24] In [25] the fuzzy controller andthe fuzzy observer are given which guarantees the closedloop T-S fuzzy descriptor system to be admissibility and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 639168 12 pageshttpdxdoiorg1011552014639168

2 Mathematical Problems in Engineering

strict (119876 119881 119878) minus 120572 dissipativity In [26] the problem of delay-dependent120572-dissipativity analysis is investigated for a generalclass of descriptor systems with Markovian jump parametersand mode-dependent mixed time-delays

In fact dissipative systems theory is linked to Lyapunovstability theory There exist some tools from the dissipativityapproach that can be used to construct some Lyapunovfunctions which are related to119867

infincontrol positive realness

and so forth So far in [20ndash25] the dissipative analysisfor descriptor systems is only for impulse-free systems Itis known that impulsive behavior is one main feature ofdescriptor systems being different from normal systems Infact these results are essentially the mild generalization ofdissipativity theorems for normal systems which do notreveal the nature of descriptor systems significantly In [18]a new method of image space is proposed and a necessaryand sufficient condition for passivity and positive realnessof descriptor system is given As motivated by [18] we alsogive a new image space expression to characterize the stateresponses of descriptor systems with impulsive behaviorThough the idea is similar the derived condition in this paperis much more explicit with expression of smaller matrix oflower dimensions Also we solve the dissipativity problemhere instead of the positive realness problem addressed in[18]

This image space presents a new approach for the expres-sion of descriptor systems As impulsive behavior is involvedin the system we deal with the delta function as an approx-imation of a series of normal functions for mathematicalsimplicity As such the integrability of a supply function119903(119906(119905) 119910(119905)) is considered for descriptor systems with impul-sive behavior One can find that there are some constraints forthe supply function in this case which are different from theexisting results Then a design approach which ensures theintegrability of a supply function 119903(119906(119905) 119910(119905)) is constructedConsequently a necessary and sufficient condition for dis-sipativity of descriptor systems is derived in terms of LMIsThis result is valid regardless of the existence of impulsivebehavior which is significantly different from the existingresults Thirdly an algorithm on how to effectively solvethese nonstrict LMIs is presented meanwhile the earlierrelated results are discussedwith obtained results Finally onepractical example and one numerical example are given todemonstrate the effectiveness of the results in this paper

The layout of this paper is organized as below Wewill present some preliminary results on characterization ofdescriptor systems based on image space in Section 2 Thedissipativity analysis is presented in Section 3 The relation-ship with typical previous results is investigated in Section 4Some illustrative examples are given in Section 5 and the finalconclusions are made in Section 6

For convenience of presentation we first give the follow-ing notations used in this paper

Notations Let R R+ and C denote the sets of real nonneg-ative real and complex numbers respectively The notationR119898times119899 denotes the set of 119898 times 119899 matrices with real elementsLet 119872 and 119873 be matrices The notation col(119872119873) denotesthe matrix obtained by stacking119872 over119873The image of119872 is

denoted by im119872 and kernel of119872 by ker119872 The determinantof119872 is denoted by det(119872) and rank of119872 by rank(119872) Let 119875be a square matrix it is said to be symmetric if 119875 = 119875

T Wesay that 119875 (necessarily symmetric) is positive semidefinite ifVT119875V ge 0 for all vectors V We write 119875 ge 0 by meaning that 119875is positive semidefinite Negative semidefiniteness is definedin a similar fashion The notation deg(lowast) is the degree of apolynomial For a symmetric matrix represented blockwiseoffdiagonal blocks are abbreviated with lowast that is

[11988311

11988312

119883T12

11988322

] = [11988311

11988312

lowast 11988322

] = [11988311

lowast

119883T12

11988322

] (1)

The notation 119872120596

le 0 with 120596 being a vector space stands for984858T119872984858 le 0 for all 984858 isin 120596 Let 119860119862 119862infin and 119862

ℎminus1

119901denote

respectively the sets of absolutely continuous smooth and(ℎ minus 1) times piecewise continuously differentiable functions

2 Preliminaries

Consider the following descriptor system

119864 (119905) = 119860119909 (119905) + 119861119906 (119905) (2a)

119910 (119905) = 119862119909 (119905) + 119863119906 (119905) (2b)

where 119909(119905) isin R119899 is the state 119906(119905) isin R119898 is the input 119910(119905) isin R119903is the output the matrices 119864119860 119861 119862119863 are of appropriatesizes and rank119864 = 119903 le 119899 The following concepts fordescriptor systems are essential and will be used throughoutthis paper

Definition 1 (see [11]) System (2a) and (2b) is said to beregular if det(119904119864 minus 119860) is not identically zero It is said to beimpulse-free if deg(det(119904119864 minus 119860)) = rank119864

As we know from [11] only when system (2a) and (2b)is regular it has unique solution Usually this regularitycondition is required for all control problems of descriptorsystems In this case there exist two invertible matrices119872 isin

R119899times119899 and 119879 isin R119899times119899 such that

119864 = MET = [119868 0

0 119873] 119860 = MAT = [

11986010

0 119868]

119861 = MB = [1198611

1198612

] 119862 = CT = [11986211198622]

(3)

where 1198601isin R1198991times1198991 119861

119894isin R119899119894times119898 and 119862

119894isin R119903times119899119894 119894 = 1 2 and

1198991+ 1198992= 119899 119873 is nilpotent that is 119873ℎ = 0 for some integer

ℎ ge 1 Then the restricted equivalent form of system (2a) and(2b) can be respectively expressed as

1(119905) = 119860

11199091(119905) + 119861

1119906 (119905) (4a)

1198732(119905) = 119909

2(119905) + 119861

2119906 (119905) (4b)

which is called the Weierstrass canonical form In fact theinvertible matrices 119872 and 119879 may be not unique and thetransformed system is unique in this canonical form

Mathematical Problems in Engineering 3

Throughout the paper we are interested in the solutions ofsystem (2a) and (2b) at an initial value119909

0Then any trajectory

of (2a) and (2b) is given by (see [27])

1199091(119905) = 119890

1198601119905

1199091(0) + int

119905

0

1198901198601(119905minus120591)

1198611119906 (120591) 119889120591 (5a)

1199092(119905) = minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905) minus 11990920(119905) (5b)

where

11990920(119905) =

ℎminus1

sum

119894=0

120575(119894minus1)

(119905)119873119894

11990920+

ℎminus1

sum

119894=0

119894minus1

sum

119896=0

119873119894

1198612120575(119896)

119906(119894minus119896minus1)

0 (6)

One can see that if119873 = 0 this system will have impulsivebehavior for nonzero initial values 119909

20 119906119894minus119896minus10

and discon-tinuity 119906(119894)(119905) Meanwhile we know there are no impulsiveterms in substate 119909

1(119905) when 119906(119905) isin 119862

ℎminus1

119901 For simplicity we

assume consistent initial values in this paper that is 11990920= 0

and 119906(119894minus119896minus1)

0= 0 In this case the main impulsive nature of

the system is characterized by 119873 = 0 in (5b) That indicatesthat if 119906(119894)(119905) has a discontinuity point at 119905 = 120591 119909

2(119905) will

have impulsive behavior In fact when119873 = 0 the system willhave no impulsive behavior and the initial jump may appearas discussed in [28]

In order to investigate the solution of systems (2a) and(2b) from a new perspective we follow the technique in [18]and define the following two sets

S = (119909 (119905) 119906 (119905)) | (119909 (119905) 119906 (119905)) is a solution of system

(2a) (2b) 119906 (119905) isin 119860119862 119906 (119905) isin 119862ℎ119901

S119904= (119909 (119905) 119906 (119905)) | (119909 (119905) 119906 (119905)) is a solution of system

(2a) (2b) 119906 (119905) isin 119862infin (7)

One can see that S119904sub S Further we have the following

results

Lemma 2 For system (2a) and (2b) there exist matrices 119881 isin

R119899times119897 119880 isin R119898times119897 and 119882 isin R119899times119897 with 119897 = 119899 + 119898 and EW =

AV + BU such that the following statements hold(i) If (119909(119905) 119906(119905)) isin S is a solution

col ( (119905) 119909 (119905) 119906 (119905)) isin im col (119882119881119880) (8)

(ii) If 120585 isin imcol(119882119881119880) there exists a smooth solution(119909(119905) 119906(119905)) isin S

119904such that

col ( (1) 119909 (1) 119906 (1)) = 120585 (9)

where

119881 = [119868 0 0

0 minus1198612119873]

119882 = [119860111986110

0 0 119868]

119880 = [0 119868 0]

(10)

Proof This proof is similar to Lemma 32 in [18] We omitthem here

Remark 3 The above lemma describes descriptor systemswith impulsive behavior in terms of an image space Theimage space imcol(119882119881119880) can include all impulsive solu-tions of system (2a) and (2b) with consistent initial values Ifwe take

2(119905) = minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894+1)

(119905) (11)

into (24) we can obtain Lemma 32 in [18] In terms of theexpression formulae Lemma 2 in this paper is more explicitthan description of Lemma 32 in [18] In detail the matricesinvolved in Lemma 2 have lower dimensions and thus aremore computationally efficient One can see from the mainresults of this paper in next section that this new characteri-zation is more powerful in solving the dissipativity problemfor descriptor systems Instead [18] is only concerned withthe passivity problem which is a special case of this paper

The next two lemmas will be used in the proof of mainresults in next section

Lemma 4 (see [29]) Given a nilpotent matrix 119873 isin R119899times119899and a symmetric matrix 119882 isin R119899times119899 119882 ge 0 If there exists asymmetric matrix 119881 isin R119899times119899 119881 ge 0 such that

119873119879

119881 + 119881119873 = minus119873119879

119882119873 (12)

and then

119873T119882119873 = 0 119873

T119881119873 = 0 (13)

Definition 5 A state space model (119864 119860 119861 119862119863) is a realiza-tion of a transfer function119867(119904) if 119862(119904119864 minus 119860)

minus1

119861 + 119863 = 119867(119904)A realization (119864 119860 119861 119862119863) of a transfer function119867(119904) is saidto be minimal if no other realization of 119867(119904) has smallerdimension

In order to investigate the dissipativity of descriptorsystems we need one integral property of delta functionAs we know delta function is a generalized function and itsproperties can be investigated along the generalized functiontheory as in [30] For mathematical simplicity and rigorous-ness we treat this problem from engineering intuition bytaking the delta function as an approximation of a series ofnormal functions as was done in [31] We will investigate itsgeneric correctness in the future

Lemma 6 (see [31]) If the Dirac delta function 120575(119905) is definedas a limit of the following 119892

120576(119909) as 120576 goes to zero then it is not

square integrable


Proof As 120575(119905) is defined as the limit of the following 119892120576(119909) as

120576 goes to zero then for any continuous function119891(119905) one hasthe following properties

120575 (119905) = 0 119905 = 0

int

infin

minusinfin

120575 (119905) 119889119905 = 1

int

infin

minusinfin

119891 (119905) 120575 (119905) 119889119905 = 119891 (0)

(14)

where 119891(0) is the function 119891(119905) at 119905 = 0 and

119892120576(119909) =

1

1205762(119909 + 120576) minus120576 le 119909 le 0

1

1205762(120576 minus 119909) 0 le 119909 le 120576

(15)

One can prove easily that as 120576 goes to zero 119892120576(119909) will satisfy

the above properties and it can approximate the Dirac deltafunction This 119892

120576(119909) function is integrable and the integral

of this function is 1 which is independent of the value of 120576In fact this integral can be computed explicitly (noting theintegral symmetry) as

int

infin

minusinfin

119892120576(119909) 119889119909 =

2

1205762int

120576

0

(120576 minus 119909) 119889119909 =2

1205762[119909120576 minus

1199092

2]

120576

0

= 1

(16)

Now we can compute the integral of the square of thisfunction as follows (again noting the symmetry of thisfunction)

int

infin

minusinfin

1198922

120576(119909) 119889119909

=2

1205764int

120576

0

(120576 minus 119909)2

119889119909 =2

1205764[1199091205762

minus 1199092

120576 +1199093

3]

120576

0

=2

3120576

(17)

In the limit as 120576 rarr 0 the integral of the square of thisfunction approaches infinity Thus the Dirac delta functionis not square integrable This completes the proof

There are different types of approximations for deltafunction by using different normal functions Another typicalone is using Gaussian function [32] In fact Lemma 6 isalso true for the case of Gaussian approximation Withthe approximation technique we can obtain the followingcorollary

Corollary 7 If 120572 is a constant variable and the function120575(119905)120572120575(119905) with 120575(119905) defined as a limit of 119892

120576(119909) is Lebesgue

integrable then we have 120572 = 0

With the above property we can prove the followinglemma which will play an important role in the proof of ourmain result in next section

Lemma 8 If the function 120578TΓ120578 is Lebesgue integrable on

(minusinfin +infin) for forall119906 isin 119862ℎminus1119901

where

120578 =

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

Γ =

[[[[

[

Γ0

0 sdot sdot sdot 0

0 Γ1sdot sdot sdot 0

d

0 0 sdot sdot sdot Γ

ℎminus1

]]]]

]

(18)

then we have

Γ119894= 0 119894 = 1 2 ℎ minus 1 (19)

Proof Because 120578TΓ120578 is Lebesgue integrable on (minusinfin +infin) forall 119906 isin 119862ℎminus1

119901 we can specially choose 119906(119905) to meet that 119906(119894)(119905)

119894 = 0 1 ℎ minus 2 are the Lebesgue integrable functions and119906(ℎminus1)

(119905) = 120575(119905) By Corollary 7 we can obtain Γℎminus1

= 0Then we can specially choose 119906(119905) iteratively such that 119906(119894)(119905)119894 = 0 1 ℎ minus 3 are the Lebesgue integrable functions and119906(ℎminus2)

(119905) = 120575(119905) We will have Γℎminus2

= 0 Iteratively we canprove that Γ

119894= 0 119894 = 1 2 ℎ minus 1 This completes the

proof

3 Dissipativity Analysis

Following the intuition of dissipativity for a dynamic systemgiven in [1] one can see that dissipativity of a dynamic impliesthat the internal energy storage of a system never exceedsthe energy storage supplied to the system In this paper weformulate the notion of dissipativity for the system (2a) and(2b) via the so-called dissipativity inequality

Definition 9 System (2a) and (2b) with supply rate 119903(119906(119905)119910(119905)) which is locally Lebesgue integrable independently ofthe input and the initial conditions is said to be dissipativeif there exists a nonnegative function 119881(119909(119905)) R119899 rarr R+(called the storage function) such that

119881 (119909 (1199051)) le 119881 (119909 (119905

0)) + int

1199051

1199050

119903 (119906 (119904) 119910 (119904)) 119889119904 (20)

for all 1199050 1199051with 119905

1ge 1199050and the solution (119909(119905) 119906(119905)) of system

(2a) and (2b) If the above inequality is strict then the systemis said to be strictly dissipative

Furthermore the energy supply function 119903(119906(119905) 119910(119905)) ofsystems (2a) and (2b) is usually defined as

119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119910

T(119905) 119878119906 (119905) + 119906

T(119905) 119877119906 (119905)

(21)

where 119876 119878 and 119877 are real matrices with 119876 and 119877 beingsymmetric In this case the dissipative system (2a) and (2b)is said to be (119876 119878 119877) dissipative In practice the followingassumptions are usually put on for the study of dissipativity

Assumption 10 Consider 119876 le 0

Remark 11 From Definition 9 we know that the above strict(119876 119878 119877) dissipativity includes119867

infinperformance and passivity


as special cases For example if 119876 = minus119868 119878 = 0 and 119877 =

1205742

119868 (20) reduces to an119867infin

performance index If 119876 = 0 119878 =119868 and 119877 = 0 (20) corresponds to the passivity or positiverealness for systems as discussed in [18]

One can see from the above remark that dissipativityanalysis is more general in comparison to 119867

infinperformance

positive realness and passivity analysis Though there aresome results on other problems for descriptor systems withimpulsive behavior to the best of our knowledge the analysisfor dissipativity is still not available in the existing literature

In order to give the main result in the following theoremwe use the following notations for simplicity

