research article delay-dependent finite-time filtering for...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 269091 13 pageshttpdxdoiorg1011552013269091

Research ArticleDelay-Dependent Finite-Time119867

infinFiltering for Markovian Jump

Systems with Different System Modes

Yong Zeng1 Jun Cheng1 Shouming Zhong2 and Xiucheng Dong3

1 School of Automation Engineering University of Electronic Science and Technology of China Chengdu Sichuan 611731 China2 School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu Sichuan 611731 China3 Key Laboratory on Signal and Information Processing Xihua University Chengdu Sichuan 610039 China

Correspondence should be addressed to Jun Cheng jcheng6819126com

Received 4 March 2013 Accepted 16 April 2013

Academic Editor Qiankun Song

Copyright copy 2013 Yong Zeng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with the problem of delay-dependent finite-time119867infinfiltering for Markovian jump systems with different

system modes By using the new augmented multiple mode-dependent Lyapunov-Krasovskii functional and employing theproposed integrals inequalities in the derivation of our results a novel sufficient condition for finite-time boundness with an119867

infin

performance index is derived Particularly two different Markov processes have been considered for modeling the randomness ofsystem matrix and the state delay Based on the derived condition the 119867

infinfiltering problem is solved and an explicit expression

of the desired filter is also given the system trajectory stays within a prescribed bound during a specified time interval Finally anumerical example is given to illustrate the effectiveness and the potential of the proposed techniques

1 Introduction

Markovian jump systems were introduced by Krasovskiı andLidskiı [1] which can be described by a set of systemswith the transitions in a finite mode set In the past fewdecades there has been increasing interest in Markovianjump systems because this class of systems is appropriateto many physical systems which always go with randomfailures repairs and sudden environment disturbance [2ndash5]Such class of systems is a special class of stochastic hybridsystems with finite operation modes which may switch fromone to another at different time such as component failuressudden environmental disturbance and abrupt variationsof the operating points of a nonlinear system As a crucialfactor it is shown that such jumping can be determined by aMarkovian chain [6] For linear Markovian jumping systemsmany important issues have been studied extensively such asstability stabilization control synthesis and filter design [6ndash12] In finite operation modes Markovian jump systems are aspecial class of stochastic systems that can switch from one toanother at different time

It is worth pointing out that time delay is of interest tomany researchers because of the fact that time delay is often

encountered in various systems such as networked controlsystems chemical processes and communication systems Itis worth pointing out that time delay is one of the instabil-ity sources for dynamical systems and is a common phenom-enon inmany industrial and engineering systems Hence it isnot surprising that much effort has been made to investigateMarkovian jump systems with time delay during the last twodecades [13ndash15] The exponential stabilization of Markovianjump systems with time delay was firstly studied in [16]where the decay rate was estimated by solving linear matrixinequalities [17] However in the aforementioned works thenetwork-induced delays have been commonly assumed to bedeterministic which is fairly unrealistic since delays resultingfrom network transmissions are typically time varying [18ndash24]

Generally speaking the delay-dependent criterions areless conservative than delay-independent ones especiallywhen the time delay is small enough in Markovian jumpsystems Thus recent efforts were devoted to the delay-dependent Markovian jump systems stability analysis byemploying Lyapunov-Krasovskii functionals [25ndash33] How-ever in most thesis the time delay to be arbitrarily largeare allowed in criterion it always tends to be conservative

2 Journal of Applied Mathematics

Furthermore though the decay rate can be computed it is afixed value that one cannot adjust to deduce if a larger de-cay rate is possible Therefore how to obtain the improvedresults without increasing the computational burden hasgreatly improved the current study On the other hand thepractical problems which system described does not exceeda certain threshold over some finite time interval are con-sidered In finite-time interval finite-time stability is inves-tigated to address these transient performances of controlsystems Recently the concept of finite-time stability has beenrevisited in the light of linear matrix inequalities (LMIs) andLyapunov function theory and some results are obtained toensure that systems are finite-time stability or finite-timeboundness [34ndash50] To the best of our knowledge in mostof the works about Markovian jump systems with mode-dependent delay the delay mode is always assumed to be thesame as the system matrices mode However in real systemsthe delay mode may not be the same as that for jump inother system parameters In other words variations of delayusually depend on phenomena which may not cause abruptchanges in other systems parameters Therefore the work ofMarkovian jump systems with different system modes is notonly theoretically interesting and challenging but also veryimportant in practical applications

Motivated by the previous above discussions in this pa-per we present a new augmented Lyapunov functional fora class of Markovian jump systems with different systemmodes in order to reduce the possible conservativeness andcomputational burden some slack matrices are introduced[32] Several sufficient conditions are derived to guarantee thefinite-time stability and boundedness of the resulting closed-loop system We find that finite-time stability is an inde-pendent concept from Lyapunov stability and always can beaffected by switching behavior significantly and the finite-time boundness criteria can be tackled in the form of LMIsFinally a numerical example is presented to illustrate theeffectiveness of the developed techniques

Notations Throughout this paper we let 119875 gt 0 (119875 ge 0 119875 lt0 and 119875 le 0) denote a symmetric positive definite matrix 119875(positive semidefinite negative definite and negative semi-definite) For any symmetric matrix 119875 120582max(119875) and 120582min(119875)denote the maximum andminimum eigenvalues of matrix 119875respectivelyR119899 denotes the 119899-dimensional Euclidean spaceand R119899times119898 refers to the set of all 119899 times 119898 real matrices andN = 1 2 119873 The identity matrix of order 119899 is denotedas 119868

119899 lowast represents the elements below the main diagonal of a

symmetricmatrixThe superscripts ⊺ and minus1 stand formatrixtransposition and matrix inverse respectively

2 Preliminaries

In this paper we consider the followingMarkov jump systemdescribed by

(119905) = 119860119903119905

119909 (119905) + 119860120591119903119905

119909 (119905 minus 120591119904119905

(119905)) + 119863119903119905

120596 (119905)

119910 (119905) = 119862119910119903119905

119909 (119905) + 119862119910120591119903119905

119909 (119905 minus 120591119904119905

(119905)) + 119863119910119903119905

120596 (119905)

119911 (119905) = 119862119911119903119905

119909 (119905) + 119862119911120591119903119905

119909 (119905 minus 120591119904119905

(119905)) + 119863119911119903119905

120596 (119905)

119909 (119905) = 120593 (119905) 119905 = [minusℎ 0]

(1)

where 119909(119905) isinR119899 is the state vector of the system 119910(119905) isinR119897 isthemeasured output 119911(119905) isinR119902 is the controlled output120593(119905)119905 isin [minusℎ 0] are initial conditions of continuous state and 119903

0isin

N 1199040isin M = 1 2 119872 are initial conditions of mode

120596(119905) isin R119902 is the disturbance input satisfying the followingcondition

int

infin

0

120596⊺

(119905) 120596 (119905) 119889119905 le 119889 (2)

Let the random form processes 119903119905 119904

119905be the Markov sto-

chastic processes taking values on finite sets N = 1 2

119873 and M = 1 2 119872 with probability transition ratematrices Λ = 120582

119894119895 119894 119895 isin N and Π = 120587

119898119899 119898 119899 isin M The

transition probabilities from mode 119895 to mode 119895 for Markovprocess 119903

119905and from mode 119898 to mode 119899 for the Markov

process 119904119905in time ℎ are described as

Pr (119903119905+Δ

= 119895 | 119903119905= 119894) = 984858

119894119895+ 120582

119894119895Δ + 119900 (Δ)

Pr (119904119905+Δ

= 119899 | 119904119905= 119898) = 120577

119898119899+ 120587

119898119899Δ + 119900 (Δ)

(3)

where

984858119894119895=

0 if 119894 = 119895

1 if 119894 = 119895120577119898119899=

0 if 119898 = 119899

1 if 119898 = 119899

(4)

and Δ gt 0 120582119894119895ge 0 for 119894 = 119895 is the transition rate from mode

119894 at time 119905 to mode 119895 at time 119905 + Δ and

minus120582119894119894=

119873

sum

119895=1119895 = 119894

120582119894119895 (5)

for each mode 119894 isin N limΔrarr0

+

(119900(Δ)Δ) = 0 120587119898119899

ge 0 for119898 = 119899 is the transition rate from mode 119898 to mode 119899 at time119905 + Δ and

minus120587119898119898

=

119872

sum

119899=1119898 = 119899

120587119898119899 (6)

for each mode 119894 isin M limΔrarr0

+

(119900(Δ)Δ) = 0 For conve-nience we denote the Markov process 119903

119905and 119904

119905by 119894 and

119898 indices respectively 120591119898(119905) denotes the mode-dependent

time-varying state delay in the system and satisfies thefollowing condition

0 lt 120591119898(119905) le ℎ

119898lt infin

120591119898(119905) le 120583

119898 forall119898 isinM

(7)

where ℎ = maxℎ119894 119894 isin M is prescribed integer representing

the upper bounds of time-varying delay 120591119898(119905) Similarly 120583 =

max120583119898 119898 isin M is prescribed integer representing the up-

per bounds of time-varying delay 120591119898(119905) 119860

119903119905

119860120591119903119905

119863119903119905

119862119910119903119905

119862119910120591119903119905

119863119910119903119905

119862119911119903119905

119862119911120591119903119905

and 119863119911119903119905

are known mode-dependent

Journal of Applied Mathematics 3

matrices with appropriate dimensions functions of the ran-dom jumping process 119903

119905 and represent the nominal systems

for each 119903119905isin N For notation simplicity when the system

operates in the 119894-th mode (119903119905= 119894) 119860

119903119905

119860120591119903119905

119863119903119905

119862119910119903119905

119862119910120591119903119905

119863

119910119903119905

119862119911119903119905

119862119911120591119903119905

and119863119911119903119905

are denoted as119860119894119860

120591119894119863

119894119862

119910119894119862

119910120591119894

119863119910119894 119862

119911119894 119862

119911120591119894 and119863

119911119894 respectively

Here we are interested in designing a full-order filterdescribed by

119891(119905) = 119860

119891119903119905119904119905

119909119891(119905) + 119861

119891119903119905119904119905

119910 (119905)

119911119891(119905) = 119862

119891119903119905119904119905

119909119891(119905)

(8)

where 119909119891(119905) isin R119899 is the filter state 119911

119891(119896) isin R119902 and the

matrices 119860119891119903119905119904119905

119861119891119903119905119904119905

and 119862119891119903119905119904119905

are unknown filter param-eters to be designed

Augmenting the model of (1) to include the filter (8) weobtain the following filtering error system

120578 (119905) = 119860119903119905

120578 (119905) + 119861119903119905

120578 (119905 minus 120591119898(119905)) + 119863

119903119905

120596 (119905)

119890 (119905) = 119862119903119905

120578 (119905) + 119864119903119905

120578 (119905 minus 120591119898(119905))

(9)

where

120578 (119905) = [

119909 (119905)

119909119891(119905)] 119890 (119905) = 119911 (119905) minus 119911

119891(119905)

119860119903119905

= [

119860119903119905

0

119861119891119903119905119904119905

119862119910119903119905

119860119891119903119905119904119905

] 119861119903119905

= [

119860120591119903119905

119861119891119903119905119904119905

119862119910120591119903119905

]

119863119903119905

= [

119863119903119905

119861119891119903119905119904119905

119863119910119903119905

]

119862119903119905

= [119862119911119903119905

minus119862119891119903119905119904119905

] 119864119903119905

= [119862119911120591119903119905

0]

(10)

In order to more precisely describe the main objectivewe introduce the following definitions and Lemmas for theunderlying system

Definition 1 System (1) is said to be finite-time bounded withrespect to (119888

1 119888

2 119879 119877 119889) if condition (2) and the following

inequality hold

supminusℎle120592le0

E 120578⊺

(120592) 119877120578 (120592) 120578⊺

(120592) 119877 120578 (120592)

le 1198881997904rArr E 120578

⊺

(119905) 119877120578 (119905) lt 1198882

forall119905 isin [0 119879]

(11)

where 1198882gt 119888

1ge 0 and 119877 gt 0

Definition 2 Consider 119881(120578119905 119903

119905 119904

119905 119905 gt 0) as the stochastic

positive Lyapunov function its weak infinitesimal operatoris defined as

pound119881 (120578119905 119903

119905 119904

119905)

= limΔrarr0

1

Δ

[E 119881 (120578119905+Δ 119903

119905+Δ 119904

119905+Δ) | 120578

119905 119903

119905 119904

119905

minus119881 (120578119905 119903

119905 119904

119905)]

(12)

Definition 3 Given a constant 119879 gt 0 for all admissible120596(119905) subject to condition (2) under zero initial conditionsif the closed-loop Markovian jump system (1) is finite-timebounded and the control outputs satisfy condition (8) withattenuation 120574 gt 0

Eint119879

0

119890⊺

(119905) 119890 (119905) 119889119905 le 1205742

119890120578119879

Eint119879

0

120596⊺

(119905) 120596 (119905) 119889119905 (13)

Then the controller system (1) finite-time bounded withdisturbance attenuation 120574

Remark 4 It should be pointed out that the assumption ofzero initial condition in system (1) is only for the purposeof technical simplification in the derivation and it does notcause loss of generality In fact if this assumption is lost thesame control result can be obtained along the same lineexcept for adding extra manipulations in the derivation andextra terms in the control presentation However in real-world applications the initial condition of the underlyingsystem is generally not zero

Lemma 5 (see [32]) Let 119891119894 R119898

rarr R (119894 = 1 2 119873)

have positive values in an open subset D of R119898 Then the re-ciprocally convex combination of 119891

119894overD satisfies

min120573119894|120573119894gt0sum119894120573119894=1

sum

119894

1

120573119894

119891119894(119905) = sum

119894

119891119894(119905) +max

119892119894119895(119905)

sum

119894 = 119895

119892119894119895(119905)

119904119906119887119895119890119888119905 119905119900 119892119894119895R

119898

997888rarrR 119892119895119894(119905) = 119892

119894119895(119905)

[

119891119894(119905) 119892

119894119895(119905)

119892119894119895(119905) 119891

119895(119905)] ge 0

(14)

Lemma 6 (Schur Complement [17]) Given constant matrices119883119884119885 where119883 = 119883

⊺ and 0 lt 119884 = 119884⊺ then119883+119885⊺

119884minus1

119885 lt 0

if and only if

[

119883 119885⊺

lowast minus119884] lt 0 119900119903 [

minus119884 119885

lowast 119883] lt 0 (15)

3 Finite-Time 119867infin

Performance Analysis

Theorem 7 System (9) is finite-time bounded with respect to(1198881 119888

2 119889 119877 119879) if there exist matrices 119875

119894119898gt 0 119876

119897119894gt 0 (119897 = 1

2) 119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0119867 gt 0 119878

119894119898 scalars

1198881lt 119888

2 119879 gt 0120582

119904gt 0 (119904 = 1 2 10) 120578 gt 0 and Λ gt 0 such

that for all 119894 119895 isin N and 119898 119899 isin M the following inequalitieshold

sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895) + sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876 lt 0 (16)

sum

119899isinM119898 = 119899

120587119898119899119883

119894119899+ sum

119895isinN119894 = 119895

120582119894119895119883

119895119898minus 119883 lt 0 (17)


sum

119899isinM119898 = 119899

120587119898119899119884119894119899+ sum

119895isinN119894 = 119895

120582119894119895119884119895119898minus 119884 lt 0 (18)

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

gt 0 (19)

Ξ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980

lowast lowast minus1198761119894+

119884119894119898

ℎ

0 0 0

lowast lowast lowast Ξ44119894119898

minusℎ119872⊺

119894119898119881119894119898119863119894

lowast lowast lowast lowast minusℎ119883119894119898

ℎ119872119894119898119863119894

lowast lowast lowast lowast lowast minus120575119867

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(20)

Ξ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980


119884119894119898

ℎ

0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894


ℎ119873119894119898119863119894


]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(21)

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (22)

where

Ξ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898

+ 120575119875119894119898+ 119875

119894119898119860

119894+ 119860

⊺

119894119875119894119898+ 119890

120575ℎ

1198761119894

+ 119890120575ℎ

1198762119894+ ℎ119876

119890120575ℎ

minus 1

120575

119883119894119898

+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883 +

119884119894119898

ℎ

Ξ12119894119898

= 119875119894119898119861119894minus

119884119894119898

ℎ

+ 119878119894119898

Ξ22119894119898

= 119903 (120583119898) 119876

2119894+

2119884119894119898

ℎ

minus 119878119894119898minus 119878

⊺

119894119898

Ξ44119894119898

= minus119881119894119898minus 119881

⊺

119894119898+

119890120575ℎ

minus 1

120575

119884119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119884

119903 (120591119898) =

minus (1 minus 120583119898) 119890

120575ℎ119894 if 120583

119898gt 1

minus (1 minus 120583119898) if 120583

119898le 1

Λ = 1205822+ ℎ119890

120575ℎ

(1205823+ 120582

4) + ℎ

2

119890120575ℎ

(1205825+ 120582

6+ 120582

8)

+

1

2

ℎ3

119890120575ℎ

(1205827+ 120582

9)

1205821= min

119894isinN119898isinM120582min (119894119898) 120582

2= max

119894isinN119898isinM120582max (119894119898)

1205823= max

119894isinN120582max (1198761119894

)

1205824= max

119894isinN120582max (1198762119894

) 1205825= 120582max (119876)

1205826= max

119894isinN119898isinM120582max (119883119894119898

) 1205827= 120582max (119883)

1205828= max

119894isinN119898isinM120582max (119894119898) 120582

9= 120582max ()

12058210= 120582max (119867)

119894119898= 119877

minus(12)

119875119894119898119877minus(12)

119876119897119894= 119877

minus(12)

119876119897119894119877minus(12)

(119897 = 1 2)

119876 = 119877minus(12)

119876119877minus(12)

119883119894119898= 119877

minus(12)

119883119894119898119877minus(12)

119883 = 119877minus(12)

119883119877minus(12)

119894119898= 119877

minus(12)

119884119894119898119877minus(12)

= 119877minus(12)

119884119877minus(12)

(23)

Proof First in order to cast our model into the framework oftheMarkov processes we define a new process (120578

119905 119903

119905 119904

119905) 119905 ge

0 by

120578119905(119904) = 120578 (119905 + 119904) 119904 isin [minusℎ 0] (24)

Now we consider the following Lyapunov-Krasovskiifunctional

119881 (120578119905 119903

119905 119904

119905) =

4

sum

119897=1

119881119897(120578

119905 119903

119905 119905) (25)

where

1198811(120578

119905 119903

119905 119904

119905) = 120578(119905)

⊺

119890120575119905

119875119903119905119904119905

120578 (119905)

1198812(120578

119905 119903

119905 119904

119905) = int

119905

119905minusℎ

119890120575(119904+ℎ)

120578⊺

(119904) 1198761119903119905

120578 (119904) 119889119904

+ int

119905

119905minus120591119904119905(119905)

119890120575(119904+ℎ)

120578⊺

(119904) 1198762119903119905

120578 (119904) 119889119904

+ int

0

minusℎ

int

119905

119905+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

1198813(120578

119905 119903

119905 119904

119905) = int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)119883119903119905119904119905

120578 (119904) 119889119904 119889120579

+ int

0

minusℎ

int

0

120579

int

119905

119905+120592

119890120575(119904minus120579)

120578⊺

(119904)119883120578 (119904) 119889119904 119889120592 119889120579

1198814(120578

119905 119903

119905 119904

119905) = int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904) 119884119903119905119904119905

120578 (119904) 119889119904 119889120579

+ int

0

minusℎ

int

0

120579

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904) 119884 120578 (119904) 119889119904 119889120592 119889120579

(26)


Then for each 119903119905= 119894 119904

119905= 119898 we have

pound1198811(120578

119905 119894 119898)

= limΔrarr0

+

1

Δ

E 120578⊺

(119905 + Δ) 119890120575(119905+Δ)

119875119903119905+Δ

119904119905+Δ

120578 (119905 + Δ)

minus120578⊺

(119905) 119890120575119905

119875119894119898120578 (119905)

= limΔrarr0

+

1

Δ

E

120578⊺

(119905 + Δ)[

[

(1 + 120582119894119894Δ + 119900 (Δ))

times (1 + 120587119898119898Δ + 119900 (Δ)) 119890

120575(119905+Δ)

119875119894119898

+ (1 + 120582119894119894Δ + 119900 (Δ))

times ( sum

119899isinM

(120587119898119899Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119894119899

+ (1 + 120587119898119898Δ + 119900 (Δ))

times (sum

119895isinN

(120582119894119895Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119895119898

+ ( sum

119899isinM

(120587119898119899Δ + 119900 (Δ)))

times(sum

119895isinN

(120582119894119895Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119895119899

]

]

times 120578 (119905 + Δ)

minus 120578⊺

(119905) 119890120575119905

119875119894119898120578 (119905)

= 120575119890120575119905

120578⊺

(119905) 119875119894119898120578 (119905) + 2119890

120575119905

120578⊺

(119905) 119875119894119898120578 (119905)

+ 119890120575119905

120578⊺

(119905)( sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898)120578 (119905)

= 119890120575119905

120578⊺

(119905)( sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 120575119875

119894119898

+ 119875119894119898119860

119894+ 119860

⊺

119894119875119894119898)120578 (119905)

+ 2119890120575119905

120578⊺

(119905) 119875119894119898119861119894120578 (119905 minus 120591

119898(119905)) + 2119890

120575119905

120578⊺

(119905) 119875119894119898119863

119894120596 (119905)

(27)pound119881

2(120578

119905 119894 119898)

= limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119903119905+Δ

120578 (119904) 119889119904

minusint

119905

119905minusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminus120591119904119905+Δ

(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119903119905+Δ

120578 (119904) 119889119904

minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

Eint0

minusℎ

int

119905+Δ

119905+Δ+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

minusint

0

minusℎ

int

119905

119905+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

= limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

+ int

119905+Δ

119905+Δminusℎ

120578⊺

(119904)

times sum

119895isinN

(120582119894119895Δ + 119900 (Δ))

times 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

minusint

119905

119905minusℎ

120578⊺

(119904) 1198761119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

E[int119905+Δ

119905+Δminus120591119895(119905+Δ)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ sum

119899isinM

(120587119898119899Δ + 119900 (Δ))

times int

119905

119905+Δminus120591119899(119905+Δ)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ int

119905+Δ

minus120591119895(119905 + Δ)

119905+Δ

120578⊺

(119904)

times sum

119895isinN

(120582119894119895Δ + 119900 (Δ))

times 119890120575(119904+ℎ)

1198762119895120578 (119904) 119889119904

minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904]

+ 119890120575119905

120578⊺

(119905) ℎ119876120578 (119905) minus int

119905minusℎ

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904

le 119890120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

minus (1 minus 120591119898(119905)) 119890

120575119905

120578⊺

(119905 minus 120591119898(119905)) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)


+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(28)

Since

0 le 120591119898(119905) le ℎ

119898 (29)

we define

119903 (120591119898) =

minus (1 minus 120583119898) 119890

120575ℎ119898 if 120583

119898gt 1

minus (1 minus 120583119898) if 120583

119898le 1

(30)

Then

pound1198812(120578

119905 119894 119898) le 119890

120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

+ 119890120575119905

120578⊺

(119905 minus 120591119898(119905)) 119903 (120591

119898) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)

times ( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)

+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(31)

Similar to the previous process we can obtain

pound1198813(120578

119905 119894 119898) le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119883

119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119883

119895119898minus 119883)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119883119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119883120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579119889120592

(32)

By using Lemma 5 it yields that

minusint

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904 = minusint

119905

119905minus120591119898(119905)

