complete periodic synchronization of memristor-based neural

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 140153 12 pageshttpdxdoiorg1011552013140153

Research ArticleComplete Periodic Synchronization of Memristor-Based NeuralNetworks with Time-Varying Delays

Huaiqin Wu Luying Zhang Sanbo Ding Xueqing Guo and Lingling Wang

Department of Applied Mathematics Yanshan University Qinhuangdao 066001 China

Correspondence should be addressed to Huaiqin Wu huaiqinwuysueducn

Received 6 April 2013 Revised 4 June 2013 Accepted 8 June 2013

Academic Editor Zhengqiu Zhang

Copyright copy 2013 Huaiqin Wu 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 investigates the complete periodic synchronization of memristor-based neural networks with time-varying delaysFirstly under the framework of Filippov solutions by using M-matrix theory and the Mawhin-like coincidence theorem in set-valued analysis the existence of the periodic solution for the network system is proved Secondly complete periodic synchronizationis considered for memristor-based neural networks According to the state-dependent switching feature of the memristor the errorsystem is divided into four cases Adaptive controller is designed such that the considered model can realize global asymptoticalsynchronization Finally an illustrative example is given to demonstrate the validity of the theoretical results

1 Introduction

Memristor as the fourth fundamental passive circuit wasfirstly postulated by Chua [1] in 1971 On May 1 2008 theHewlett-Packard (HP) research team announced their real-ization of a memristor prototype with an official publicationin Nature [2 3] This new circuit element of memristorshares many properties of resistors and shares the sameunit of measurement Recently memristor has received agreat deal of attention because of its potential applications innext generation computer and powerful brain-like ldquoneuralrdquocomputer The papers [4ndash21] have given a detailed introduc-tion on the memristor so readers can consult [4ndash21] to getmore explanation As noted in [10] from a systems-theoreticpoint of view and a mathematical point of view memris-tor dynamics strictly obey Bernoullirsquos nonlinear differentialequation so the mathematical framework and its useful-ness are worth studying The paper [10] by Wu and Zengdiscussed the exponential stabilization of memristive neuralnetworks with time delays The papers [11ndash14] investigatedthe synchronization and antisynchronization control of aclass of memristor-based recurrent neural networks A seriesof results on stability analysis of memristor-based recurrentneural networks were presented in [15ndash18] The papers [19ndash21] dealt with the existence and stability of periodic solution

of almost periodic of a class of memristor-based recurrentneural networks

Different from the previous works in this paper wewill study complete periodic synchronization of memristor-based neural networks described by the following differentialequation

119894 (119905) = minus 119889

119894(119909

119894 (119905)) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895(119909

119894 (119905)) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

b119894119895(119909

119894 (119905)) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

119905 ge 0 119894 = 1 2 119899

(1)

where

119889

119894(119909

119894 (119905)) =

119889

119894

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119889

119894

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119886

119894119895(119909

119894 (119905)) =

119886

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119886

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119887

119894119895(119909

119894 (119905)) =

119887

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119887

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

(2)

2 Discrete Dynamics in Nature and Society

in which switching jumps 119879119894gt 0

119889

119894gt 0

119889

119894gt 0 119886

119894119895 119886

119894119895

119887

119894119895 119887119894119895 119894 119895 = 1 2 119899 are constants 119891

119895and 119892

119895are feedback

functions 120591119894119895(119905) is the time delay with 0 le 120591

119894119895(119905) le 120591 and

120591

119894119895le 120583

119894119895lt 1 (120591 and 120583

119894119895are negative constants) At first

glance one might intuitively believe that the chaotic motionis more complicated compared with the periodic motionthe synchronization of chaotic oscillators is also complicatedthan those of periodic oscillators [22] However this is notalways true just as indicated in [23 24] where an oppositeresult was given

The rest of this paper is organized as follows In Section 2some preliminaries are introduced In Section 3 the proofof the existence of periodic solutions is presented Com-plete periodic synchronization is discussed in Section 4 InSection 5 a numerical example is presented to demonstratethe validity of the proposed results Some conclusions aredrawn in Section 6

Notation R denotes the set of real numbers R119899 denotes the119899-dimensional Euclidean space and R119898times119899 denotes the set ofall 119898 times 119899 real matrices Given the vectors 119909 = (119909