Π = 119877 + 119863T119878 + 119878

T119863 + 119863

T119876119863

Ω = 119862T119878 + 119862

T119876119863

(22)

31 Design Energy Supply Function 119903(119906(119905)119910(119905)) In orderto investigate the impulsive behavior for system (2a) and(2b) we need to discuss the integrability of supply function119903(119906(119905) 119910(119905)) There is a significant difference for descriptorsystems with or without impulsive behavior First we give thefollowing theorem based on Lemma 8

Theorem 12 The supply function 119903(119906(119905) 119910(119905)) of system (2a)and (2b) is Lebesgue integrable for all 119906 isin 119862ℎminus1

119901if and only if

1198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (23)

1198781198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (24)

Proof To prove this theorem we rewrite

119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119910

T(119905) 119878119906 (119905) + 119906

T(119905) 119877119906 (119905)

= 119909T1(119905) 119862

T111987611986211199091(119905) + 119909

T2(119905) 119862

T211987611986221199092(119905)

+ 2119909T1(119905) 119862

T111987611986221199092(119905) + 119906

T(119905) Π119906 (119905)

+ 2119909T2(119905) 119862

T2(119876119863 + 119878) 119906 (119905)

+ 2119909T1(119905) 119862

T1(119876119863 + 119878) 119906 (119905)

(25)

Sufficiency From (23)-(24) and 1199092= minussum

ℎminus1

119894=0119873119894

1198612119906(119894)

(119905) wehave119903 (119906 (119905) 119910 (119905))

= [

[

1199091(119905)

119906 (119905)

(119905)

]

]

T

[

[

119862T11198761198621Ξ1

0

lowast Ξ2119878T11986221198731198612

lowast lowast 0

]

]

[

[

1199091(119905)

119906 (119905)

(119905)

]

]

Ξ1= 119862

T1119878 + 119862

T1119876 (119863 minus 119862

21198612)

Ξ2= 119877 + (119863 minus 119862

21198612)T119876 (119863 minus 119862

21198612) + (119863 minus 119862

21198612)T119878

+ 119878T(119863 minus 119862

21198612)

(26)

And 1199091(119905) and 119906(119905) are locally integrable Lebesgue func-

tions so the supply function is Lebesgue integrableNecessity Assume that the supply function 119903(119906(119905) 119910(119905)) ofsystem (2a) and (2b) is Lebesgue integrable for all 119906 isin 119862

ℎminus1

119901

and then we have that

119909T2(119905) 119862

T211987611986221199092(119905)

= (minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

T

119862T21198761198622(minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

=

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

T

Γ

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

(27)

is Lebesgue integrable where

Γ =

[[[[[[[[[[

[

119861T2119862T211987611986221198612

119861T2119862T211987611987311986221198612

sdot sdot sdot 119861T2119862T21198761198622119873ℎminus1

1198612

119861T2119873

T119862T211987611986221198612

119861T2119873

T119862T211987611986221198731198612sdot sdot sdot 119861

T2119873

T119862T21198761198622119873ℎminus1

1198612

d

119861T2(119873ℎminus1

)T119862T211987611986221198612

sdot sdot sdot sdot sdot sdot 119861T2(119873ℎminus1

)T119862T21198761198622119873ℎminus1

1198612

]]]]]]]]]]

]

(28)

By Lemma 8 we can have

119861T2119873119894T119862T21198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (29)

Recall that 119876 le 0 we can obtain

1198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (30)

With similar reasoning we can derive that

119906T(119905) 119878

T11986221199092(119905) = minus119906

T(119905) 119878

T1198622(

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

= minus

ℎminus1

sum

119894=0

119906T(119905) 119878

T1198622119873119894

1198612119906(119894)

(119905)

(31)


is Lebesgue integrable Then we can obtain

119878T1198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (32)

This completes the proof

Remark 13 One can see from the above proof that if thedescriptor system is impulse-free (ie 119873 = 0) the supplyfunction is always Lebesgue integrable Otherwise the selec-tion of parameters 119876 and 119878 has some restrictions as stated inthe above theoremHowever if the descriptor system (2a) and(2b) is output impulsive-free (ie 119862

2119873 = 0) the selection for

parameters (119876 le 0 119878 119877) would be very flexible If the deltafunction is in general case we can put these constraints as arequirement to eliminate impulse in the storage function

Based on the observations from previous sections we canstate our main results in this paper


Theorem 14 Consider system (2a) and (2b) the followingstatements are equivalent

(1) System (2a) and (2b) is (119876 119878 119877) dissipative

(2) The following LMIs

119875 = 119875Tge 0 (33)

[

[

0 119875 0

lowast minus119862T119876119862 minus119862

T119878 minus 119862

T119876119863

lowast lowast minusΠ

]

]

120596

le 0 (34)

have a common solution 119875 where 120596 = im[119882119879 119881119879 119880119879]119879

Proof (1) rArr (2)Suppose that system (2a) and (2b) is (119876 119878 119877) dissipative

with the storage function given below

119881 (119909) = 119909T(119905) 119875119909 (119905) (35)

where 119875 isin R119899times119899 119875 ge 0 Let (119909(119905) 119906(119905)) isin S119904 From the

dissipativity inequality we can obtain

int

1199051

1199050

119889119881 (119905)

119889119905119889119904 le int

1199051

1199050

119903 (119906 (119904) 119910 (119904)) 119889119904 (36)

for all 1199051ge 1199050 Since all involved functions are smooth we can

derive

119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (37)

Let

119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119876119910 (119905) + 119906

T(119905) 119876119906 (119905)

(38)

Then we can have

119889119881 (119905)

119889119905minus 119910

T(119905) 119876119910 (119905) minus 2119906

T(119905) 119876119910 (119905) minus 119906

T(119905) 119876119906 (119905)

= T(119905) 119875119909 (119905) + 119909

T(119905) 119875 (119905) minus 119909

T(119905) 119862

T119876119862119909 (119905)

minus 2119909T(119905) (119862

T119878 + 119862

T119876119863)119906 (119905) minus 119906

T(119905) Π119906 (119905)

= [

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0

(39)

for 119905 isin R+ Substituting 119905 = 1 we obtain

[

[

(1)

119909 (1)

119906 (1)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(1)

119909 (1)

119906 (1)

]

]

le 0 (40)

It follows from the claim (ii) of Lemma 2 that 119875 is a solutionto the LMIs in (33)-(34)(2) rArr (1)

Suppose that the LMIs (33)-(34) admit a solution119875 Fromthe claim (i) of Lemma 2 we can derive the following

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0 (41)

whenever (119909(119905) 119906(119905)) isin S Clearly this yields

119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (42)

By integrating (42) from 1199050to 1199051 we obtain the dissipativity

inequality The proof is completed

Remark 15 It should be mentioned that Theorem 14 is validfor descriptor system with impulsive behavior Furthermorethose LMIs conditions aim to seek symmetric solutionswhich is different from the typical lower triangular matrixsolutions for dissipativity of descriptor systems withoutimpulsive behavior as reported in [21]

Remark 16 We can also consider the stability of system (2a)and (2b) by using our approach If 119906(119905) = 0 we can obtain2(119905) = 0 In this case (8) will change to the following

[[[

[

1(119905)

2(119905)

1199091(119905)

1199092(119905)

]]]

]

=

[[[

[

1198601

0

119868

0

]]]

]

1199091(119905) (43)

Then we can choose the Lyapunov function

119881 (119909 (119905)) = 119909T(119905) 119875119909 (119905) (44)

with

119875 = [11987511

11987512

119875T12

11987522

] gt 0 (45)


By this way we can obtain the stability condition of system(2a) and (2b) as follows

[[[

[

1198601

0

119868

0

]]]

]

T

[[[

[

0 0 11987511

11987512

0 0 119875T12

11987522

11987511

11987512

0 0

119875T12

11987522

0 0

]]]

]

[[[

[

1198601

0

119868

0

]]]

]

= 119860T111987511+ 119875

T111198601lt 0

(46)

This is a known stability result for descriptor systems asreported in [29]

Next we discuss a parametrization form for all possiblesolutions of the LMIs (33)-(34)

Theorem 17 If the LMIs (33)-(34) are solvable all possiblesolutions must be given by

119875 = [11987511

0

0 11987522

] (47)

where 11987511

isin R1198991times1198991 and 11987522

isin R1198992times1198992 are with the followingrequirements

(1) 11987511is a solution to the following LMIs

11987511= 119875

T11ge 0

[119860

T111987511+ 119875111198601minus 119862

T1119876119862111987511198611minus Ξ1

lowast minusΞ2

] le 0

(48)

(2) 11987522is a solution of

11987522= 119875

T22ge 0

11987522[119873

1198612

] = [0

minus119873T119862T2119878]

(49)

Proof Suppose that the LMIs (33)-(34) admit a solution119875 Let 119875 be a symmetric positive semidefinite matrix andpartitioned as

119875 = [11987511

11987512

119875T12

11987522

] = 119875Tge 0 (50)

It can be verified that the following equation holds

[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

= [

[

11987211

11987212

11987213

lowast 11987222

11987223

lowast lowast 11987233

]

]

(51)

where

11987211= 119860

T111987511+ 119875

T111198601minus 119862

T11198761198621

11987212= 119875111198611minus 119860

T1119875121198612+ 119862

T111987611986221198612minus 119862

T1119878 minus 119862

T1119876119863

11987213= 11987512minus 119860

T111987512119873 minus 119862

T111987611986221198731198612

11987222= minus119861

T2119875T121198611minus 119861

T1119875121198612+ (119878

T+ 119863

T119876)11986221198612

+ 119861T2119862T2(119878 + 119876119863) minus 119861

T2119862T211987611986221198612+ Π

11987223= minus119861

T211987522+ 119861

T111987512119873 + 119861

T2119862T21198761198622119873

minus 119878T1198622119873 minus 119863

T1198761198622119873

11987233= 119873

T11987522+ 11987522119873 minus119873

T119862T21198761198622119873

(52)

Suppose that (51) is negative semidefinite we will have11987233le 0 Since 119873 is nilpotent matrix 119875

22ge 0 and minus119876 ge 0

we can obtain11987233= 0 that is119873T

11987522+11987522119873 = 119873

T119862T21198761198622119873

By Lemma 4 we obtain

119873T11987522119873 = 0 minus119873

T119862T21198761198622119873 = 0 (53)

With simple manipulations we can see that 11987522119873 = 0 and

1198761198622119873 = 0 Since119872

33= 0 it follows frommatrix theory that

all the elements on the corresponding row and column blocksmust be zero In other words

11987213= 0 119872

23= 0 (54)

Then we can derive that

11987512= 0 119861

T211987522= minus119878

T1198622119873 (55)

Finally the right-hand side of (51) with combination of(33)-(34) will give the following conclusion

[[[

[

119860T111987511+ 119875

T111198601minus 119862

T1119876119862111987212

0 0

lowast 11987222

0 0

0 0 0 0

0 0 0 0

]]]

]

le 0 (56)

We can see that 11987511is a solution to LMIs (48)This completes

the proof

Remark 18 ByTheorem 17 we can obtain

119881 (119909) = 119909T119875119909 = 119909

T1119875111199091+ 119909

T1119875121199092+ 119909

T2119875T121199091+ 119909

T2119875221199092

= 119909T1119875111199091+ 119906

T119861T2119875221198612119906

(57)

So storage function of system (2a) and (2b) can bedivided into two parts storage function of slow subsystem(4a) and storage function of fast subsystems (4b) Thisconclusion is in agreement with actual states that is theoverall descriptor system is dissipative can be regarded as twosubsystems are dissipative respectively


Since 1199091(119905) and 119906(119905) are absolutely continuous functions

119881(119909) is absolutely continuous function and satisfies

119881 (119909 (1199051)) minus 119881 (119909 (119905

0)) = int

1199051

1199050

119889119881 (119905)

119889119905119889119905 (58)

This reveals that (42) is Lebesgue integrable By the abovediscussion and Theorem 12 if suitable matrices are chosenaccording to these theorems the impulsive behavior doesnot display out the influence in the running process of thesystem and the descriptor system is dissipative That impliesthat the impulses have to be eliminated from (some parts of)the signals related to the storage function

One will see that the main result in this paper is quitedifferent from the existing results

4 Comparison with Existing Results

In this section we will compare the obtained results in thelast section with some important existing results

First we compare with the results for normal systems If119864 = 119868 one can verify the following equation easily

120596 = im[

[

119882

119881

119880

]

]

= im[

[

119860 119861

119868 0

0 119868

]

]

(59)

This will lead to the following result

Corollary 19 The continuous normal system is (119876 119878 119877)

dissipative if and only if there exists 119875 isin R119899times119899 with 119875 = 119875Tge 0

such that the following LMI is satisfied

[119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω

lowast minusΠ] le 0 (60)

Proof When 119864 = 119868 by usingTheorem 14 and (59) we have

[

[

119860 119861

119868 0

0 119868

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 minusΩ


]

]

[

[

119860 119861

119868 0

0 119868

]

]

= [119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω

lowast minusΠ] le 0

(61)

Corollary 19 is proved

Remark 20 If 119864 = 119868 our theorems become necessary andsufficient conditions for dissipativity of continuous normalsystems In fact Corollary 19 is equivalent to the dissipativelemma reported in [10]

Next we compare with some results for descriptor sys-tems First we derive the following results which are actuallynew for descriptor systems

Corollary 21 System (2a) and (2b) is (119876 119878 119877) dissipative ifthere exists119883 isin R119899times119899 such that the following LMIs are satisfied

119864T119883 = 119883

T119864 ge 0

[119860

T119883 + 119883

T119860 minus 119862

T119876119862 119883

T119861 minus Ω


(62)

Proof When system (2a) and (2b) is (119876 119878 119877) dissipative bytaking 119875 = 119864

T119883 and using EW = AV+BU we can obtain the

following

[

[

119882

119881

119880

]

]

T

[

[

0 119875 0



]

]

[

[

119882

119881

119880

]

]

= [119881

119880]

T[

[

119860T119883 + 119883

T119860

minus119862T119876119862

119883T119861 minus Ω

lowast minusΠ

]

]

[119881

119880] le 0

(63)

From Theorem 14 one can see that Corollary 21 is proved

Similarly by taking 119875 = 119864T119883119864 and EW = AV + BU we

can obtain the following results

Corollary 22 System (2a) and (2b) is (119876 119878 119877) dissipative ifthere exists symmetric matrix119883 isin R119899times119899 such that the followingLMIs are satisfied

119864T119883119864 ge 0

[119860

T119883119864 + 119864

T119883119860 minus 119862

T119876119862 119864

T119883

T119861 minus Ω


(64)

Though the LMIs (33) and (34) are investigated in generalin the last section we will give a special case in the nextcorollary which can be used easily in practical computation

Corollary 23 Suppose that(1) (119864 119860 119861 119862) is (119876 119878 119877) dissipative(2) (119864 119860 119861 119862) is minimal(3) (119864 119860 119861 119862) is given in the Weierstrass form and 119873 =

[0 119868

0 0]

Then the LMIs (33)-(34) are solvable and all the solutions canbe given by

= [

[

11

0 0

0 0 0

0 0 33

]

]

(65)

with the following requirements


11=

T11ge 0

[119860

T111+ 111198601minus 119862

T1119876119862111198611minus Ξ1

lowast minusΞ2

] le 0

(66)

where

Ξ1= 119862

T1(119878 + 119876 (119863 minus 119862

21198612minus 11986231198613))

Ξ2= 119877 + (119863 minus 119862

21198612minus 11986231198613)T119876(119863 minus 119862

21198612minus 11986231198613)

+ (119863 minus 11986221198612minus 11986231198613)T119878 + 119878

T(119863 minus 119862

21198612minus 11986231198613)

(67)


(2) 33is the unique solution of 119861T

333= minus119878

T1198622

Proof By (2)-(3) we have

(119864 119860 119861 119862)

= ([

[

119868 0 0

0 0 119868

0 0 0

]

]

[

[

11986010 0

0 119868 0

0 0 119868

]

]

[

[

1198611

1198612

1198613

]

]