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

le minus120591119898(119905) 119880

⊺

1119883

1198941198981198801

minus (ℎ minus 120591119898(119905)) 119880

⊺

2119883

1198941198981198802

(33)

where

1198801=

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

1198802=

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

lim120591119898(119905)rarr0

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904 = 120578 (119905)

lim120591119898(119905)rarrℎ

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904 = 120578 (119905 minus ℎ)

(34)

From the Newton-Leibniz formula the following equa-tion is true for any matrices119872

119894119898 119873

119894119898 and 119881

119894119898with appro-

priate dimensions

(2120591119898(119905) 119880

⊺

1119872

119894119898+ 2 (ℎ minus 120591

119898(119905)) 119880

⊺

2119873

119894119898+ 2 120578

⊺

(119905) 119881119894119898)

times [minus 120578 (119905) + 119860119894120578 (119905) + 119861

120591119894120578 (119905 minus 120591

119898(119905)) + 119863

119894120596 (119905)] = 0

(35)

pound1198814(120578

119905 119894 119898)

le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119884119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119884119895119898minus 119884)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119884119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119884 120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579 119889120592

(36)


From Lemma 5 it yields that

minus int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

= minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

le minus

ℎ

120591119898(119905)

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

minus

ℎ

ℎ minus 120591119898(119905)

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

le minus

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

⊺

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

(37)

From (25)ndash(37) we can eventually obtain

pound119881 (120578119905 119903

119905 119904

119905) minus 120575120596

⊺

(119905)119867120596 (119905) le 119890120575119905

120585⊺

(119905) Ξ119894119898120585 (119905) (38)

where

120585⊺

(119905) = [120578⊺

(119905) 120578⊺

(119905 minus 120591119898(119905)) 120578

⊺

(119905 minus ℎ) 120578⊺

(119905) 119880⊺

1 119880

⊺

2 120596

⊺

(119905)]

Ξ119894119898=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860⊺

119894119881

⊺

119894119898120591119898(119905) 119860

⊺

119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

119894119873

⊺

119894119898119875119894119898119863

119894

lowast Ξ22119894119898

119878119894119898minus

119884119894119898

ℎ

119860⊺

120591119894119881

⊺

119894119898120591119898(119905) 119860

⊺

120591119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

120591119894119873

⊺

1198941198980


119884119894119898

ℎ

0 0 0 0


minus120591119898(119905)119872

⊺

119894119898minus (ℎ minus 120591

119898(119905))119873

⊺

119894119898119881119894119898119863

119894

lowast lowast lowast lowast minus120591119898(119905) 119883

1198941198980 120591

119898(119905)119872

119894119898119863

119894

lowast lowast lowast lowast lowast minus (ℎ minus 120591119898(119905))119883

119894119898(ℎ minus 120591

119898(119905))119873

119894119898119863

119894

lowast lowast lowast lowast lowast lowast minus120575119867

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(39)

The LMIs (20) and (21) lead to 120591119898(119905) rarr ℎ and 120591

119898(119905) rarr

0 respectively It is easy to see that Ξ1119894119898

results from Ξ119894119898|

120591119898(119905) = ℎ and Ξ

119894119898|120591119898(119905) = 0 Thus we obtain

E pound119881 (120578119905 119903

119905 119904

119905) le 120578E [119881 (120578

119905 119903

119905 119904

119905)] + 120575120596

⊺

(119905)119867120596 (119905)

(40)

Multiplying the aforementioned inequality by 119890minus120578119905 we canget

E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

120575120596⊺

(119905)119867120596 (119905) (41)

By integrating the aforementioned inequality between 0and 119905 it follows that

119890minus120578119905

E [119881 (120578119905 119903

119905 119904

119905)] minus E [119881 (120578

0 119903

0 119904

0)]

le 120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

(42)

Denote that 119894119898

= 119877minus(12)

119875119894119898119877minus(12) 119876

119904119894=

119877minus(12)

119876119904119894119877minus(12)

(119904 = 1 2) 119876 = 119877minus(12)

119876119877minus(12)

119883119894119898

= 119877minus(12)

119883119894119898119877minus(12) 119883 = 119877

minus(12)

119883119877minus(12)

119894119898

= 119877minus(12)

119884119894119898119877minus(12) and = 119877

minus(12)

119884119877minus(12) it yields

that

E [119881 (1205780 119903

0 119904

0)]

le max119894isinN119898isinM

120582max (119894119898) 120578⊺

(0) 119877120578 (0)

+ (max119894isinN

120582max (1198761119894) +max

119894isinN120582max (1198762119894

)) 119890120575ℎ

times int

0

minusℎ

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ

120582max (119876)int0

minusℎ

int

0

120579

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ max

119894isinN119898isinM120582max (119883119894119898

)

times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max (119883)

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579 119889120592

+ 119890120575ℎ max

119894isinN119898isinM120582max (119894119898)


times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max ()

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579 119889120592


120582max (119894119898)

+ ℎ119890120575ℎ

(max119894isinN

120582max (1198761119894+

119894isinNmax120582max (1198762119894

)))

+ ℎ2

119890120575ℎ

( max119894isinN119898isinM

120582max (119883119894119898) + 120582max (119876)

+ max119894isinN119898isinM

120582max (119894119898))

+

1

2

ℎ3

119890120575ℎ

(120582max (119883) + 120582max ())

times supminusℎle119904le0

120578⊺

(119904) 119877120578 (119904) 120578⊺

(119904) 119877 120578 (119904)

= 1198881Λ

(43)

For given 120578 gt 0 and 0 le 119905 le 119879 we have

E [119881 (120578119905 119903

119905 119904

119905)] le E [119890

120578119905

119881 (1205780 119903

0 119904

0)]

+ 119890120578119905

120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

le 119890120578119879

1198881Λ + 119889120575119890

120578119879

120582max (119867)int119879

0

119890minus120578119904

119889119904

le 119890120578119879

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

)

(44)

On the other hand it follows from (25) that

E [119881 (120578119905 119903

119905 119904

119905)] ge E [120578

⊺

(119905) 119890120582119905

119875119894119898120578 (119905)]

ge max119894isinN

120582min (119875119894119898)E [120578⊺

(119905) 119877120578 (119905)]

= 1205821E [120578

⊺

(119905) 119877120578 (119905)]

(45)

It can be derived from (43)ndash(45) that

E [120578⊺

(119905) 119877120578 (119905)] le

119890120578119879

1205821

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) (46)

From (22) and (46) we have

E [120578⊺

(119905) 119877120578 (119905)] lt 1198882 (47)

Then the system is finite-time bounded with respect to(1198881 119888

2 119889 119877 119879)

Remark 8 In this paper 120591119898(119905) and 120591

119898(119905) may have different

upper bounds in various delay intervals satisfying (7) respec-tively However in previous work such as [20 21] 120591

119898(119905) and

120591119898(119905) are enlarged to 120591

119898(119905) le ℎ = maxℎ

119894 119898 isin M and

120591119898(119905) le 120583 = max120583

119898 119898 isin M respectively which may lead

to conservativeness inevitably However the previouse casecan be taken fully into account by employing the Lyapunov-Krasovskii functional (25)

Remark 9 When dealing with termminusint

119905

119905minusℎ

120578⊺

(119904)119884119894119898120578(119904)119889119904 the

convex combination is not employed Lemma 5 is used in thispaper and then the free-weighting matrices-dependent nulladd items are necessary to be introduced in our proof whichlead to the decrease of the number of LMIs and LMIs scalardecision variables

Remark 10 The feature of this paper is the way to dealwith the integral term Many researchers have enlarged thederivative of the Lyapunov functional in order to deal with theintegral term in mathematical operations In this paper wepropose a novel delay-dependent sufficient criterion whichensures that the Markovian jump system with different modesystems is finite-time stable

Remark 11 It should be pointed out that the novelty of theLyapunov functional (25) lies in distinct Lyapunov matrices(119875

119894119898 119883

119894119898 119884

119894119898) which is chosen for different system modes

119894 (119894 = 1 2 119873) and119898 (119898 = 1 2 119872)



119894gt 0 119876

119897119894gt 0 (119897 = 1 2)

119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0 119878

119894119898 scalars 119888

1lt 119888

2

119879 gt 0 120582119904gt 0 (119904 = 1 2 9) 120578 gt 0 120574 gt 0 and Λ gt 0 such

that for all 119894 119895 isinN and119898 119899 isinM and (16)ndash(19) the followinginequalities hold

Σ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894

0


ℎ119872119894119898119863119894

0

lowast lowast lowast lowast lowast minus1205742119868 0


]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(48)

Σ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119873⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894

0


ℎ119873119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(49)

1198881Λ + 119889120575120574

21

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (50)


Proof We now consider the 119867infin

performance of system (9)Select the same Lyapunov-Krasovskii functional as Theo-rem 7 it yields that

pound119881 (120578119905 119903

119905 119904

119905) + 119890

⊺

(119905) 119890 (119905) minus 1205742

120596⊺

(119905) 120596 (119905) le 120585⊺

(119905) Σ119897119894119898120585 (119905)

(119897 = 1 2)

(51)

It follows from (49)-(50) that

E pound119881 (120578119905 119903

119905 119904

119905)

le E [120578119881 (120578119905 119903

119905 119904

119905)] + 120574

2

120596⊺

(119905) 120596 (119905) minus E [119890⊺

(119905) 119890 (119905)]

(52)

Multiplying the aforementioned inequality by 119890minus120578119905 onehas

E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)]

(53)

In zero initial condition and E[119881(120578119905 119903

119905 119904

119905)] gt 0 by inte-

grating the aforementioned inequality between 0 and 119879 wecan get

int

119879

0

119890minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)] 119889119905

le Eint119879

0

pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] 119889119905 le 119881 (120578

0 119903

0 119904

0) = 0

(54)

Using Dynkins formula it results that

E [int119879

0

119890minus120578120592

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

E [int119879

0

119890minus120578120592

120596⊺

(120592) 120596 (120592) 119889120592]

(55)

Then it yields that

E [int119879

0

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

119890120578119879

E [int119879

0

120596⊺

(120592) 120596 (120592) 119889120592] (56)

Thus it is concluded by Definition 3 that system (9) is fi-nite-time bounded with an 119867

infinperformance 120574 This com-

pletes the proof

Remark 13 From the proof process of Theorems 7 and 12it is easy to see that neither bounding technique for cross-terms nor model transformation is involved In other wordsthe obtained result is expected to be less conservative

Remark 14 The Lyapunov asymptotic stability and finite-time stability of a class of system are independent conceptsA Lyapunov asymptotic stability system may not be finite-time stability Moreover finite-time stability system may alsonot be Lyapunov asymptotic stabilityThere exist some resultson Lyapunov stability while finite-time stability also needsour full investigation which was neglected by most previouswork


Filtering



119894119898gt 0119872

119894119898 119873

119894119898 119881

119894119898

119876119897119894gt 0 (119897 = 1 2) 119876 gt 0 119883

119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0

119860119891119894119898

119861119891119894119898

119862119891119894119898

119872119860119894119898

119872119861119894119898

119873119860119894119898

119873119861119894119898

119881119860119894119898

119881119861119894119898

119878119894119898

scalars 1198881lt 119888

2 119879 gt 0 120590

119904gt 0 (119904 = 1 2 9) 120575 gt 0 120578 gt 0

and Λ gt 0 such that for all 119894 119895 isin N and 119898 119899 isin M thefollowing inequalities hold

119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] gt 0 119872119894119898= [

1198721119894119898

1198722119894119898

1198722119894119898

1198722119894119898

] (57)

119873119894119898= [

1198731119894119898

1198732119894119898

1198732119894119898

1198732119894119898

] 119881119894119898= [

1198811119894119898

1198812119894119898

1198812119894119898

1198812119894119898

] (58)

Γ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ15119894119898

Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ25119894119898

0 [119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898Γ46119894119898

0


Γ56119894119898

0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(59)

Γ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ1015840

15119894119898Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ1015840

251198941198980 [

119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898Γ1015840

461198941198980


Γ1015840

561198941198980



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(60)where

Γ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 119890

120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876 +

119890120575ℎ

minus 1

120575

119883119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883

+

119884119894119898

ℎ

+ [

1205751198751119894119898

+ 1198751119894119898119860

119894+ 119860

⊺

1198941198751119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

1205751198752119894119898

+ 1198752119894119898119860

119894+ 119860

⊺

1198941198752119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

]

(61)


Γ12119894119898

= minus

119884119894119898

ℎ

+ 119878119894119898+ [

1198751119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

]

Γ14119894119898

= [

119860⊺

119894119881

⊺

1119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

119860⊺

119894119881

⊺

2119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

]

Γ15119894119898

= [

ℎ119860⊺

119894119872

⊺

1119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

ℎ119860⊺

119894119872

⊺

2119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

]

Γ1015840

15119894119898= [

ℎ119860⊺

119894119873

⊺

1119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

ℎ119860⊺

119894119873

⊺

2119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

]

Γ16119894119898

= [

1198751119894119898119863

119894+ 119861

119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119861

119891119894119898119863

119910119894

]

Γ17119894119898

= [

119862⊺

119911119894

minus119862119891119894119898

]

Γ24119894119898

= [

119860⊺

120591119894119881

⊺

1119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

119860⊺

120591119894119881

⊺

2119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

]

Γ25119894119898

= [

ℎ119860⊺

120591119894119872

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

ℎ119860⊺

120591119894119872

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

]

Γ1015840

25119894119898= [

ℎ119860⊺

120591119894119873

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

ℎ119860⊺

120591119894119873

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

]

Γ46119894119898

= [

1198811119894119898119863

119894+ 119881

119861119894119898119863

119910119894

1198812119894119898119863

119894+ 119881

119861119894119898119863

119910119894

]

Γ56119894119898

= [

ℎ1198721119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

ℎ1198722119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

]

Γ1015840

56119894119898= [

ℎ1198731119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

ℎ1198732119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

]

(62)

Then a desired filter can be chosen with parameters as

119860119891119894119898

= 119875minus1

2119894119898119860

119891119894119898 119861

119891119894119898= 119875

minus1

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

(63)

Proof We denote that

119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] (64)

The term 119875119894119898119860

119894can be rewritten as

119875119894119898119860

119894= [

1198751119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

1198752119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

] (65)

Similarly we have

119875119894119898119861119894= [

1198751119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

]

119875119894119898119863

119894= [

1198751119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

]

(66)

Define 119860119891119894119898

= 1198752119894119898119860

119891119894119898 119861

119891119894119898= 119875

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

119872

119860119894119898= 119872

2119894119898119860

119891119894119898119872

119861119894119898= 119872

2119894119898119861119891119894119898

119873119860119894119898

= 1198732119894119898119860

119891119894119898

119873119861119894119898

= 1198732119894119898119861119891119894119898

119881119860119894119898

= 1198812119894119898119860

119891119894119898 and 119881

119861119894119898= 119881

2119894119898119861119891119894119898

Therefore if (59) and (60) hold system (9) is finite-timebounded with a prescribed 119867

infinperformance index 120574 The

proof is completed

Remark 16 In many actual applications the minimum valueof 1205742min is of interest In Theorem 12 with a fixed 120582 120574min canbe obtained through the following optimization procedure

min 1205742

st (48)ndash(50) (67)

In Theorem 15 as for finite-time stability and bounded-ness once the state bound 119888

2is not ascertained theminimum

value 1198882min is of interest With a fixed 120582 define 120582

1= 1 then

the following optimization problem can be formulated to getminimum value 119888

2min

min 1205891205742

+ (1 minus 120589) 1198882

st (59)-(60) and (50)

(68)

where 120589 is weighted factor and 120589 isin [0 1]

5 Illustrative Example

Example 17 Consider that the Markovian jump system andthe delay mode switching are governed by a Markov processwith the following transition rates

Λ = [

minus07 07

09 minus09] Π = [

minus1 1

12 minus12] (69)

as well as with the following parameters

1198601= [

minus35 086

minus064 minus325] 119860

2= [

minus25 034

14 minus002]

1198601205911= [

minus08 minus13


1205912= [

minus28 05

minus08 minus14]

1198631= 119863

2= [

01

02] 119862

1199101= [04 07]

1198621199102= [05 08] 119862

1199101205911= [01 02]

1198621199101205912

= [01 01]

1198631199101= 119863

1199102= 01 119862

1199111= [01 005]

1198621199112= [02 009] 119862

1199111205911= [03 06]

1198621199111205912

= [04 08] 1198631199111= 119863

1199112= 005

(70)


Then we choose 119877 = 119868 119879 = 2 1198881= 1 and 119889 = 001

throughTheorem 15 it yields that themode-dependent filtersare as follows

11986011989111

= [

minus61254 15565

minus02347 minus03763]

11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)

This paper deals with the finite-time filter design problemfor a class of Markovian jump systems particularly two dif-ferentMarkov processes are considered formodeling the ran-domness of system matrix and the state delay Then throughthe numerical example we can see that results in this paperare feasible which further verified the correctness of our the-ory Therefore the paper shorten this gap

6 Conclusions

In this paper we have examined the problems of finite-time119867

infinfiltering for a class of Markovian jump systems with dif-

ferent system modes Based on a novel approach a sufficientcondition is derived such that the closed-loop Markovianjump system is finite-time bounded and satisfies a prescribedlevel of119867

infindisturbance attenuation in a finite time interval

Finally a numerical example is also given to illustrate theeffectiveness of the proposed design approach It should benoted that one of future research topics would be to investi-gate the problems of fault detection and fault tolerant controlfor time-varying Markovian jump systems with incompleteinformation over a finite-time horizon

Acknowledgments

The authors would like to thank the associate editor and theanonymous reviewers for their detailed comments and sug-gestions This work was supported by the Fund of SichuanProvincial Key Laboratory of Signal and Information Pro-cessing Xihua University (SZJJ2009-002 and SGXZD0101-10-1) and National Basic Research Program of China(2010CB732501)

References

[1] N N Krasovskiı and E A Lidskiı ldquoAnalytical design ofcontrollers in systems with random attributesmdashI Statementof the problem method of solvingrdquo Automation and RemoteControl vol 22 pp 1021ndash1025 1961

[2] J Tian Y Li J Zhao and S Zhong ldquoDelay-dependent stochasticstability criteria for Markovian jumping neural networks withmode-dependent time-varying delays and partially knowntransition ratesrdquo Applied Mathematics and Computation vol218 no 9 pp 5769ndash5781 2012

[3] L Wu X Su and P Shi ldquoSliding mode control with bounded1198712gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[4] H Gao Z Fei J Lam and B Du ldquoFurther results on expo-

nential estimates of Markovian jump systems with mode-de-pendent time-varying delaysrdquo IEEE Transactions on AutomaticControl vol 56 no 1 pp 223ndash229 2011

[5] Z Wu H Su and J Chu ldquo119867infinfiltering for singular Markovian

jump systems with time delayrdquo International Journal of Robustand Nonlinear Control vol 20 no 8 pp 939ndash957 2010

[6] P Tino M Cernansky and L Benuskova ldquoMarkovian architec-tural bias of recurrent neural networksrdquo IEEE Transactions onNeural Networks vol 15 no 1 pp 6ndash15 2004

[7] H Dong Z Wang D W C Ho and H Gao ldquoRobust 119867infin

filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011

[8] Q Ma S Xu and Y Zou ldquoStability and synchronization forMarkovian jump neural networks with partly unknown tran-sition probabilitiesrdquo Neurocomputing vol 74 pp 3404ndash33412011

[9] H Zhao S Xu and Y Zou ldquoRobust119867infinfiltering for uncertain

Markovian jump systems with mode-dependent distributeddelaysrdquo International Journal of Adaptive Control and SignalProcessing vol 24 no 1 pp 83ndash94 2010

[10] Y Zhang S Xu and B Zhang ldquoDiscrete-time fuzzy Markovianjump systems with time-varying delaysrdquo IEEE Transactions onFuzzy Systems vol 17 no 2 pp 411ndash420 2009

[11] Z Wu H Su and J Chu ldquoState estimation for discreteMarkovian jumping neural networks with time delayrdquo Neuro-computing vol 73 no 10ndash12 pp 2247ndash2254 2010

[12] Q Zhu and J Cao ldquoRobust exponential stability of markovianjump impulsive stochastic Cohen-Grossberg neural networkswithmixed time delaysrdquo IEEE Transactions onNeural Networksvol 21 no 8 pp 1314ndash1325 2010

[13] H Huang G Feng and X Chen ldquoStability and stabilizationof Markovian jump systems with time delay via new Lyapunovfunctionalsrdquo IEEE Transactions on Circuits and Systems I vol59 no 10 pp 2413ndash2421 2012


[14] Q Zhu and J Cao ldquoStability of Markovian jump neural net-works with impulse control and time varying delaysrdquoNonlinearAnalysis Real World Applications vol 13 no 5 pp 2259ndash22702012

[15] M Sun J Lam S Xu andY Zou ldquoRobust exponential stabiliza-tion for Markovian jump systems with mode-dependent inputdelayrdquo Automatica vol 43 no 10 pp 1799ndash1807 2007

[16] M Mohammadian A Abolmasoumi and H Momeni ldquo119867infin

mode-independent filter design for Markovian jump geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 87 pp 10ndash18 2012

[17] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in SystemandControlTheory vol 15 of SIAMStudies in Applied Mathematics SIAM Philadelphia Pa USA1994

[18] S Mou H GaoW Qiang and K Chen ldquoNew delay-dependentexponential stability for neural networks with time delayrdquo IEEETransactions on Systems Man and Cybernetics Part B vol 38no 2 pp 571ndash576 2008

[19] O M Kwon J Park S Lee and E Cha ldquoAnalysis on delay-dependent stability for neural networks with time-varyingdelaysrdquo Neurocomputing vol 103 pp 114ndash120 2013

[20] L Hu H Gao andW X Zheng ldquoNovel stability of cellular neu-ral networkswith interval time-varying delayrdquoNeuralNetworksvol 21 no 10 pp 1458ndash1463 2008

[21] H Zhang Z Liu G B Huang and ZWang ldquoNovel weighting-delay-based stability criteria for recurrent neural networks withtime-varying delayrdquo IEEE Transactions on Neural Networks vol21 no 1 pp 91ndash106 2010

[22] Z Zuo C Yang and YWang ldquoA newmethod for stability anal-ysis of recurrent neural networks with interval time-varyingdelayrdquo IEEE Transactions on Neural Networks vol 21 no 2 pp339ndash344 2010

[23] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[24] D Zhang L Yu and W Zhang ldquoDelay-dependent fault detec-tion for switched linear systems with time-varying delays-theaverage dwell time approachrdquo Signal Processing vol 91 pp 832ndash840 2011

[25] Z Fei H Gao and P Shi ldquoNew results on stabilization ofMark-ovian jump systems with time delayrdquoAutomatica vol 45 no 10pp 2300ndash2306 2009

[26] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[27] M Luo G Liu and S Zhong ldquoRobust fault detection ofMarkovian jump systems with different systemmodesrdquoAppliedMathematical Modelling vol 37 no 7 pp 5001ndash5012 2013

[28] B Zhang andY Li ldquoExponential1198712minus119871

infinfiltering for distributed

delay systems with Markovian jumping parametersrdquo SignalProcessing vol 93 no 1 pp 206ndash216 2013

[29] A Abolmasoumi and H Momeni ldquoRobust observer-based119867infin

control of a Markovian jump system with different delay andsystem modesrdquo International Journal of Control Automationand Systems vol 9 no 4 pp 768ndash776 2011

[30] L Zhang E-K Boukas and J Lam ldquoAnalysis and synthesisof Markov jump linear systems with time-varying delays andpartially known transition probabilitiesrdquo IEEE Transactions onAutomatic Control vol 53 no 10 pp 2458ndash2464 2008