1 119909

119899)

119879119910 = (119910

1 119910

119899)

119879isin R119899 119909 = (sum

119899

119894=1119909

2

119894)

12 119909119879119910 =

sum

119899

119894=1119909

119894119910

119894 For 119903 gt 0 119862([minus119903 0]R119899) denotes the family of

continuous function 120593 from [minus119903 0] to R119899 with the norm120593 = sup

minus119903le119904le0|120593(119904)| [sdot sdot] represents the interval co(119876)

denotes the closure of the convex hull of 119876 119864119899denotes

the identity matrix of size 119899 A vector or matrix 119860 ge 0

means that all entries of 119860 are greater than or equal to zero119860 gt 0 can be defined similarly For vectors or matrices 119860and 119861 119860 ge 119861 (or 119860 gt 119861) means that 119860 minus 119861 ge 0 (or119860 minus 119861 gt 0) 119870(R119899) denotes the collection of all nonemptycompact subsets of R with the Hausdorff metric 120588 definedby 120588(119860 119861) = max120573(119860 119861) 120573(119861 119860) 119860 119861 isin 119870(R119899) and120588(119860 119861) = supdist(119909 119861) 119909 isin 119860 120588(119861 119860) = supdist(119910 119860) 119910 isin 119861 119870120592(R119899) = 119860 isin 119870(R119899) 119860 is convex

2 Preliminaries

In this section we give some definitions and propertieswhich are needed later

Definition 1 (see [25]) Suppose 119864 sube R119899 then 119909 rarr 119865(119909) iscalled a set-valued map from 119864 rarr R119899 if for each point 119909 isin119864 there exists a nonempty set 119865(119909) sube R119899 A set-valued map119865 with nonempty values is said to be upper semicontinuous(USC) at 119909

0isin 119864 if for any open set119873 containing 119865(119909

0) there

exists a neighborhood119872 of 1199090such that 119865(119872) sube 119873Themap

119865(119909) is said to have a closed (convex compact) image if foreach 119909 isin 119864 119865(119909) is closed (convex compact)

Definition 2 (see [26]) For the system (119905) = 119891(119905 119909) 119909 isin

R119899 with discontinuous right-hand sides a set-valued map isdefined as

119865 (119905 119909) = ⋂

120575gt0

⋂

120583(119873)=0

co [119891 (119861 (119909 120575) 119873)] (3)

where co[119864] is the closure of the convex hull of set119864119861(119909 120575) =119910 119910 minus 119909 le 120575 and 120583(119873) is Lebesgue measure of set

119873 A solution in Filippovrsquos sense of the Cauchy problem forthis system with initial condition 119909(0) = 119909

0is an absolutely

continuous function 119909(119905) 119905 isin [0 119879] which satisfies 119909(0) = 1199090

and differential inclusion

(119905) isin 119865 (119905 119909) for aa 119905 isin [0 119879] (4)

The initial value associated with system (1) is 119909119894(119905) =

120601(119905) isin 119862([minus120591 0]R) 119894 = 1 2 119899 Let 119889119894= min 119889

119894

119889

119894

119889

119894= max 119889

119894

119889

119894 119886119894119895= min 119886

119894119895 119886

119894119895 119886119894119895= max 119886

119894119895 119886

119894119895

119887

119894119895= min 119887

119894119895

119887

119894119895 119887119894119895= max 119887

119894119895

119887

119894119895 119886119894119895= max|119886

119894119895| |119886

119894119895|

and 119887119894119895= max|119887

119894119895| |119887

119894119895| We define

co (119889119894(119909

119894 (119905))) =

119889

119894

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119879

119894

[119889

119894 119889

119894]

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

= 119879

119894

119889

119894

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

co (119886119894119895(119909

119894 (119905))) =

119886

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119879

119894

[119886

119894119895 119886

119894119895]

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

= 119879

119894

119886

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

co (119887119894119895(119909

119894 (119905))) =

119887

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119879

119894

[119887

119894119895 119887

119894119895]

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

= 119879

119894

119887

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

(5)

Clearly system (1) is a differential equation with discontin-uous right-hand side its solution in the conventional sensedoes not exist Inspired by [10 20 21 27ndash30] we adopt thefollowing definition of the solution in the sense of Filippovfor system (1)