[119862111986211198623])

(68)

Let be a symmetric positive semidefinite matrix andpartitioned as

= [

[

11

12

13

T12

22

23

T13

T23

33

]

]

(69)

where 11isin R1198991times1198991

22isin R1198972times1198972

33isin R1198973times1198973 and 119897

2+ 1198973= 1198992

By Theorem 14 we have

11987511= 11

11987512= [12

13] = 0

11987522119873 = [

22

23

T23

33

] [0 119868

0 0] = 0

119861T211987522= [

1198612

1198613

]

T

[22

23

T23

33

]

= 119878T1198622119873 = 119878

T[11986221198623] [0 119868

0 0] = 0

(70)

Then straightforward calculations yield

13= 0

13= 0

23= 0

22= 0 119861

T333= minus119878

T1198622

(71)

Due to minimality 1198613must be full row rank Hence

33

is the unique solution of 119861T333= minus119878

T1198622

Remark 24 By particularly choosing119876 = 0 119878 = 119868 and119877 = 0we can obtain corresponding results for Corollaries 21ndash23 andthey are in fact the Theorems 64(1) 64(2) and 54 in [18]

Before we conclude this section we discuss how toeffectively solve the nonstrict LMIs (33)-(34) We can see thatthey are given with an image space and not the conventionalstrict LMIs By (34) we know that

984858T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

984858 le 0 (72)

for all 984858 isin 120596 = im [119882

119881

119880

] This implies that LMI (34) isequivalent to

[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

le 0 (73)

According to the discussion above LMIs (33) and (73)are also not the standard LMIs which can be solved by theLMI toolbox in MATLAB directly Luckily LMIs in (33) and(73) problem are still a semidefinite programming (SDP)problem So we can use MATLAB toolbox YALMIP [33]which was initially developed to model SDP and solved byinterfacing external solver-SeDuMi [34] YALMIP toolboxsolves optimization problems in general and can be usedfor control oriented SDP problems Next we will give analgorithm for solving the nonstrict LMIs (33) and (73) indetail

Step 1 Find two invertible matrices 119872 and 119879 and changesystem (2a) and (2b) into Weierstrass canonical forms

Step 2 Compute the matrices119882 119881 119880 and 119884 = [119882

119881

119880

]

Step 3 Solve LMIs (33) and (73) as follows Define decisionvariable and constraints

119875 = sdpvar (119899 119899) 119865 = set (119875 ge 0) (74)

And let

119865 = 119865 + 119884T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

119884 ge 0 (75)

Then solve problem solvesdp(119865) and 119875 = double(119875)

5 Illustrative Examples

In this section we will present two examples to demonstratethe effectiveness and the merits of the proposed results

Example 1 (see [30]) Consider a mechanical system shownin Figure 1 This system consists of two one-mass oscillatorsconnected by a dashpot element which is described by

119872 + 119863 + 119870119911 = 119871119891 + 119869120583

119866 + 119867119911 = 0

(76)

where 119911 isin 119877119899 is the displacement vector 119891 isin 119877

119899 is thevector of known input forces 120583 isin 119877

119902 is the vector ofLagrangian multipliers 119872 is the inertial matrix which isusually symmetric and positive definite119863 is the damping andgyroscopic matrix 119870 is the stiffness and circulator matrix119871 is the force distribution matrix 119869 is the Jacobian of theconstraint equation and119866 and119867 are the coefficient matricesof the constraint equation All matrices in (76) are known andconstant with appropriate dimensions

Assume that a linear combination of displacements andvelocities is measurable that is the output equation is of theform

119910 = 119862119901119911 + 119862V (77)


k1 k2

m1m2

d

f

z1 z2

Dashpot

Rigid connection

Figure 1 Two connected one-mass oscillators

By further choosing the state vector and the input vectoras

119909 = (

119911

120583

) 119906 = 119891 (78)

respectively we can rewrite the above (76)-(77) in thefollowing descriptor system form Let119898

1= 1119898

2= 1 119889 = 1

1198961= 2 and 119896

2= 1 Thus we obtain a descriptor system with

the following parameters

119864 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(79)

where

119864 =

[[[[[

[

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

]]]]]

]

119860 =

[[[[[

[

0 0 1 0 0

0 0 0 1 0

minus2 0 minus1 minus1 1

0 minus1 minus1 minus1 1

1 1 0 0 0

]]]]]

]

119861 =

[[[[[

[

0

0

minus1

1

0

]]]]]

]

119862 = [1 1 0 0 0]

(80)

One can verify that deg(det(119904119864 minus 119860)) = 2 = 4 = rank119864so this descriptor system is not impulse-free If we choose119876 = minus1 119878 = 1 and 119877 = 5 in the energy supply functionone solution to the LMIs of Theorem 14 is as follows

119875 =

[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00001 00149

0 0 0 00149 82588

]]]]]

]

(81)

Therefore this system is dissipative although it is not impulse-free

Remark 25 In the above example we know that this descrip-tor system consists of the physical equations

1+ 1+ 2+ 21199111= minus119891 + 120583

2+ 1+ 2+ 1199112= 119891 + 120583

(82)

with a constraint equation 1199111+ 1199112= 0 After substituting

the constraint equation into the physical equations whileignoring its initial conditions we can obtain the followingsystem

1+ 21199111= minus119891 + 120583

2+ 1199112= 119891 + 120583

(83)

This becomes a normal system According to [2] we knowthat this normal system is dissipative with energy function119881(119911 ) = (12)119898

2

+ (12)1198701199112 since

119881 [119911 (1199051) (119905

1)] le 119881 [119911 (119905

0) (119905

0)] + int

1199051

1199050

119865 (119905) 120592 (119905) 119889119905

(84)

where the force 119865(119905) = 119871119891 + 119869120583 and the velocity 120592(119905) = (119905)Therefore the original descriptor system is dissipativeThis isconsistent withTheorem 14

The above example shows that a dynamic system canbe modeled with different modelsequations with differentsimplification requirements The results obtained in thispaper can be used to investigate the dissipativity problemfor descriptor system in (76)-(77) while the existing resultscannot be used due to the impulsive behavior When weuse the model (76)-(77) in fact we ignored the impulsivebehavior due to the inconsistent initial conditions

Next we will demonstrate how to appropriately choose(119876 119878 119877) to ensure that the supply function 119903(119910(119905) 119906(119905)) isintegrable

Example 2 Consider a descriptor system

119873 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(85)


where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)

According to (3) its solution is given by

119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)

In this system impulsive behavior may also appear inthe state response although the input function is continuousAnd such behavior appears due to the discontinuous propertyin input derivatives If we choose

119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)

which satisfy the condition ofTheorem 12 thenwe can obtain

119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)

Because 119906(119905) is a continuous function 119903(119906(119905) 119910(119905)) isLebesgue integrable However if we choose

119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)

they will not satisfy conditions inTheorem 12 In this case wecan verify that

119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)

is not Lebesgue integrable as it will contain 120575(119905 minus 120573)120575(119905 minus 120573)

which is not integrableExample 2 verifies the validity of Theorem 12 When a

descriptor systemhas impulsive behavior the supply functionunlike impulse-free system is not always Lebesgue integrableIn this case we need to choose the parameters (119876 119878 119877)

carefully in order to guarantee dissipativity

6 Conclusions

In this paper we investigate the dissipativity problem forcontinuous descriptor systems A new image space is usedto characterize the state responses of descriptor systemswith impulsive behavior Following it we give a dissipativitytheorem for descriptor systems in general This result is validregardless of the impulsive behavior which is different fromthe existing resultsThe integrability of a supply function andthe parametrization form of all solutions for the proposedLMIs are discussed Meanwhile our results include someprevious ones as special cases Finally some examples aregiven to show validity of the results obtained in this paper

The results in this paper are mainly based on analysisIt should be pointed out that the image space approachcan also be used to solve controlsynthesis problems fordescriptor systems Some related problems are under ourcurrent investigation

Another significant issue in this paper is the choice ofdelta function using the limit of a series of normal functionsIt is our task in the future to fill this gap by using thegeneralized function theory though we believe all the resultsderived in Lemma 8 andTheorem 12 are true in general

Conflict of Interests

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

Acknowledgments

The authors thank all the reviewers and the Associate Editorfor their constructive comments as the quality of this paperis improved significantly after significant revision basedon their comments This work was supported by NationalNatural Science Foundation of China (61273008 60974004and 61104003)

References

[1] J C Willems ldquoDissipative dynamical systems part I GeneraltheoryrdquoArchive for RationalMechanics and Analysis vol 45 no5 pp 321ndash351 1972

[2] D J Hill and P JMoylan ldquoConnections between finite-gain andasymptotic stabilityrdquo IEEE Transactions on Automatic Controlvol 25 no 5 pp 931ndash936 1980

[3] D J Hill and P J Moylan ldquoStability results for nonlinearfeedback systemsrdquo Automatica vol 13 no 4 pp 377ndash382 1977

[4] B Brogliato R Lozano and O Egeland Dissipative SystemsAnalysis and Control Theory and application Communications


and Control Engineering Series Springer London UK 2ndedition 2000

[5] J C Willems ldquoDissipative dynamical systems part II linearsystems with quadratic supply ratesrdquo Archive for RationalMechanics and Analysis vol 45 no 5 pp 352ndash393 1972

[6] S Xie L Xie and C E de Souza ldquoRobust dissipative controlfor linear systems with dissipative uncertaintyrdquo InternationalJournal of Control vol 70 no 2 pp 169ndash191 1998

[7] Z Q Tan Y C Soh and LH Xie ldquoDissipative control for lineardiscrete-time systemsrdquoAutomatica vol 35 no 9 pp 1557ndash15641999

[8] Z Li J Wang and H Shao ldquoDelay-dependent dissipativecontrol for linear time-delay systemsrdquo Journal of the FranklinInstitute vol 339 no 6-7 pp 529ndash542 2002

[9] M Meisami-Azad J Mohammadpour and K M GrigoriadisldquoDissipative analysis and control of state-space symmetricsystemsrdquo Automatica vol 45 no 6 pp 1574ndash1579 2009

[10] S Gupta ldquoRobust stabilization of uncertain systems based onenergy dissipation conceptsrdquo National Aeronautics and SpaceAdministration Contractor Report 4713 1996

[11] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer 1989

[12] C Y Yang Q L Zhang Y P Lin and L Zhou ldquoPositive realnessand absolute stability problem of descriptor systemsrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 54no 5 pp 1142ndash1149 2007

[13] D Cobb ldquoControllability observability and duality in singularsystemsrdquo IEEE Transactions on Automatic Control vol 29 no12 pp 1076ndash1082 1984

[14] M S Mahmoud and H E Emara-Shabaik ldquoNew 1198672filter for

uncertain singular systems using strict LMIsrdquo Circuits Systemsand Signal Processing vol 28 no 5 pp 665ndash677 2009

[15] S Y Xu J Lam and C Yang ldquoRobust 119867infin

control for discretesingular systems with state delay and parameter uncertaintyrdquoDynamics of Continuous Discrete and Impulsive Systems BApplications amp Algorithms vol 9 no 4 pp 539ndash554 2002

[16] J Ren and Q Zhang ldquoRobust 119867infin

control for uncertaindescriptor systems by proportional-derivative state feedbackrdquoInternational Journal of Control vol 83 no 1 pp 89ndash96 2010

[17] E Zeheb R Shorten and S S K Sajja ldquoStrict positive realnessof descriptor systems in state spacerdquo International Journal ofControl vol 83 no 9 pp 1799ndash1809 2010

[18] M K Camlibel and R Frasca ldquoExtension of Kalman-Yakubovich-Popov lemma to descriptor systemsrdquo Systems ampControl Letters vol 58 no 12 pp 795ndash803 2009

[19] Y Gao G Lu and Z Wang ldquoPassive control for continuoussingular systems with non-linear perturbationsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2554ndash2564 2010

[20] X Z Dong and Q L Zhang ldquoRobust strictly dissipative controlfor linear singular systemsrdquo Control and Decision vol 20 no 2pp 195ndash198 2005

[21] I Masubuchi ldquoDissipativity inequalities for continuous-timedescriptor systems with applications to synthesis of controlgainsrdquo Systems amp Control Letters vol 55 no 2 pp 158ndash1642006

[22] I Masubuchi ldquoOutput feedback controller synthesis for descri-ptor systems satisfying closed-loop dissipativityrdquo Automaticavol 43 no 2 pp 339ndash345 2007

[23] X Z Dong ldquoRobust strictly dissipative control for discretesingular systemsrdquo IET Control Theory amp Applications vol 1 no4 pp 1060ndash1067 2007

[24] C Han L Wu P Shi and Q Zeng ldquoOn dissipativity of Takagi-Sugeno fuzzy descriptor systems with time-delayrdquo Journal of theFranklin Institute vol 349 no 10 pp 3170ndash3184 2012

[25] H Gassara A El Hajjaji M Kchaou and M ChaabaneldquoObserver based (119876 119881 119877)-120572-dissipative control for TS fuzzydescriptor systems with time delayrdquo Journal of the FranklinInstitute vol 351 no 1 pp 187ndash206 2014

[26] W X Cui J A Fang Y L Shen andW B Zhang ldquoDissipativityanalysis of singular systems with Markovian jump parametersandmode-dependentmixed time-delaysrdquoNeurocomputing vol110 pp 121ndash127 2013

[27] D Chu and R C E Tan ldquoAlgebraic characterizations forpositive realness of descriptor systemsrdquo SIAM Journal onMatrixAnalysis and Applications vol 30 no 1 pp 197ndash222 2008

[28] W Q Liu W Y Yan and K L Teo ldquoOn initial instantaneousjumps of singular systemsrdquo IEEE Transactions on AutomaticControl vol 40 no 9 pp 1650ndash1655 1995

[29] Q L Zhang and D M Yang Analysis and Synthesis forUncertain Descrptor Systems Northeastern University PressShenyang China 2003

[30] G-R Duan Analysis and Design of Descriptor Linear Systemsvol 23 of Advances in Mechanics and Mathematics SpringerNew York NY USA 2010

[31] K D Hjelmstad Fundamentals of Structural Mechanics Spri-nger 2nd edition 2004

[32] G B Arfken and H J Weber Mathematical Methods forPhysicists Academic Press Boston Mass USA 5th edition2000

[33] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the IEEE International Sympo-sium on Computer Aided Control System Design pp 284ndash289Taipei Taiwan September 2004

[34] J F Sturm ldquoUsing SeDuMi 102 a Matlab toolbox for opti-mization over symmetric conesrdquo Optimization Methods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


strict (119876 119881 119878) minus 120572 dissipativity In [26] the problem of delay-dependent120572-dissipativity analysis is investigated for a generalclass of descriptor systems with Markovian jump parametersand mode-dependent mixed time-delays

In fact dissipative systems theory is linked to Lyapunovstability theory There exist some tools from the dissipativityapproach that can be used to construct some Lyapunovfunctions which are related to119867

infincontrol positive realness

and so forth So far in [20ndash25] the dissipative analysisfor descriptor systems is only for impulse-free systems Itis known that impulsive behavior is one main feature ofdescriptor systems being different from normal systems Infact these results are essentially the mild generalization ofdissipativity theorems for normal systems which do notreveal the nature of descriptor systems significantly In [18]a new method of image space is proposed and a necessaryand sufficient condition for passivity and positive realnessof descriptor system is given As motivated by [18] we alsogive a new image space expression to characterize the stateresponses of descriptor systems with impulsive behaviorThough the idea is similar the derived condition in this paperis much more explicit with expression of smaller matrix oflower dimensions Also we solve the dissipativity problemhere instead of the positive realness problem addressed in[18]

This image space presents a new approach for the expres-sion of descriptor systems As impulsive behavior is involvedin the system we deal with the delta function as an approx-imation of a series of normal functions for mathematicalsimplicity As such the integrability of a supply function119903(119906(119905) 119910(119905)) is considered for descriptor systems with impul-sive behavior One can find that there are some constraints forthe supply function in this case which are different from theexisting results Then a design approach which ensures theintegrability of a supply function 119903(119906(119905) 119910(119905)) is constructedConsequently a necessary and sufficient condition for dis-sipativity of descriptor systems is derived in terms of LMIsThis result is valid regardless of the existence of impulsivebehavior which is significantly different from the existingresults Thirdly an algorithm on how to effectively solvethese nonstrict LMIs is presented meanwhile the earlierrelated results are discussedwith obtained results Finally onepractical example and one numerical example are given todemonstrate the effectiveness of the results in this paper