[31] S Ma and E K Boukas ldquoRobust 119867infin

filtering for uncertaindiscrete Markov jump singular systems with mode-dependenttime delayrdquo IET Control Theory amp Applications vol 3 no 3 pp351ndash361 2009

[32] Y ChenW Bi andW Li ldquoStability analysis for neural networkswith time-varying delay a more general delay decompositionapproachrdquo Neurocomputing vol 73 no 4ndash6 pp 853ndash857 2010

[33] Z Shu J Lam and S Xu ldquoRobust stabilization of Markoviandelay systems with delay-dependent exponential estimatesrdquoAutomatica vol 42 no 11 pp 2001ndash2008 2006

[34] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[35] C Qian and J Li ldquoGlobal finite-time stabilization by outputfeedback for planar systems without observable linearizationrdquoIEEE Transactions on Automatic Control vol 50 no 6 pp 885ndash890 2005

[36] F Amato R Ambrosino C Cosentino and G De TommasildquoInput-output finite time stabilization of linear systemsrdquo Auto-matica vol 46 no 9 pp 1558ndash1562 2010

[37] Y Hong ldquoFinite-time stabilization and stabilizability of a classof controllable systemsrdquo Systems amp Control Letters vol 46 no4 pp 231ndash236 2002

[38] F Amato and M Ariola ldquoFinite-time control of discrete-timelinear systemsrdquo IEEETransactions onAutomatic Control vol 50no 5 pp 724ndash729 2005

[39] Z Zuo Y Liu Y Wang and H Li ldquoFinite-time stochasticstability and stabilisation of linear discrete-time Markovianjump systems with partly unknown transition probabilitiesrdquoIET Control Theory amp Applications vol 6 no 10 pp 1522ndash15262012

[40] X Luan F Liu and P Shi ldquoFinite-time filtering for non-linearstochastic systems with partially known transition jump ratesrdquoIET Control Theory amp Applications vol 4 no 5 pp 735ndash7452010

[41] F Amato M Ariola and C Cosentino ldquoFinite-time control ofdiscrete-time linear systems analysis and design conditionsrdquoAutomatica vol 46 no 5 pp 919ndash924 2010

[42] X Lin H Du and S Li ldquoFinite-time boundedness and 1198712-

gain analysis for switched delay systems with norm-boundeddisturbancerdquo Applied Mathematics and Computation vol 217no 12 pp 5982ndash5993 2011

[43] J Lin S Fei and Z Gao ldquoStabilization of discrete-time switchedsingular time-delay systems under asynchronous switchingrdquoJournal of the Franklin Institute vol 349 no 5 pp 1808ndash18272012

[44] H Song L Yu D Zhang and W-A Zhang ldquoFinite-time 119867infin

control for a class of discrete-time switched time-delay systemswith quantized feedbackrdquoCommunications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4802ndash4814 2012

[45] Z Zuo H Li Y Liu and Y Wang ldquoOn finite-time stochasticstability and stabilization ofMarkovian jump systems subject topartial information on transition probabilitiesrdquoCircuits Systemsand Signal Processing 2012

[46] H Liu Y Shen and X Zhao ldquoDelay-dependent observer-based119867

infinfinite-time control for switched systems with time-varying

delayrdquoNonlinear Analysis Hybrid Systems vol 6 no 3 pp 885ndash898 2012

[47] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012


[48] J Cheng H Zhu S Zhong Y Zhang and Y Zeng ldquoFinite-timeStabilization of 119867

infinfiltering for switched stochastic systemsrdquo

Circuits Systems and Signal Processing 2012[49] J Cheng H Zhu S Zhong and Y Zhang ldquoFinite-time bound-

ness of 119867infin

filtering for switching discrete-time systemsrdquo In-ternational Journal of Control Automation and Systems vol 10pp 1129ndash1135 2012

[50] Z Xiang C Qiao and M S Mahmoud ldquoFinite-time analysisand119867

infincontrol for switched stochastic systemsrdquo Journal of the

Franklin Institute vol 349 no 3 pp 915ndash927 2012

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


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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


Furthermore though the decay rate can be computed it is afixed value that one cannot adjust to deduce if a larger de-cay rate is possible Therefore how to obtain the improvedresults without increasing the computational burden hasgreatly improved the current study On the other hand thepractical problems which system described does not exceeda certain threshold over some finite time interval are con-sidered In finite-time interval finite-time stability is inves-tigated to address these transient performances of controlsystems Recently the concept of finite-time stability has beenrevisited in the light of linear matrix inequalities (LMIs) andLyapunov function theory and some results are obtained toensure that systems are finite-time stability or finite-timeboundness [34ndash50] To the best of our knowledge in mostof the works about Markovian jump systems with mode-dependent delay the delay mode is always assumed to be thesame as the system matrices mode However in real systemsthe delay mode may not be the same as that for jump inother system parameters In other words variations of delayusually depend on phenomena which may not cause abruptchanges in other systems parameters Therefore the work ofMarkovian jump systems with different system modes is notonly theoretically interesting and challenging but also veryimportant in practical applications

Motivated by the previous above discussions in this pa-per we present a new augmented Lyapunov functional fora class of Markovian jump systems with different systemmodes in order to reduce the possible conservativeness andcomputational burden some slack matrices are introduced[32] Several sufficient conditions are derived to guarantee thefinite-time stability and boundedness of the resulting closed-loop system We find that finite-time stability is an inde-pendent concept from Lyapunov stability and always can beaffected by switching behavior significantly and the finite-time boundness criteria can be tackled in the form of LMIsFinally a numerical example is presented to illustrate theeffectiveness of the developed techniques

Notations Throughout this paper we let 119875 gt 0 (119875 ge 0 119875 lt0 and 119875 le 0) denote a symmetric positive definite matrix 119875(positive semidefinite negative definite and negative semi-definite) For any symmetric matrix 119875 120582max(119875) and 120582min(119875)denote the maximum andminimum eigenvalues of matrix 119875respectivelyR119899 denotes the 119899-dimensional Euclidean spaceand R119899times119898 refers to the set of all 119899 times 119898 real matrices andN = 1 2 119873 The identity matrix of order 119899 is denotedas 119868

119899 lowast represents the elements below the main diagonal of a

symmetricmatrixThe superscripts ⊺ and minus1 stand formatrixtransposition and matrix inverse respectively

2 Preliminaries

In this paper we consider the followingMarkov jump systemdescribed by

(119905) = 119860119903119905

119909 (119905) + 119860120591119903119905

119909 (119905 minus 120591119904119905

(119905)) + 119863119903119905

120596 (119905)

119910 (119905) = 119862119910119903119905

119909 (119905) + 119862119910120591119903119905

119909 (119905 minus 120591119904119905

(119905)) + 119863119910119903119905

120596 (119905)

119911 (119905) = 119862119911119903119905

119909 (119905) + 119862119911120591119903119905

119909 (119905 minus 120591119904119905

(119905)) + 119863119911119903119905

120596 (119905)

119909 (119905) = 120593 (119905) 119905 = [minusℎ 0]

(1)

where 119909(119905) isinR119899 is the state vector of the system 119910(119905) isinR119897 isthemeasured output 119911(119905) isinR119902 is the controlled output120593(119905)119905 isin [minusℎ 0] are initial conditions of continuous state and 119903

0isin

N 1199040isin M = 1 2 119872 are initial conditions of mode

120596(119905) isin R119902 is the disturbance input satisfying the followingcondition

int

infin

0

120596⊺

(119905) 120596 (119905) 119889119905 le 119889 (2)

Let the random form processes 119903119905 119904

119905be the Markov sto-

chastic processes taking values on finite sets N = 1 2

119873 and M = 1 2 119872 with probability transition ratematrices Λ = 120582

119894119895 119894 119895 isin N and Π = 120587

119898119899 119898 119899 isin M The

transition probabilities from mode 119895 to mode 119895 for Markovprocess 119903

119905and from mode 119898 to mode 119899 for the Markov

process 119904119905in time ℎ are described as

Pr (119903119905+Δ

= 119895 | 119903119905= 119894) = 984858

119894119895+ 120582

119894119895Δ + 119900 (Δ)

Pr (119904119905+Δ

= 119899 | 119904119905= 119898) = 120577

119898119899+ 120587

119898119899Δ + 119900 (Δ)

(3)

where

984858119894119895=

0 if 119894 = 119895

1 if 119894 = 119895120577119898119899=

0 if 119898 = 119899

1 if 119898 = 119899

(4)

and Δ gt 0 120582119894119895ge 0 for 119894 = 119895 is the transition rate from mode

119894 at time 119905 to mode 119895 at time 119905 + Δ and

minus120582119894119894=

119873

sum

119895=1119895 = 119894

120582119894119895 (5)

for each mode 119894 isin N limΔrarr0

+

(119900(Δ)Δ) = 0 120587119898119899

ge 0 for119898 = 119899 is the transition rate from mode 119898 to mode 119899 at time119905 + Δ and

minus120587119898119898

=

119872

sum

119899=1119898 = 119899

120587119898119899 (6)

for each mode 119894 isin M limΔrarr0

+

(119900(Δ)Δ) = 0 For conve-nience we denote the Markov process 119903

119905and 119904

119905by 119894 and

119898 indices respectively 120591119898(119905) denotes the mode-dependent

time-varying state delay in the system and satisfies thefollowing condition

0 lt 120591119898(119905) le ℎ

119898lt infin

120591119898(119905) le 120583

119898 forall119898 isinM

(7)

where ℎ = maxℎ119894 119894 isin M is prescribed integer representing

the upper bounds of time-varying delay 120591119898(119905) Similarly 120583 =

max120583119898 119898 isin M is prescribed integer representing the up-

per bounds of time-varying delay 120591119898(119905) 119860

119903119905

119860120591119903119905

119863119903119905

119862119910119903119905

119862119910120591119903119905

119863119910119903119905

119862119911119903119905

119862119911120591119903119905

and 119863119911119903119905

are known mode-dependent






119903119905

119860120591119903119905

119863119903119905

119862119910119903119905

119862119910120591119903119905

119863

119910119903119905

119862119911119903119905

119862119911120591119903119905

and119863119911119903119905

are denoted as119860119894119860

120591119894119863

119894119862

119910119894119862

119910120591119894

119863119910119894 119862

119911119894 119862

119911120591119894 and119863



119891(119905) = 119860

119891119903119905119904119905

119909119891(119905) + 119861

119891119903119905119904119905

119910 (119905)

119911119891(119905) = 119862

119891119903119905119904119905

119909119891(119905)

(8)


119891(119896) isin R119902 and the

matrices 119860119891119903119905119904119905

119861119891119903119905119904119905

and 119862119891119903119905119904119905



120578 (119905) = 119860119903119905

120578 (119905) + 119861119903119905

120578 (119905 minus 120591119898(119905)) + 119863

119903119905

120596 (119905)

119890 (119905) = 119862119903119905

120578 (119905) + 119864119903119905

120578 (119905 minus 120591119898(119905))

(9)

where

120578 (119905) = [

119909 (119905)

119909119891(119905)] 119890 (119905) = 119911 (119905) minus 119911

119891(119905)

119860119903119905

= [

119860119903119905

0

119861119891119903119905119904119905

119862119910119903119905

119860119891119903119905119904119905

] 119861119903119905

= [

119860120591119903119905

119861119891119903119905119904119905

119862119910120591119903119905

]

119863119903119905

= [

119863119903119905

119861119891119903119905119904119905

119863119910119903119905

]

119862119903119905

= [119862119911119903119905

minus119862119891119903119905119904119905

] 119864119903119905

= [119862119911120591119903119905

0]

(10)



1 119888


inequality hold


E 120578⊺

(120592) 119877120578 (120592) 120578⊺

(120592) 119877 120578 (120592)

le 1198881997904rArr E 120578

⊺

(119905) 119877120578 (119905) lt 1198882

forall119905 isin [0 119879]

(11)

where 1198882gt 119888

1ge 0 and 119877 gt 0


119905 119904



pound119881 (120578119905 119903

119905 119904

119905)

= limΔrarr0

1

Δ

[E 119881 (120578119905+Δ 119903

119905+Δ 119904

119905+Δ) | 120578

119905 119903

119905 119904

119905

minus119881 (120578119905 119903

119905 119904

119905)]

(12)


Eint119879

0

119890⊺

(119905) 119890 (119905) 119889119905 le 1205742

119890120578119879

Eint119879

0

120596⊺

(119905) 120596 (119905) 119889119905 (13)



Lemma 5 (see [32]) Let 119891119894 R119898

rarr R (119894 = 1 2 119873)



min120573119894|120573119894gt0sum119894120573119894=1

sum

119894

1

120573119894

119891119894(119905) = sum

119894

119891119894(119905) +max

119892119894119895(119905)

sum

119894 = 119895

119892119894119895(119905)

119904119906119887119895119890119888119905 119905119900 119892119894119895R

119898

997888rarrR 119892119895119894(119905) = 119892

119894119895(119905)

[

119891119894(119905) 119892

119894119895(119905)

119892119894119895(119905) 119891

119895(119905)] ge 0

(14)


⊺ and 0 lt 119884 = 119884⊺ then119883+119885⊺

119884minus1

119885 lt 0

if and only if

[

119883 119885⊺

lowast minus119884] lt 0 119900119903 [

minus119884 119885

lowast 119883] lt 0 (15)





119894119898gt 0 119876

119897119894gt 0 (119897 = 1

2) 119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0119867 gt 0 119878

119894119898 scalars

1198881lt 119888

2 119879 gt 0120582

119904gt 0 (119904 = 1 2 10) 120578 gt 0 and Λ gt 0 such


sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895) + sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876 lt 0 (16)

sum

119899isinM119898 = 119899

120587119898119899119883

119894119899+ sum

119895isinN119894 = 119895

120582119894119895119883

119895119898minus 119883 lt 0 (17)


sum

119899isinM119898 = 119899

120587119898119899119884119894119899+ sum

119895isinN119894 = 119895

120582119894119895119884119895119898minus 119884 lt 0 (18)

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

gt 0 (19)

Ξ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980


119884119894119898

ℎ

0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894


ℎ119872119894119898119863119894


]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(20)

Ξ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980


119884119894119898

ℎ

0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894


ℎ119873119894119898119863119894


]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(21)

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (22)

where

Ξ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898

+ 120575119875119894119898+ 119875

119894119898119860

119894+ 119860

⊺

119894119875119894119898+ 119890

120575ℎ

1198761119894

+ 119890120575ℎ

1198762119894+ ℎ119876

119890120575ℎ

minus 1

120575

119883119894119898

+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883 +

119884119894119898

ℎ

Ξ12119894119898

= 119875119894119898119861119894minus

119884119894119898

ℎ

+ 119878119894119898

Ξ22119894119898

= 119903 (120583119898) 119876

2119894+

2119884119894119898

ℎ

minus 119878119894119898minus 119878

⊺

119894119898

Ξ44119894119898

= minus119881119894119898minus 119881

⊺

119894119898+

119890120575ℎ

minus 1

120575

119884119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119884

119903 (120591119898) =

minus (1 minus 120583119898) 119890

120575ℎ119894 if 120583

119898gt 1

minus (1 minus 120583119898) if 120583

119898le 1

Λ = 1205822+ ℎ119890

120575ℎ

(1205823+ 120582

4) + ℎ

2

119890120575ℎ

(1205825+ 120582

6+ 120582

8)

+

1

2

ℎ3

119890120575ℎ

(1205827+ 120582

9)

1205821= min

119894isinN119898isinM120582min (119894119898) 120582

2= max

119894isinN119898isinM120582max (119894119898)

1205823= max

119894isinN120582max (1198761119894

)

1205824= max

119894isinN120582max (1198762119894

) 1205825= 120582max (119876)

1205826= max

119894isinN119898isinM120582max (119883119894119898

) 1205827= 120582max (119883)

1205828= max

119894isinN119898isinM120582max (119894119898) 120582

9= 120582max ()

12058210= 120582max (119867)

119894119898= 119877

minus(12)

119875119894119898119877minus(12)

119876119897119894= 119877

minus(12)

119876119897119894119877minus(12)

(119897 = 1 2)

119876 = 119877minus(12)

119876119877minus(12)

119883119894119898= 119877

minus(12)

119883119894119898119877minus(12)

119883 = 119877minus(12)

119883119877minus(12)

119894119898= 119877

minus(12)

119884119894119898119877minus(12)

= 119877minus(12)

119884119877minus(12)

(23)


119905 119903

119905 119904

119905) 119905 ge

0 by

120578119905(119904) = 120578 (119905 + 119904) 119904 isin [minusℎ 0] (24)


119881 (120578119905 119903

119905 119904

119905) =

4

sum

119897=1

119881119897(120578

119905 119903

119905 119905) (25)

where

1198811(120578

119905 119903

119905 119904

119905) = 120578(119905)

⊺

119890120575119905

119875119903119905119904119905

120578 (119905)

1198812(120578

119905 119903

119905 119904

119905) = int

119905

119905minusℎ

119890120575(119904+ℎ)

120578⊺

(119904) 1198761119903119905

120578 (119904) 119889119904

+ int

119905

119905minus120591119904119905(119905)

119890120575(119904+ℎ)

120578⊺

(119904) 1198762119903119905

120578 (119904) 119889119904

+ int

0

minusℎ

int

119905

119905+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

1198813(120578

119905 119903

119905 119904

119905) = int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)119883119903119905119904119905

120578 (119904) 119889119904 119889120579

+ int

0

minusℎ

int

0

120579

int

119905

119905+120592

119890120575(119904minus120579)

120578⊺

(119904)119883120578 (119904) 119889119904 119889120592 119889120579

1198814(120578

119905 119903

119905 119904

119905) = int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904) 119884119903119905119904119905

120578 (119904) 119889119904 119889120579

+ int

0

minusℎ

int

0

120579

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904) 119884 120578 (119904) 119889119904 119889120592 119889120579

(26)


Then for each 119903119905= 119894 119904

119905= 119898 we have

pound1198811(120578

119905 119894 119898)

= limΔrarr0

+

1

Δ

E 120578⊺

(119905 + Δ) 119890120575(119905+Δ)

119875119903119905+Δ

119904119905+Δ

120578 (119905 + Δ)

minus120578⊺

(119905) 119890120575119905

119875119894119898120578 (119905)

= limΔrarr0

+

1

Δ

E

120578⊺

(119905 + Δ)[

[

(1 + 120582119894119894Δ + 119900 (Δ))

times (1 + 120587119898119898Δ + 119900 (Δ)) 119890

120575(119905+Δ)

119875119894119898

+ (1 + 120582119894119894Δ + 119900 (Δ))

times ( sum

119899isinM

(120587119898119899Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119894119899

+ (1 + 120587119898119898Δ + 119900 (Δ))

times (sum

119895isinN

(120582119894119895Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119895119898

+ ( sum

119899isinM

(120587119898119899Δ + 119900 (Δ)))

times(sum

119895isinN

(120582119894119895Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119895119899

]

]

times 120578 (119905 + Δ)

minus 120578⊺

(119905) 119890120575119905

119875119894119898120578 (119905)

= 120575119890120575119905

120578⊺

(119905) 119875119894119898120578 (119905) + 2119890

120575119905

120578⊺

(119905) 119875119894119898120578 (119905)

+ 119890120575119905

120578⊺

(119905)( sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898)120578 (119905)

= 119890120575119905

120578⊺

(119905)( sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 120575119875

119894119898

+ 119875119894119898119860

119894+ 119860

⊺

119894119875119894119898)120578 (119905)

+ 2119890120575119905

120578⊺

(119905) 119875119894119898119861119894120578 (119905 minus 120591

119898(119905)) + 2119890

120575119905

120578⊺

(119905) 119875119894119898119863

119894120596 (119905)

(27)pound119881

2(120578

119905 119894 119898)

= limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119903119905+Δ

120578 (119904) 119889119904

minusint

119905

119905minusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminus120591119904119905+Δ

(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119903119905+Δ

120578 (119904) 119889119904

minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

Eint0

minusℎ

int

119905+Δ

119905+Δ+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

minusint

0

minusℎ

int

119905

119905+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

= limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

+ int

119905+Δ

119905+Δminusℎ

120578⊺

(119904)

times sum

119895isinN

(120582119894119895Δ + 119900 (Δ))

times 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

minusint

119905

119905minusℎ

120578⊺

(119904) 1198761119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

E[int119905+Δ

119905+Δminus120591119895(119905+Δ)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ sum

119899isinM

(120587119898119899Δ + 119900 (Δ))

times int

119905

119905+Δminus120591119899(119905+Δ)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ int

119905+Δ

minus120591119895(119905 + Δ)

119905+Δ

120578⊺

(119904)

times sum

119895isinN

(120582119894119895Δ + 119900 (Δ))

times 119890120575(119904+ℎ)

1198762119895120578 (119904) 119889119904

minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904]

+ 119890120575119905

120578⊺

(119905) ℎ119876120578 (119905) minus int

119905minusℎ

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904

le 119890120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

minus (1 minus 120591119898(119905)) 119890

120575119905

120578⊺

(119905 minus 120591119898(119905)) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)


+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(28)

Since

0 le 120591119898(119905) le ℎ

119898 (29)

we define

119903 (120591119898) =

minus (1 minus 120583119898) 119890

120575ℎ119898 if 120583

119898gt 1

minus (1 minus 120583119898) if 120583

119898le 1

(30)

Then

pound1198812(120578

119905 119894 119898) le 119890

120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

+ 119890120575119905

120578⊺

(119905 minus 120591119898(119905)) 119903 (120591

119898) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)

times ( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)

+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(31)


pound1198813(120578

119905 119894 119898) le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119883

119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119883

119895119898minus 119883)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119883119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119883120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579119889120592

(32)


minusint

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904 = minusint

119905

119905minus120591119898(119905)

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

le minus120591119898(119905) 119880

⊺

1119883

1198941198981198801

minus (ℎ minus 120591119898(119905)) 119880

⊺

2119883

1198941198981198802

(33)

where

1198801=

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

1198802=

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

lim120591119898(119905)rarr0

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904 = 120578 (119905)

lim120591119898(119905)rarrℎ

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904 = 120578 (119905 minus ℎ)

(34)


119894119898 119873

119894119898 and 119881

119894119898with appro-

priate dimensions

(2120591119898(119905) 119880

⊺

1119872

119894119898+ 2 (ℎ minus 120591

119898(119905)) 119880

⊺

2119873

119894119898+ 2 120578

⊺

(119905) 119881119894119898)

times [minus 120578 (119905) + 119860119894120578 (119905) + 119861

120591119894120578 (119905 minus 120591

119898(119905)) + 119863

119894120596 (119905)] = 0

(35)

pound1198814(120578

119905 119894 119898)

le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119884119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119884119895119898minus 119884)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119884119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119884 120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579 119889120592

(36)



minus int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

= minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

le minus

ℎ

120591119898(119905)

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

minus

ℎ

ℎ minus 120591119898(119905)

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

le minus

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

⊺

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

(37)


pound119881 (120578119905 119903

119905 119904

119905) minus 120575120596

⊺

(119905)119867120596 (119905) le 119890120575119905

120585⊺

(119905) Ξ119894119898120585 (119905) (38)

where

120585⊺

(119905) = [120578⊺

(119905) 120578⊺

(119905 minus 120591119898(119905)) 120578

⊺

(119905 minus ℎ) 120578⊺

(119905) 119880⊺

1 119880

⊺

2 120596

⊺

(119905)]