Definition 3 Suppose that 120601(119904) = (1206011(119904) 120601

2(119904) 120601

119899(119904))

119879isin

119862([minus120591 0]R119899) is a continuous function An absolutely con-tinuous function 119909(119905) is said to be a solution with initial data120601(119904) of system (1) if 119909(119905) satisfies the differential inclusion

119894 (119905) isin minus co (119889119894 (119909119894 (119905))) 119909119894 (119905)

+

119899

sum

119895=1

co (119886119894119895(119909

119894 (119905))) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

co (119887119894119895(119909

119894 (119905))) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(6)

for all 119894 = 1 2 119899 or equivalently there exist 119889119894(119905) isin

co(119889119894(119909

119894(119905))) 119886

119894119895(119905) isin co(119886

119894119895(119909

119894(119905))) and 119887

119894119895(119905) isin co(119887

119894119895(119909

119894(119905)))

such that

119894 (119905) = minus 119889

119894 (119905) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895 (119905) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

119887

119894119895 (119905) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(7)

Remark 4 From the theoretical point of view the aboveparameters 119889

119894(119905) 119886

119894119895(119905) and 119887

119894119895(119905) in (7) are measurable

functions and depend on the state 119909119894(119905) and time 119905

Discrete Dynamics in Nature and Society 3

Definition 5 We say that real matrix119872 = (119898

119894119895) isin R119899times119899 is an

119872-matrix if and only if we have 119898119894119895le 0 119894 119895 = 1 2 119899

119894 = 119895 and all successive principal minors of119872 are positive

Lemma6 (see [31]) Let119872 be an 119899times119899matrix with nonpositiveoff-diagonal elements Then 119872 is an 119872-matrix if and only ifone of the following conditions hold

(1) there exists a vector 120585 = (1205851 120585

2 120585

119899) gt (0 0 0)

such that 120585119872 gt 0(2) there exists a vector 120578 = (120578

1 120578

2 120578

119899)

119879gt (0 0

0)

119879 such that119872120578 gt 0

Lemma 7 (Mawhin-Like Coincidence Theorem [32]) Sup-pose that 119865R timesR119899 rarr 119870120592(R119899) is USC and 120596-periodic in 119905 Ifthe following conditions hold

(1) there exists a bounded open set Ω sube 119862

120596 the set of all

continuous 120596-periodic functions R rarr R119899 such thatfor any 120582 isin (0 1) each 120596-periodic function 119909(119905) of theinclusion

(119905) isin 120582119865 (119905 119909) (8)

satisfies 119909 notin 120597Ω if it exists(2) each solution 119906 isin R119899 of the inclusion

0 isin

1

120596

int

120596

0

119865 (119905 119906) 119889119905 = 120603 (119906) (9)

satisfies 119906 notin 120597Ω⋂R119899(3) deg(120603Ω⋂R119899 0) = 0

then differential inclusion (4) has at least one 120596-periodic solution 119909(119905) with 119909 isin Ω

To proceed with our analysis we need the followingassumptions for system (1)(A1) 119868

119894(119905) and 120591

119894119895(119905) are continuous 120596-periodic functions

(A2) For 119894 = 1 2 119899 for all 119904

1 119904

2isin R 119904

1= 119904

2 the neural

activation function 119892119894is bounded and 119891

119894 119892

119894satisfy

Lipschitz condition that is there exist 984858119894gt 0 120588

119894gt 0

and 119866119894 such that1003816

1003816

1003816

1003816

119891

119894(119904

1) minus 119891

119894(119904

2)

1003816

1003816

1003816

1003816

le 984858

119894

1003816

1003816

1003816

1003816

119904

1minus 119904

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119894(119904

1) minus 119892

119894(119904

2)

1003816

1003816

1003816

1003816

le 120588

119894

1003816

1003816

1003816

1003816

119904

1minus 119904

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119894 (sdot)

1003816

1003816

1003816

1003816

le 119866

119894

(10)

3 Existence of Periodic Solution

In this section we will give a sufficient condition whichensures the existence of periodic solution ofmemristor-basedneural network (1)