The layout of this paper is organized as below Wewill present some preliminary results on characterization ofdescriptor systems based on image space in Section 2 Thedissipativity analysis is presented in Section 3 The relation-ship with typical previous results is investigated in Section 4Some illustrative examples are given in Section 5 and the finalconclusions are made in Section 6

For convenience of presentation we first give the follow-ing notations used in this paper

Notations Let R R+ and C denote the sets of real nonneg-ative real and complex numbers respectively The notationR119898times119899 denotes the set of 119898 times 119899 matrices with real elementsLet 119872 and 119873 be matrices The notation col(119872119873) denotesthe matrix obtained by stacking119872 over119873The image of119872 is

denoted by im119872 and kernel of119872 by ker119872 The determinantof119872 is denoted by det(119872) and rank of119872 by rank(119872) Let 119875be a square matrix it is said to be symmetric if 119875 = 119875

T Wesay that 119875 (necessarily symmetric) is positive semidefinite ifVT119875V ge 0 for all vectors V We write 119875 ge 0 by meaning that 119875is positive semidefinite Negative semidefiniteness is definedin a similar fashion The notation deg(lowast) is the degree of apolynomial For a symmetric matrix represented blockwiseoffdiagonal blocks are abbreviated with lowast that is

[11988311

11988312

119883T12

11988322

] = [11988311

11988312

lowast 11988322

] = [11988311

lowast

119883T12

11988322

] (1)

The notation 119872120596

le 0 with 120596 being a vector space stands for984858T119872984858 le 0 for all 984858 isin 120596 Let 119860119862 119862infin and 119862

ℎminus1

119901denote

respectively the sets of absolutely continuous smooth and(ℎ minus 1) times piecewise continuously differentiable functions

2 Preliminaries

Consider the following descriptor system

119864 (119905) = 119860119909 (119905) + 119861119906 (119905) (2a)

119910 (119905) = 119862119909 (119905) + 119863119906 (119905) (2b)

where 119909(119905) isin R119899 is the state 119906(119905) isin R119898 is the input 119910(119905) isin R119903is the output the matrices 119864119860 119861 119862119863 are of appropriatesizes and rank119864 = 119903 le 119899 The following concepts fordescriptor systems are essential and will be used throughoutthis paper

Definition 1 (see [11]) System (2a) and (2b) is said to beregular if det(119904119864 minus 119860) is not identically zero It is said to beimpulse-free if deg(det(119904119864 minus 119860)) = rank119864

As we know from [11] only when system (2a) and (2b)is regular it has unique solution Usually this regularitycondition is required for all control problems of descriptorsystems In this case there exist two invertible matrices119872 isin

R119899times119899 and 119879 isin R119899times119899 such that

119864 = MET = [119868 0

0 119873] 119860 = MAT = [

11986010

0 119868]

119861 = MB = [1198611

1198612

] 119862 = CT = [11986211198622]

(3)

where 1198601isin R1198991times1198991 119861

119894isin R119899119894times119898 and 119862

119894isin R119903times119899119894 119894 = 1 2 and

1198991+ 1198992= 119899 119873 is nilpotent that is 119873ℎ = 0 for some integer

ℎ ge 1 Then the restricted equivalent form of system (2a) and(2b) can be respectively expressed as

1(119905) = 119860

11199091(119905) + 119861

1119906 (119905) (4a)

1198732(119905) = 119909

2(119905) + 119861

2119906 (119905) (4b)

which is called the Weierstrass canonical form In fact theinvertible matrices 119872 and 119879 may be not unique and thetransformed system is unique in this canonical form





1199091(119905) = 119890

1198601119905

1199091(0) + int

119905

0

1198901198601(119905minus120591)

1198611119906 (120591) 119889120591 (5a)

1199092(119905) = minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905) minus 11990920(119905) (5b)

where

11990920(119905) =

ℎminus1

sum

119894=0

120575(119894minus1)

(119905)119873119894

11990920+

ℎminus1

sum

119894=0

119894minus1

sum

119896=0

119873119894

1198612120575(119896)

119906(119894minus119896minus1)

0 (6)


20 119906119894minus119896minus10


1(119905) when 119906(119905) isin 119862

ℎminus1



and 119906(119894minus119896minus1)



2(119905) will




(2a) (2b) 119906 (119905) isin 119860119862 119906 (119905) isin 119862ℎ119901


(2a) (2b) 119906 (119905) isin 119862infin (7)


results




col ( (119905) 119909 (119905) 119906 (119905)) isin im col (119882119881119880) (8)


119904such that

col ( (1) 119909 (1) 119906 (1)) = 120585 (9)

where

119881 = [119868 0 0

0 minus1198612119873]

119882 = [119860111986110

0 0 119868]

119880 = [0 119868 0]

(10)



2(119905) = minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894+1)

(119905) (11)




119873119879

119881 + 119881119873 = minus119873119879

119882119873 (12)

and then

119873T119882119873 = 0 119873

T119881119873 = 0 (13)


minus1





square integrable




120575 (119905) = 0 119905 = 0

int

infin

minusinfin

120575 (119905) 119889119905 = 1

int

infin

minusinfin

119891 (119905) 120575 (119905) 119889119905 = 119891 (0)

(14)


119892120576(119909) =

1

1205762(119909 + 120576) minus120576 le 119909 le 0

1

1205762(120576 minus 119909) 0 le 119909 le 120576

(15)





int

infin

minusinfin

119892120576(119909) 119889119909 =

2

1205762int

120576

0

(120576 minus 119909) 119889119909 =2

1205762[119909120576 minus

1199092

2]

120576

0

= 1

(16)


int

infin

minusinfin

1198922

120576(119909) 119889119909

=2

1205764int

120576

0

(120576 minus 119909)2

119889119909 =2

1205764[1199091205762

minus 1199092

120576 +1199093

3]

120576

0

=2

3120576

(17)




120576(119909) is Lebesgue





where

120578 =

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

Γ =

[[[[

[

Γ0

0 sdot sdot sdot 0


d


ℎminus1

]]]]

]

(18)

then we have

Γ119894= 0 119894 = 1 2 ℎ minus 1 (19)









proof




119881 (119909 (1199051)) le 119881 (119909 (119905

0)) + int

1199051

1199050

119903 (119906 (119904) 119910 (119904)) 119889119904 (20)

for all 1199050 1199051with 119905




119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119910

T(119905) 119878119906 (119905) + 119906

T(119905) 119877119906 (119905)

(21)







1205742




infinperformance



Π = 119877 + 119863T119878 + 119878

T119863 + 119863

T119876119863

Ω = 119862T119878 + 119862

T119876119863

(22)




1198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (23)

1198781198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (24)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119910

T(119905) 119878119906 (119905) + 119906

T(119905) 119877119906 (119905)

= 119909T1(119905) 119862

T111987611986211199091(119905) + 119909

T2(119905) 119862

T211987611986221199092(119905)

+ 2119909T1(119905) 119862

T111987611986221199092(119905) + 119906

T(119905) Π119906 (119905)

+ 2119909T2(119905) 119862

T2(119876119863 + 119878) 119906 (119905)

+ 2119909T1(119905) 119862

T1(119876119863 + 119878) 119906 (119905)

(25)


ℎminus1

119894=0119873119894

1198612119906(119894)

(119905) wehave119903 (119906 (119905) 119910 (119905))

= [

[

1199091(119905)

119906 (119905)

(119905)

]

]

T

[

[

119862T11198761198621Ξ1

0

lowast Ξ2119878T11986221198731198612

lowast lowast 0

]

]

[

[

1199091(119905)

119906 (119905)

(119905)

]

]

Ξ1= 119862

T1119878 + 119862

T1119876 (119863 minus 119862

21198612)

Ξ2= 119877 + (119863 minus 119862

21198612)T119876 (119863 minus 119862

21198612) + (119863 minus 119862

21198612)T119878

+ 119878T(119863 minus 119862

21198612)

(26)



ℎminus1

119901


119909T2(119905) 119862

T211987611986221199092(119905)

= (minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

T

119862T21198761198622(minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

=

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

T

Γ

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

(27)


Γ =

[[[[[[[[[[

[

119861T2119862T211987611986221198612

119861T2119862T211987611987311986221198612


1198612

119861T2119873

T119862T211987611986221198612

119861T2119873

T119862T211987611986221198731198612sdot sdot sdot 119861

T2119873

T119862T21198761198622119873ℎminus1

1198612

d

119861T2(119873ℎminus1

)T119862T211987611986221198612


)T119862T21198761198622119873ℎminus1

1198612

]]]]]]]]]]

]

(28)


119861T2119873119894T119862T21198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (29)


1198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (30)


119906T(119905) 119878

T11986221199092(119905) = minus119906

T(119905) 119878

T1198622(

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

= minus

ℎminus1

sum

119894=0

119906T(119905) 119878

T1198622119873119894

1198612119906(119894)

(119905)

(31)



119878T1198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (32)










119875 = 119875Tge 0 (33)

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

120596

le 0 (34)




119881 (119909) = 119909T(119905) 119875119909 (119905) (35)



int

1199051

1199050

119889119881 (119905)

119889119905119889119904 le int

1199051

1199050

119903 (119906 (119904) 119910 (119904)) 119889119904 (36)


derive

119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (37)

Let

119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119876119910 (119905) + 119906

T(119905) 119876119906 (119905)

(38)

Then we can have

119889119881 (119905)

119889119905minus 119910

T(119905) 119876119910 (119905) minus 2119906

T(119905) 119876119910 (119905) minus 119906

T(119905) 119876119906 (119905)

= T(119905) 119875119909 (119905) + 119909

T(119905) 119875 (119905) minus 119909

T(119905) 119862

T119876119862119909 (119905)

minus 2119909T(119905) (119862

T119878 + 119862

T119876119863)119906 (119905) minus 119906

T(119905) Π119906 (119905)

= [

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0

(39)


[

[

(1)

119909 (1)

119906 (1)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(1)

119909 (1)

119906 (1)

]

]

le 0 (40)



[

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0 (41)


119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (42)





[[[

[

1(119905)

2(119905)

1199091(119905)

1199092(119905)

]]]

]

=

[[[

[

1198601

0

119868

0

]]]

]

1199091(119905) (43)


119881 (119909 (119905)) = 119909T(119905) 119875119909 (119905) (44)

with

119875 = [11987511

11987512

119875T12

11987522

] gt 0 (45)



[[[

[

1198601

0

119868

0

]]]

]

T

[[[

[

0 0 11987511

11987512

0 0 119875T12

11987522

11987511

11987512

0 0

119875T12

11987522

0 0

]]]

]

[[[

[

1198601

0

119868

0

]]]

]

= 119860T111987511+ 119875

T111198601lt 0

(46)




119875 = [11987511

0

0 11987522

] (47)

where 11987511

isin R1198991times1198991 and 11987522



11987511= 119875

T11ge 0

[119860

T111987511+ 119875111198601minus 119862

T1119876119862111987511198611minus Ξ1

lowast minusΞ2

] le 0

(48)


11987522= 119875

T22ge 0

11987522[119873

1198612

] = [0

minus119873T119862T2119878]

(49)


119875 = [11987511

11987512

119875T12

11987522

] = 119875Tge 0 (50)


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

= [

[

11987211

11987212

11987213

lowast 11987222

11987223


]

]

(51)

where

11987211= 119860

T111987511+ 119875

T111198601minus 119862

T11198761198621

11987212= 119875111198611minus 119860

T1119875121198612+ 119862

T111987611986221198612minus 119862

T1119878 minus 119862

T1119876119863

11987213= 11987512minus 119860

T111987512119873 minus 119862

T111987611986221198731198612

11987222= minus119861

T2119875T121198611minus 119861

T1119875121198612+ (119878

T+ 119863

T119876)11986221198612

+ 119861T2119862T2(119878 + 119876119863) minus 119861

T2119862T211987611986221198612+ Π

11987223= minus119861

T211987522+ 119861

T111987512119873 + 119861

T2119862T21198761198622119873

minus 119878T1198622119873 minus 119863

T1198761198622119873

11987233= 119873

T11987522+ 11987522119873 minus119873

T119862T21198761198622119873

(52)




11987522+11987522119873 = 119873

T119862T21198761198622119873


119873T11987522119873 = 0 minus119873

T119862T21198761198622119873 = 0 (53)


1198761198622119873 = 0 Since119872



11987213= 0 119872

23= 0 (54)


11987512= 0 119861

T211987522= minus119878

T1198622119873 (55)


[[[

[

119860T111987511+ 119875

T111198601minus 119862

T1119876119862111987212

0 0

lowast 11987222

0 0

0 0 0 0

0 0 0 0

]]]

]

le 0 (56)


the proof


119881 (119909) = 119909T119875119909 = 119909

T1119875111199091+ 119909

T1119875121199092+ 119909

T2119875T121199091+ 119909

T2119875221199092

= 119909T1119875111199091+ 119906

T119861T2119875221198612119906

(57)





119881 (119909 (1199051)) minus 119881 (119909 (119905

0)) = int

1199051

1199050

119889119881 (119905)

119889119905119889119905 (58)






120596 = im[

[

119882

119881

119880

]

]

= im[

[

119860 119861

119868 0

0 119868

]

]

(59)





[119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω



[

[

119860 119861

119868 0

0 119868

]

]

T

[

[

0 119875 0



]

]

[

[

119860 119861

119868 0

0 119868

]

]

= [119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω


(61)





119864T119883 = 119883

T119864 ge 0

[119860

T119883 + 119883

T119860 minus 119862

T119876119862 119883

T119861 minus Ω


(62)



following

[

[

119882

119881

119880

]

]

T

[

[

0 119875 0



]

]

[

[

119882

119881

119880

]

]

= [119881

119880]

T[

[

119860T119883 + 119883

T119860

minus119862T119876119862

119883T119861 minus Ω

lowast minusΠ

]

]

[119881

119880] le 0

(63)





119864T119883119864 ge 0

[119860

T119883119864 + 119864

T119883119860 minus 119862

T119876119862 119864

T119883

T119861 minus Ω


(64)



[0 119868

0 0]


= [

[

11

0 0

0 0 0

0 0 33

]

]

(65)



11=

T11ge 0

[119860

T111+ 111198601minus 119862

T1119876119862111198611minus Ξ1

lowast minusΞ2

] le 0

(66)

where

Ξ1= 119862

T1(119878 + 119876 (119863 minus 119862

21198612minus 11986231198613))

Ξ2= 119877 + (119863 minus 119862

21198612minus 11986231198613)T119876(119863 minus 119862

21198612minus 11986231198613)

+ (119863 minus 11986221198612minus 11986231198613)T119878 + 119878

T(119863 minus 119862

21198612minus 11986231198613)

(67)



333= minus119878

T1198622


(119864 119860 119861 119862)

= ([

[

119868 0 0

0 0 119868

0 0 0

]

]

[

[

11986010 0

0 119868 0

0 0 119868

]

]

[

[

1198611

1198612

1198613

]

]

[119862111986211198623])

(68)


= [

[

11

12

13

T12

22

23

T13

T23

33

]

]

(69)


22isin R1198972times1198972

33isin R1198973times1198973 and 119897

2+ 1198973= 1198992


11987511= 11

11987512= [12

13] = 0

11987522119873 = [

22

23

T23

33

] [0 119868

0 0] = 0

119861T211987522= [

1198612

1198613

]

T

[22

23

T23

33

]

= 119878T1198622119873 = 119878

T[11986221198623] [0 119868

0 0] = 0

(70)


13= 0

13= 0

23= 0

22= 0 119861

T333= minus119878

T1198622

(71)


33


T1198622



984858T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

984858 le 0 (72)

for all 984858 isin 120596 = im [119882

119881

119880


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

le 0 (73)