Ξ119894119898=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860⊺

119894119881

⊺

119894119898120591119898(119905) 119860

⊺

119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

119894119873

⊺

119894119898119875119894119898119863

119894

lowast Ξ22119894119898

119878119894119898minus

119884119894119898

ℎ

119860⊺

120591119894119881

⊺

119894119898120591119898(119905) 119860

⊺

120591119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

120591119894119873

⊺

1198941198980


119884119894119898

ℎ

0 0 0 0


minus120591119898(119905)119872

⊺

119894119898minus (ℎ minus 120591

119898(119905))119873

⊺

119894119898119881119894119898119863

119894


1198941198980 120591

119898(119905)119872

119894119898119863

119894


119894119898(ℎ minus 120591

119898(119905))119873

119894119898119863

119894


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(39)


119898(119905) rarr



120591119898(119905) = ℎ and Ξ

119894119898|120591119898(119905) = 0 Thus we obtain

E pound119881 (120578119905 119903

119905 119904

119905) le 120578E [119881 (120578

119905 119903

119905 119904

119905)] + 120575120596

⊺

(119905)119867120596 (119905)

(40)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

120575120596⊺

(119905)119867120596 (119905) (41)


119890minus120578119905

E [119881 (120578119905 119903

119905 119904

119905)] minus E [119881 (120578

0 119903

0 119904

0)]

le 120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

(42)


= 119877minus(12)

119875119894119898119877minus(12) 119876

119904119894=

119877minus(12)

119876119904119894119877minus(12)

(119904 = 1 2) 119876 = 119877minus(12)

119876119877minus(12)

119883119894119898

= 119877minus(12)

119883119894119898119877minus(12) 119883 = 119877

minus(12)

119883119877minus(12)

119894119898

= 119877minus(12)

119884119894119898119877minus(12) and = 119877

minus(12)


that

E [119881 (1205780 119903

0 119904

0)]


120582max (119894119898) 120578⊺

(0) 119877120578 (0)

+ (max119894isinN

120582max (1198761119894) +max

119894isinN120582max (1198762119894

)) 119890120575ℎ

times int

0

minusℎ

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ

120582max (119876)int0

minusℎ

int

0

120579

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ max

119894isinN119898isinM120582max (119883119894119898

)

times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max (119883)

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579 119889120592

+ 119890120575ℎ max

119894isinN119898isinM120582max (119894119898)


times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max ()

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579 119889120592


120582max (119894119898)

+ ℎ119890120575ℎ

(max119894isinN

120582max (1198761119894+

119894isinNmax120582max (1198762119894

)))

+ ℎ2

119890120575ℎ


120582max (119883119894119898) + 120582max (119876)


120582max (119894119898))

+

1

2

ℎ3

119890120575ℎ

(120582max (119883) + 120582max ())


120578⊺

(119904) 119877120578 (119904) 120578⊺

(119904) 119877 120578 (119904)

= 1198881Λ

(43)


E [119881 (120578119905 119903

119905 119904

119905)] le E [119890

120578119905

119881 (1205780 119903

0 119904

0)]

+ 119890120578119905

120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

le 119890120578119879

1198881Λ + 119889120575119890

120578119879

120582max (119867)int119879

0

119890minus120578119904

119889119904

le 119890120578119879

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

)

(44)


E [119881 (120578119905 119903

119905 119904

119905)] ge E [120578

⊺

(119905) 119890120582119905

119875119894119898120578 (119905)]

ge max119894isinN

120582min (119875119894119898)E [120578⊺

(119905) 119877120578 (119905)]

= 1205821E [120578

⊺

(119905) 119877120578 (119905)]

(45)


E [120578⊺

(119905) 119877120578 (119905)] le

119890120578119879

1205821

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) (46)


E [120578⊺

(119905) 119877120578 (119905)] lt 1198882 (47)


2 119889 119877 119879)




119898(119905) and

120591119898(119905) are enlarged to 120591

119898(119905) le ℎ = maxℎ

119894 119898 isin M and

120591119898(119905) le 120583 = max120583




119905

119905minusℎ

120578⊺

(119904)119884119894119898120578(119904)119889119904 the




119894119898 119883

119894119898 119884


119894 (119894 = 1 2 119873) and119898 (119898 = 1 2 119872)



119894gt 0 119876

119897119894gt 0 (119897 = 1 2)

119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0 119878

119894119898 scalars 119888

1lt 119888

2



Σ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894

0


ℎ119872119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(48)

Σ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119873⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894

0


ℎ119873119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(49)

1198881Λ + 119889120575120574

21

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (50)




pound119881 (120578119905 119903

119905 119904

119905) + 119890

⊺

(119905) 119890 (119905) minus 1205742

120596⊺

(119905) 120596 (119905) le 120585⊺

(119905) Σ119897119894119898120585 (119905)

(119897 = 1 2)

(51)


E pound119881 (120578119905 119903

119905 119904

119905)

le E [120578119881 (120578119905 119903

119905 119904

119905)] + 120574

2

120596⊺

(119905) 120596 (119905) minus E [119890⊺

(119905) 119890 (119905)]

(52)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)]

(53)


119905 119904

119905)] gt 0 by inte-


int

119879

0

119890minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)] 119889119905

le Eint119879

0

pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] 119889119905 le 119881 (120578

0 119903

0 119904

0) = 0

(54)


E [int119879

0

119890minus120578120592

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

E [int119879

0

119890minus120578120592

120596⊺

(120592) 120596 (120592) 119889120592]

(55)

Then it yields that

E [int119879

0

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

119890120578119879

E [int119879

0

120596⊺

(120592) 120596 (120592) 119889120592] (56)



pletes the proof




Filtering



119894119898gt 0119872

119894119898 119873

119894119898 119881

119894119898

119876119897119894gt 0 (119897 = 1 2) 119876 gt 0 119883

119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0

119860119891119894119898

119861119891119894119898

119862119891119894119898

119872119860119894119898

119872119861119894119898

119873119860119894119898

119873119861119894119898

119881119860119894119898

119881119861119894119898

119878119894119898

scalars 1198881lt 119888

2 119879 gt 0 120590

119904gt 0 (119904 = 1 2 9) 120575 gt 0 120578 gt 0


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] gt 0 119872119894119898= [

1198721119894119898

1198722119894119898

1198722119894119898

1198722119894119898

] (57)

119873119894119898= [

1198731119894119898

1198732119894119898

1198732119894119898

1198732119894119898

] 119881119894119898= [

1198811119894119898

1198812119894119898

1198812119894119898

1198812119894119898

] (58)

Γ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ15119894119898

Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ25119894119898

0 [119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898Γ46119894119898

0


Γ56119894119898

0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(59)

Γ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ1015840

15119894119898Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ1015840

251198941198980 [

119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898Γ1015840

461198941198980


Γ1015840

561198941198980



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(60)where

Γ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 119890

120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876 +

119890120575ℎ

minus 1

120575

119883119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883

+

119884119894119898

ℎ

+ [

1205751198751119894119898

+ 1198751119894119898119860

119894+ 119860

⊺

1198941198751119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

1205751198752119894119898

+ 1198752119894119898119860

119894+ 119860

⊺

1198941198752119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

]

(61)


Γ12119894119898

= minus

119884119894119898

ℎ

+ 119878119894119898+ [

1198751119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

]

Γ14119894119898

= [

119860⊺

119894119881

⊺

1119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

119860⊺

119894119881

⊺

2119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

]

Γ15119894119898

= [

ℎ119860⊺

119894119872

⊺

1119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

ℎ119860⊺

119894119872

⊺

2119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

]

Γ1015840

15119894119898= [

ℎ119860⊺

119894119873

⊺

1119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

ℎ119860⊺

119894119873

⊺

2119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

]

Γ16119894119898

= [

1198751119894119898119863

119894+ 119861

119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119861

119891119894119898119863

119910119894

]

Γ17119894119898

= [

119862⊺

119911119894

minus119862119891119894119898

]

Γ24119894119898

= [

119860⊺

120591119894119881

⊺

1119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

119860⊺

120591119894119881

⊺

2119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

]

Γ25119894119898

= [

ℎ119860⊺

120591119894119872

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

ℎ119860⊺

120591119894119872

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

]

Γ1015840

25119894119898= [

ℎ119860⊺

120591119894119873

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

ℎ119860⊺

120591119894119873

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

]

Γ46119894119898

= [

1198811119894119898119863

119894+ 119881

119861119894119898119863

119910119894

1198812119894119898119863

119894+ 119881

119861119894119898119863

119910119894

]

Γ56119894119898

= [

ℎ1198721119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

ℎ1198722119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

]

Γ1015840

56119894119898= [

ℎ1198731119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

ℎ1198732119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

]

(62)


119860119891119894119898

= 119875minus1

2119894119898119860

119891119894119898 119861

119891119894119898= 119875

minus1

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

(63)


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] (64)

The term 119875119894119898119860


119875119894119898119860

119894= [

1198751119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

1198752119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

] (65)

Similarly we have

119875119894119898119861119894= [

1198751119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

]

119875119894119898119863

119894= [

1198751119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

]

(66)

Define 119860119891119894119898

= 1198752119894119898119860

119891119894119898 119861

119891119894119898= 119875

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

119872

119860119894119898= 119872

2119894119898119860

119891119894119898119872

119861119894119898= 119872

2119894119898119861119891119894119898

119873119860119894119898

= 1198732119894119898119860

119891119894119898

119873119861119894119898

= 1198732119894119898119861119891119894119898

119881119860119894119898

= 1198812119894119898119860

119891119894119898 and 119881

119861119894119898= 119881

2119894119898119861119891119894119898



proof is completed


min 1205742

st (48)ndash(50) (67)




1= 1 then


2min

min 1205891205742

+ (1 minus 120589) 1198882

st (59)-(60) and (50)

(68)




Λ = [

minus07 07

09 minus09] Π = [

minus1 1

12 minus12] (69)


1198601= [

minus35 086


2= [

minus25 034

14 minus002]

1198601205911= [

minus08 minus13


1205912= [

minus28 05

minus08 minus14]

1198631= 119863

2= [

01

02] 119862

1199101= [04 07]

1198621199102= [05 08] 119862

1199101205911= [01 02]

1198621199101205912

= [01 01]

1198631199101= 119863

1199102= 01 119862

1199111= [01 005]

1198621199112= [02 009] 119862

1199111205911= [03 06]

1198621199111205912

= [04 08] 1198631199111= 119863

1199112= 005

(70)




11986011989111

= [

minus61254 15565


11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)


6 Conclusions






Acknowledgments


References




































































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119903119905

119860120591119903119905

119863119903119905

119862119910119903119905

119862119910120591119903119905

119863

119910119903119905

119862119911119903119905

119862119911120591119903119905

and119863119911119903119905

are denoted as119860119894119860

120591119894119863

119894119862

119910119894119862

119910120591119894

119863119910119894 119862

119911119894 119862

119911120591119894 and119863



119891(119905) = 119860

119891119903119905119904119905

119909119891(119905) + 119861

119891119903119905119904119905

119910 (119905)

119911119891(119905) = 119862

119891119903119905119904119905

119909119891(119905)

(8)


119891(119896) isin R119902 and the

matrices 119860119891119903119905119904119905

119861119891119903119905119904119905

and 119862119891119903119905119904119905



120578 (119905) = 119860119903119905

120578 (119905) + 119861119903119905

120578 (119905 minus 120591119898(119905)) + 119863

119903119905

120596 (119905)

119890 (119905) = 119862119903119905

120578 (119905) + 119864119903119905

120578 (119905 minus 120591119898(119905))

(9)

where

120578 (119905) = [

119909 (119905)

119909119891(119905)] 119890 (119905) = 119911 (119905) minus 119911

119891(119905)

119860119903119905

= [

119860119903119905

0

119861119891119903119905119904119905

119862119910119903119905

119860119891119903119905119904119905

] 119861119903119905

= [

119860120591119903119905

119861119891119903119905119904119905

119862119910120591119903119905

]

119863119903119905

= [

119863119903119905

119861119891119903119905119904119905

119863119910119903119905

]

119862119903119905

= [119862119911119903119905

minus119862119891119903119905119904119905

] 119864119903119905

= [119862119911120591119903119905

0]

(10)



1 119888


inequality hold


E 120578⊺

(120592) 119877120578 (120592) 120578⊺

(120592) 119877 120578 (120592)

le 1198881997904rArr E 120578

⊺

(119905) 119877120578 (119905) lt 1198882

forall119905 isin [0 119879]

(11)

where 1198882gt 119888

1ge 0 and 119877 gt 0


119905 119904



pound119881 (120578119905 119903

119905 119904

119905)

= limΔrarr0

1

Δ

[E 119881 (120578119905+Δ 119903

119905+Δ 119904

119905+Δ) | 120578

119905 119903

119905 119904

119905

minus119881 (120578119905 119903

119905 119904

119905)]

(12)


Eint119879

0

119890⊺

(119905) 119890 (119905) 119889119905 le 1205742

119890120578119879

Eint119879

0

120596⊺

(119905) 120596 (119905) 119889119905 (13)



Lemma 5 (see [32]) Let 119891119894 R119898

rarr R (119894 = 1 2 119873)



min120573119894|120573119894gt0sum119894120573119894=1

sum

119894

1

120573119894

119891119894(119905) = sum

119894

119891119894(119905) +max

119892119894119895(119905)

sum

119894 = 119895

119892119894119895(119905)

119904119906119887119895119890119888119905 119905119900 119892119894119895R

119898

997888rarrR 119892119895119894(119905) = 119892

119894119895(119905)

[

119891119894(119905) 119892

119894119895(119905)

119892119894119895(119905) 119891

119895(119905)] ge 0

(14)


⊺ and 0 lt 119884 = 119884⊺ then119883+119885⊺

119884minus1

119885 lt 0

if and only if

[

119883 119885⊺

lowast minus119884] lt 0 119900119903 [

minus119884 119885

lowast 119883] lt 0 (15)





119894119898gt 0 119876

119897119894gt 0 (119897 = 1

2) 119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0119867 gt 0 119878

119894119898 scalars

1198881lt 119888

2 119879 gt 0120582

119904gt 0 (119904 = 1 2 10) 120578 gt 0 and Λ gt 0 such


sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895) + sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876 lt 0 (16)

sum

119899isinM119898 = 119899

120587119898119899119883

119894119899+ sum

119895isinN119894 = 119895

120582119894119895119883

119895119898minus 119883 lt 0 (17)


sum

119899isinM119898 = 119899

120587119898119899119884119894119899+ sum

119895isinN119894 = 119895

120582119894119895119884119895119898minus 119884 lt 0 (18)

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

gt 0 (19)

Ξ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980


119884119894119898

ℎ

0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894


ℎ119872119894119898119863119894


]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(20)

Ξ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980


119884119894119898

ℎ

0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894


ℎ119873119894119898119863119894


]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(21)

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (22)

where

Ξ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898

+ 120575119875119894119898+ 119875

119894119898119860

119894+ 119860

⊺

119894119875119894119898+ 119890

120575ℎ

1198761119894

+ 119890120575ℎ

1198762119894+ ℎ119876

119890120575ℎ

minus 1

120575

119883119894119898

+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883 +

119884119894119898

ℎ

Ξ12119894119898

= 119875119894119898119861119894minus

119884119894119898

ℎ

+ 119878119894119898

Ξ22119894119898

= 119903 (120583119898) 119876

2119894+

2119884119894119898

ℎ

minus 119878119894119898minus 119878

⊺

119894119898

Ξ44119894119898

= minus119881119894119898minus 119881

⊺

119894119898+

119890120575ℎ

minus 1

120575

119884119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119884

119903 (120591119898) =

minus (1 minus 120583119898) 119890

120575ℎ119894 if 120583

119898gt 1

minus (1 minus 120583119898) if 120583

119898le 1

Λ = 1205822+ ℎ119890

120575ℎ

(1205823+ 120582

4) + ℎ

2

119890120575ℎ

(1205825+ 120582

6+ 120582

8)

+

1

2

ℎ3

119890120575ℎ

(1205827+ 120582

9)

1205821= min

119894isinN119898isinM120582min (119894119898) 120582

2= max

119894isinN119898isinM120582max (119894119898)

1205823= max

119894isinN120582max (1198761119894

)

1205824= max

119894isinN120582max (1198762119894

) 1205825= 120582max (119876)

1205826= max

119894isinN119898isinM120582max (119883119894119898

) 1205827= 120582max (119883)

1205828= max

119894isinN119898isinM120582max (119894119898) 120582

9= 120582max ()

12058210= 120582max (119867)

119894119898= 119877

minus(12)

119875119894119898119877minus(12)

119876119897119894= 119877

minus(12)

119876119897119894119877minus(12)

(119897 = 1 2)

119876 = 119877minus(12)

119876119877minus(12)

119883119894119898= 119877

minus(12)

119883119894119898119877minus(12)

119883 = 119877minus(12)

119883119877minus(12)

119894119898= 119877

minus(12)

119884119894119898119877minus(12)

= 119877minus(12)

119884119877minus(12)

(23)


119905 119903

119905 119904

119905) 119905 ge

0 by

120578119905(119904) = 120578 (119905 + 119904) 119904 isin [minusℎ 0] (24)


119881 (120578119905 119903

119905 119904

119905) =

4

sum

119897=1

119881119897(120578

119905 119903

119905 119905) (25)

where

1198811(120578

119905 119903

119905 119904

119905) = 120578(119905)

⊺

119890120575119905

119875119903119905119904119905

120578 (119905)

1198812(120578

119905 119903

119905 119904

119905) = int

119905

119905minusℎ

119890120575(119904+ℎ)

120578⊺

(119904) 1198761119903119905

120578 (119904) 119889119904

+ int

119905

119905minus120591119904119905(119905)

119890120575(119904+ℎ)

120578⊺

(119904) 1198762119903119905

120578 (119904) 119889119904

+ int

0

minusℎ

int

119905

119905+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

1198813(120578

119905 119903

119905 119904

119905) = int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)119883119903119905119904119905

120578 (119904) 119889119904 119889120579

+ int

0

minusℎ

int

0

120579

int

119905

119905+120592

119890120575(119904minus120579)

120578⊺

(119904)119883120578 (119904) 119889119904 119889120592 119889120579

1198814(120578

119905 119903

119905 119904

119905) = int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904) 119884119903119905119904119905

120578 (119904) 119889119904 119889120579

+ int

0

minusℎ

int

0

120579

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904) 119884 120578 (119904) 119889119904 119889120592 119889120579

(26)


Then for each 119903119905= 119894 119904

119905= 119898 we have

pound1198811(120578

119905 119894 119898)

= limΔrarr0

+

1

Δ

E 120578⊺

(119905 + Δ) 119890120575(119905+Δ)

119875119903119905+Δ

119904119905+Δ

120578 (119905 + Δ)

minus120578⊺

(119905) 119890120575119905

119875119894119898120578 (119905)

= limΔrarr0

+

1

Δ

E

120578⊺

(119905 + Δ)[

[

(1 + 120582119894119894Δ + 119900 (Δ))

times (1 + 120587119898119898Δ + 119900 (Δ)) 119890

120575(119905+Δ)

119875119894119898

+ (1 + 120582119894119894Δ + 119900 (Δ))

times ( sum

119899isinM

(120587119898119899Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119894119899

+ (1 + 120587119898119898Δ + 119900 (Δ))

times (sum

119895isinN

(120582119894119895Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119895119898

+ ( sum

119899isinM

(120587119898119899Δ + 119900 (Δ)))

times(sum

119895isinN

(120582119894119895Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119895119899

]

]

times 120578 (119905 + Δ)

minus 120578⊺

(119905) 119890120575119905

119875119894119898120578 (119905)

= 120575119890120575119905

120578⊺

(119905) 119875119894119898120578 (119905) + 2119890

120575119905

120578⊺

(119905) 119875119894119898120578 (119905)

+ 119890120575119905

120578⊺

(119905)( sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898)120578 (119905)

= 119890120575119905

120578⊺

(119905)( sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 120575119875

119894119898

+ 119875119894119898119860

119894+ 119860

⊺

119894119875119894119898)120578 (119905)

+ 2119890120575119905

120578⊺

(119905) 119875119894119898119861119894120578 (119905 minus 120591

119898(119905)) + 2119890

120575119905

120578⊺

(119905) 119875119894119898119863

119894120596 (119905)

(27)pound119881

2(120578

119905 119894 119898)

= limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119903119905+Δ

120578 (119904) 119889119904

minusint

119905

119905minusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminus120591119904119905+Δ

(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119903119905+Δ

120578 (119904) 119889119904

minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

Eint0

minusℎ

int

119905+Δ

119905+Δ+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

minusint

0

minusℎ

int

119905

119905+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

= limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

+ int

119905+Δ

119905+Δminusℎ

120578⊺

(119904)

times sum

119895isinN

(120582119894119895Δ + 119900 (Δ))

times 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

minusint

119905

119905minusℎ

120578⊺

(119904) 1198761119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

E[int119905+Δ

119905+Δminus120591119895(119905+Δ)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ sum

119899isinM

(120587119898119899Δ + 119900 (Δ))

times int

119905

119905+Δminus120591119899(119905+Δ)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ int

119905+Δ

minus120591119895(119905 + Δ)

119905+Δ

120578⊺

(119904)

times sum

119895isinN

(120582119894119895Δ + 119900 (Δ))

times 119890120575(119904+ℎ)

1198762119895120578 (119904) 119889119904

minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904]

+ 119890120575119905

120578⊺

(119905) ℎ119876120578 (119905) minus int

119905minusℎ

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904

le 119890120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

minus (1 minus 120591119898(119905)) 119890

120575119905

120578⊺

(119905 minus 120591119898(119905)) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)


+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(28)

Since

0 le 120591119898(119905) le ℎ

119898 (29)

we define

119903 (120591119898) =

minus (1 minus 120583119898) 119890

120575ℎ119898 if 120583

119898gt 1

minus (1 minus 120583119898) if 120583

119898le 1

(30)

Then

pound1198812(120578

119905 119894 119898) le 119890

120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

+ 119890120575119905

120578⊺

(119905 minus 120591119898(119905)) 119903 (120591

119898) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)

times ( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)

+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(31)


pound1198813(120578

119905 119894 119898) le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119883

119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119883

119895119898minus 119883)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119883119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119883120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579119889120592

(32)


minusint

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904 = minusint

119905

119905minus120591119898(119905)

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

le minus120591119898(119905) 119880

⊺

1119883

1198941198981198801

minus (ℎ minus 120591119898(119905)) 119880

⊺

2119883

1198941198981198802

(33)

where

1198801=

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

1198802=

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

lim120591119898(119905)rarr0

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904 = 120578 (119905)

lim120591119898(119905)rarrℎ

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904 = 120578 (119905 minus ℎ)

(34)


119894119898 119873

119894119898 and 119881

119894119898with appro-

priate dimensions

(2120591119898(119905) 119880

⊺

1119872

119894119898+ 2 (ℎ minus 120591

119898(119905)) 119880

⊺

2119873

119894119898+ 2 120578

⊺

(119905) 119881119894119898)

times [minus 120578 (119905) + 119860119894120578 (119905) + 119861

120591119894120578 (119905 minus 120591

119898(119905)) + 119863

119894120596 (119905)] = 0

(35)

pound1198814(120578

119905 119894 119898)

le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119884119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119884119895119898minus 119884)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119884119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119884 120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579 119889120592

(36)



minus int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

= minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

le minus

ℎ

120591119898(119905)

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

minus

ℎ

ℎ minus 120591119898(119905)

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

le minus

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

⊺

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

(37)


pound119881 (120578119905 119903

119905 119904

119905) minus 120575120596

⊺

(119905)119867120596 (119905) le 119890120575119905

120585⊺

(119905) Ξ119894119898120585 (119905) (38)

where

120585⊺

(119905) = [120578⊺

(119905) 120578⊺

(119905 minus 120591119898(119905)) 120578

⊺

(119905 minus ℎ) 120578⊺

(119905) 119880⊺

1 119880

⊺

2 120596

⊺

(119905)]