Theorem 8 Under assumptions (A1) and (A

2) if 119864119899minus119876 is an

119872-matrix where 119876 = (119902119894119895)

119899times119899and

119902

119894119895=

1

119889

119894

(984858

119895119886

119894119895+

120588

119895

119887

119894119895

radic1 minus 120583

119894119895

) 119894 119895 = 1 2 119899 (11)

then system (1) has at least one 120596-periodic solution

Proof Set 119862120596= 119909(119905) isin 119862(RR119899) 119909(119905 + 120596) = 119909(119905)

119909(119905)

119862120596

= sum

119899

119894=1max119905isin[0120596]

|119909

119894(119905)| Then 119862

120596is a Banach space

with the norm sdot

119862120596

Let 119909(119905) = (1199091(119905) 119909

2(119905) 119909

119899(119905))

119879isin

119862

120596 119865(119905 119909) = (119865

1(119905 119909) 119865

2(119905 119909) 119865

119899(119905 119909))

119879 where

119865

119894 (119905 119909) = minus co (119889

119894(119909

119894 (119905))) 119909119894 (

119905)

+

119899

sum

119895=1

co (119886119894119895(119909

119894 (119905))) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

co (119887119894119895(119909

119894 (119905))) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(12)

119894 = 1 2 119899 It is obvious that the set-valuedmap119865(119905 119909) hasnonempty compact convex values Futhermore it is USC

Based on the conditions of Lemma 7 the proof will bedivided into three steps

Step 1 We need to search for appropriate open boundedsubset Ω Assume that 119909(119905) = (119909

1(119905) 119909

2(119905) 119909

119899(119905))

119879 is anarbitrary 120596-periodic solution of differential inclusion (119905) isin120582119865(119905 119909) for a certain 120582 isin (0 1) Then one has

119894 (119905) isin 120582

[

[

minus co (119889119894(119909

119894 (119905))) 119909119894 (

119905)

+

119899

sum

119895=1

co (119886119894119895(119909

119894 (119905))) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

co (119887119894119895(119909

119894 (119905))) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

]

]

(13)

or equivalently there exist measurable functions 119889119894(119905) isin

co(119889119894(119909

119894(119905))) 119886

119894119895(119905) isin co(119886

119894119895(119909

119894(119905))) and 119887

119894119895(119905) isin co(119887

119894119895(119909

119894(119905)))

such that

119894 (119905) = 120582

[

[

minus119889

119894 (119905) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895 (119905) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

119887

119894119895 (119905) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

]

]

(14)

for 119894 = 1 2 119899 Multiplying both sides of (14) by 119909119894(119905) and

integrating over the interval [0 120596] one has

int

120596

0

119889

119894 (119905) 119909

2

119894(119905) 119889119905

= int

120596

0

119909

119894 (119905)

[

[

119899

sum

119895=1

119886

119894119895 (119905) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

119887

119894119895 (119905) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

]

]

119889119905

(15)


Noting that

int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

= int

120596minus120591119894119895(120596)

minus120591119894119895(0)

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

1 minus 120591

119894119895(120581

minus1

119894119895(119905))

119889119905

= int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

1 minus 120591

119894119895(120581

minus1

119894119895(119905))

119889119905 le

1

1 minus 120583

119894119895

int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

(16)

where 120581minus1119894119895

is the inverse function of 120581119894119895(119905) = 119905 minus 120591

119894119895(119905) 119894 119895 =

1 2 119899 from (15) and (16) it yields

119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905

le

119899

sum

119895=1

119886

119894119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

119886

119894119895984858

119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895120588

119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895

1 minus 120583

119894119895

)int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895

1 minus 120583

119894119895

)(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

times (int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

2

119889119905)

12

+

119868

119894radic120596(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

(17)

where 119868119894= sup

119905isin[0120596]|119868

119894(119905)| This means

(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

le

119899

sum

119895=1

119902

119894119895(int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

2

119889119905)

12

+

radic120596

119868

119894

119889

119894

(18)

Define 119909119894

120596

2= (int

120596

0|119909

119894(119905)|

2119889119905)

12119909119894isin 119862(RR) 119894 = 1 2 119899

and 119868119873= (

119868

1119889

1

119868

2119889

2

119868

119899119889

119899)