119881

119880

]


119875 = sdpvar (119899 119899) 119865 = set (119875 ge 0) (74)

And let

119865 = 119865 + 119884T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

119884 ge 0 (75)





119872 + 119863 + 119870119911 = 119871119891 + 119869120583

119866 + 119867119911 = 0

(76)





119910 = 119862119901119911 + 119862V (77)


k1 k2

m1m2

d

f

z1 z2

Dashpot

Rigid connection



119909 = (

119911

120583

) 119906 = 119891 (78)


1= 1119898

2= 1 119889 = 1

1198961= 2 and 119896



119864 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(79)

where

119864 =

[[[[[

[

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

]]]]]

]

119860 =

[[[[[

[

0 0 1 0 0

0 0 0 1 0



1 1 0 0 0

]]]]]

]

119861 =

[[[[[

[

0

0

minus1

1

0

]]]]]

]

119862 = [1 1 0 0 0]

(80)


119875 =

[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00001 00149

0 0 0 00149 82588

]]]]]

]

(81)



1+ 1+ 2+ 21199111= minus119891 + 120583

2+ 1+ 2+ 1199112= 119891 + 120583

(82)



1+ 21199111= minus119891 + 120583

2+ 1199112= 119891 + 120583

(83)


2

+ (12)1198701199112 since

119881 [119911 (1199051) (119905

1)] le 119881 [119911 (119905

0) (119905

0)] + int

1199051

1199050

119865 (119905) 120592 (119905) 119889119905

(84)





119873 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(85)


where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)


119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)


119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)


119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)





6 Conclusions






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1199091(119905) = 119890

1198601119905

1199091(0) + int

119905

0

1198901198601(119905minus120591)

1198611119906 (120591) 119889120591 (5a)

1199092(119905) = minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905) minus 11990920(119905) (5b)

where

11990920(119905) =

ℎminus1

sum

119894=0

120575(119894minus1)

(119905)119873119894

11990920+

ℎminus1

sum

119894=0

119894minus1

sum

119896=0

119873119894

1198612120575(119896)

119906(119894minus119896minus1)

0 (6)


20 119906119894minus119896minus10


1(119905) when 119906(119905) isin 119862

ℎminus1



and 119906(119894minus119896minus1)



2(119905) will




(2a) (2b) 119906 (119905) isin 119860119862 119906 (119905) isin 119862ℎ119901


(2a) (2b) 119906 (119905) isin 119862infin (7)


results




col ( (119905) 119909 (119905) 119906 (119905)) isin im col (119882119881119880) (8)


119904such that

col ( (1) 119909 (1) 119906 (1)) = 120585 (9)

where

119881 = [119868 0 0

0 minus1198612119873]

119882 = [119860111986110

0 0 119868]

119880 = [0 119868 0]

(10)



2(119905) = minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894+1)

(119905) (11)




119873119879

119881 + 119881119873 = minus119873119879

119882119873 (12)

and then

119873T119882119873 = 0 119873

T119881119873 = 0 (13)


minus1





square integrable




120575 (119905) = 0 119905 = 0

int

infin

minusinfin

120575 (119905) 119889119905 = 1

int

infin

minusinfin

119891 (119905) 120575 (119905) 119889119905 = 119891 (0)

(14)


119892120576(119909) =

1

1205762(119909 + 120576) minus120576 le 119909 le 0

1

1205762(120576 minus 119909) 0 le 119909 le 120576

(15)





int

infin

minusinfin

119892120576(119909) 119889119909 =

2

1205762int

120576

0

(120576 minus 119909) 119889119909 =2

1205762[119909120576 minus

1199092

2]

120576

0

= 1

(16)


int

infin

minusinfin

1198922

120576(119909) 119889119909

=2

1205764int

120576

0

(120576 minus 119909)2

119889119909 =2

1205764[1199091205762

minus 1199092

120576 +1199093

3]

120576

0

=2

3120576

(17)




120576(119909) is Lebesgue





where

120578 =

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

Γ =

[[[[

[

Γ0

0 sdot sdot sdot 0


d


ℎminus1

]]]]

]

(18)

then we have

Γ119894= 0 119894 = 1 2 ℎ minus 1 (19)









proof




119881 (119909 (1199051)) le 119881 (119909 (119905

0)) + int

1199051

1199050

119903 (119906 (119904) 119910 (119904)) 119889119904 (20)

for all 1199050 1199051with 119905




119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119910

T(119905) 119878119906 (119905) + 119906

T(119905) 119877119906 (119905)

(21)







1205742




infinperformance



Π = 119877 + 119863T119878 + 119878

T119863 + 119863

T119876119863

Ω = 119862T119878 + 119862

T119876119863

(22)




1198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (23)

1198781198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (24)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119910

T(119905) 119878119906 (119905) + 119906

T(119905) 119877119906 (119905)

= 119909T1(119905) 119862

T111987611986211199091(119905) + 119909

T2(119905) 119862

T211987611986221199092(119905)

+ 2119909T1(119905) 119862

T111987611986221199092(119905) + 119906

T(119905) Π119906 (119905)

+ 2119909T2(119905) 119862

T2(119876119863 + 119878) 119906 (119905)

+ 2119909T1(119905) 119862

T1(119876119863 + 119878) 119906 (119905)

(25)


ℎminus1

119894=0119873119894

1198612119906(119894)

(119905) wehave119903 (119906 (119905) 119910 (119905))

= [

[

1199091(119905)

119906 (119905)

(119905)

]

]

T

[

[

119862T11198761198621Ξ1

0

lowast Ξ2119878T11986221198731198612

lowast lowast 0

]

]

[

[

1199091(119905)

119906 (119905)

(119905)

]

]

Ξ1= 119862

T1119878 + 119862

T1119876 (119863 minus 119862

21198612)

Ξ2= 119877 + (119863 minus 119862

21198612)T119876 (119863 minus 119862

21198612) + (119863 minus 119862

21198612)T119878

+ 119878T(119863 minus 119862

21198612)

(26)



ℎminus1

119901


119909T2(119905) 119862

T211987611986221199092(119905)

= (minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

T

119862T21198761198622(minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

=

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

T

Γ

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

(27)


Γ =

[[[[[[[[[[

[

119861T2119862T211987611986221198612

119861T2119862T211987611987311986221198612


1198612

119861T2119873

T119862T211987611986221198612

119861T2119873

T119862T211987611986221198731198612sdot sdot sdot 119861

T2119873

T119862T21198761198622119873ℎminus1

1198612

d

119861T2(119873ℎminus1

)T119862T211987611986221198612


)T119862T21198761198622119873ℎminus1

1198612

]]]]]]]]]]

]

(28)


119861T2119873119894T119862T21198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (29)


1198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (30)


119906T(119905) 119878

T11986221199092(119905) = minus119906

T(119905) 119878

T1198622(

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

= minus

ℎminus1

sum

119894=0

119906T(119905) 119878

T1198622119873119894

1198612119906(119894)

(119905)

(31)



119878T1198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (32)










119875 = 119875Tge 0 (33)

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

120596

le 0 (34)




119881 (119909) = 119909T(119905) 119875119909 (119905) (35)



int

1199051

1199050

119889119881 (119905)

119889119905119889119904 le int

1199051

1199050

119903 (119906 (119904) 119910 (119904)) 119889119904 (36)


derive

119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (37)

Let

119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119876119910 (119905) + 119906

T(119905) 119876119906 (119905)

(38)

Then we can have

119889119881 (119905)

119889119905minus 119910

T(119905) 119876119910 (119905) minus 2119906

T(119905) 119876119910 (119905) minus 119906

T(119905) 119876119906 (119905)

= T(119905) 119875119909 (119905) + 119909

T(119905) 119875 (119905) minus 119909

T(119905) 119862

T119876119862119909 (119905)

minus 2119909T(119905) (119862

T119878 + 119862

T119876119863)119906 (119905) minus 119906

T(119905) Π119906 (119905)

= [

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0

(39)


[

[

(1)

119909 (1)

119906 (1)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(1)

119909 (1)

119906 (1)

]

]

le 0 (40)



[

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0 (41)


119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (42)





[[[

[

1(119905)

2(119905)

1199091(119905)

1199092(119905)

]]]

]

=

[[[

[

1198601

0

119868

0

]]]

]

1199091(119905) (43)


119881 (119909 (119905)) = 119909T(119905) 119875119909 (119905) (44)

with

119875 = [11987511

11987512

119875T12

11987522

] gt 0 (45)



[[[

[

1198601

0

119868

0

]]]

]

T

[[[

[

0 0 11987511

11987512

0 0 119875T12

11987522

11987511

11987512

0 0

119875T12

11987522

0 0

]]]

]

[[[

[

1198601

0

119868

0

]]]

]

= 119860T111987511+ 119875

T111198601lt 0

(46)




119875 = [11987511

0

0 11987522

] (47)

where 11987511

isin R1198991times1198991 and 11987522



11987511= 119875

T11ge 0

[119860

T111987511+ 119875111198601minus 119862

T1119876119862111987511198611minus Ξ1

lowast minusΞ2

] le 0

(48)


11987522= 119875

T22ge 0

11987522[119873

1198612

] = [0

minus119873T119862T2119878]

(49)


119875 = [11987511

11987512

119875T12

11987522

] = 119875Tge 0 (50)


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

= [

[

11987211

11987212

11987213

lowast 11987222

11987223


]

]

(51)

where

11987211= 119860

T111987511+ 119875

T111198601minus 119862

T11198761198621

11987212= 119875111198611minus 119860

T1119875121198612+ 119862

T111987611986221198612minus 119862

T1119878 minus 119862

T1119876119863

11987213= 11987512minus 119860

T111987512119873 minus 119862

T111987611986221198731198612

11987222= minus119861

T2119875T121198611minus 119861

T1119875121198612+ (119878

T+ 119863

T119876)11986221198612

+ 119861T2119862T2(119878 + 119876119863) minus 119861

T2119862T211987611986221198612+ Π

11987223= minus119861

T211987522+ 119861

T111987512119873 + 119861

T2119862T21198761198622119873

minus 119878T1198622119873 minus 119863

T1198761198622119873

11987233= 119873

T11987522+ 11987522119873 minus119873

T119862T21198761198622119873

(52)




11987522+11987522119873 = 119873

T119862T21198761198622119873


119873T11987522119873 = 0 minus119873

T119862T21198761198622119873 = 0 (53)


1198761198622119873 = 0 Since119872



11987213= 0 119872

23= 0 (54)


11987512= 0 119861

T211987522= minus119878

T1198622119873 (55)


[[[

[

119860T111987511+ 119875

T111198601minus 119862

T1119876119862111987212

0 0

lowast 11987222

0 0

0 0 0 0

0 0 0 0

]]]

]

le 0 (56)


the proof


119881 (119909) = 119909T119875119909 = 119909

T1119875111199091+ 119909

T1119875121199092+ 119909

T2119875T121199091+ 119909

T2119875221199092

= 119909T1119875111199091+ 119906

T119861T2119875221198612119906

(57)





119881 (119909 (1199051)) minus 119881 (119909 (119905

0)) = int

1199051

1199050

119889119881 (119905)

119889119905119889119905 (58)






120596 = im[

[

119882

119881

119880

]

]

= im[

[

119860 119861

119868 0

0 119868

]

]

(59)





[119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω



[

[

119860 119861

119868 0

0 119868

]

]

T

[

[

0 119875 0



]

]

[

[

119860 119861

119868 0

0 119868

]

]

= [119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω


(61)





119864T119883 = 119883

T119864 ge 0

[119860

T119883 + 119883

T119860 minus 119862

T119876119862 119883

T119861 minus Ω


(62)



following

[

[

119882

119881

119880

]

]

T

[

[

0 119875 0



]

]

[

[

119882

119881

119880

]

]

= [119881

119880]

T[

[

119860T119883 + 119883

T119860

minus119862T119876119862

119883T119861 minus Ω

lowast minusΠ

]

]

[119881

119880] le 0

(63)





119864T119883119864 ge 0

[119860

T119883119864 + 119864

T119883119860 minus 119862

T119876119862 119864

T119883

T119861 minus Ω


(64)



[0 119868

0 0]


= [

[

11

0 0

0 0 0

0 0 33

]

]

(65)



11=

T11ge 0

[119860

T111+ 111198601minus 119862

T1119876119862111198611minus Ξ1

lowast minusΞ2

] le 0

(66)

where

Ξ1= 119862

T1(119878 + 119876 (119863 minus 119862

21198612minus 11986231198613))

Ξ2= 119877 + (119863 minus 119862

21198612minus 11986231198613)T119876(119863 minus 119862

21198612minus 11986231198613)

+ (119863 minus 11986221198612minus 11986231198613)T119878 + 119878

T(119863 minus 119862

21198612minus 11986231198613)

(67)



333= minus119878

T1198622


(119864 119860 119861 119862)

= ([

[

119868 0 0

0 0 119868

0 0 0

]

]

[

[

11986010 0

0 119868 0

0 0 119868

]

]

[

[

1198611

1198612

1198613

]

]

[119862111986211198623])

(68)


= [

[

11

12

13

T12

22

23

T13

T23

33

]

]

(69)


22isin R1198972times1198972

33isin R1198973times1198973 and 119897

2+ 1198973= 1198992


11987511= 11

11987512= [12

13] = 0

11987522119873 = [

22

23

T23

33

] [0 119868

0 0] = 0

119861T211987522= [

1198612

1198613

]

T

[22

23

T23

33

]

= 119878T1198622119873 = 119878

T[11986221198623] [0 119868

0 0] = 0

(70)


13= 0

13= 0

23= 0

22= 0 119861

T333= minus119878

T1198622

(71)


33


T1198622



984858T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

984858 le 0 (72)

for all 984858 isin 120596 = im [119882

119881

119880


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

le 0 (73)




119881

119880

]


119875 = sdpvar (119899 119899) 119865 = set (119875 ge 0) (74)

And let

119865 = 119865 + 119884T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

119884 ge 0 (75)





119872 + 119863 + 119870119911 = 119871119891 + 119869120583

119866 + 119867119911 = 0

(76)





119910 = 119862119901119911 + 119862V (77)


k1 k2

m1m2

d

f

z1 z2

Dashpot

Rigid connection



119909 = (

119911

120583

) 119906 = 119891 (78)


1= 1119898

2= 1 119889 = 1

1198961= 2 and 119896



119864 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(79)

where

119864 =

[[[[[

[

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

]]]]]

]

119860 =

[[[[[

[

0 0 1 0 0

0 0 0 1 0



1 1 0 0 0

]]]]]

]

119861 =

[[[[[

[

0

0

minus1

1

0

]]]]]

]

119862 = [1 1 0 0 0]

(80)


119875 =

[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00001 00149

0 0 0 00149 82588

]]]]]

]

(81)



1+ 1+ 2+ 21199111= minus119891 + 120583

2+ 1+ 2+ 1199112= 119891 + 120583

(82)



1+ 21199111= minus119891 + 120583

2+ 1199112= 119891 + 120583

(83)


2

+ (12)1198701199112 since

119881 [119911 (1199051) (119905

1)] le 119881 [119911 (119905

0) (119905

0)] + int

1199051

1199050

119865 (119905) 120592 (119905) 119889119905

(84)





119873 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(85)


where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)


119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)


119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)


119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)





6 Conclusions






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120575 (119905) = 0 119905 = 0

int

infin

minusinfin

120575 (119905) 119889119905 = 1

int

infin

minusinfin

119891 (119905) 120575 (119905) 119889119905 = 119891 (0)

(14)


119892120576(119909) =

1

1205762(119909 + 120576) minus120576 le 119909 le 0

1

1205762(120576 minus 119909) 0 le 119909 le 120576

(15)





int

infin

minusinfin

119892120576(119909) 119889119909 =

2

1205762int

120576

0

(120576 minus 119909) 119889119909 =2

1205762[119909120576 minus

1199092

2]

120576

0

= 1

(16)


int

infin

minusinfin

1198922

120576(119909) 119889119909

=2

1205764int

120576

0

(120576 minus 119909)2

119889119909 =2

1205764[1199091205762

minus 1199092

120576 +1199093

3]