Ξ119894119898=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860⊺

119894119881

⊺

119894119898120591119898(119905) 119860

⊺

119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

119894119873

⊺

119894119898119875119894119898119863

119894

lowast Ξ22119894119898

119878119894119898minus

119884119894119898

ℎ

119860⊺

120591119894119881

⊺

119894119898120591119898(119905) 119860

⊺

120591119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

120591119894119873

⊺

1198941198980


119884119894119898

ℎ

0 0 0 0


minus120591119898(119905)119872

⊺

119894119898minus (ℎ minus 120591

119898(119905))119873

⊺

119894119898119881119894119898119863

119894


1198941198980 120591

119898(119905)119872

119894119898119863

119894


119894119898(ℎ minus 120591

119898(119905))119873

119894119898119863

119894


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(39)


119898(119905) rarr



120591119898(119905) = ℎ and Ξ

119894119898|120591119898(119905) = 0 Thus we obtain

E pound119881 (120578119905 119903

119905 119904

119905) le 120578E [119881 (120578

119905 119903

119905 119904

119905)] + 120575120596

⊺

(119905)119867120596 (119905)

(40)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

120575120596⊺

(119905)119867120596 (119905) (41)


119890minus120578119905

E [119881 (120578119905 119903

119905 119904

119905)] minus E [119881 (120578

0 119903

0 119904

0)]

le 120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

(42)


= 119877minus(12)

119875119894119898119877minus(12) 119876

119904119894=

119877minus(12)

119876119904119894119877minus(12)

(119904 = 1 2) 119876 = 119877minus(12)

119876119877minus(12)

119883119894119898

= 119877minus(12)

119883119894119898119877minus(12) 119883 = 119877

minus(12)

119883119877minus(12)

119894119898

= 119877minus(12)

119884119894119898119877minus(12) and = 119877

minus(12)


that

E [119881 (1205780 119903

0 119904

0)]


120582max (119894119898) 120578⊺

(0) 119877120578 (0)

+ (max119894isinN

120582max (1198761119894) +max

119894isinN120582max (1198762119894

)) 119890120575ℎ

times int

0

minusℎ

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ

120582max (119876)int0

minusℎ

int

0

120579

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ max

119894isinN119898isinM120582max (119883119894119898

)

times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max (119883)

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579 119889120592

+ 119890120575ℎ max

119894isinN119898isinM120582max (119894119898)


times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max ()

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579 119889120592


120582max (119894119898)

+ ℎ119890120575ℎ

(max119894isinN

120582max (1198761119894+

119894isinNmax120582max (1198762119894

)))

+ ℎ2

119890120575ℎ


120582max (119883119894119898) + 120582max (119876)


120582max (119894119898))

+

1

2

ℎ3

119890120575ℎ

(120582max (119883) + 120582max ())


120578⊺

(119904) 119877120578 (119904) 120578⊺

(119904) 119877 120578 (119904)

= 1198881Λ

(43)


E [119881 (120578119905 119903

119905 119904

119905)] le E [119890

120578119905

119881 (1205780 119903

0 119904

0)]

+ 119890120578119905

120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

le 119890120578119879

1198881Λ + 119889120575119890

120578119879

120582max (119867)int119879

0

119890minus120578119904

119889119904

le 119890120578119879

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

)

(44)


E [119881 (120578119905 119903

119905 119904

119905)] ge E [120578

⊺

(119905) 119890120582119905

119875119894119898120578 (119905)]

ge max119894isinN

120582min (119875119894119898)E [120578⊺

(119905) 119877120578 (119905)]

= 1205821E [120578

⊺

(119905) 119877120578 (119905)]

(45)


E [120578⊺

(119905) 119877120578 (119905)] le

119890120578119879

1205821

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) (46)


E [120578⊺

(119905) 119877120578 (119905)] lt 1198882 (47)


2 119889 119877 119879)




119898(119905) and

120591119898(119905) are enlarged to 120591

119898(119905) le ℎ = maxℎ

119894 119898 isin M and

120591119898(119905) le 120583 = max120583




119905

119905minusℎ

120578⊺

(119904)119884119894119898120578(119904)119889119904 the




119894119898 119883

119894119898 119884


119894 (119894 = 1 2 119873) and119898 (119898 = 1 2 119872)



119894gt 0 119876

119897119894gt 0 (119897 = 1 2)

119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0 119878

119894119898 scalars 119888

1lt 119888

2



Σ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894

0


ℎ119872119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(48)

Σ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119873⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894

0


ℎ119873119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(49)

1198881Λ + 119889120575120574

21

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (50)




pound119881 (120578119905 119903

119905 119904

119905) + 119890

⊺

(119905) 119890 (119905) minus 1205742

120596⊺

(119905) 120596 (119905) le 120585⊺

(119905) Σ119897119894119898120585 (119905)

(119897 = 1 2)

(51)


E pound119881 (120578119905 119903

119905 119904

119905)

le E [120578119881 (120578119905 119903

119905 119904

119905)] + 120574

2

120596⊺

(119905) 120596 (119905) minus E [119890⊺

(119905) 119890 (119905)]

(52)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)]

(53)


119905 119904

119905)] gt 0 by inte-


int

119879

0

119890minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)] 119889119905

le Eint119879

0

pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] 119889119905 le 119881 (120578

0 119903

0 119904

0) = 0

(54)


E [int119879

0

119890minus120578120592

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

E [int119879

0

119890minus120578120592

120596⊺

(120592) 120596 (120592) 119889120592]

(55)

Then it yields that

E [int119879

0

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

119890120578119879

E [int119879

0

120596⊺

(120592) 120596 (120592) 119889120592] (56)



pletes the proof




Filtering



119894119898gt 0119872

119894119898 119873

119894119898 119881

119894119898

119876119897119894gt 0 (119897 = 1 2) 119876 gt 0 119883

119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0

119860119891119894119898

119861119891119894119898

119862119891119894119898

119872119860119894119898

119872119861119894119898

119873119860119894119898

119873119861119894119898

119881119860119894119898

119881119861119894119898

119878119894119898

scalars 1198881lt 119888

2 119879 gt 0 120590

119904gt 0 (119904 = 1 2 9) 120575 gt 0 120578 gt 0


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] gt 0 119872119894119898= [

1198721119894119898

1198722119894119898

1198722119894119898

1198722119894119898

] (57)

119873119894119898= [

1198731119894119898

1198732119894119898

1198732119894119898

1198732119894119898

] 119881119894119898= [

1198811119894119898

1198812119894119898

1198812119894119898

1198812119894119898

] (58)

Γ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ15119894119898

Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ25119894119898

0 [119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898Γ46119894119898

0


Γ56119894119898

0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(59)

Γ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ1015840

15119894119898Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ1015840

251198941198980 [

119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898Γ1015840

461198941198980


Γ1015840

561198941198980



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(60)where

Γ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 119890

120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876 +

119890120575ℎ

minus 1

120575

119883119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883

+

119884119894119898

ℎ

+ [

1205751198751119894119898

+ 1198751119894119898119860

119894+ 119860

⊺

1198941198751119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

1205751198752119894119898

+ 1198752119894119898119860

119894+ 119860

⊺

1198941198752119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

]

(61)


Γ12119894119898

= minus

119884119894119898

ℎ

+ 119878119894119898+ [

1198751119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

]

Γ14119894119898

= [

119860⊺

119894119881

⊺

1119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

119860⊺

119894119881

⊺

2119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

]

Γ15119894119898

= [

ℎ119860⊺

119894119872

⊺

1119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

ℎ119860⊺

119894119872

⊺

2119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

]

Γ1015840

15119894119898= [

ℎ119860⊺

119894119873

⊺

1119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

ℎ119860⊺

119894119873

⊺

2119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

]

Γ16119894119898

= [

1198751119894119898119863

119894+ 119861

119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119861

119891119894119898119863

119910119894

]

Γ17119894119898

= [

119862⊺

119911119894

minus119862119891119894119898

]

Γ24119894119898

= [

119860⊺

120591119894119881

⊺

1119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

119860⊺

120591119894119881

⊺

2119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

]

Γ25119894119898

= [

ℎ119860⊺

120591119894119872

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

ℎ119860⊺

120591119894119872

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

]

Γ1015840

25119894119898= [

ℎ119860⊺

120591119894119873

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

ℎ119860⊺

120591119894119873

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

]

Γ46119894119898

= [

1198811119894119898119863

119894+ 119881

119861119894119898119863

119910119894

1198812119894119898119863

119894+ 119881

119861119894119898119863

119910119894

]

Γ56119894119898

= [

ℎ1198721119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

ℎ1198722119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

]

Γ1015840

56119894119898= [

ℎ1198731119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

ℎ1198732119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

]

(62)


119860119891119894119898

= 119875minus1

2119894119898119860

119891119894119898 119861

119891119894119898= 119875

minus1

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

(63)


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] (64)

The term 119875119894119898119860


119875119894119898119860

119894= [

1198751119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

1198752119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

] (65)

Similarly we have

119875119894119898119861119894= [

1198751119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

]

119875119894119898119863

119894= [

1198751119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

]

(66)

Define 119860119891119894119898

= 1198752119894119898119860

119891119894119898 119861

119891119894119898= 119875

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

119872

119860119894119898= 119872

2119894119898119860

119891119894119898119872

119861119894119898= 119872

2119894119898119861119891119894119898

119873119860119894119898

= 1198732119894119898119860

119891119894119898

119873119861119894119898

= 1198732119894119898119861119891119894119898

119881119860119894119898

= 1198812119894119898119860

119891119894119898 and 119881

119861119894119898= 119881

2119894119898119861119891119894119898



proof is completed


min 1205742

st (48)ndash(50) (67)




1= 1 then


2min

min 1205891205742

+ (1 minus 120589) 1198882

st (59)-(60) and (50)

(68)




Λ = [

minus07 07

09 minus09] Π = [

minus1 1

12 minus12] (69)


1198601= [

minus35 086


2= [

minus25 034

14 minus002]

1198601205911= [

minus08 minus13


1205912= [

minus28 05

minus08 minus14]

1198631= 119863

2= [

01

02] 119862

1199101= [04 07]

1198621199102= [05 08] 119862

1199101205911= [01 02]

1198621199101205912

= [01 01]

1198631199101= 119863

1199102= 01 119862

1199111= [01 005]

1198621199112= [02 009] 119862

1199111205911= [03 06]

1198621199111205912

= [04 08] 1198631199111= 119863

1199112= 005

(70)




11986011989111

= [

minus61254 15565


11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)


6 Conclusions






Acknowledgments


References




































































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










sum

119899isinM119898 = 119899

120587119898119899119884119894119899+ sum

119895isinN119894 = 119895

120582119894119895119884119895119898minus 119884 lt 0 (18)

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

gt 0 (19)

Ξ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980


119884119894119898

ℎ

0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894


ℎ119872119894119898119863119894


]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(20)

Ξ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980


119884119894119898

ℎ

0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894


ℎ119873119894119898119863119894


]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(21)

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (22)

where

Ξ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898

+ 120575119875119894119898+ 119875

119894119898119860

119894+ 119860

⊺

119894119875119894119898+ 119890

120575ℎ

1198761119894

+ 119890120575ℎ

1198762119894+ ℎ119876

119890120575ℎ

minus 1

120575

119883119894119898

+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883 +

119884119894119898

ℎ

Ξ12119894119898

= 119875119894119898119861119894minus

119884119894119898

ℎ

+ 119878119894119898

Ξ22119894119898

= 119903 (120583119898) 119876

2119894+

2119884119894119898

ℎ

minus 119878119894119898minus 119878

⊺

119894119898

Ξ44119894119898

= minus119881119894119898minus 119881

⊺

119894119898+

119890120575ℎ

minus 1

120575

119884119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119884

119903 (120591119898) =

minus (1 minus 120583119898) 119890

120575ℎ119894 if 120583

119898gt 1

minus (1 minus 120583119898) if 120583

119898le 1

Λ = 1205822+ ℎ119890

120575ℎ

(1205823+ 120582

4) + ℎ

2

119890120575ℎ

(1205825+ 120582

6+ 120582

8)

+

1

2

ℎ3

119890120575ℎ

(1205827+ 120582

9)

1205821= min

119894isinN119898isinM120582min (119894119898) 120582

2= max

119894isinN119898isinM120582max (119894119898)

1205823= max

119894isinN120582max (1198761119894

)

1205824= max

119894isinN120582max (1198762119894

) 1205825= 120582max (119876)

1205826= max

119894isinN119898isinM120582max (119883119894119898

) 1205827= 120582max (119883)

1205828= max

119894isinN119898isinM120582max (119894119898) 120582

9= 120582max ()

12058210= 120582max (119867)

119894119898= 119877

minus(12)

119875119894119898119877minus(12)

119876119897119894= 119877

minus(12)

119876119897119894119877minus(12)

(119897 = 1 2)

119876 = 119877minus(12)

119876119877minus(12)

119883119894119898= 119877

minus(12)

119883119894119898119877minus(12)

119883 = 119877minus(12)

119883119877minus(12)

119894119898= 119877

minus(12)

119884119894119898119877minus(12)

= 119877minus(12)

119884119877minus(12)

(23)


119905 119903

119905 119904

119905) 119905 ge

0 by

120578119905(119904) = 120578 (119905 + 119904) 119904 isin [minusℎ 0] (24)


119881 (120578119905 119903

119905 119904

119905) =

4

sum

119897=1

119881119897(120578

119905 119903

119905 119905) (25)

where

1198811(120578

119905 119903

119905 119904

119905) = 120578(119905)

⊺

119890120575119905

119875119903119905119904119905

120578 (119905)

1198812(120578

119905 119903

119905 119904

119905) = int

119905

119905minusℎ

119890120575(119904+ℎ)

120578⊺

(119904) 1198761119903119905

120578 (119904) 119889119904

+ int

119905

119905minus120591119904119905(119905)

119890120575(119904+ℎ)

120578⊺

(119904) 1198762119903119905

120578 (119904) 119889119904

+ int

0

minusℎ

int

119905

119905+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

1198813(120578

119905 119903

119905 119904

119905) = int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)119883119903119905119904119905

120578 (119904) 119889119904 119889120579

+ int

0

minusℎ

int

0

120579

int

119905

119905+120592

119890120575(119904minus120579)

120578⊺

(119904)119883120578 (119904) 119889119904 119889120592 119889120579

1198814(120578

119905 119903

119905 119904

119905) = int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904) 119884119903119905119904119905

120578 (119904) 119889119904 119889120579

+ int

0

minusℎ

int

0

120579

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904) 119884 120578 (119904) 119889119904 119889120592 119889120579

(26)


Then for each 119903119905= 119894 119904

119905= 119898 we have

pound1198811(120578

119905 119894 119898)

= limΔrarr0

+

1

Δ

E 120578⊺

(119905 + Δ) 119890120575(119905+Δ)

119875119903119905+Δ

119904119905+Δ

120578 (119905 + Δ)

minus120578⊺

(119905) 119890120575119905

119875119894119898120578 (119905)

= limΔrarr0

+

1

Δ

E

120578⊺

(119905 + Δ)[

[

(1 + 120582119894119894Δ + 119900 (Δ))

times (1 + 120587119898119898Δ + 119900 (Δ)) 119890

120575(119905+Δ)

119875119894119898

+ (1 + 120582119894119894Δ + 119900 (Δ))

times ( sum

119899isinM

(120587119898119899Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119894119899

+ (1 + 120587119898119898Δ + 119900 (Δ))

times (sum

119895isinN

(120582119894119895Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119895119898

+ ( sum

119899isinM

(120587119898119899Δ + 119900 (Δ)))

times(sum

119895isinN

(120582119894119895Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119895119899

]

]

times 120578 (119905 + Δ)

minus 120578⊺

(119905) 119890120575119905

119875119894119898120578 (119905)

= 120575119890120575119905

120578⊺

(119905) 119875119894119898120578 (119905) + 2119890

120575119905

120578⊺

(119905) 119875119894119898120578 (119905)

+ 119890120575119905

120578⊺

(119905)( sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898)120578 (119905)

= 119890120575119905

120578⊺

(119905)( sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 120575119875

119894119898

+ 119875119894119898119860

119894+ 119860

⊺

119894119875119894119898)120578 (119905)

+ 2119890120575119905

120578⊺

(119905) 119875119894119898119861119894120578 (119905 minus 120591

119898(119905)) + 2119890

120575119905

120578⊺

(119905) 119875119894119898119863

119894120596 (119905)

(27)pound119881

2(120578

119905 119894 119898)

= limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119903119905+Δ

120578 (119904) 119889119904

minusint

119905

119905minusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminus120591119904119905+Δ

(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119903119905+Δ

120578 (119904) 119889119904

minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

Eint0

minusℎ

int

119905+Δ

119905+Δ+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

minusint

0

minusℎ

int

119905

119905+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

= limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

+ int

119905+Δ

119905+Δminusℎ

120578⊺

(119904)

times sum

119895isinN

(120582119894119895Δ + 119900 (Δ))

times 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

minusint

119905

119905minusℎ

120578⊺

(119904) 1198761119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

E[int119905+Δ

119905+Δminus120591119895(119905+Δ)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ sum

119899isinM

(120587119898119899Δ + 119900 (Δ))

times int

119905

119905+Δminus120591119899(119905+Δ)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ int

119905+Δ

minus120591119895(119905 + Δ)

119905+Δ

120578⊺

(119904)

times sum

119895isinN

(120582119894119895Δ + 119900 (Δ))

times 119890120575(119904+ℎ)

1198762119895120578 (119904) 119889119904

minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904]

+ 119890120575119905

120578⊺

(119905) ℎ119876120578 (119905) minus int

119905minusℎ

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904

le 119890120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

minus (1 minus 120591119898(119905)) 119890

120575119905

120578⊺

(119905 minus 120591119898(119905)) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)


+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(28)

Since

0 le 120591119898(119905) le ℎ

119898 (29)

we define

119903 (120591119898) =

minus (1 minus 120583119898) 119890

120575ℎ119898 if 120583

119898gt 1

minus (1 minus 120583119898) if 120583

119898le 1

(30)

Then

pound1198812(120578

119905 119894 119898) le 119890

120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

+ 119890120575119905

120578⊺

(119905 minus 120591119898(119905)) 119903 (120591

119898) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)

times ( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)

+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(31)


pound1198813(120578

119905 119894 119898) le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119883

119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119883

119895119898minus 119883)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119883119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119883120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579119889120592

(32)


minusint

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904 = minusint

119905

119905minus120591119898(119905)

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

le minus120591119898(119905) 119880

⊺

1119883

1198941198981198801

minus (ℎ minus 120591119898(119905)) 119880

⊺

2119883

1198941198981198802

(33)

where

1198801=

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

1198802=

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

lim120591119898(119905)rarr0

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904 = 120578 (119905)

lim120591119898(119905)rarrℎ

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904 = 120578 (119905 minus ℎ)

(34)


119894119898 119873

119894119898 and 119881

119894119898with appro-

priate dimensions

(2120591119898(119905) 119880

⊺

1119872

119894119898+ 2 (ℎ minus 120591

119898(119905)) 119880

⊺

2119873

119894119898+ 2 120578

⊺

(119905) 119881119894119898)

times [minus 120578 (119905) + 119860119894120578 (119905) + 119861

120591119894120578 (119905 minus 120591

119898(119905)) + 119863

119894120596 (119905)] = 0

(35)

pound1198814(120578

119905 119894 119898)

le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119884119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119884119895119898minus 119884)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119884119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119884 120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579 119889120592

(36)



minus int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

= minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

le minus

ℎ

120591119898(119905)

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

minus

ℎ

ℎ minus 120591119898(119905)

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

le minus

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

⊺

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

(37)


pound119881 (120578119905 119903

119905 119904

119905) minus 120575120596

⊺

(119905)119867120596 (119905) le 119890120575119905

120585⊺

(119905) Ξ119894119898120585 (119905) (38)

where

120585⊺

(119905) = [120578⊺

(119905) 120578⊺

(119905 minus 120591119898(119905)) 120578

⊺

(119905 minus ℎ) 120578⊺

(119905) 119880⊺

1 119880

⊺

2 120596

⊺

(119905)]

Ξ119894119898=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860⊺

119894119881

⊺

119894119898120591119898(119905) 119860

⊺

119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

119894119873

⊺

119894119898119875119894119898119863

119894

lowast Ξ22119894119898

119878119894119898minus

119884119894119898

ℎ

119860⊺

120591119894119881

⊺

119894119898120591119898(119905) 119860

⊺

120591119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

120591119894119873

⊺

1198941198980


119884119894119898

ℎ

0 0 0 0


minus120591119898(119905)119872

⊺

119894119898minus (ℎ minus 120591

119898(119905))119873

⊺

119894119898119881119894119898119863

119894


1198941198980 120591

119898(119905)119872

119894119898119863

119894


119894119898(ℎ minus 120591

119898(119905))119873

119894119898119863

119894


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(39)


119898(119905) rarr



120591119898(119905) = ℎ and Ξ

119894119898|120591119898(119905) = 0 Thus we obtain

E pound119881 (120578119905 119903

119905 119904

119905) le 120578E [119881 (120578

119905 119903

119905 119904

119905)] + 120575120596

⊺

(119905)119867120596 (119905)

(40)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

120575120596⊺

(119905)119867120596 (119905) (41)


119890minus120578119905

E [119881 (120578119905 119903

119905 119904

119905)] minus E [119881 (120578

0 119903

0 119904

0)]

le 120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

(42)


= 119877minus(12)

119875119894119898119877minus(12) 119876

119904119894=

119877minus(12)

119876119904119894119877minus(12)

(119904 = 1 2) 119876 = 119877minus(12)

119876119877minus(12)

119883119894119898

= 119877minus(12)

119883119894119898119877minus(12) 119883 = 119877

minus(12)

119883119877minus(12)

119894119898

= 119877minus(12)

119884119894119898119877minus(12) and = 119877

minus(12)


that

E [119881 (1205780 119903

0 119904

0)]


120582max (119894119898) 120578⊺

(0) 119877120578 (0)

+ (max119894isinN

120582max (1198761119894) +max

119894isinN120582max (1198762119894

)) 119890120575ℎ

times int

0

minusℎ

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ

120582max (119876)int0

minusℎ

int

0

120579

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ max

119894isinN119898isinM120582max (119883119894119898

)

times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max (119883)

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579 119889120592

+ 119890120575ℎ max

119894isinN119898isinM120582max (119894119898)


times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max ()

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579 119889120592


120582max (119894119898)

+ ℎ119890120575ℎ

(max119894isinN

120582max (1198761119894+

119894isinNmax120582max (1198762119894

)))

+ ℎ2

119890120575ℎ


120582max (119883119894119898) + 120582max (119876)


120582max (119894119898))

+

1

2

ℎ3

119890120575ℎ

(120582max (119883) + 120582max ())


120578⊺

(119904) 119877120578 (119904) 120578⊺

(119904) 119877 120578 (119904)

= 1198881Λ

(43)


E [119881 (120578119905 119903

119905 119904

119905)] le E [119890

120578119905

119881 (1205780 119903

0 119904

0)]

+ 119890120578119905

120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

le 119890120578119879

1198881Λ + 119889120575119890

120578119879

120582max (119867)int119879

0

119890minus120578119904

119889119904

le 119890120578119879

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

)

(44)


E [119881 (120578119905 119903

119905 119904

119905)] ge E [120578

⊺

(119905) 119890120582119905

119875119894119898120578 (119905)]

ge max119894isinN

120582min (119875119894119898)E [120578⊺

(119905) 119877120578 (119905)]

= 1205821E [120578

⊺

(119905) 119877120578 (119905)]