119879 From (18) we have

(119864

119899minus 119876) (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

119879

leradic120596119868

119873

(19)

Since 119864119899minus 119876 is an 119872-matrix we can choose a vector 120585 =

(120585

1 120585

2 120585

119899) gt (0 0 0) such that

120585

lowast= (120585

lowast

1 120585

lowast

2 120585

lowast

119899) = 120585 (119864

119899minus 119876) gt (0 0 0) (20)

By combining (19) and (20) together one can derive

min 120585lowast1 120585

lowast

2 120585

lowast

119899 (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2+

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2+ sdot sdot sdot +

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

le 120585

lowast

1

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2+ 120585

lowast

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2+ sdot sdot sdot + 120585

lowast

119899

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2

= 120585 (119864

119899minus 119876) (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

119879

le 120585radic120596(

119868

1

119889

1

119868

2

119889

2

119868

119899

119889

119899

)

119879

=radic120596

119899

sum

119894=1

120585

119894

119868

119894

119889

119894

(21)

Thus we can easily get that

1003817

1003817

1003817

1003817

119909

119894

1003817

1003817

1003817

1003817

120596

2= (int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

leradic120596119873 119894 = 1 2 119899

(22)

where 119873 = sum

119899

119894=1(120585

119894

119868

119894119889

119894)min120585lowast

1 120585

lowast

2 120585

lowast

119899 Then there

exists 119905lowast isin [0 120596] such that

1003816

1003816

1003816

1003816

119909

119894(119905

lowast)

1003816

1003816

1003816

1003816

le 119873 119894 = 1 2 119899 (23)

Obviously for 119905 isin [0 120596] 119909119894(119905) = 119909

119894(119905

lowast)+int

119905

119905lowast

119894(119904)119889119904 It follows

from (23) that

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

le 119873 + int

120596

0

1003816

1003816

1003816

1003816

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 119894 = 1 2 119899 (24)

On the other hand from (14) one easily obtains that

int

120596

0

1003816

1003816

1003816

1003816

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 lt int

120596

0

1003816

1003816

1003816

1003816

119889

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

int

120596

0

1003816

1003816

1003816

1003816

1003816

119886

119894119895 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

int

120596

0

1003816

1003816

1003816

1003816

1003816

119887

119894119895 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

119889119905

+ int

120596

0

1003816

1003816

1003816

1003816

119868

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le 119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 +

119899

sum

119895=1

119886

119894119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895(119905)))

1003816

1003816

1003816

1003816

1003816

119889119905 + 120596

119868

119894


le 119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 +

119899

sum

119895=1

119886

119894119895984858

119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895120588

119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905 + 120596

119868

119894

le 119889

119894radic120596

1003817

1003817

1003817

1003817

119909

119894

1003817

1003817

1003817

1003817

120596

2

+

119899

sum

119895=1

(984858

119895119886

119894119895+

120588

119895

119887

119894119895

radic1 minus 120583

119894119895

)radic120596

1003817

1003817

1003817

1003817

1003817

119909

119895

1003817

1003817

1003817

1003817

1003817

120596

2+ 120596

119868

119894

leradic120596119873(119889

119894+ 119889

119894

119899

sum

119895=1

119902

119894119895) + 120596

119868

119894

(25)

Let119872119894= radic120596119873(119889

119894+ 119889

119894sum

119899

119895=1119902

119894119895) + 120596

119868

119894 combining (24) and

(25) we can derive

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119873 +119872

119894= 119877

119894 119894 = 1 2 119899 (26)

Clearly 119877119894is independent of 120582 In addition since 119864

119899minus 119876 is

an 119872-matrix there exists a vector 120578 = (120578

1 120578

2 120578

119899)

119879gt

(0 0 0)

119879 such that (119864119899minus 119876)120578 gt (0 0 0)

119879 Thus wecan choose a sufficiently large constant 120579 such that 120578lowast =(120578

lowast

1 120578

lowast

2 120578

lowast

119899)

119879= (120579120578

1 120579120578

2 120579120578

119899)

119879= 120579120578 120578lowast

119894= 120579120578

119894gt 119877

119894

and

(119864

119899minus 119876) 120578

lowastgt 119868

119873 (27)