120576

0

=2

3120576

(17)




120576(119909) is Lebesgue





where

120578 =

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

Γ =

[[[[

[

Γ0

0 sdot sdot sdot 0


d


ℎminus1

]]]]

]

(18)

then we have

Γ119894= 0 119894 = 1 2 ℎ minus 1 (19)









proof




119881 (119909 (1199051)) le 119881 (119909 (119905

0)) + int

1199051

1199050

119903 (119906 (119904) 119910 (119904)) 119889119904 (20)

for all 1199050 1199051with 119905




119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119910

T(119905) 119878119906 (119905) + 119906

T(119905) 119877119906 (119905)

(21)







1205742




infinperformance



Π = 119877 + 119863T119878 + 119878

T119863 + 119863

T119876119863

Ω = 119862T119878 + 119862

T119876119863

(22)




1198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (23)

1198781198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (24)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119910

T(119905) 119878119906 (119905) + 119906

T(119905) 119877119906 (119905)

= 119909T1(119905) 119862

T111987611986211199091(119905) + 119909

T2(119905) 119862

T211987611986221199092(119905)

+ 2119909T1(119905) 119862

T111987611986221199092(119905) + 119906

T(119905) Π119906 (119905)

+ 2119909T2(119905) 119862

T2(119876119863 + 119878) 119906 (119905)

+ 2119909T1(119905) 119862

T1(119876119863 + 119878) 119906 (119905)

(25)


ℎminus1

119894=0119873119894

1198612119906(119894)

(119905) wehave119903 (119906 (119905) 119910 (119905))

= [

[

1199091(119905)

119906 (119905)

(119905)

]

]

T

[

[

119862T11198761198621Ξ1

0

lowast Ξ2119878T11986221198731198612

lowast lowast 0

]

]

[

[

1199091(119905)

119906 (119905)

(119905)

]

]

Ξ1= 119862

T1119878 + 119862

T1119876 (119863 minus 119862

21198612)

Ξ2= 119877 + (119863 minus 119862

21198612)T119876 (119863 minus 119862

21198612) + (119863 minus 119862

21198612)T119878

+ 119878T(119863 minus 119862

21198612)

(26)



ℎminus1

119901


119909T2(119905) 119862

T211987611986221199092(119905)

= (minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

T

119862T21198761198622(minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

=

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

T

Γ

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

(27)


Γ =

[[[[[[[[[[

[

119861T2119862T211987611986221198612

119861T2119862T211987611987311986221198612


1198612

119861T2119873

T119862T211987611986221198612

119861T2119873

T119862T211987611986221198731198612sdot sdot sdot 119861

T2119873

T119862T21198761198622119873ℎminus1

1198612

d

119861T2(119873ℎminus1

)T119862T211987611986221198612


)T119862T21198761198622119873ℎminus1

1198612

]]]]]]]]]]

]

(28)


119861T2119873119894T119862T21198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (29)


1198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (30)


119906T(119905) 119878

T11986221199092(119905) = minus119906

T(119905) 119878

T1198622(

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

= minus

ℎminus1

sum

119894=0

119906T(119905) 119878

T1198622119873119894

1198612119906(119894)

(119905)

(31)



119878T1198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (32)










119875 = 119875Tge 0 (33)

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

120596

le 0 (34)




119881 (119909) = 119909T(119905) 119875119909 (119905) (35)



int

1199051

1199050

119889119881 (119905)

119889119905119889119904 le int

1199051

1199050

119903 (119906 (119904) 119910 (119904)) 119889119904 (36)


derive

119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (37)

Let

119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119876119910 (119905) + 119906

T(119905) 119876119906 (119905)

(38)

Then we can have

119889119881 (119905)

119889119905minus 119910

T(119905) 119876119910 (119905) minus 2119906

T(119905) 119876119910 (119905) minus 119906

T(119905) 119876119906 (119905)

= T(119905) 119875119909 (119905) + 119909

T(119905) 119875 (119905) minus 119909

T(119905) 119862

T119876119862119909 (119905)

minus 2119909T(119905) (119862

T119878 + 119862

T119876119863)119906 (119905) minus 119906

T(119905) Π119906 (119905)

= [

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0

(39)


[

[

(1)

119909 (1)

119906 (1)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(1)

119909 (1)

119906 (1)

]

]

le 0 (40)



[

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0 (41)


119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (42)





[[[

[

1(119905)

2(119905)

1199091(119905)

1199092(119905)

]]]

]

=

[[[

[

1198601

0

119868

0

]]]

]

1199091(119905) (43)


119881 (119909 (119905)) = 119909T(119905) 119875119909 (119905) (44)

with

119875 = [11987511

11987512

119875T12

11987522

] gt 0 (45)



[[[

[

1198601

0

119868

0

]]]

]

T

[[[

[

0 0 11987511

11987512

0 0 119875T12

11987522

11987511

11987512

0 0

119875T12

11987522

0 0

]]]

]

[[[

[

1198601

0

119868

0

]]]

]

= 119860T111987511+ 119875

T111198601lt 0

(46)




119875 = [11987511

0

0 11987522

] (47)

where 11987511

isin R1198991times1198991 and 11987522



11987511= 119875

T11ge 0

[119860

T111987511+ 119875111198601minus 119862

T1119876119862111987511198611minus Ξ1

lowast minusΞ2

] le 0

(48)


11987522= 119875

T22ge 0

11987522[119873

1198612

] = [0

minus119873T119862T2119878]

(49)


119875 = [11987511

11987512

119875T12

11987522

] = 119875Tge 0 (50)


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

= [

[

11987211

11987212

11987213

lowast 11987222

11987223


]

]

(51)

where

11987211= 119860

T111987511+ 119875

T111198601minus 119862

T11198761198621

11987212= 119875111198611minus 119860

T1119875121198612+ 119862

T111987611986221198612minus 119862

T1119878 minus 119862

T1119876119863

11987213= 11987512minus 119860

T111987512119873 minus 119862

T111987611986221198731198612

11987222= minus119861

T2119875T121198611minus 119861

T1119875121198612+ (119878

T+ 119863

T119876)11986221198612

+ 119861T2119862T2(119878 + 119876119863) minus 119861

T2119862T211987611986221198612+ Π

11987223= minus119861

T211987522+ 119861

T111987512119873 + 119861

T2119862T21198761198622119873

minus 119878T1198622119873 minus 119863

T1198761198622119873

11987233= 119873

T11987522+ 11987522119873 minus119873

T119862T21198761198622119873

(52)




11987522+11987522119873 = 119873

T119862T21198761198622119873


119873T11987522119873 = 0 minus119873

T119862T21198761198622119873 = 0 (53)


1198761198622119873 = 0 Since119872



11987213= 0 119872

23= 0 (54)


11987512= 0 119861

T211987522= minus119878

T1198622119873 (55)


[[[

[

119860T111987511+ 119875

T111198601minus 119862

T1119876119862111987212

0 0

lowast 11987222

0 0

0 0 0 0

0 0 0 0

]]]

]

le 0 (56)


the proof


119881 (119909) = 119909T119875119909 = 119909

T1119875111199091+ 119909

T1119875121199092+ 119909

T2119875T121199091+ 119909

T2119875221199092

= 119909T1119875111199091+ 119906

T119861T2119875221198612119906

(57)





119881 (119909 (1199051)) minus 119881 (119909 (119905

0)) = int

1199051

1199050

119889119881 (119905)

119889119905119889119905 (58)






120596 = im[

[

119882

119881

119880

]

]

= im[

[

119860 119861

119868 0

0 119868

]

]

(59)





[119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω



[

[

119860 119861

119868 0

0 119868

]

]

T

[

[

0 119875 0



]

]

[

[

119860 119861

119868 0

0 119868

]

]

= [119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω


(61)





119864T119883 = 119883

T119864 ge 0

[119860

T119883 + 119883

T119860 minus 119862

T119876119862 119883

T119861 minus Ω


(62)



following

[

[

119882

119881

119880

]

]

T

[

[

0 119875 0



]

]

[

[

119882

119881

119880

]

]

= [119881

119880]

T[

[

119860T119883 + 119883

T119860

minus119862T119876119862

119883T119861 minus Ω

lowast minusΠ

]

]

[119881

119880] le 0

(63)





119864T119883119864 ge 0

[119860

T119883119864 + 119864

T119883119860 minus 119862

T119876119862 119864

T119883

T119861 minus Ω


(64)



[0 119868

0 0]


= [

[

11

0 0

0 0 0

0 0 33

]

]

(65)



11=

T11ge 0

[119860

T111+ 111198601minus 119862

T1119876119862111198611minus Ξ1

lowast minusΞ2

] le 0

(66)

where

Ξ1= 119862

T1(119878 + 119876 (119863 minus 119862

21198612minus 11986231198613))

Ξ2= 119877 + (119863 minus 119862

21198612minus 11986231198613)T119876(119863 minus 119862

21198612minus 11986231198613)

+ (119863 minus 11986221198612minus 11986231198613)T119878 + 119878

T(119863 minus 119862

21198612minus 11986231198613)

(67)



333= minus119878

T1198622


(119864 119860 119861 119862)

= ([

[

119868 0 0

0 0 119868

0 0 0

]

]

[

[

11986010 0

0 119868 0

0 0 119868

]

]

[

[

1198611

1198612

1198613

]

]

[119862111986211198623])

(68)


= [

[

11

12

13

T12

22

23

T13

T23

33

]

]

(69)


22isin R1198972times1198972

33isin R1198973times1198973 and 119897

2+ 1198973= 1198992


11987511= 11

11987512= [12

13] = 0

11987522119873 = [

22

23

T23

33

] [0 119868

0 0] = 0

119861T211987522= [

1198612

1198613

]

T

[22

23

T23

33

]

= 119878T1198622119873 = 119878

T[11986221198623] [0 119868

0 0] = 0

(70)


13= 0

13= 0

23= 0

22= 0 119861

T333= minus119878

T1198622

(71)


33


T1198622



984858T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

984858 le 0 (72)

for all 984858 isin 120596 = im [119882

119881

119880


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

le 0 (73)




119881

119880

]


119875 = sdpvar (119899 119899) 119865 = set (119875 ge 0) (74)

And let

119865 = 119865 + 119884T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

119884 ge 0 (75)





119872 + 119863 + 119870119911 = 119871119891 + 119869120583

119866 + 119867119911 = 0

(76)





119910 = 119862119901119911 + 119862V (77)


k1 k2

m1m2

d

f

z1 z2

Dashpot

Rigid connection



119909 = (

119911

120583

) 119906 = 119891 (78)


1= 1119898

2= 1 119889 = 1

1198961= 2 and 119896



119864 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(79)

where

119864 =

[[[[[

[

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

]]]]]

]

119860 =

[[[[[

[

0 0 1 0 0

0 0 0 1 0



1 1 0 0 0

]]]]]

]

119861 =

[[[[[

[

0

0

minus1

1

0

]]]]]

]

119862 = [1 1 0 0 0]

(80)


119875 =

[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00001 00149

0 0 0 00149 82588

]]]]]

]

(81)



1+ 1+ 2+ 21199111= minus119891 + 120583

2+ 1+ 2+ 1199112= 119891 + 120583

(82)



1+ 21199111= minus119891 + 120583

2+ 1199112= 119891 + 120583

(83)


2

+ (12)1198701199112 since

119881 [119911 (1199051) (119905

1)] le 119881 [119911 (119905

0) (119905

0)] + int

1199051

1199050

119865 (119905) 120592 (119905) 119889119905

(84)





119873 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(85)


where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)


119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)


119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)


119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)





6 Conclusions






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205742




infinperformance



Π = 119877 + 119863T119878 + 119878

T119863 + 119863

T119876119863

Ω = 119862T119878 + 119862

T119876119863

(22)




1198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (23)

1198781198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (24)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119910

T(119905) 119878119906 (119905) + 119906

T(119905) 119877119906 (119905)

= 119909T1(119905) 119862

T111987611986211199091(119905) + 119909

T2(119905) 119862

T211987611986221199092(119905)

+ 2119909T1(119905) 119862

T111987611986221199092(119905) + 119906

T(119905) Π119906 (119905)

+ 2119909T2(119905) 119862

T2(119876119863 + 119878) 119906 (119905)

+ 2119909T1(119905) 119862

T1(119876119863 + 119878) 119906 (119905)

(25)


ℎminus1

119894=0119873119894

1198612119906(119894)

(119905) wehave119903 (119906 (119905) 119910 (119905))

= [

[

1199091(119905)

119906 (119905)

(119905)

]

]

T

[

[

119862T11198761198621Ξ1

0

lowast Ξ2119878T11986221198731198612

lowast lowast 0

]

]

[

[

1199091(119905)

119906 (119905)

(119905)

]

]

Ξ1= 119862

T1119878 + 119862

T1119876 (119863 minus 119862

21198612)

Ξ2= 119877 + (119863 minus 119862

21198612)T119876 (119863 minus 119862

21198612) + (119863 minus 119862

21198612)T119878

+ 119878T(119863 minus 119862

21198612)

(26)



ℎminus1

119901


119909T2(119905) 119862

T211987611986221199092(119905)

= (minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

T

119862T21198761198622(minus

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

=

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

T

Γ

[[[[

[

119906 (119905)

(119905)

119906(ℎminus1)

(119905)

]]]]

]

(27)


Γ =

[[[[[[[[[[

[

119861T2119862T211987611986221198612

119861T2119862T211987611987311986221198612


1198612

119861T2119873

T119862T211987611986221198612

119861T2119873

T119862T211987611986221198731198612sdot sdot sdot 119861

T2119873

T119862T21198761198622119873ℎminus1

1198612

d

119861T2(119873ℎminus1

)T119862T211987611986221198612


)T119862T21198761198622119873ℎminus1

1198612

]]]]]]]]]]

]

(28)


119861T2119873119894T119862T21198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (29)


1198761198622119873119894

1198612= 0 119894 = 1 2 ℎ minus 1 (30)


119906T(119905) 119878

T11986221199092(119905) = minus119906

T(119905) 119878

T1198622(

ℎminus1

sum

119894=0

119873119894

1198612119906(119894)

(119905))

= minus

ℎminus1

sum

119894=0

119906T(119905) 119878

T1198622119873119894

1198612119906(119894)

(119905)

(31)



119878T1198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (32)










119875 = 119875Tge 0 (33)

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

120596

le 0 (34)




119881 (119909) = 119909T(119905) 119875119909 (119905) (35)



int

1199051

1199050

119889119881 (119905)

119889119905119889119904 le int

1199051

1199050

119903 (119906 (119904) 119910 (119904)) 119889119904 (36)


derive

119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (37)

Let

119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119876119910 (119905) + 119906

T(119905) 119876119906 (119905)

(38)

Then we can have

119889119881 (119905)

119889119905minus 119910

T(119905) 119876119910 (119905) minus 2119906

T(119905) 119876119910 (119905) minus 119906

T(119905) 119876119906 (119905)

= T(119905) 119875119909 (119905) + 119909

T(119905) 119875 (119905) minus 119909

T(119905) 119862

T119876119862119909 (119905)

minus 2119909T(119905) (119862

T119878 + 119862

T119876119863)119906 (119905) minus 119906

T(119905) Π119906 (119905)

= [

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0

(39)


[

[

(1)

119909 (1)

119906 (1)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(1)

119909 (1)

119906 (1)

]

]

le 0 (40)



[

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0 (41)


119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (42)





[[[

[

1(119905)

2(119905)

1199091(119905)

1199092(119905)

]]]

]

=

[[[

[

1198601

0

119868

0

]]]

]

1199091(119905) (43)


119881 (119909 (119905)) = 119909T(119905) 119875119909 (119905) (44)

with

119875 = [11987511

11987512

119875T12

11987522

] gt 0 (45)



[[[

[

1198601

0

119868

0

]]]

]

T

[[[

[

0 0 11987511

11987512

0 0 119875T12

11987522

11987511

11987512

0 0

119875T12

11987522

0 0

]]]

]

[[[

[

1198601

0

119868

0

]]]

]