(45)


E [120578⊺

(119905) 119877120578 (119905)] le

119890120578119879

1205821

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) (46)


E [120578⊺

(119905) 119877120578 (119905)] lt 1198882 (47)


2 119889 119877 119879)




119898(119905) and

120591119898(119905) are enlarged to 120591

119898(119905) le ℎ = maxℎ

119894 119898 isin M and

120591119898(119905) le 120583 = max120583




119905

119905minusℎ

120578⊺

(119904)119884119894119898120578(119904)119889119904 the




119894119898 119883

119894119898 119884


119894 (119894 = 1 2 119873) and119898 (119898 = 1 2 119872)



119894gt 0 119876

119897119894gt 0 (119897 = 1 2)

119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0 119878

119894119898 scalars 119888

1lt 119888

2



Σ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894

0


ℎ119872119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(48)

Σ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119873⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894

0


ℎ119873119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(49)

1198881Λ + 119889120575120574

21

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (50)




pound119881 (120578119905 119903

119905 119904

119905) + 119890

⊺

(119905) 119890 (119905) minus 1205742

120596⊺

(119905) 120596 (119905) le 120585⊺

(119905) Σ119897119894119898120585 (119905)

(119897 = 1 2)

(51)


E pound119881 (120578119905 119903

119905 119904

119905)

le E [120578119881 (120578119905 119903

119905 119904

119905)] + 120574

2

120596⊺

(119905) 120596 (119905) minus E [119890⊺

(119905) 119890 (119905)]

(52)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)]

(53)


119905 119904

119905)] gt 0 by inte-


int

119879

0

119890minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)] 119889119905

le Eint119879

0

pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] 119889119905 le 119881 (120578

0 119903

0 119904

0) = 0

(54)


E [int119879

0

119890minus120578120592

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

E [int119879

0

119890minus120578120592

120596⊺

(120592) 120596 (120592) 119889120592]

(55)

Then it yields that

E [int119879

0

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

119890120578119879

E [int119879

0

120596⊺

(120592) 120596 (120592) 119889120592] (56)



pletes the proof




Filtering



119894119898gt 0119872

119894119898 119873

119894119898 119881

119894119898

119876119897119894gt 0 (119897 = 1 2) 119876 gt 0 119883

119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0

119860119891119894119898

119861119891119894119898

119862119891119894119898

119872119860119894119898

119872119861119894119898

119873119860119894119898

119873119861119894119898

119881119860119894119898

119881119861119894119898

119878119894119898

scalars 1198881lt 119888

2 119879 gt 0 120590

119904gt 0 (119904 = 1 2 9) 120575 gt 0 120578 gt 0


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] gt 0 119872119894119898= [

1198721119894119898

1198722119894119898

1198722119894119898

1198722119894119898

] (57)

119873119894119898= [

1198731119894119898

1198732119894119898

1198732119894119898

1198732119894119898

] 119881119894119898= [

1198811119894119898

1198812119894119898

1198812119894119898

1198812119894119898

] (58)

Γ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ15119894119898

Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ25119894119898

0 [119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898Γ46119894119898

0


Γ56119894119898

0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(59)

Γ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ1015840

15119894119898Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ1015840

251198941198980 [

119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898Γ1015840

461198941198980


Γ1015840

561198941198980



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(60)where

Γ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 119890

120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876 +

119890120575ℎ

minus 1

120575

119883119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883

+

119884119894119898

ℎ

+ [

1205751198751119894119898

+ 1198751119894119898119860

119894+ 119860

⊺

1198941198751119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

1205751198752119894119898

+ 1198752119894119898119860

119894+ 119860

⊺

1198941198752119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

]

(61)


Γ12119894119898

= minus

119884119894119898

ℎ

+ 119878119894119898+ [

1198751119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

]

Γ14119894119898

= [

119860⊺

119894119881

⊺

1119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

119860⊺

119894119881

⊺

2119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

]

Γ15119894119898

= [

ℎ119860⊺

119894119872

⊺

1119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

ℎ119860⊺

119894119872

⊺

2119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

]

Γ1015840

15119894119898= [

ℎ119860⊺

119894119873

⊺

1119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

ℎ119860⊺

119894119873

⊺

2119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

]

Γ16119894119898

= [

1198751119894119898119863

119894+ 119861

119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119861

119891119894119898119863

119910119894

]

Γ17119894119898

= [

119862⊺

119911119894

minus119862119891119894119898

]

Γ24119894119898

= [

119860⊺

120591119894119881

⊺

1119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

119860⊺

120591119894119881

⊺

2119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

]

Γ25119894119898

= [

ℎ119860⊺

120591119894119872

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

ℎ119860⊺

120591119894119872

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

]

Γ1015840

25119894119898= [

ℎ119860⊺

120591119894119873

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

ℎ119860⊺

120591119894119873

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

]

Γ46119894119898

= [

1198811119894119898119863

119894+ 119881

119861119894119898119863

119910119894

1198812119894119898119863

119894+ 119881

119861119894119898119863

119910119894

]

Γ56119894119898

= [

ℎ1198721119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

ℎ1198722119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

]

Γ1015840

56119894119898= [

ℎ1198731119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

ℎ1198732119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

]

(62)


119860119891119894119898

= 119875minus1

2119894119898119860

119891119894119898 119861

119891119894119898= 119875

minus1

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

(63)


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] (64)

The term 119875119894119898119860


119875119894119898119860

119894= [

1198751119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

1198752119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

] (65)

Similarly we have

119875119894119898119861119894= [

1198751119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

]

119875119894119898119863

119894= [

1198751119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

]

(66)

Define 119860119891119894119898

= 1198752119894119898119860

119891119894119898 119861

119891119894119898= 119875

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

119872

119860119894119898= 119872

2119894119898119860

119891119894119898119872

119861119894119898= 119872

2119894119898119861119891119894119898

119873119860119894119898

= 1198732119894119898119860

119891119894119898

119873119861119894119898

= 1198732119894119898119861119891119894119898

119881119860119894119898

= 1198812119894119898119860

119891119894119898 and 119881

119861119894119898= 119881

2119894119898119861119891119894119898



proof is completed


min 1205742

st (48)ndash(50) (67)




1= 1 then


2min

min 1205891205742

+ (1 minus 120589) 1198882

st (59)-(60) and (50)

(68)




Λ = [

minus07 07

09 minus09] Π = [

minus1 1

12 minus12] (69)


1198601= [

minus35 086


2= [

minus25 034

14 minus002]

1198601205911= [

minus08 minus13


1205912= [

minus28 05

minus08 minus14]

1198631= 119863

2= [

01

02] 119862

1199101= [04 07]

1198621199102= [05 08] 119862

1199101205911= [01 02]

1198621199101205912

= [01 01]

1198631199101= 119863

1199102= 01 119862

1199111= [01 005]

1198621199112= [02 009] 119862

1199111205911= [03 06]

1198621199111205912

= [04 08] 1198631199111= 119863

1199112= 005

(70)




11986011989111

= [

minus61254 15565


11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)


6 Conclusions






Acknowledgments


References




































































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Then for each 119903119905= 119894 119904

119905= 119898 we have

pound1198811(120578

119905 119894 119898)

= limΔrarr0

+

1

Δ

E 120578⊺

(119905 + Δ) 119890120575(119905+Δ)

119875119903119905+Δ

119904119905+Δ

120578 (119905 + Δ)

minus120578⊺

(119905) 119890120575119905

119875119894119898120578 (119905)

= limΔrarr0

+

1

Δ

E

120578⊺

(119905 + Δ)[

[

(1 + 120582119894119894Δ + 119900 (Δ))

times (1 + 120587119898119898Δ + 119900 (Δ)) 119890

120575(119905+Δ)

119875119894119898

+ (1 + 120582119894119894Δ + 119900 (Δ))

times ( sum

119899isinM

(120587119898119899Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119894119899

+ (1 + 120587119898119898Δ + 119900 (Δ))

times (sum

119895isinN

(120582119894119895Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119895119898

+ ( sum

119899isinM

(120587119898119899Δ + 119900 (Δ)))

times(sum

119895isinN

(120582119894119895Δ + 119900 (Δ))) 119890

120575(119905+Δ)

119875119895119899

]

]

times 120578 (119905 + Δ)

minus 120578⊺

(119905) 119890120575119905

119875119894119898120578 (119905)

= 120575119890120575119905

120578⊺

(119905) 119875119894119898120578 (119905) + 2119890

120575119905

120578⊺

(119905) 119875119894119898120578 (119905)

+ 119890120575119905

120578⊺

(119905)( sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898)120578 (119905)

= 119890120575119905

120578⊺

(119905)( sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 120575119875

119894119898

+ 119875119894119898119860

119894+ 119860

⊺

119894119875119894119898)120578 (119905)

+ 2119890120575119905

120578⊺

(119905) 119875119894119898119861119894120578 (119905 minus 120591

119898(119905)) + 2119890

120575119905

120578⊺

(119905) 119875119894119898119863

119894120596 (119905)

(27)pound119881

2(120578

119905 119894 119898)

= limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119903119905+Δ

120578 (119904) 119889119904

minusint

119905

119905minusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminus120591119904119905+Δ

(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119903119905+Δ

120578 (119904) 119889119904

minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

Eint0

minusℎ

int

119905+Δ

119905+Δ+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

minusint

0

minusℎ

int

119905

119905+120579

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904 119889120579

= limΔrarr0

+

1

Δ

Eint119905+Δ

119905+Δminusℎ

120578⊺

(119904) 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

+ int

119905+Δ

119905+Δminusℎ

120578⊺

(119904)

times sum

119895isinN

(120582119894119895Δ + 119900 (Δ))

times 119890120575(119904+ℎ)

1198761119894120578 (119904) 119889119904

minusint

119905

119905minusℎ

120578⊺

(119904) 1198761119894120578 (119904) 119889119904

+ limΔrarr0

+

1

Δ

E[int119905+Δ

119905+Δminus120591119895(119905+Δ)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ sum

119899isinM

(120587119898119899Δ + 119900 (Δ))

times int

119905

119905+Δminus120591119899(119905+Δ)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904

+ int

119905+Δ

minus120591119895(119905 + Δ)

119905+Δ

120578⊺

(119904)

times sum

119895isinN

(120582119894119895Δ + 119900 (Δ))

times 119890120575(119904+ℎ)

1198762119895120578 (119904) 119889119904

minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119890120575(119904+ℎ)

1198762119894120578 (119904) 119889119904]

+ 119890120575119905

120578⊺

(119905) ℎ119876120578 (119905) minus int

119905minusℎ

119890120575(119904+ℎ)

120578⊺

(119904) 119876120578 (119904) 119889119904

le 119890120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

minus (1 minus 120591119898(119905)) 119890

120575119905

120578⊺

(119905 minus 120591119898(119905)) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)


+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(28)

Since

0 le 120591119898(119905) le ℎ

119898 (29)

we define

119903 (120591119898) =

minus (1 minus 120583119898) 119890

120575ℎ119898 if 120583

119898gt 1

minus (1 minus 120583119898) if 120583

119898le 1

(30)

Then

pound1198812(120578

119905 119894 119898) le 119890

120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

+ 119890120575119905

120578⊺

(119905 minus 120591119898(119905)) 119903 (120591

119898) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)

times ( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)

+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(31)


pound1198813(120578

119905 119894 119898) le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119883

119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119883

119895119898minus 119883)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119883119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119883120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579119889120592

(32)


minusint

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904 = minusint

119905

119905minus120591119898(119905)

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

le minus120591119898(119905) 119880

⊺

1119883

1198941198981198801

minus (ℎ minus 120591119898(119905)) 119880

⊺

2119883

1198941198981198802

(33)

where

1198801=

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

1198802=

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

lim120591119898(119905)rarr0

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904 = 120578 (119905)

lim120591119898(119905)rarrℎ

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904 = 120578 (119905 minus ℎ)

(34)


119894119898 119873

119894119898 and 119881

119894119898with appro-

priate dimensions

(2120591119898(119905) 119880

⊺

1119872

119894119898+ 2 (ℎ minus 120591

119898(119905)) 119880

⊺

2119873

119894119898+ 2 120578

⊺

(119905) 119881119894119898)

times [minus 120578 (119905) + 119860119894120578 (119905) + 119861

120591119894120578 (119905 minus 120591

119898(119905)) + 119863

119894120596 (119905)] = 0

(35)

pound1198814(120578

119905 119894 119898)

le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119884119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119884119895119898minus 119884)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119884119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119884 120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579 119889120592

(36)



minus int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

= minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

le minus

ℎ

120591119898(119905)

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

minus

ℎ

ℎ minus 120591119898(119905)

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

le minus

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

⊺

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

(37)


pound119881 (120578119905 119903

119905 119904

119905) minus 120575120596

⊺

(119905)119867120596 (119905) le 119890120575119905

120585⊺

(119905) Ξ119894119898120585 (119905) (38)

where

120585⊺

(119905) = [120578⊺

(119905) 120578⊺

(119905 minus 120591119898(119905)) 120578

⊺

(119905 minus ℎ) 120578⊺

(119905) 119880⊺

1 119880

⊺

2 120596

⊺

(119905)]

Ξ119894119898=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860⊺

119894119881

⊺

119894119898120591119898(119905) 119860

⊺

119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

119894119873

⊺

119894119898119875119894119898119863

119894

lowast Ξ22119894119898

119878119894119898minus

119884119894119898

ℎ

119860⊺

120591119894119881

⊺

119894119898120591119898(119905) 119860

⊺

120591119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

120591119894119873

⊺

1198941198980


119884119894119898

ℎ

0 0 0 0


minus120591119898(119905)119872

⊺

119894119898minus (ℎ minus 120591

119898(119905))119873

⊺

119894119898119881119894119898119863

119894


1198941198980 120591

119898(119905)119872

119894119898119863

119894


119894119898(ℎ minus 120591

119898(119905))119873

119894119898119863

119894


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(39)


119898(119905) rarr



120591119898(119905) = ℎ and Ξ

119894119898|120591119898(119905) = 0 Thus we obtain

E pound119881 (120578119905 119903

119905 119904

119905) le 120578E [119881 (120578

119905 119903

119905 119904

119905)] + 120575120596

⊺

(119905)119867120596 (119905)

(40)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

120575120596⊺

(119905)119867120596 (119905) (41)


119890minus120578119905

E [119881 (120578119905 119903

119905 119904

119905)] minus E [119881 (120578

0 119903

0 119904

0)]

le 120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

(42)


= 119877minus(12)

119875119894119898119877minus(12) 119876

119904119894=

119877minus(12)

119876119904119894119877minus(12)

(119904 = 1 2) 119876 = 119877minus(12)

119876119877minus(12)

119883119894119898

= 119877minus(12)

119883119894119898119877minus(12) 119883 = 119877

minus(12)

119883119877minus(12)

119894119898

= 119877minus(12)

119884119894119898119877minus(12) and = 119877

minus(12)


that

E [119881 (1205780 119903

0 119904

0)]


120582max (119894119898) 120578⊺

(0) 119877120578 (0)

+ (max119894isinN

120582max (1198761119894) +max

119894isinN120582max (1198762119894

)) 119890120575ℎ

times int

0

minusℎ

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ

120582max (119876)int0

minusℎ

int

0

120579

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ max

119894isinN119898isinM120582max (119883119894119898

)

times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max (119883)

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579 119889120592

+ 119890120575ℎ max

119894isinN119898isinM120582max (119894119898)


times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max ()

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579 119889120592


120582max (119894119898)

+ ℎ119890120575ℎ

(max119894isinN

120582max (1198761119894+

119894isinNmax120582max (1198762119894

)))

+ ℎ2

119890120575ℎ


120582max (119883119894119898) + 120582max (119876)


120582max (119894119898))

+

1

2

ℎ3

119890120575ℎ

(120582max (119883) + 120582max ())


120578⊺

(119904) 119877120578 (119904) 120578⊺

(119904) 119877 120578 (119904)

= 1198881Λ

(43)


E [119881 (120578119905 119903

119905 119904

119905)] le E [119890

120578119905

119881 (1205780 119903

0 119904

0)]

+ 119890120578119905

120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

le 119890120578119879

1198881Λ + 119889120575119890

120578119879

120582max (119867)int119879

0

119890minus120578119904

119889119904

le 119890120578119879

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

)

(44)


E [119881 (120578119905 119903

119905 119904

119905)] ge E [120578

⊺

(119905) 119890120582119905

119875119894119898120578 (119905)]

ge max119894isinN

120582min (119875119894119898)E [120578⊺

(119905) 119877120578 (119905)]

= 1205821E [120578

⊺

(119905) 119877120578 (119905)]

(45)


E [120578⊺

(119905) 119877120578 (119905)] le

119890120578119879

1205821

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) (46)


E [120578⊺

(119905) 119877120578 (119905)] lt 1198882 (47)


2 119889 119877 119879)




119898(119905) and

120591119898(119905) are enlarged to 120591

119898(119905) le ℎ = maxℎ

119894 119898 isin M and

120591119898(119905) le 120583 = max120583




119905

119905minusℎ

120578⊺

(119904)119884119894119898120578(119904)119889119904 the




119894119898 119883

119894119898 119884


119894 (119894 = 1 2 119873) and119898 (119898 = 1 2 119872)



119894gt 0 119876

119897119894gt 0 (119897 = 1 2)

119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0 119878

119894119898 scalars 119888

1lt 119888

2



Σ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894

0


ℎ119872119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(48)

Σ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119873⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894

0


ℎ119873119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(49)

1198881Λ + 119889120575120574

21

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (50)




pound119881 (120578119905 119903

119905 119904

119905) + 119890

⊺

(119905) 119890 (119905) minus 1205742

120596⊺

(119905) 120596 (119905) le 120585⊺

(119905) Σ119897119894119898120585 (119905)

(119897 = 1 2)

(51)


E pound119881 (120578119905 119903

119905 119904

119905)

le E [120578119881 (120578119905 119903

119905 119904

119905)] + 120574

2

120596⊺

(119905) 120596 (119905) minus E [119890⊺

(119905) 119890 (119905)]

(52)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)]

(53)


119905 119904

119905)] gt 0 by inte-


int

119879

0

119890minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)] 119889119905

le Eint119879

0

pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] 119889119905 le 119881 (120578

0 119903

0 119904

0) = 0

(54)


E [int119879

0

119890minus120578120592

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

E [int119879

0

119890minus120578120592

120596⊺

(120592) 120596 (120592) 119889120592]

(55)

Then it yields that

E [int119879

0

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

119890120578119879

E [int119879

0

120596⊺

(120592) 120596 (120592) 119889120592] (56)



pletes the proof




Filtering



119894119898gt 0119872

119894119898 119873

119894119898 119881

119894119898

119876119897119894gt 0 (119897 = 1 2) 119876 gt 0 119883

119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0

119860119891119894119898

119861119891119894119898

119862119891119894119898

119872119860119894119898

119872119861119894119898

119873119860119894119898

119873119861119894119898

119881119860119894119898

119881119861119894119898

119878119894119898

scalars 1198881lt 119888

2 119879 gt 0 120590

119904gt 0 (119904 = 1 2 9) 120575 gt 0 120578 gt 0


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] gt 0 119872119894119898= [

1198721119894119898

1198722119894119898

1198722119894119898

1198722119894119898

] (57)

119873119894119898= [

1198731119894119898

1198732119894119898

1198732119894119898

1198732119894119898

] 119881119894119898= [

1198811119894119898

1198812119894119898

1198812119894119898

1198812119894119898

] (58)

Γ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ15119894119898

Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ25119894119898

0 [119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898Γ46119894119898

0


Γ56119894119898

0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(59)

Γ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ1015840

15119894119898Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ1015840

251198941198980 [

119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898Γ1015840

461198941198980


Γ1015840

561198941198980



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(60)where

Γ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 119890

120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876 +

119890120575ℎ

minus 1

120575

119883119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883

+

119884119894119898

ℎ

+ [

1205751198751119894119898

+ 1198751119894119898119860

119894+ 119860

⊺

1198941198751119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

1205751198752119894119898

+ 1198752119894119898119860

119894+ 119860

⊺

1198941198752119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

]

(61)


Γ12119894119898

= minus

119884119894119898

ℎ

+ 119878119894119898+ [

1198751119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

]

Γ14119894119898

= [

119860⊺

119894119881

⊺

1119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

119860⊺

119894119881

⊺

2119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

]

Γ15119894119898

= [

ℎ119860⊺

119894119872

⊺

1119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

ℎ119860⊺

119894119872

⊺

2119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

]

Γ1015840

15119894119898= [

ℎ119860⊺

119894119873

⊺

1119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

ℎ119860⊺

119894119873

⊺

2119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

]

Γ16119894119898

= [

1198751119894119898119863

119894+ 119861

119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119861

119891119894119898119863

119910119894

]

Γ17119894119898

= [

119862⊺

119911119894

minus119862119891119894119898

]

Γ24119894119898

= [

119860⊺

120591119894119881

⊺

1119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

119860⊺

120591119894119881

⊺

2119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

]

Γ25119894119898

= [

ℎ119860⊺

120591119894119872

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

ℎ119860⊺

120591119894119872

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

]

Γ1015840

25119894119898= [

ℎ119860⊺

120591119894119873

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

ℎ119860⊺

120591119894119873

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

]

Γ46119894119898

= [

1198811119894119898119863

119894+ 119881

119861119894119898119863

119910119894

1198812119894119898119863

119894+ 119881

119861119894119898119863

119910119894

]

Γ56119894119898

= [

ℎ1198721119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

ℎ1198722119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

]

Γ1015840

56119894119898= [

ℎ1198731119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

ℎ1198732119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

]

(62)


119860119891119894119898

= 119875minus1

2119894119898119860

119891119894119898 119861

119891119894119898= 119875

minus1

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

(63)


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] (64)

The term 119875119894119898119860


119875119894119898119860

119894= [

1198751119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

1198752119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

] (65)

Similarly we have

119875119894119898119861119894= [

1198751119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

]

119875119894119898119863

119894= [

1198751119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

]

(66)

Define 119860119891119894119898

= 1198752119894119898119860

119891119894119898 119861

119891119894119898= 119875

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

119872

119860119894119898= 119872

2119894119898119860

119891119894119898119872

119861119894119898= 119872

2119894119898119861119891119894119898

119873119860119894119898

= 1198732119894119898119860

119891119894119898

119873119861119894119898

= 1198732119894119898119861119891119894119898

119881119860119894119898

= 1198812119894119898119860

119891119894119898 and 119881

119861119894119898= 119881

2119894119898119861119891119894119898



proof is completed


min 1205742

st (48)ndash(50) (67)




1= 1 then


2min

min 1205891205742

+ (1 minus 120589) 1198882

st (59)-(60) and (50)

(68)




Λ = [

minus07 07

09 minus09] Π = [

minus1 1

12 minus12] (69)


1198601= [

minus35 086


2= [

minus25 034

14 minus002]

1198601205911= [

minus08 minus13


1205912= [

minus28 05

minus08 minus14]

1198631= 119863

2= [

01

02] 119862

1199101= [04 07]

1198621199102= [05 08] 119862

1199101205911= [01 02]

1198621199101205912

= [01 01]

1198631199101= 119863

1199102= 01 119862

1199111= [01 005]

1198621199112= [02 009] 119862

1199111205911= [03 06]

1198621199111205912

= [04 08] 1198631199111= 119863

1199112= 005

(70)




11986011989111

= [

minus61254 15565


11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)


6 Conclusions






Acknowledgments


References




































































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(28)

Since

0 le 120591119898(119905) le ℎ

119898 (29)

we define

119903 (120591119898) =

minus (1 minus 120583119898) 119890

120575ℎ119898 if 120583

119898gt 1

minus (1 minus 120583119898) if 120583

119898le 1

(30)