Taking Ω = 119909(119905) isin 119862

120596| minus120578

lowastlt 119909(119905) lt 120578

lowast then Ω is an

open bounded set of 119862120596and 119909 notin 120597Ω for any 120582 isin (0 1) This

proves that condition (1) in Lemma 7 is satisfied

Step 2 Suppose that there exists a solution 119906 = (1199061 119906

2

119906

119899)

119879isin 120597Ω⋂R119899 of the inclusion 0 isin (1120596) int120596

0119865(119905 119906)119889119905 =

120603(119906) then 119906 is a constant vector onR119899 such that |119906119894| = 120578

lowast

119894for

some 119894 isin 1 2 119899 Therefore one has

0 isin (120603 (119906))119894

= minus

119906

119894

120596

int

120596

0

co (119889119894(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

1

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119891119895

(119906

119895) 119889119905

+

119899

sum

119895=1

1

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119892119895

(119906

119895) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

(28)

or equivalently there exist 119889119894(119905) isin co(119889

119894(119909

119894(119905))) 119886

119894119895(119905) isin

co(119886119894119895(119909

119894(119905))) and 119887

119894119895(119905) isin co(119887

119894119895(119909

119894(119905))) such that

0 = minus

119906

119894

120596

int

120596

0

119889

119894 (119905) 119889119905 +

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

(29)

Then there exists 119905lowastisin [0 120596] such that

0 = minus 119906

119894119889

119894(119905

lowast) +

119899

sum

119895=1

119886

119894119895(119905

lowast) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

lowast) 119892

119895(119906

119895) + 119868

119894(119905

lowast)

(30)

It follows from (30) that

120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119895+

119868

119894

119889

119894

(31)

This means (119864119899minus 119876)120578

lowastle 119868

119873 which contradicts (27)

Step 3 We define a homotopic set-valued map 120601 Ω⋂R119899 times

[0 1] rarr 119862

120596by 120601(119906 ℎ) = minus diag(ℎ119889

1 ℎ119889

2 ℎ119889

119899)119906 + (1 minus

ℎ)120603(119906)If 119906 = (119906

1 119906

2 119906

119899)

119879isin 120597Ω⋂R119899 then 119906 is a constant

vector on R119899 such that |119906119894| = 120578

lowast

119894for some 119894 isin 1 2 119899 It

follows that

(120601 (119906 ℎ))

119894

= minusℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(32)


In fact we have 0 notin (120601(119906 ℎ))

119894 119894 isin 1 2 119899 If 0 isin

(120601(119906 ℎ))

119894 that is

0 isin minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(33)

or

0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894 (119905) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(34)

There exists 1199050isin [0 120596] such that

0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894(119905

0) 119906

119894+

119899

sum

119895=1

119886

119894119895(119905

0) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

0) 119892

119895(119906

119895) + 119868

119894(119905

0)

]

]

(35)

that is

ℎ (119889

119894minus 119889

119894(119905

0)) 119906

119894+ 119889

119894(119905

0) 119906

119894

= (1 minus ℎ)

[

[

119899

sum

119895=1

(119886

119894119895(119905

0) 119891

119895(119906

119895) + 119887

119894119895(119905

0) 119892

119895(119906

119895)) + 119868

119894(119905

0)

]

]

(36)

Therefore we have

120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1 minus ℎ

ℎ (119889

119894minus 119889

119894(119905

0)) + 119889

119894(119905

0)

times

[

[

119899

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

119886

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119887

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

)

+

1003816

1003816

1003816

1003816

119868

119894(119905

0)

1003816

1003816

1003816

1003816

]

]

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119894+

119868

119894

119889

119894

(37)

This means that (119864119899minus 119876)120578

lowastle 119868

119873 which contradicts

(27) Thus 0 notin (120601(119906 ℎ))

119894 119894 = 1 2 119899 It follows that

(0 0 0)

119879notin 120601(119906 ℎ) for any 119906 = (119906

1 119906

2 119906

119899)

119879isin

120597Ω⋂R119899 ℎ isin [0 1] Therefore by the homotopy invarianceand the solution properties of the topological degree one has

deg 120603Ω⋂R119899 0 = deg 120601 (119906 0) Ω⋂R

119899 0

= deg 120601 (119906 1) Ω⋂R119899 0

= deg (minus1198891119906

1 minus119889

2119906

2 minus119889

119899119906

119899)