= 119860T111987511+ 119875

T111198601lt 0

(46)




119875 = [11987511

0

0 11987522

] (47)

where 11987511

isin R1198991times1198991 and 11987522



11987511= 119875

T11ge 0

[119860

T111987511+ 119875111198601minus 119862

T1119876119862111987511198611minus Ξ1

lowast minusΞ2

] le 0

(48)


11987522= 119875

T22ge 0

11987522[119873

1198612

] = [0

minus119873T119862T2119878]

(49)


119875 = [11987511

11987512

119875T12

11987522

] = 119875Tge 0 (50)


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

= [

[

11987211

11987212

11987213

lowast 11987222

11987223


]

]

(51)

where

11987211= 119860

T111987511+ 119875

T111198601minus 119862

T11198761198621

11987212= 119875111198611minus 119860

T1119875121198612+ 119862

T111987611986221198612minus 119862

T1119878 minus 119862

T1119876119863

11987213= 11987512minus 119860

T111987512119873 minus 119862

T111987611986221198731198612

11987222= minus119861

T2119875T121198611minus 119861

T1119875121198612+ (119878

T+ 119863

T119876)11986221198612

+ 119861T2119862T2(119878 + 119876119863) minus 119861

T2119862T211987611986221198612+ Π

11987223= minus119861

T211987522+ 119861

T111987512119873 + 119861

T2119862T21198761198622119873

minus 119878T1198622119873 minus 119863

T1198761198622119873

11987233= 119873

T11987522+ 11987522119873 minus119873

T119862T21198761198622119873

(52)




11987522+11987522119873 = 119873

T119862T21198761198622119873


119873T11987522119873 = 0 minus119873

T119862T21198761198622119873 = 0 (53)


1198761198622119873 = 0 Since119872



11987213= 0 119872

23= 0 (54)


11987512= 0 119861

T211987522= minus119878

T1198622119873 (55)


[[[

[

119860T111987511+ 119875

T111198601minus 119862

T1119876119862111987212

0 0

lowast 11987222

0 0

0 0 0 0

0 0 0 0

]]]

]

le 0 (56)


the proof


119881 (119909) = 119909T119875119909 = 119909

T1119875111199091+ 119909

T1119875121199092+ 119909

T2119875T121199091+ 119909

T2119875221199092

= 119909T1119875111199091+ 119906

T119861T2119875221198612119906

(57)





119881 (119909 (1199051)) minus 119881 (119909 (119905

0)) = int

1199051

1199050

119889119881 (119905)

119889119905119889119905 (58)






120596 = im[

[

119882

119881

119880

]

]

= im[

[

119860 119861

119868 0

0 119868

]

]

(59)





[119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω



[

[

119860 119861

119868 0

0 119868

]

]

T

[

[

0 119875 0



]

]

[

[

119860 119861

119868 0

0 119868

]

]

= [119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω


(61)





119864T119883 = 119883

T119864 ge 0

[119860

T119883 + 119883

T119860 minus 119862

T119876119862 119883

T119861 minus Ω


(62)



following

[

[

119882

119881

119880

]

]

T

[

[

0 119875 0



]

]

[

[

119882

119881

119880

]

]

= [119881

119880]

T[

[

119860T119883 + 119883

T119860

minus119862T119876119862

119883T119861 minus Ω

lowast minusΠ

]

]

[119881

119880] le 0

(63)





119864T119883119864 ge 0

[119860

T119883119864 + 119864

T119883119860 minus 119862

T119876119862 119864

T119883

T119861 minus Ω


(64)



[0 119868

0 0]


= [

[

11

0 0

0 0 0

0 0 33

]

]

(65)



11=

T11ge 0

[119860

T111+ 111198601minus 119862

T1119876119862111198611minus Ξ1

lowast minusΞ2

] le 0

(66)

where

Ξ1= 119862

T1(119878 + 119876 (119863 minus 119862

21198612minus 11986231198613))

Ξ2= 119877 + (119863 minus 119862

21198612minus 11986231198613)T119876(119863 minus 119862

21198612minus 11986231198613)

+ (119863 minus 11986221198612minus 11986231198613)T119878 + 119878

T(119863 minus 119862

21198612minus 11986231198613)

(67)



333= minus119878

T1198622


(119864 119860 119861 119862)

= ([

[

119868 0 0

0 0 119868

0 0 0

]

]

[

[

11986010 0

0 119868 0

0 0 119868

]

]

[

[

1198611

1198612

1198613

]

]

[119862111986211198623])

(68)


= [

[

11

12

13

T12

22

23

T13

T23

33

]

]

(69)


22isin R1198972times1198972

33isin R1198973times1198973 and 119897

2+ 1198973= 1198992


11987511= 11

11987512= [12

13] = 0

11987522119873 = [

22

23

T23

33

] [0 119868

0 0] = 0

119861T211987522= [

1198612

1198613

]

T

[22

23

T23

33

]

= 119878T1198622119873 = 119878

T[11986221198623] [0 119868

0 0] = 0

(70)


13= 0

13= 0

23= 0

22= 0 119861

T333= minus119878

T1198622

(71)


33


T1198622



984858T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

984858 le 0 (72)

for all 984858 isin 120596 = im [119882

119881

119880


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

le 0 (73)




119881

119880

]


119875 = sdpvar (119899 119899) 119865 = set (119875 ge 0) (74)

And let

119865 = 119865 + 119884T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

119884 ge 0 (75)





119872 + 119863 + 119870119911 = 119871119891 + 119869120583

119866 + 119867119911 = 0

(76)





119910 = 119862119901119911 + 119862V (77)


k1 k2

m1m2

d

f

z1 z2

Dashpot

Rigid connection



119909 = (

119911

120583

) 119906 = 119891 (78)


1= 1119898

2= 1 119889 = 1

1198961= 2 and 119896



119864 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(79)

where

119864 =

[[[[[

[

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

]]]]]

]

119860 =

[[[[[

[

0 0 1 0 0

0 0 0 1 0



1 1 0 0 0

]]]]]

]

119861 =

[[[[[

[

0

0

minus1

1

0

]]]]]

]

119862 = [1 1 0 0 0]

(80)


119875 =

[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00001 00149

0 0 0 00149 82588

]]]]]

]

(81)



1+ 1+ 2+ 21199111= minus119891 + 120583

2+ 1+ 2+ 1199112= 119891 + 120583

(82)



1+ 21199111= minus119891 + 120583

2+ 1199112= 119891 + 120583

(83)


2

+ (12)1198701199112 since

119881 [119911 (1199051) (119905

1)] le 119881 [119911 (119905

0) (119905

0)] + int

1199051

1199050

119865 (119905) 120592 (119905) 119889119905

(84)





119873 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(85)


where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)


119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)


119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)


119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)





6 Conclusions






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119878T1198622119873119894

1198612= 0 119894 = 2 3 ℎ minus 1 (32)










119875 = 119875Tge 0 (33)

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

120596

le 0 (34)




119881 (119909) = 119909T(119905) 119875119909 (119905) (35)



int

1199051

1199050

119889119881 (119905)

119889119905119889119904 le int

1199051

1199050

119903 (119906 (119904) 119910 (119904)) 119889119904 (36)


derive

119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (37)

Let

119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119876119910 (119905) + 119906

T(119905) 119876119906 (119905)

(38)

Then we can have

119889119881 (119905)

119889119905minus 119910

T(119905) 119876119910 (119905) minus 2119906

T(119905) 119876119910 (119905) minus 119906

T(119905) 119876119906 (119905)

= T(119905) 119875119909 (119905) + 119909

T(119905) 119875 (119905) minus 119909

T(119905) 119862

T119876119862119909 (119905)

minus 2119909T(119905) (119862

T119878 + 119862

T119876119863)119906 (119905) minus 119906

T(119905) Π119906 (119905)

= [

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0

(39)


[

[

(1)

119909 (1)

119906 (1)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(1)

119909 (1)

119906 (1)

]

]

le 0 (40)



[

[

(119905)

119909 (119905)

119906 (119905)

]

]

T

[

[

0 119875 0

lowast minus119862T119876119862 Ω


]

]

[

[

(119905)

119909 (119905)

119906 (119905)

]

]

le 0 (41)


119903 (119906 (119905) 119910 (119905)) ge119889119881 (119905)

119889119905 (42)





[[[

[

1(119905)

2(119905)

1199091(119905)

1199092(119905)

]]]

]

=

[[[

[

1198601

0

119868

0

]]]

]

1199091(119905) (43)


119881 (119909 (119905)) = 119909T(119905) 119875119909 (119905) (44)

with

119875 = [11987511

11987512

119875T12

11987522

] gt 0 (45)



[[[

[

1198601

0

119868

0

]]]

]

T

[[[

[

0 0 11987511

11987512

0 0 119875T12

11987522

11987511

11987512

0 0

119875T12

11987522

0 0

]]]

]

[[[

[

1198601

0

119868

0

]]]

]

= 119860T111987511+ 119875

T111198601lt 0

(46)




119875 = [11987511

0

0 11987522

] (47)

where 11987511

isin R1198991times1198991 and 11987522



11987511= 119875

T11ge 0

[119860

T111987511+ 119875111198601minus 119862

T1119876119862111987511198611minus Ξ1

lowast minusΞ2

] le 0

(48)


11987522= 119875

T22ge 0

11987522[119873

1198612

] = [0

minus119873T119862T2119878]

(49)


119875 = [11987511

11987512

119875T12

11987522

] = 119875Tge 0 (50)


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

= [

[

11987211

11987212

11987213

lowast 11987222

11987223


]

]

(51)

where

11987211= 119860

T111987511+ 119875

T111198601minus 119862

T11198761198621

11987212= 119875111198611minus 119860

T1119875121198612+ 119862

T111987611986221198612minus 119862

T1119878 minus 119862

T1119876119863

11987213= 11987512minus 119860

T111987512119873 minus 119862

T111987611986221198731198612

11987222= minus119861

T2119875T121198611minus 119861

T1119875121198612+ (119878

T+ 119863

T119876)11986221198612

+ 119861T2119862T2(119878 + 119876119863) minus 119861

T2119862T211987611986221198612+ Π

11987223= minus119861

T211987522+ 119861

T111987512119873 + 119861

T2119862T21198761198622119873

minus 119878T1198622119873 minus 119863

T1198761198622119873

11987233= 119873

T11987522+ 11987522119873 minus119873

T119862T21198761198622119873

(52)




11987522+11987522119873 = 119873

T119862T21198761198622119873


119873T11987522119873 = 0 minus119873

T119862T21198761198622119873 = 0 (53)


1198761198622119873 = 0 Since119872



11987213= 0 119872

23= 0 (54)


11987512= 0 119861

T211987522= minus119878

T1198622119873 (55)


[[[

[

119860T111987511+ 119875

T111198601minus 119862

T1119876119862111987212

0 0

lowast 11987222

0 0

0 0 0 0

0 0 0 0

]]]

]

le 0 (56)


the proof


119881 (119909) = 119909T119875119909 = 119909

T1119875111199091+ 119909

T1119875121199092+ 119909

T2119875T121199091+ 119909

T2119875221199092

= 119909T1119875111199091+ 119906

T119861T2119875221198612119906

(57)





119881 (119909 (1199051)) minus 119881 (119909 (119905

0)) = int

1199051

1199050

119889119881 (119905)

119889119905119889119905 (58)






120596 = im[

[

119882

119881

119880

]

]

= im[

[

119860 119861

119868 0

0 119868

]

]

(59)





[119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω



[

[

119860 119861

119868 0

0 119868

]

]

T

[

[

0 119875 0



]

]

[

[

119860 119861

119868 0

0 119868

]

]

= [119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω


(61)





119864T119883 = 119883

T119864 ge 0

[119860

T119883 + 119883

T119860 minus 119862

T119876119862 119883

T119861 minus Ω


(62)



following

[

[

119882

119881

119880

]

]

T

[

[

0 119875 0



]

]

[

[

119882

119881

119880

]

]

= [119881

119880]

T[

[

119860T119883 + 119883

T119860

minus119862T119876119862

119883T119861 minus Ω

lowast minusΠ

]

]

[119881

119880] le 0

(63)





119864T119883119864 ge 0

[119860

T119883119864 + 119864

T119883119860 minus 119862

T119876119862 119864

T119883

T119861 minus Ω


(64)



[0 119868

0 0]


= [

[

11

0 0

0 0 0

0 0 33

]

]

(65)



11=

T11ge 0

[119860

T111+ 111198601minus 119862

T1119876119862111198611minus Ξ1

lowast minusΞ2

] le 0

(66)

where

Ξ1= 119862

T1(119878 + 119876 (119863 minus 119862

21198612minus 11986231198613))

Ξ2= 119877 + (119863 minus 119862

21198612minus 11986231198613)T119876(119863 minus 119862

21198612minus 11986231198613)

+ (119863 minus 11986221198612minus 11986231198613)T119878 + 119878

T(119863 minus 119862

21198612minus 11986231198613)

(67)



333= minus119878

T1198622


(119864 119860 119861 119862)

= ([

[

119868 0 0

0 0 119868

0 0 0

]

]

[

[

11986010 0

0 119868 0

0 0 119868

]

]

[

[

1198611

1198612

1198613

]

]

[119862111986211198623])

(68)


= [

[

11

12

13

T12

22

23

T13

T23

33

]

]

(69)


22isin R1198972times1198972

33isin R1198973times1198973 and 119897

2+ 1198973= 1198992


11987511= 11

11987512= [12

13] = 0

11987522119873 = [

22

23

T23

33

] [0 119868

0 0] = 0

119861T211987522= [

1198612

1198613

]

T

[22

23

T23

33

]

= 119878T1198622119873 = 119878

T[11986221198623] [0 119868

0 0] = 0

(70)


13= 0

13= 0

23= 0

22= 0 119861

T333= minus119878

T1198622

(71)


33


T1198622



984858T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

984858 le 0 (72)

for all 984858 isin 120596 = im [119882

119881

119880


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

le 0 (73)




119881

119880

]


119875 = sdpvar (119899 119899) 119865 = set (119875 ge 0) (74)

And let

119865 = 119865 + 119884T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

119884 ge 0 (75)





119872 + 119863 + 119870119911 = 119871119891 + 119869120583

119866 + 119867119911 = 0

(76)





119910 = 119862119901119911 + 119862V (77)


k1 k2

m1m2

d

f

z1 z2

Dashpot

Rigid connection



119909 = (

119911

120583

) 119906 = 119891 (78)


1= 1119898

2= 1 119889 = 1

1198961= 2 and 119896



119864 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(79)

where

119864 =

[[[[[

[

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

]]]]]

]

119860 =

[[[[[

[

0 0 1 0 0

0 0 0 1 0



1 1 0 0 0

]]]]]

]

119861 =

[[[[[

[

0

0

minus1

1

0

]]]]]

]

119862 = [1 1 0 0 0]

(80)


119875 =

[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00001 00149

0 0 0 00149 82588

]]]]]

]

(81)



1+ 1+ 2+ 21199111= minus119891 + 120583

2+ 1+ 2+ 1199112= 119891 + 120583

(82)



1+ 21199111= minus119891 + 120583

2+ 1199112= 119891 + 120583

(83)


2

+ (12)1198701199112 since

119881 [119911 (1199051) (119905

1)] le 119881 [119911 (119905

0) (119905

0)] + int

1199051

1199050

119865 (119905) 120592 (119905) 119889119905

(84)





119873 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(85)


where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)


119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)


119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)


119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)





6 Conclusions






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[[[

[

1198601

0

119868

0

]]]

]

T

[[[

[

0 0 11987511

11987512

0 0 119875T12

11987522

11987511

11987512

0 0

119875T12

11987522

0 0

]]]

]

[[[

[

1198601

0

119868

0

]]]

]

= 119860T111987511+ 119875

T111198601lt 0

(46)




119875 = [11987511

0

0 11987522

] (47)

where 11987511

isin R1198991times1198991 and 11987522



11987511= 119875

T11ge 0

[119860

T111987511+ 119875111198601minus 119862

T1119876119862111987511198611minus Ξ1

lowast minusΞ2

] le 0

(48)