Then

pound1198812(120578

119905 119894 119898) le 119890

120575119905

120578⊺

(119905) (119890120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876) 120578 (119905)

minus 119890120575119905

120578⊺

(119905 minus ℎ)1198761119894120578 (119905 minus ℎ)

+ 119890120575119905

120578⊺

(119905 minus 120591119898(119905)) 119903 (120591

119898) 119876

2119894120578 (119905 minus 120591

119898(119905))

+ int

119905

119905minusℎ

119890120590(119904+ℎ)

120578⊺

(119904)

times ( sum

119895isinN119895 = 119894

120582119894119895(119876

1119895+ 119876

2119895)

+ sum

119899isinM119898 = 119899

120587119898119899119876

2119894minus 119876)

times 120578 (119904) 119889119904

(31)


pound1198813(120578

119905 119894 119898) le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119883

119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119883

119895119898minus 119883)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119883119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119883120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579119889120592

(32)


minusint

119905

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904 = minusint

119905

119905minus120591119898(119905)

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904)119883119894119898120578 (119904) 119889119904

le minus120591119898(119905) 119880

⊺

1119883

1198941198981198801

minus (ℎ minus 120591119898(119905)) 119880

⊺

2119883

1198941198981198802

(33)

where

1198801=

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

1198802=

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

lim120591119898(119905)rarr0

1

120591119898(119905)

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904 = 120578 (119905)

lim120591119898(119905)rarrℎ

1

ℎ minus 120591119898(119905)

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904 = 120578 (119905 minus ℎ)

(34)


119894119898 119873

119894119898 and 119881

119894119898with appro-

priate dimensions

(2120591119898(119905) 119880

⊺

1119872

119894119898+ 2 (ℎ minus 120591

119898(119905)) 119880

⊺

2119873

119894119898+ 2 120578

⊺

(119905) 119881119894119898)

times [minus 120578 (119905) + 119860119894120578 (119905) + 119861

120591119894120578 (119905 minus 120591

119898(119905)) + 119863

119894120596 (119905)] = 0

(35)

pound1198814(120578

119905 119894 119898)

le int

0

minusℎ

int

119905

119905+120579

119890120575(119904minus120579)

120578⊺

(119904)

times ( sum

119899isinM119898 = 119899

120587119898119899119884119894119899

+ sum

119895isinN119894 = 119895

120582119894119895119884119895119898minus 119884)

times 120578 (119904) 119889119904 119889120579

+ 119890120575119905

120578⊺

(119905) 119884119894119898120578 (119905) int

0

minusℎ

119890minus120575120592

119889120592

minus 119890120575119905

int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

+ 119890120575119905

120578⊺

(119905) 119884 120578 (119905) int

0

minusℎ

int

0

120592

119890minus120575120592

119889120579 119889120592

(36)



minus int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

= minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

le minus

ℎ

120591119898(119905)

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

minus

ℎ

ℎ minus 120591119898(119905)

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

le minus

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

⊺

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

(37)


pound119881 (120578119905 119903

119905 119904

119905) minus 120575120596

⊺

(119905)119867120596 (119905) le 119890120575119905

120585⊺

(119905) Ξ119894119898120585 (119905) (38)

where

120585⊺

(119905) = [120578⊺

(119905) 120578⊺

(119905 minus 120591119898(119905)) 120578

⊺

(119905 minus ℎ) 120578⊺

(119905) 119880⊺

1 119880

⊺

2 120596

⊺

(119905)]

Ξ119894119898=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860⊺

119894119881

⊺

119894119898120591119898(119905) 119860

⊺

119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

119894119873

⊺

119894119898119875119894119898119863

119894

lowast Ξ22119894119898

119878119894119898minus

119884119894119898

ℎ

119860⊺

120591119894119881

⊺

119894119898120591119898(119905) 119860

⊺

120591119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

120591119894119873

⊺

1198941198980


119884119894119898

ℎ

0 0 0 0


minus120591119898(119905)119872

⊺

119894119898minus (ℎ minus 120591

119898(119905))119873

⊺

119894119898119881119894119898119863

119894


1198941198980 120591

119898(119905)119872

119894119898119863

119894


119894119898(ℎ minus 120591

119898(119905))119873

119894119898119863

119894


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(39)


119898(119905) rarr



120591119898(119905) = ℎ and Ξ

119894119898|120591119898(119905) = 0 Thus we obtain

E pound119881 (120578119905 119903

119905 119904

119905) le 120578E [119881 (120578

119905 119903

119905 119904

119905)] + 120575120596

⊺

(119905)119867120596 (119905)

(40)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

120575120596⊺

(119905)119867120596 (119905) (41)


119890minus120578119905

E [119881 (120578119905 119903

119905 119904

119905)] minus E [119881 (120578

0 119903

0 119904

0)]

le 120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

(42)


= 119877minus(12)

119875119894119898119877minus(12) 119876

119904119894=

119877minus(12)

119876119904119894119877minus(12)

(119904 = 1 2) 119876 = 119877minus(12)

119876119877minus(12)

119883119894119898

= 119877minus(12)

119883119894119898119877minus(12) 119883 = 119877

minus(12)

119883119877minus(12)

119894119898

= 119877minus(12)

119884119894119898119877minus(12) and = 119877

minus(12)


that

E [119881 (1205780 119903

0 119904

0)]


120582max (119894119898) 120578⊺

(0) 119877120578 (0)

+ (max119894isinN

120582max (1198761119894) +max

119894isinN120582max (1198762119894

)) 119890120575ℎ

times int

0

minusℎ

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ

120582max (119876)int0

minusℎ

int

0

120579

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ max

119894isinN119898isinM120582max (119883119894119898

)

times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max (119883)

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579 119889120592

+ 119890120575ℎ max

119894isinN119898isinM120582max (119894119898)


times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max ()

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579 119889120592


120582max (119894119898)

+ ℎ119890120575ℎ

(max119894isinN

120582max (1198761119894+

119894isinNmax120582max (1198762119894

)))

+ ℎ2

119890120575ℎ


120582max (119883119894119898) + 120582max (119876)


120582max (119894119898))

+

1

2

ℎ3

119890120575ℎ

(120582max (119883) + 120582max ())


120578⊺

(119904) 119877120578 (119904) 120578⊺

(119904) 119877 120578 (119904)

= 1198881Λ

(43)


E [119881 (120578119905 119903

119905 119904

119905)] le E [119890

120578119905

119881 (1205780 119903

0 119904

0)]

+ 119890120578119905

120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

le 119890120578119879

1198881Λ + 119889120575119890

120578119879

120582max (119867)int119879

0

119890minus120578119904

119889119904

le 119890120578119879

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

)

(44)


E [119881 (120578119905 119903

119905 119904

119905)] ge E [120578

⊺

(119905) 119890120582119905

119875119894119898120578 (119905)]

ge max119894isinN

120582min (119875119894119898)E [120578⊺

(119905) 119877120578 (119905)]

= 1205821E [120578

⊺

(119905) 119877120578 (119905)]

(45)


E [120578⊺

(119905) 119877120578 (119905)] le

119890120578119879

1205821

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) (46)


E [120578⊺

(119905) 119877120578 (119905)] lt 1198882 (47)


2 119889 119877 119879)




119898(119905) and

120591119898(119905) are enlarged to 120591

119898(119905) le ℎ = maxℎ

119894 119898 isin M and

120591119898(119905) le 120583 = max120583




119905

119905minusℎ

120578⊺

(119904)119884119894119898120578(119904)119889119904 the




119894119898 119883

119894119898 119884


119894 (119894 = 1 2 119873) and119898 (119898 = 1 2 119872)



119894gt 0 119876

119897119894gt 0 (119897 = 1 2)

119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0 119878

119894119898 scalars 119888

1lt 119888

2



Σ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894

0


ℎ119872119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(48)

Σ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119873⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894

0


ℎ119873119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(49)

1198881Λ + 119889120575120574

21

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (50)




pound119881 (120578119905 119903

119905 119904

119905) + 119890

⊺

(119905) 119890 (119905) minus 1205742

120596⊺

(119905) 120596 (119905) le 120585⊺

(119905) Σ119897119894119898120585 (119905)

(119897 = 1 2)

(51)


E pound119881 (120578119905 119903

119905 119904

119905)

le E [120578119881 (120578119905 119903

119905 119904

119905)] + 120574

2

120596⊺

(119905) 120596 (119905) minus E [119890⊺

(119905) 119890 (119905)]

(52)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)]

(53)


119905 119904

119905)] gt 0 by inte-


int

119879

0

119890minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)] 119889119905

le Eint119879

0

pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] 119889119905 le 119881 (120578

0 119903

0 119904

0) = 0

(54)


E [int119879

0

119890minus120578120592

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

E [int119879

0

119890minus120578120592

120596⊺

(120592) 120596 (120592) 119889120592]

(55)

Then it yields that

E [int119879

0

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

119890120578119879

E [int119879

0

120596⊺

(120592) 120596 (120592) 119889120592] (56)



pletes the proof




Filtering



119894119898gt 0119872

119894119898 119873

119894119898 119881

119894119898

119876119897119894gt 0 (119897 = 1 2) 119876 gt 0 119883

119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0

119860119891119894119898

119861119891119894119898

119862119891119894119898

119872119860119894119898

119872119861119894119898

119873119860119894119898

119873119861119894119898

119881119860119894119898

119881119861119894119898

119878119894119898

scalars 1198881lt 119888

2 119879 gt 0 120590

119904gt 0 (119904 = 1 2 9) 120575 gt 0 120578 gt 0


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] gt 0 119872119894119898= [

1198721119894119898

1198722119894119898

1198722119894119898

1198722119894119898

] (57)

119873119894119898= [

1198731119894119898

1198732119894119898

1198732119894119898

1198732119894119898

] 119881119894119898= [

1198811119894119898

1198812119894119898

1198812119894119898

1198812119894119898

] (58)

Γ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ15119894119898

Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ25119894119898

0 [119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898Γ46119894119898

0


Γ56119894119898

0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(59)

Γ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ1015840

15119894119898Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ1015840

251198941198980 [

119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898Γ1015840

461198941198980


Γ1015840

561198941198980



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(60)where

Γ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 119890

120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876 +

119890120575ℎ

minus 1

120575

119883119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883

+

119884119894119898

ℎ

+ [

1205751198751119894119898

+ 1198751119894119898119860

119894+ 119860

⊺

1198941198751119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

1205751198752119894119898

+ 1198752119894119898119860

119894+ 119860

⊺

1198941198752119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

]

(61)


Γ12119894119898

= minus

119884119894119898

ℎ

+ 119878119894119898+ [

1198751119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

]

Γ14119894119898

= [

119860⊺

119894119881

⊺

1119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

119860⊺

119894119881

⊺

2119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

]

Γ15119894119898

= [

ℎ119860⊺

119894119872

⊺

1119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

ℎ119860⊺

119894119872

⊺

2119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

]

Γ1015840

15119894119898= [

ℎ119860⊺

119894119873

⊺

1119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

ℎ119860⊺

119894119873

⊺

2119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

]

Γ16119894119898

= [

1198751119894119898119863

119894+ 119861

119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119861

119891119894119898119863

119910119894

]

Γ17119894119898

= [

119862⊺

119911119894

minus119862119891119894119898

]

Γ24119894119898

= [

119860⊺

120591119894119881

⊺

1119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

119860⊺

120591119894119881

⊺

2119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

]

Γ25119894119898

= [

ℎ119860⊺

120591119894119872

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

ℎ119860⊺

120591119894119872

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

]

Γ1015840

25119894119898= [

ℎ119860⊺

120591119894119873

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

ℎ119860⊺

120591119894119873

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

]

Γ46119894119898

= [

1198811119894119898119863

119894+ 119881

119861119894119898119863

119910119894

1198812119894119898119863

119894+ 119881

119861119894119898119863

119910119894

]

Γ56119894119898

= [

ℎ1198721119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

ℎ1198722119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

]

Γ1015840

56119894119898= [

ℎ1198731119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

ℎ1198732119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

]

(62)


119860119891119894119898

= 119875minus1

2119894119898119860

119891119894119898 119861

119891119894119898= 119875

minus1

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

(63)


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] (64)

The term 119875119894119898119860


119875119894119898119860

119894= [

1198751119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

1198752119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

] (65)

Similarly we have

119875119894119898119861119894= [

1198751119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

]

119875119894119898119863

119894= [

1198751119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

]

(66)

Define 119860119891119894119898

= 1198752119894119898119860

119891119894119898 119861

119891119894119898= 119875

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

119872

119860119894119898= 119872

2119894119898119860

119891119894119898119872

119861119894119898= 119872

2119894119898119861119891119894119898

119873119860119894119898

= 1198732119894119898119860

119891119894119898

119873119861119894119898

= 1198732119894119898119861119891119894119898

119881119860119894119898

= 1198812119894119898119860

119891119894119898 and 119881

119861119894119898= 119881

2119894119898119861119891119894119898



proof is completed


min 1205742

st (48)ndash(50) (67)




1= 1 then


2min

min 1205891205742

+ (1 minus 120589) 1198882

st (59)-(60) and (50)

(68)




Λ = [

minus07 07

09 minus09] Π = [

minus1 1

12 minus12] (69)


1198601= [

minus35 086


2= [

minus25 034

14 minus002]

1198601205911= [

minus08 minus13


1205912= [

minus28 05

minus08 minus14]

1198631= 119863

2= [

01

02] 119862

1199101= [04 07]

1198621199102= [05 08] 119862

1199101205911= [01 02]

1198621199101205912

= [01 01]

1198631199101= 119863

1199102= 01 119862

1199111= [01 005]

1198621199112= [02 009] 119862

1199111205911= [03 06]

1198621199111205912

= [04 08] 1198631199111= 119863

1199112= 005

(70)




11986011989111

= [

minus61254 15565


11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)


6 Conclusions






Acknowledgments


References




































































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus int

119905

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

= minusint

119905

119905minus120591119898(119905)

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

minus int

119905minus120591119898(119905)

119905minusℎ

120578⊺

(119904) 119884119894119898120578 (119904) 119889119904

le minus

ℎ

120591119898(119905)

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904]

minus

ℎ

ℎ minus 120591119898(119905)

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

⊺

119884119894119898

ℎ

[int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904]

le minus

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

⊺

[

[

[

119884119894119898

ℎ

119878119894119898

lowast

119884119894119898

ℎ

]

]

]

[

[

[

[

int

119905

119905minus120591119898(119905)

120578 (119904) 119889119904

int

119905minus120591119898(119905)

119905minusℎ

120578 (119904) 119889119904

]

]

]

]

(37)


pound119881 (120578119905 119903

119905 119904

119905) minus 120575120596

⊺

(119905)119867120596 (119905) le 119890120575119905

120585⊺

(119905) Ξ119894119898120585 (119905) (38)

where

120585⊺

(119905) = [120578⊺

(119905) 120578⊺

(119905 minus 120591119898(119905)) 120578

⊺

(119905 minus ℎ) 120578⊺

(119905) 119880⊺

1 119880

⊺

2 120596

⊺

(119905)]

Ξ119894119898=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860⊺

119894119881

⊺

119894119898120591119898(119905) 119860

⊺

119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

119894119873

⊺

119894119898119875119894119898119863

119894

lowast Ξ22119894119898

119878119894119898minus

119884119894119898

ℎ

119860⊺

120591119894119881

⊺

119894119898120591119898(119905) 119860

⊺

120591119894119872

⊺

119894119898(ℎ minus 120591

119898(119905)) 119860

⊺

120591119894119873

⊺

1198941198980


119884119894119898

ℎ

0 0 0 0


minus120591119898(119905)119872

⊺

119894119898minus (ℎ minus 120591

119898(119905))119873

⊺

119894119898119881119894119898119863

119894


1198941198980 120591

119898(119905)119872

119894119898119863

119894


119894119898(ℎ minus 120591

119898(119905))119873

119894119898119863

119894


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(39)


119898(119905) rarr



120591119898(119905) = ℎ and Ξ

119894119898|120591119898(119905) = 0 Thus we obtain

E pound119881 (120578119905 119903

119905 119904

119905) le 120578E [119881 (120578

119905 119903

119905 119904

119905)] + 120575120596

⊺

(119905)119867120596 (119905)

(40)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

120575120596⊺

(119905)119867120596 (119905) (41)


119890minus120578119905

E [119881 (120578119905 119903

119905 119904

119905)] minus E [119881 (120578

0 119903

0 119904

0)]

le 120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

(42)


= 119877minus(12)

119875119894119898119877minus(12) 119876

119904119894=

119877minus(12)

119876119904119894119877minus(12)

(119904 = 1 2) 119876 = 119877minus(12)

119876119877minus(12)

119883119894119898

= 119877minus(12)

119883119894119898119877minus(12) 119883 = 119877

minus(12)

119883119877minus(12)

119894119898

= 119877minus(12)

119884119894119898119877minus(12) and = 119877

minus(12)


that

E [119881 (1205780 119903

0 119904

0)]


120582max (119894119898) 120578⊺

(0) 119877120578 (0)

+ (max119894isinN

120582max (1198761119894) +max

119894isinN120582max (1198762119894

)) 119890120575ℎ

times int

0

minusℎ

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ

120582max (119876)int0

minusℎ

int

0

120579

119890120575119904

120578⊺

(119904) 119877120578 (119904) 119889119904

+ 119890120575ℎ max

119894isinN119898isinM120582max (119883119894119898

)

times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max (119883)

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877120578 (119904) 119889119904 119889120579 119889120592

+ 119890120575ℎ max

119894isinN119898isinM120582max (119894119898)


times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max ()

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579 119889120592


120582max (119894119898)

+ ℎ119890120575ℎ

(max119894isinN

120582max (1198761119894+

119894isinNmax120582max (1198762119894

)))

+ ℎ2

119890120575ℎ


120582max (119883119894119898) + 120582max (119876)


120582max (119894119898))

+

1

2

ℎ3

119890120575ℎ

(120582max (119883) + 120582max ())


120578⊺

(119904) 119877120578 (119904) 120578⊺

(119904) 119877 120578 (119904)

= 1198881Λ

(43)


E [119881 (120578119905 119903

119905 119904

119905)] le E [119890

120578119905

119881 (1205780 119903

0 119904

0)]

+ 119890120578119905

120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

le 119890120578119879

1198881Λ + 119889120575119890

120578119879

120582max (119867)int119879

0

119890minus120578119904

119889119904

le 119890120578119879

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

)

(44)


E [119881 (120578119905 119903

119905 119904

119905)] ge E [120578

⊺

(119905) 119890120582119905

119875119894119898120578 (119905)]

ge max119894isinN

120582min (119875119894119898)E [120578⊺

(119905) 119877120578 (119905)]

= 1205821E [120578

⊺

(119905) 119877120578 (119905)]

(45)


E [120578⊺

(119905) 119877120578 (119905)] le

119890120578119879

1205821

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) (46)


E [120578⊺

(119905) 119877120578 (119905)] lt 1198882 (47)


2 119889 119877 119879)




119898(119905) and

120591119898(119905) are enlarged to 120591

119898(119905) le ℎ = maxℎ

119894 119898 isin M and

120591119898(119905) le 120583 = max120583




119905

119905minusℎ

120578⊺

(119904)119884119894119898120578(119904)119889119904 the




119894119898 119883

119894119898 119884


119894 (119894 = 1 2 119873) and119898 (119898 = 1 2 119872)



119894gt 0 119876

119897119894gt 0 (119897 = 1 2)

119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0 119878

119894119898 scalars 119888

1lt 119888

2



Σ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894

0


ℎ119872119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(48)

Σ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119873⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894

0


ℎ119873119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(49)

1198881Λ + 119889120575120574

21

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (50)




pound119881 (120578119905 119903

119905 119904

119905) + 119890

⊺

(119905) 119890 (119905) minus 1205742

120596⊺

(119905) 120596 (119905) le 120585⊺

(119905) Σ119897119894119898120585 (119905)

(119897 = 1 2)

(51)


E pound119881 (120578119905 119903

119905 119904

119905)

le E [120578119881 (120578119905 119903

119905 119904

119905)] + 120574

2

120596⊺

(119905) 120596 (119905) minus E [119890⊺

(119905) 119890 (119905)]

(52)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)]

(53)


119905 119904

119905)] gt 0 by inte-


int

119879

0

119890minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)] 119889119905

le Eint119879

0

pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] 119889119905 le 119881 (120578

0 119903

0 119904

0) = 0

(54)


E [int119879

0

119890minus120578120592

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

E [int119879

0

119890minus120578120592

120596⊺

(120592) 120596 (120592) 119889120592]

(55)

Then it yields that

E [int119879

0

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

119890120578119879

E [int119879

0

120596⊺

(120592) 120596 (120592) 119889120592] (56)



pletes the proof




Filtering



119894119898gt 0119872

119894119898 119873

119894119898 119881

119894119898

119876119897119894gt 0 (119897 = 1 2) 119876 gt 0 119883

119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0

119860119891119894119898

119861119891119894119898

119862119891119894119898

119872119860119894119898

119872119861119894119898

119873119860119894119898

119873119861119894119898

119881119860119894119898

119881119861119894119898

119878119894119898

scalars 1198881lt 119888

2 119879 gt 0 120590

119904gt 0 (119904 = 1 2 9) 120575 gt 0 120578 gt 0


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] gt 0 119872119894119898= [

1198721119894119898

1198722119894119898

1198722119894119898

1198722119894119898

] (57)

119873119894119898= [

1198731119894119898

1198732119894119898

1198732119894119898

1198732119894119898

] 119881119894119898= [

1198811119894119898

1198812119894119898

1198812119894119898

1198812119894119898

] (58)

Γ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ15119894119898

Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ25119894119898

0 [119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898Γ46119894119898

0


Γ56119894119898

0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(59)

Γ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ1015840

15119894119898Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ1015840

251198941198980 [

119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898Γ1015840

461198941198980


Γ1015840

561198941198980



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(60)where

Γ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 119890

120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876 +

119890120575ℎ

minus 1

120575

119883119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883

+

119884119894119898

ℎ

+ [

1205751198751119894119898

+ 1198751119894119898119860

119894+ 119860

⊺

1198941198751119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

1205751198752119894119898

+ 1198752119894119898119860

119894+ 119860

⊺

1198941198752119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

]

(61)


Γ12119894119898

= minus

119884119894119898

ℎ

+ 119878119894119898+ [

1198751119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

]

Γ14119894119898

= [

119860⊺

119894119881

⊺

1119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

119860⊺

119894119881

⊺

2119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

]

Γ15119894119898

= [

ℎ119860⊺

119894119872

⊺

1119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

ℎ119860⊺

119894119872

⊺

2119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

]

Γ1015840

15119894119898= [

ℎ119860⊺

119894119873

⊺

1119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

ℎ119860⊺

119894119873

⊺

2119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

]

Γ16119894119898

= [

1198751119894119898119863

119894+ 119861

119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119861

119891119894119898119863

119910119894

]

Γ17119894119898

= [

119862⊺

119911119894

minus119862119891119894119898

]

Γ24119894119898

= [

119860⊺

120591119894119881

⊺

1119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

119860⊺

120591119894119881

⊺

2119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

]

Γ25119894119898

= [

ℎ119860⊺

120591119894119872

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

ℎ119860⊺

120591119894119872

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

]

Γ1015840

25119894119898= [

ℎ119860⊺

120591119894119873

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

ℎ119860⊺

120591119894119873

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

]

Γ46119894119898

= [

1198811119894119898119863

119894+ 119881

119861119894119898119863

119910119894

1198812119894119898119863

119894+ 119881

119861119894119898119863

119910119894

]

Γ56119894119898

= [

ℎ1198721119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

ℎ1198722119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

]