119879

Ω⋂R119899 (0 0 0)

119879

= sign

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus119889

1sdot sdot sdot 0

d

0 minus119889

119899

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= (minus1)

119899= 0

(38)

where deg(sdot sdot sdot) denotes the topological degree for USC set-valued maps with compact convex values

Up to now we have proved that Ω satisfies all theconditions in Lemma 7 then system (1) has at least one 120596-periodic solution This completes the proof

Notice that a constant function can be regarded asa special periodic function with arbitrary period or zeroamplitude Hence we can obtain the following result

Corollary 9 Suppose assumption (A2) holds and 119868

119894(119905) = 119868

119894

120591

119894119895(119905) = 120591

119894119895 if 119864119899minus 119876 is an119872-matrix where 119876 = (119902

119894119895)


119902

119894119895=

1

119889

119894

(984858

119895119886

119894119895+ 120588

119895

119887

119894119895) 119894 119895 = 1 2 119899 (39)

then system (1) exists at least one equilibrium point

Remark 10 By employing the method based on the 119872-matrix theory our results can be easily verified and are muchdifferent from these in the literature [20 21] It is also worthmentioning that the119872-matrix theory is one of the effective


and important methods to deal with the existence of periodicsolution and equilibrium point for large-scale dynamicalneuron systems

4 Complete Periodic Synchronization

In this paper we consider model (1) as the master systemand a slave system for (1) can be described by the followingequation

119910

119894 (119905) = minus 119889

119894(119910

119894 (119905)) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895(119910

119894 (119905)) 119891119895

(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895(119910

119894 (119905)) 119892119895

(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

119905 ge 0 119894 = 1 2 119899

(40)

where 119906119894(119905) is the controller to be designed and

119889

119894(119910

119894 (119905)) =

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119886

119894119895(119910

119894 (119905)) =

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119887

119894119895(119910

119894 (119905)) =

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

(41)

Let 119890119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 119899 one can obtain the

following result

Theorem 11 Suppose that all the conditions of Theorem 8 aresatisfied then the salve system (40) can globally synchronizewith the master system (1) under the following adaptivecontroller

119906

119894 (119905) = minus120572119894 (

119905) 119890119894 (119905) minus 120575119894

120573

119894 (119905) sign (119890119894 (119905))

119894 (119905) = minus120576119894

119890

2

119894(119905) 119894 = 1 2 119899

120573

119894 (119905) = minus120578119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

119894 = 1 2 119899

(42)

where 120572119894 120578119894are arbitrary positive constants and 120575

119894gt 1

Proof Consider the following Lyapunov functional

119881 (119905) =

1

2

119899

sum

119894=1

119890

2

119894(119905) +

119899

sum

119894119895=1

1

2 (1 minus 120583

119894119895)

int

119905

119905minus120591119894119895(119905)

119890

2

119894(119904) 119889119904

+

119899

sum

119894=1

1

2120576

119894

(120572

119894 (119905) minus 119870119894

)

2+

119899

sum

119894=1

1

2120578

119894

(119872

119894minus 120573

119894 (119905))

2

(43)

where

119870

119894ge max

minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

minus

119889

119894

+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

119872

119894ge

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895

(44)

The master system (1) and the slave system (40) are state-dependent switching systems then the four casesmay appearin the following at time 119905

Case 1 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| le 119879

119894at time 119905 then the master

system (1) and the slave system (40) reduce to the followingsystems respectively

119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(45)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(t minus 120591119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(46)

Correspondingly the error system can be written as

119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(47)

where 119891119895(119890

119895(119905)) = 119891

119895(119910

119895(119905)) minus 119891

119895(119909

119895(119905)) 119892

119895(119890

119895(119905 minus 120591

119894119895(119905))) =

119892

119895(119910

119895(119905 minus 120591

119894119895(119905))) minus 119892

119895(119909

119895(119905 minus 120591

119894119895(119905))) Under assumption (A

2)

evaluating the upper right derivation 119863+119881(119905) of 119881(119905) alongthe trajectory of (47) gives

119863

+119881 (119905)

le

119899

sum

119894=1

119890

119894 (119905)