11987522= 119875

T22ge 0

11987522[119873

1198612

] = [0

minus119873T119862T2119878]

(49)


119875 = [11987511

11987512

119875T12

11987522

] = 119875Tge 0 (50)


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

= [

[

11987211

11987212

11987213

lowast 11987222

11987223


]

]

(51)

where

11987211= 119860

T111987511+ 119875

T111198601minus 119862

T11198761198621

11987212= 119875111198611minus 119860

T1119875121198612+ 119862

T111987611986221198612minus 119862

T1119878 minus 119862

T1119876119863

11987213= 11987512minus 119860

T111987512119873 minus 119862

T111987611986221198731198612

11987222= minus119861

T2119875T121198611minus 119861

T1119875121198612+ (119878

T+ 119863

T119876)11986221198612

+ 119861T2119862T2(119878 + 119876119863) minus 119861

T2119862T211987611986221198612+ Π

11987223= minus119861

T211987522+ 119861

T111987512119873 + 119861

T2119862T21198761198622119873

minus 119878T1198622119873 minus 119863

T1198761198622119873

11987233= 119873

T11987522+ 11987522119873 minus119873

T119862T21198761198622119873

(52)




11987522+11987522119873 = 119873

T119862T21198761198622119873


119873T11987522119873 = 0 minus119873

T119862T21198761198622119873 = 0 (53)


1198761198622119873 = 0 Since119872



11987213= 0 119872

23= 0 (54)


11987512= 0 119861

T211987522= minus119878

T1198622119873 (55)


[[[

[

119860T111987511+ 119875

T111198601minus 119862

T1119876119862111987212

0 0

lowast 11987222

0 0

0 0 0 0

0 0 0 0

]]]

]

le 0 (56)


the proof


119881 (119909) = 119909T119875119909 = 119909

T1119875111199091+ 119909

T1119875121199092+ 119909

T2119875T121199091+ 119909

T2119875221199092

= 119909T1119875111199091+ 119906

T119861T2119875221198612119906

(57)





119881 (119909 (1199051)) minus 119881 (119909 (119905

0)) = int

1199051

1199050

119889119881 (119905)

119889119905119889119905 (58)






120596 = im[

[

119882

119881

119880

]

]

= im[

[

119860 119861

119868 0

0 119868

]

]

(59)





[119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω



[

[

119860 119861

119868 0

0 119868

]

]

T

[

[

0 119875 0



]

]

[

[

119860 119861

119868 0

0 119868

]

]

= [119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω


(61)





119864T119883 = 119883

T119864 ge 0

[119860

T119883 + 119883

T119860 minus 119862

T119876119862 119883

T119861 minus Ω


(62)



following

[

[

119882

119881

119880

]

]

T

[

[

0 119875 0



]

]

[

[

119882

119881

119880

]

]

= [119881

119880]

T[

[

119860T119883 + 119883

T119860

minus119862T119876119862

119883T119861 minus Ω

lowast minusΠ

]

]

[119881

119880] le 0

(63)





119864T119883119864 ge 0

[119860

T119883119864 + 119864

T119883119860 minus 119862

T119876119862 119864

T119883

T119861 minus Ω


(64)



[0 119868

0 0]


= [

[

11

0 0

0 0 0

0 0 33

]

]

(65)



11=

T11ge 0

[119860

T111+ 111198601minus 119862

T1119876119862111198611minus Ξ1

lowast minusΞ2

] le 0

(66)

where

Ξ1= 119862

T1(119878 + 119876 (119863 minus 119862

21198612minus 11986231198613))

Ξ2= 119877 + (119863 minus 119862

21198612minus 11986231198613)T119876(119863 minus 119862

21198612minus 11986231198613)

+ (119863 minus 11986221198612minus 11986231198613)T119878 + 119878

T(119863 minus 119862

21198612minus 11986231198613)

(67)



333= minus119878

T1198622


(119864 119860 119861 119862)

= ([

[

119868 0 0

0 0 119868

0 0 0

]

]

[

[

11986010 0

0 119868 0

0 0 119868

]

]

[

[

1198611

1198612

1198613

]

]

[119862111986211198623])

(68)


= [

[

11

12

13

T12

22

23

T13

T23

33

]

]

(69)


22isin R1198972times1198972

33isin R1198973times1198973 and 119897

2+ 1198973= 1198992


11987511= 11

11987512= [12

13] = 0

11987522119873 = [

22

23

T23

33

] [0 119868

0 0] = 0

119861T211987522= [

1198612

1198613

]

T

[22

23

T23

33

]

= 119878T1198622119873 = 119878

T[11986221198623] [0 119868

0 0] = 0

(70)


13= 0

13= 0

23= 0

22= 0 119861

T333= minus119878

T1198622

(71)


33


T1198622



984858T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

984858 le 0 (72)

for all 984858 isin 120596 = im [119882

119881

119880


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

le 0 (73)




119881

119880

]


119875 = sdpvar (119899 119899) 119865 = set (119875 ge 0) (74)

And let

119865 = 119865 + 119884T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

119884 ge 0 (75)





119872 + 119863 + 119870119911 = 119871119891 + 119869120583

119866 + 119867119911 = 0

(76)





119910 = 119862119901119911 + 119862V (77)


k1 k2

m1m2

d

f

z1 z2

Dashpot

Rigid connection



119909 = (

119911

120583

) 119906 = 119891 (78)


1= 1119898

2= 1 119889 = 1

1198961= 2 and 119896



119864 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(79)

where

119864 =

[[[[[

[

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

]]]]]

]

119860 =

[[[[[

[

0 0 1 0 0

0 0 0 1 0



1 1 0 0 0

]]]]]

]

119861 =

[[[[[

[

0

0

minus1

1

0

]]]]]

]

119862 = [1 1 0 0 0]

(80)


119875 =

[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00001 00149

0 0 0 00149 82588

]]]]]

]

(81)



1+ 1+ 2+ 21199111= minus119891 + 120583

2+ 1+ 2+ 1199112= 119891 + 120583

(82)



1+ 21199111= minus119891 + 120583

2+ 1199112= 119891 + 120583

(83)


2

+ (12)1198701199112 since

119881 [119911 (1199051) (119905

1)] le 119881 [119911 (119905

0) (119905

0)] + int

1199051

1199050

119865 (119905) 120592 (119905) 119889119905

(84)





119873 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(85)


where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)


119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)


119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)


119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)





6 Conclusions






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119881 (119909 (1199051)) minus 119881 (119909 (119905

0)) = int

1199051

1199050

119889119881 (119905)

119889119905119889119905 (58)






120596 = im[

[

119882

119881

119880

]

]

= im[

[

119860 119861

119868 0

0 119868

]

]

(59)





[119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω



[

[

119860 119861

119868 0

0 119868

]

]

T

[

[

0 119875 0



]

]

[

[

119860 119861

119868 0

0 119868

]

]

= [119860

T119875 + 119875119860 minus 119862

T119876119862 119875119861 minus Ω


(61)





119864T119883 = 119883

T119864 ge 0

[119860

T119883 + 119883

T119860 minus 119862

T119876119862 119883

T119861 minus Ω


(62)



following

[

[

119882

119881

119880

]

]

T

[

[

0 119875 0



]

]

[

[

119882

119881

119880

]

]

= [119881

119880]

T[

[

119860T119883 + 119883

T119860

minus119862T119876119862

119883T119861 minus Ω

lowast minusΠ

]

]

[119881

119880] le 0

(63)





119864T119883119864 ge 0

[119860

T119883119864 + 119864

T119883119860 minus 119862

T119876119862 119864

T119883

T119861 minus Ω


(64)



[0 119868

0 0]


= [

[

11

0 0

0 0 0

0 0 33

]

]

(65)



11=

T11ge 0

[119860

T111+ 111198601minus 119862

T1119876119862111198611minus Ξ1

lowast minusΞ2

] le 0

(66)

where

Ξ1= 119862

T1(119878 + 119876 (119863 minus 119862

21198612minus 11986231198613))

Ξ2= 119877 + (119863 minus 119862

21198612minus 11986231198613)T119876(119863 minus 119862

21198612minus 11986231198613)

+ (119863 minus 11986221198612minus 11986231198613)T119878 + 119878

T(119863 minus 119862

21198612minus 11986231198613)

(67)



333= minus119878

T1198622


(119864 119860 119861 119862)

= ([

[

119868 0 0

0 0 119868

0 0 0

]

]

[

[

11986010 0

0 119868 0

0 0 119868

]

]

[

[

1198611

1198612

1198613

]

]

[119862111986211198623])

(68)


= [

[

11

12

13

T12

22

23

T13

T23

33

]

]

(69)


22isin R1198972times1198972

33isin R1198973times1198973 and 119897

2+ 1198973= 1198992


11987511= 11

11987512= [12

13] = 0

11987522119873 = [

22

23

T23

33

] [0 119868

0 0] = 0

119861T211987522= [

1198612

1198613

]

T

[22

23

T23

33

]

= 119878T1198622119873 = 119878

T[11986221198623] [0 119868

0 0] = 0

(70)


13= 0

13= 0

23= 0

22= 0 119861

T333= minus119878

T1198622

(71)


33


T1198622



984858T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

984858 le 0 (72)

for all 984858 isin 120596 = im [119882

119881

119880


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

le 0 (73)




119881

119880

]


119875 = sdpvar (119899 119899) 119865 = set (119875 ge 0) (74)

And let

119865 = 119865 + 119884T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

119884 ge 0 (75)





119872 + 119863 + 119870119911 = 119871119891 + 119869120583

119866 + 119867119911 = 0

(76)





119910 = 119862119901119911 + 119862V (77)


k1 k2

m1m2

d

f

z1 z2

Dashpot

Rigid connection



119909 = (

119911

120583

) 119906 = 119891 (78)


1= 1119898

2= 1 119889 = 1

1198961= 2 and 119896



119864 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(79)

where

119864 =

[[[[[

[

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

]]]]]

]

119860 =

[[[[[

[

0 0 1 0 0

0 0 0 1 0



1 1 0 0 0

]]]]]

]

119861 =

[[[[[

[

0

0

minus1

1

0

]]]]]

]

119862 = [1 1 0 0 0]

(80)


119875 =

[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00001 00149

0 0 0 00149 82588

]]]]]

]

(81)



1+ 1+ 2+ 21199111= minus119891 + 120583

2+ 1+ 2+ 1199112= 119891 + 120583

(82)



1+ 21199111= minus119891 + 120583

2+ 1199112= 119891 + 120583

(83)


2

+ (12)1198701199112 since

119881 [119911 (1199051) (119905

1)] le 119881 [119911 (119905

0) (119905

0)] + int

1199051

1199050

119865 (119905) 120592 (119905) 119889119905

(84)





119873 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(85)


where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)


119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)


119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)


119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)





6 Conclusions






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











333= minus119878

T1198622


(119864 119860 119861 119862)

= ([

[

119868 0 0

0 0 119868

0 0 0

]

]

[

[

11986010 0

0 119868 0

0 0 119868

]

]

[

[

1198611

1198612

1198613

]

]

[119862111986211198623])

(68)


= [

[

11

12

13

T12

22

23

T13

T23

33

]

]

(69)


22isin R1198972times1198972

33isin R1198973times1198973 and 119897

2+ 1198973= 1198992


11987511= 11

11987512= [12

13] = 0

11987522119873 = [

22

23

T23

33

] [0 119868

0 0] = 0

119861T211987522= [

1198612

1198613

]

T

[22

23

T23

33

]

= 119878T1198622119873 = 119878

T[11986221198623] [0 119868

0 0] = 0

(70)


13= 0

13= 0

23= 0

22= 0 119861

T333= minus119878

T1198622

(71)


33


T1198622



984858T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

984858 le 0 (72)

for all 984858 isin 120596 = im [119882

119881

119880


[

[

119882

119881

119880

]

]

T

[

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

[

[

119882

119881

119880

]

]

le 0 (73)




119881

119880

]


119875 = sdpvar (119899 119899) 119865 = set (119875 ge 0) (74)

And let

119865 = 119865 + 119884T [

[

0 119875 0


T119878 minus 119862

T119876119863


]

]

119884 ge 0 (75)





119872 + 119863 + 119870119911 = 119871119891 + 119869120583

119866 + 119867119911 = 0

(76)





119910 = 119862119901119911 + 119862V (77)


k1 k2

m1m2

d

f

z1 z2

Dashpot

Rigid connection



119909 = (

119911

120583

) 119906 = 119891 (78)


1= 1119898

2= 1 119889 = 1

1198961= 2 and 119896



119864 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(79)

where

119864 =

[[[[[

[

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

]]]]]

]

119860 =

[[[[[

[

0 0 1 0 0

0 0 0 1 0



1 1 0 0 0

]]]]]

]

119861 =

[[[[[

[

0

0

minus1

1

0

]]]]]

]

119862 = [1 1 0 0 0]

(80)


119875 =

[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00001 00149

0 0 0 00149 82588

]]]]]

]

(81)



1+ 1+ 2+ 21199111= minus119891 + 120583

2+ 1+ 2+ 1199112= 119891 + 120583

(82)



1+ 21199111= minus119891 + 120583

2+ 1199112= 119891 + 120583

(83)


2

+ (12)1198701199112 since

119881 [119911 (1199051) (119905

1)] le 119881 [119911 (119905

0) (119905

0)] + int

1199051

1199050

119865 (119905) 120592 (119905) 119889119905

(84)





119873 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(85)


where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)


119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)


119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)


119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)





6 Conclusions






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










k1 k2

m1m2

d

f

z1 z2

Dashpot

Rigid connection



119909 = (

119911

120583

) 119906 = 119891 (78)


1= 1119898

2= 1 119889 = 1

1198961= 2 and 119896



119864 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(79)

where

119864 =

[[[[[

[

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

]]]]]

]

119860 =

[[[[[

[

0 0 1 0 0

0 0 0 1 0



1 1 0 0 0

]]]]]

]

119861 =

[[[[[

[

0

0

minus1

1

0

]]]]]

]

119862 = [1 1 0 0 0]

(80)


119875 =

[[[[[

[

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00001 00149

0 0 0 00149 82588

]]]]]

]

(81)



1+ 1+ 2+ 21199111= minus119891 + 120583

2+ 1+ 2+ 1199112= 119891 + 120583

(82)



1+ 21199111= minus119891 + 120583

2+ 1199112= 119891 + 120583

(83)


2

+ (12)1198701199112 since

119881 [119911 (1199051) (119905

1)] le 119881 [119911 (119905

0) (119905

0)] + int

1199051

1199050

119865 (119905) 120592 (119905) 119889119905

(84)





119873 (119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(85)


where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)


119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)


119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)


119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)





6 Conclusions






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119873 = [

[

0 1 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 0

0 0 1

]

]

119861 = [

[

minus1

minus1

minus1

]

]

119862 = [1 0 0

0 0 1]

(86)


119909 (119905) = minus

2

sum

119894=0

119873119894119861119906(119894)

(119905) (87)

which gives

1199091(119905) = 119906 (119905) + (119905) + (119905)

1199092(119905) = 119906 (119905) + (119905)

1199093(119905) = 119906 (119905)

(88)


119876 = [0 0

0 minus1] le 0 119878 = [

0

1] 119877 = 1 (89)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T3(119905) 1199093(119905) + 119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119906T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= 119906T(119905) 119906 (119905)

(90)


119876 = [minus1 0

0 0] le 0 119878 = [

0

1] 119877 = 1 (91)


119903 (119906 (119905) 119910 (119905)) = 119910T(119905) 119876119910 (119905) + 2119906(119905)

T119878119910 (119905) + 119906

T(119905) 119877119906 (119905)

= minus119909T(119905) 119862

T119876119862119909 (119905) + 2119909

T(119905) 119862

T119878119906 (119905)

+ 119906T(119905) 119877119906 (119905)

= minus119909T1(119905) 1199091(119905) + 2119909

T3(119905) 119906 (119905) + 119906

T(119905) 119906 (119905)

= minus119909T1(119905) 1199091(119905) + 3119906

T(119905) 119906 (119905)

(92)





6 Conclusions






Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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