Γ1015840

56119894119898= [

ℎ1198731119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

ℎ1198732119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

]

(62)


119860119891119894119898

= 119875minus1

2119894119898119860

119891119894119898 119861

119891119894119898= 119875

minus1

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

(63)


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] (64)

The term 119875119894119898119860


119875119894119898119860

119894= [

1198751119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

1198752119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

] (65)

Similarly we have

119875119894119898119861119894= [

1198751119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

]

119875119894119898119863

119894= [

1198751119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

]

(66)

Define 119860119891119894119898

= 1198752119894119898119860

119891119894119898 119861

119891119894119898= 119875

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

119872

119860119894119898= 119872

2119894119898119860

119891119894119898119872

119861119894119898= 119872

2119894119898119861119891119894119898

119873119860119894119898

= 1198732119894119898119860

119891119894119898

119873119861119894119898

= 1198732119894119898119861119891119894119898

119881119860119894119898

= 1198812119894119898119860

119891119894119898 and 119881

119861119894119898= 119881

2119894119898119861119891119894119898



proof is completed


min 1205742

st (48)ndash(50) (67)




1= 1 then


2min

min 1205891205742

+ (1 minus 120589) 1198882

st (59)-(60) and (50)

(68)




Λ = [

minus07 07

09 minus09] Π = [

minus1 1

12 minus12] (69)


1198601= [

minus35 086


2= [

minus25 034

14 minus002]

1198601205911= [

minus08 minus13


1205912= [

minus28 05

minus08 minus14]

1198631= 119863

2= [

01

02] 119862

1199101= [04 07]

1198621199102= [05 08] 119862

1199101205911= [01 02]

1198621199101205912

= [01 01]

1198631199101= 119863

1199102= 01 119862

1199111= [01 005]

1198621199112= [02 009] 119862

1199111205911= [03 06]

1198621199111205912

= [04 08] 1198631199111= 119863

1199112= 005

(70)




11986011989111

= [

minus61254 15565


11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)


6 Conclusions






Acknowledgments


References




































































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times int

0

minusℎ

int

0

120579

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579

+ 119890120575ℎ

120582max ()

times int

0

minusℎ

int

0

120579

int

0

120592

119890minus120575120579

120578⊺

(119904) 119877 120578 (119904) 119889119904 119889120579 119889120592


120582max (119894119898)

+ ℎ119890120575ℎ

(max119894isinN

120582max (1198761119894+

119894isinNmax120582max (1198762119894

)))

+ ℎ2

119890120575ℎ


120582max (119883119894119898) + 120582max (119876)


120582max (119894119898))

+

1

2

ℎ3

119890120575ℎ

(120582max (119883) + 120582max ())


120578⊺

(119904) 119877120578 (119904) 120578⊺

(119904) 119877 120578 (119904)

= 1198881Λ

(43)


E [119881 (120578119905 119903

119905 119904

119905)] le E [119890

120578119905

119881 (1205780 119903

0 119904

0)]

+ 119890120578119905

120575int

119905

0

119890minus120578119904

120596⊺

(119904)119867120596 (119904) 119889119904

le 119890120578119879

1198881Λ + 119889120575119890

120578119879

120582max (119867)int119879

0

119890minus120578119904

119889119904

le 119890120578119879

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

)

(44)


E [119881 (120578119905 119903

119905 119904

119905)] ge E [120578

⊺

(119905) 119890120582119905

119875119894119898120578 (119905)]

ge max119894isinN

120582min (119875119894119898)E [120578⊺

(119905) 119877120578 (119905)]

= 1205821E [120578

⊺

(119905) 119877120578 (119905)]

(45)


E [120578⊺

(119905) 119877120578 (119905)] le

119890120578119879

1205821

1198881Λ + 119889120575120582

10

1

120578

(1 minus 119890minus120578119879

) (46)


E [120578⊺

(119905) 119877120578 (119905)] lt 1198882 (47)


2 119889 119877 119879)




119898(119905) and

120591119898(119905) are enlarged to 120591

119898(119905) le ℎ = maxℎ

119894 119898 isin M and

120591119898(119905) le 120583 = max120583




119905

119905minusℎ

120578⊺

(119904)119884119894119898120578(119904)119889119904 the




119894119898 119883

119894119898 119884


119894 (119894 = 1 2 119873) and119898 (119898 = 1 2 119872)



119894gt 0 119876

119897119894gt 0 (119897 = 1 2)

119876 gt 0 119883119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0 119878

119894119898 scalars 119888

1lt 119888

2



Σ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119872⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119872⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898119881119894119898119863119894

0


ℎ119872119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(48)

Σ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

Ξ11119894119898

Ξ12119894119898

minus119878119894119898

119860

⊺

119894119881⊺

119894119898ℎ119860

⊺

119894119873⊺

119894119898119875119894119898119863119894

119862

⊺

119894

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

119861

⊺

119894119881⊺

119894119898ℎ119861

⊺

119894119873⊺

1198941198980 119864

⊺

119894


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898119881119894119898119863119894

0


ℎ119873119894119898119863119894

0



]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(49)

1198881Λ + 119889120575120574

21

120578

(1 minus 119890minus120578119879

) lt 1205821119890minus120578119879

1198882 (50)




pound119881 (120578119905 119903

119905 119904

119905) + 119890

⊺

(119905) 119890 (119905) minus 1205742

120596⊺

(119905) 120596 (119905) le 120585⊺

(119905) Σ119897119894119898120585 (119905)

(119897 = 1 2)

(51)


E pound119881 (120578119905 119903

119905 119904

119905)

le E [120578119881 (120578119905 119903

119905 119904

119905)] + 120574

2

120596⊺

(119905) 120596 (119905) minus E [119890⊺

(119905) 119890 (119905)]

(52)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)]

(53)


119905 119904

119905)] gt 0 by inte-


int

119879

0

119890minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)] 119889119905

le Eint119879

0

pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] 119889119905 le 119881 (120578

0 119903

0 119904

0) = 0

(54)


E [int119879

0

119890minus120578120592

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

E [int119879

0

119890minus120578120592

120596⊺

(120592) 120596 (120592) 119889120592]

(55)

Then it yields that

E [int119879

0

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

119890120578119879

E [int119879

0

120596⊺

(120592) 120596 (120592) 119889120592] (56)



pletes the proof




Filtering



119894119898gt 0119872

119894119898 119873

119894119898 119881

119894119898

119876119897119894gt 0 (119897 = 1 2) 119876 gt 0 119883

119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0

119860119891119894119898

119861119891119894119898

119862119891119894119898

119872119860119894119898

119872119861119894119898

119873119860119894119898

119873119861119894119898

119881119860119894119898

119881119861119894119898

119878119894119898

scalars 1198881lt 119888

2 119879 gt 0 120590

119904gt 0 (119904 = 1 2 9) 120575 gt 0 120578 gt 0


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] gt 0 119872119894119898= [

1198721119894119898

1198722119894119898

1198722119894119898

1198722119894119898

] (57)

119873119894119898= [

1198731119894119898

1198732119894119898

1198732119894119898

1198732119894119898

] 119881119894119898= [

1198811119894119898

1198812119894119898

1198812119894119898

1198812119894119898

] (58)

Γ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ15119894119898

Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ25119894119898

0 [119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898Γ46119894119898

0


Γ56119894119898

0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(59)

Γ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ1015840

15119894119898Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ1015840

251198941198980 [

119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898Γ1015840

461198941198980


Γ1015840

561198941198980



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(60)where

Γ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 119890

120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876 +

119890120575ℎ

minus 1

120575

119883119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883

+

119884119894119898

ℎ

+ [

1205751198751119894119898

+ 1198751119894119898119860

119894+ 119860

⊺

1198941198751119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

1205751198752119894119898

+ 1198752119894119898119860

119894+ 119860

⊺

1198941198752119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

]

(61)


Γ12119894119898

= minus

119884119894119898

ℎ

+ 119878119894119898+ [

1198751119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

]

Γ14119894119898

= [

119860⊺

119894119881

⊺

1119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

119860⊺

119894119881

⊺

2119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

]

Γ15119894119898

= [

ℎ119860⊺

119894119872

⊺

1119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

ℎ119860⊺

119894119872

⊺

2119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

]

Γ1015840

15119894119898= [

ℎ119860⊺

119894119873

⊺

1119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

ℎ119860⊺

119894119873

⊺

2119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

]

Γ16119894119898

= [

1198751119894119898119863

119894+ 119861

119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119861

119891119894119898119863

119910119894

]

Γ17119894119898

= [

119862⊺

119911119894

minus119862119891119894119898

]

Γ24119894119898

= [

119860⊺

120591119894119881

⊺

1119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

119860⊺

120591119894119881

⊺

2119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

]

Γ25119894119898

= [

ℎ119860⊺

120591119894119872

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

ℎ119860⊺

120591119894119872

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

]

Γ1015840

25119894119898= [

ℎ119860⊺

120591119894119873

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

ℎ119860⊺

120591119894119873

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

]

Γ46119894119898

= [

1198811119894119898119863

119894+ 119881

119861119894119898119863

119910119894

1198812119894119898119863

119894+ 119881

119861119894119898119863

119910119894

]

Γ56119894119898

= [

ℎ1198721119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

ℎ1198722119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

]

Γ1015840

56119894119898= [

ℎ1198731119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

ℎ1198732119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

]

(62)


119860119891119894119898

= 119875minus1

2119894119898119860

119891119894119898 119861

119891119894119898= 119875

minus1

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

(63)


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] (64)

The term 119875119894119898119860


119875119894119898119860

119894= [

1198751119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

1198752119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

] (65)

Similarly we have

119875119894119898119861119894= [

1198751119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

]

119875119894119898119863

119894= [

1198751119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

]

(66)

Define 119860119891119894119898

= 1198752119894119898119860

119891119894119898 119861

119891119894119898= 119875

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

119872

119860119894119898= 119872

2119894119898119860

119891119894119898119872

119861119894119898= 119872

2119894119898119861119891119894119898

119873119860119894119898

= 1198732119894119898119860

119891119894119898

119873119861119894119898

= 1198732119894119898119861119891119894119898

119881119860119894119898

= 1198812119894119898119860

119891119894119898 and 119881

119861119894119898= 119881

2119894119898119861119891119894119898



proof is completed


min 1205742

st (48)ndash(50) (67)




1= 1 then


2min

min 1205891205742

+ (1 minus 120589) 1198882

st (59)-(60) and (50)

(68)




Λ = [

minus07 07

09 minus09] Π = [

minus1 1

12 minus12] (69)


1198601= [

minus35 086


2= [

minus25 034

14 minus002]

1198601205911= [

minus08 minus13


1205912= [

minus28 05

minus08 minus14]

1198631= 119863

2= [

01

02] 119862

1199101= [04 07]

1198621199102= [05 08] 119862

1199101205911= [01 02]

1198621199101205912

= [01 01]

1198631199101= 119863

1199102= 01 119862

1199111= [01 005]

1198621199112= [02 009] 119862

1199111205911= [03 06]

1198621199111205912

= [04 08] 1198631199111= 119863

1199112= 005

(70)




11986011989111

= [

minus61254 15565


11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)


6 Conclusions






Acknowledgments


References




































































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra












pound119881 (120578119905 119903

119905 119904

119905) + 119890

⊺

(119905) 119890 (119905) minus 1205742

120596⊺

(119905) 120596 (119905) le 120585⊺

(119905) Σ119897119894119898120585 (119905)

(119897 = 1 2)

(51)


E pound119881 (120578119905 119903

119905 119904

119905)

le E [120578119881 (120578119905 119903

119905 119904

119905)] + 120574

2

120596⊺

(119905) 120596 (119905) minus E [119890⊺

(119905) 119890 (119905)]

(52)


E pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] le 119890

minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)]

(53)


119905 119904

119905)] gt 0 by inte-


int

119879

0

119890minus120578119905

[1205742

120596⊺

(119905) 120596 (119905) minus 119890⊺

(119905) 119890 (119905)] 119889119905

le Eint119879

0

pound [119890minus120578119905119881 (120578119905 119903

119905 119904

119905)] 119889119905 le 119881 (120578

0 119903

0 119904

0) = 0

(54)


E [int119879

0

119890minus120578120592

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

E [int119879

0

119890minus120578120592

120596⊺

(120592) 120596 (120592) 119889120592]

(55)

Then it yields that

E [int119879

0

119890⊺

(120592) 119890 (120592) 119889120592] le 1205742

119890120578119879

E [int119879

0

120596⊺

(120592) 120596 (120592) 119889120592] (56)



pletes the proof




Filtering



119894119898gt 0119872

119894119898 119873

119894119898 119881

119894119898

119876119897119894gt 0 (119897 = 1 2) 119876 gt 0 119883

119894119898gt 0 119883 gt 0 119884

119894119898gt 0 119884 gt 0

119860119891119894119898

119861119891119894119898

119862119891119894119898

119872119860119894119898

119872119861119894119898

119873119860119894119898

119873119861119894119898

119881119860119894119898

119881119861119894119898

119878119894119898

scalars 1198881lt 119888

2 119879 gt 0 120590

119904gt 0 (119904 = 1 2 9) 120575 gt 0 120578 gt 0


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] gt 0 119872119894119898= [

1198721119894119898

1198722119894119898

1198722119894119898

1198722119894119898

] (57)

119873119894119898= [

1198731119894119898

1198732119894119898

1198732119894119898

1198732119894119898

] 119881119894119898= [

1198811119894119898

1198812119894119898

1198812119894119898

1198812119894119898

] (58)

Γ1119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ15119894119898

Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ25119894119898

0 [119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119872⊺

119894119898Γ46119894119898

0


Γ56119894119898

0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(59)

Γ2119894119898

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Γ11119894119898

Γ12119894119898

minus119878119894119898

Γ14119894119898

Γ1015840

15119894119898Γ16119894119898

Γ17119894119898

lowast Ξ22119894119898

119878119894119898

minus

119884119894119898

ℎ

Γ24119894119898

Γ1015840

251198941198980 [

119862⊺

119911120591119894

0]


119884119894119898

ℎ

0 0 0 0


minusℎ119873⊺

119894119898Γ1015840

461198941198980


Γ1015840

561198941198980



]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0

(60)where

Γ11119894119898

= sum

119899isinM

120587119898119899119875119894119899+ sum

119895isinN

120582119894119895119875119895119898+ 119890

120575ℎ

1198761119894+ 119890

120575ℎ

1198762119894+ ℎ119876 +

119890120575ℎ

minus 1

120575

119883119894119898+

119890120575ℎ

minus 120575ℎ119890120575ℎ

minus 1

1205752

119883

+

119884119894119898

ℎ

+ [

1205751198751119894119898

+ 1198751119894119898119860

119894+ 119860

⊺

1198941198751119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

1205751198752119894119898

+ 1198752119894119898119860

119894+ 119860

⊺

1198941198752119894119898

+ 119861119891119894119898119862119910119894+ 119862

⊺

119910119894119861

⊺

119891119894119898120575119875

2119894119898+ 119860

119891119894119898+ 119860

⊺

119891119894119898

]

(61)


Γ12119894119898

= minus

119884119894119898

ℎ

+ 119878119894119898+ [

1198751119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

]

Γ14119894119898

= [

119860⊺

119894119881

⊺

1119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

119860⊺

119894119881

⊺

2119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

]

Γ15119894119898

= [

ℎ119860⊺

119894119872

⊺

1119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

ℎ119860⊺

119894119872

⊺

2119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

]

Γ1015840

15119894119898= [

ℎ119860⊺

119894119873

⊺

1119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

ℎ119860⊺

119894119873

⊺

2119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

]

Γ16119894119898

= [

1198751119894119898119863

119894+ 119861

119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119861

119891119894119898119863

119910119894

]

Γ17119894119898

= [

119862⊺

119911119894

minus119862119891119894119898

]

Γ24119894119898

= [

119860⊺

120591119894119881

⊺

1119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

119860⊺

120591119894119881

⊺

2119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

]

Γ25119894119898

= [

ℎ119860⊺

120591119894119872

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

ℎ119860⊺

120591119894119872

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

]

Γ1015840

25119894119898= [

ℎ119860⊺

120591119894119873

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

ℎ119860⊺

120591119894119873

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

]

Γ46119894119898

= [

1198811119894119898119863

119894+ 119881

119861119894119898119863

119910119894

1198812119894119898119863

119894+ 119881

119861119894119898119863

119910119894

]

Γ56119894119898

= [

ℎ1198721119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

ℎ1198722119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

]

Γ1015840

56119894119898= [

ℎ1198731119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

ℎ1198732119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

]

(62)


119860119891119894119898

= 119875minus1

2119894119898119860

119891119894119898 119861

119891119894119898= 119875

minus1

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

(63)


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] (64)

The term 119875119894119898119860


119875119894119898119860

119894= [

1198751119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

1198752119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

] (65)

Similarly we have

119875119894119898119861119894= [

1198751119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

]

119875119894119898119863

119894= [

1198751119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

]

(66)

Define 119860119891119894119898

= 1198752119894119898119860

119891119894119898 119861

119891119894119898= 119875

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

119872

119860119894119898= 119872

2119894119898119860

119891119894119898119872

119861119894119898= 119872

2119894119898119861119891119894119898

119873119860119894119898

= 1198732119894119898119860

119891119894119898

119873119861119894119898

= 1198732119894119898119861119891119894119898

119881119860119894119898

= 1198812119894119898119860

119891119894119898 and 119881

119861119894119898= 119881

2119894119898119861119891119894119898



proof is completed


min 1205742

st (48)ndash(50) (67)




1= 1 then


2min

min 1205891205742

+ (1 minus 120589) 1198882

st (59)-(60) and (50)

(68)




Λ = [

minus07 07

09 minus09] Π = [

minus1 1

12 minus12] (69)


1198601= [

minus35 086


2= [

minus25 034

14 minus002]

1198601205911= [

minus08 minus13


1205912= [

minus28 05

minus08 minus14]

1198631= 119863

2= [

01

02] 119862

1199101= [04 07]

1198621199102= [05 08] 119862

1199101205911= [01 02]

1198621199101205912

= [01 01]

1198631199101= 119863

1199102= 01 119862

1199111= [01 005]

1198621199112= [02 009] 119862

1199111205911= [03 06]

1198621199111205912

= [04 08] 1198631199111= 119863

1199112= 005

(70)




11986011989111

= [

minus61254 15565


11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)


6 Conclusions






Acknowledgments


References




































































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Γ12119894119898

= minus

119884119894119898

ℎ

+ 119878119894119898+ [

1198751119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119861

119891119894119898119862119910120591119894

]

Γ14119894119898

= [

119860⊺

119894119881

⊺

1119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

119860⊺

119894119881

⊺

2119894119898+ 119862

⊺

119910119894119881

⊺

119861119894119898119881

⊺

119860119894119898

]

Γ15119894119898

= [

ℎ119860⊺

119894119872

⊺

1119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

ℎ119860⊺

119894119872

⊺

2119894119898+ ℎ119862

⊺

119910119894119872

⊺

119861119894119898ℎ119872

⊺

119860119894119898

]

Γ1015840

15119894119898= [

ℎ119860⊺

119894119873

⊺

1119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

ℎ119860⊺

119894119873

⊺

2119894119898+ ℎ119862

⊺

119910119894119873

⊺

119861119894119898ℎ119873

⊺

119860119894119898

]

Γ16119894119898

= [

1198751119894119898119863

119894+ 119861

119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119861

119891119894119898119863

119910119894

]

Γ17119894119898

= [

119862⊺

119911119894

minus119862119891119894119898

]

Γ24119894119898

= [

119860⊺

120591119894119881

⊺

1119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

119860⊺

120591119894119881

⊺

2119894119898+ 119862

⊺

119910120591119894119881

⊺

119861119894119898

]

Γ25119894119898

= [

ℎ119860⊺

120591119894119872

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

ℎ119860⊺

120591119894119872

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119872

⊺

119861119894119898

]

Γ1015840

25119894119898= [

ℎ119860⊺

120591119894119873

⊺

1119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

ℎ119860⊺

120591119894119873

⊺

2119894119898+ ℎ119862

⊺

119910120591119894119873

⊺

119861119894119898

]

Γ46119894119898

= [

1198811119894119898119863

119894+ 119881

119861119894119898119863

119910119894

1198812119894119898119863

119894+ 119881

119861119894119898119863

119910119894

]

Γ56119894119898

= [

ℎ1198721119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

ℎ1198722119894119898119863

119894+ ℎ119872

119861119894119898119863

119910119894

]

Γ1015840

56119894119898= [

ℎ1198731119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

ℎ1198732119894119898119863

119894+ ℎ119873

119861119894119898119863

119910119894

]

(62)


119860119891119894119898

= 119875minus1

2119894119898119860

119891119894119898 119861

119891119894119898= 119875

minus1

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

(63)


119875119894119898= [

1198751119894119898

1198752119894119898

1198752119894119898

1198752119894119898

] (64)

The term 119875119894119898119860


119875119894119898119860

119894= [

1198751119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

1198752119894119898119860

119894+ 119875

21198941198981198611198911198941198981198621199101198941198752119894119898119860

119891119894119898

] (65)

Similarly we have

119875119894119898119861119894= [

1198751119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

1198752119894119898119860

120591119894+ 119875

2119894119898119861119891119894119898119862119910120591119894

]

119875119894119898119863

119894= [

1198751119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

1198752119894119898119863

119894+ 119875

2119894119898119861119891119894119898119863

119910119894

]

(66)

Define 119860119891119894119898

= 1198752119894119898119860

119891119894119898 119861

119891119894119898= 119875

2119894119898119861119891119894119898

119862119891119894119898

= 119862119891119894119898

119872

119860119894119898= 119872

2119894119898119860

119891119894119898119872

119861119894119898= 119872

2119894119898119861119891119894119898

119873119860119894119898

= 1198732119894119898119860

119891119894119898

119873119861119894119898

= 1198732119894119898119861119891119894119898

119881119860119894119898

= 1198812119894119898119860

119891119894119898 and 119881

119861119894119898= 119881

2119894119898119861119891119894119898



proof is completed


min 1205742

st (48)ndash(50) (67)




1= 1 then


2min

min 1205891205742

+ (1 minus 120589) 1198882

st (59)-(60) and (50)

(68)




Λ = [

minus07 07

09 minus09] Π = [

minus1 1

12 minus12] (69)


1198601= [

minus35 086


2= [

minus25 034

14 minus002]

1198601205911= [

minus08 minus13


1205912= [

minus28 05

minus08 minus14]

1198631= 119863

2= [

01

02] 119862

1199101= [04 07]

1198621199102= [05 08] 119862

1199101205911= [01 02]

1198621199101205912

= [01 01]

1198631199101= 119863

1199102= 01 119862

1199111= [01 005]

1198621199112= [02 009] 119862

1199111205911= [03 06]

1198621199111205912

= [04 08] 1198631199111= 119863

1199112= 005

(70)




11986011989111

= [

minus61254 15565


11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)


6 Conclusions






Acknowledgments


References




































































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11986011989111

= [

minus61254 15565


11986011989112

= [

minus62682 14338


11986011989121

= [

minus89214 24223


11986011989122

= [

minus84638 28237


11986111989111

= [

51465 minus28516

minus91216 95171]

11986111989112

= [

54203 minus23121

minus93156 96332]

11986111989121

= [

46193 minus11984

minus162397 163006]

11986111989122

= [

46512 minus10311

minus161132 162312]

11986211989111

= [00294 minus00397]

11986211989112

= [00271 minus003368]

11986211989121

= [01163 minus01432]

11986211989122

= [01342 minus03672]

(71)


6 Conclusions






Acknowledgments


References




































































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























































ness of 119867infin












Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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