[

[

minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890119895(119905 minus 120591

119894119895 (119905)) minus 120572119894 (

119905) 119890119894 (119905)

minus120575

119894120573

119894 (119905) sign (119890119894 (119905)) ]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))


+ (120572

119894 (119905) minus 119870119894

) 119890

2

119894(119905) minus (119872119894

minus 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

)

le

119899

sum

119894=1

[

[

minus

119889

119894119890

2

119894(119905) +

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895 (119905)

1003816

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

120588

119895

1003816

1003816

1003816

1003816

1003816

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

minus 120575

119894120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119870

119894119890

2

119894(119905) minus 119872119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

+ 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le

119899

sum

119894=1

[

[

minus (119870

119894+

119889

119894) 119890

2

119894(119905) +

119899

sum

119895=1

(

1

2

984858

2

119895119886

2

119894119895119890

119894(119905)

2+

1

2

119890

119895(119905)

2)

+

119899

sum

119895=1

(

1

2

120588

2

119895

119887

2

119894119895119890

119894(119905)

2+

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

=

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(48)

Considering the definition of 119870119894 and 120575

119894gt 1119872

119894gt 0 one has

119863

+119881 (119905) le 0 (49)

Case 2 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| gt 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(50)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(51)

Correspondingly the error system can be rewritten as

119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(52)

Arguing as in the proof of Case 1 we can obtain

119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(53)

Case 3 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| le 119879


system (1) and the slave system (40) reduce to (50) and (46)Correspondingly the error system can be rewritten as

119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119910

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(54)

Similarly evaluating the upper right derivation 119863+119881(119905) of119881(119905) along the trajectory of (54) we have

119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119910

119895 (119905))

1003816

1003816

1003816

1003816

1003816


+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119910

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(55)

Note that |119910119894(119905)| le 119879

119894 by using assumption (A

2) one has

119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(56)

According to the definition of119870119894119872119894 and 120575

119894gt 1 one has

119863

+119881 (119905) le 0 (57)

Case 4 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| gt 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119909

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119909

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(58)

By using |119909119894(119905)| le 119879

119894 we can also have

119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(59)

The above proving procedures clearly imply that one alwayshas 119863+119881(119905) le 0 at time 119905 Therefore the salve system (40)globally synchronizes with the master system (1) under theadaptive controller (42) This completes the proof

Remark 12 In the literature some results on stability analysisof periodic solution (or equilibrium point) or synchroniza-tion (or antisynchronization) control of memristor-basedneural network were obtained [11ndash13 16 17 20 21] A typicalassumption is that

[119889

119894 119889

119894] 119909 minus [119889

119894 119889

119894] 119910 sube [119889

119894 119889

119894] (119909 minus 119910) (60)

HoweverWe can prove that this assumption holds only when119909 and 119910 have different sign or 119909 = 0 or 119910 = 0 Without thisassumption we divide the error system into four cases in thispaper Under the adaptive controller (42) globally periodicsynchronization criterion between system (1) and (40) isderived The synchronization criterion of this paper whichdoes not solve any inequality or linear matrix inequality iseasily verified

Remark 13 As far as we know there is no work on theperiodic synchronization ofmemristor-based neural networkvia adaptive control Thus our outcomes are brand newand original compared to the existing results ([11ndash14]) Inaddition the obtained results in this paper are also applicableto the common systems without memristor or the memduc-tance of thememristor equals a constant since they are specialcases of memristor-based neural networks

5 Numerical Example

In this section one example is offered to illustrate theeffectiveness of the results obtained in this paper

Example 1 Consider the second-order memristor-basedneural network (1) with the following system parameters

119889

1(119909

1 (119905)) =

65

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

6

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4


119889

2(119909

2 (119905)) =

6

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

65

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

11(119909

1 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

22(119909

2 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

11(119909

1 (119905)) =

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

22(119909

2 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

(61)

and the activation functions are taken as follows

119891

1 (119904) = 1198912 (

119904) = 119904 119892

1 (119904) = 1198922 (

119904) =

1

2

tanh 119904 (62)

It can be verified that 1198891= 119889

2= 6 119886

11= 119886

22= 2 119886

12= 119886

21=

1 119887119894119895= 1 119894 119895 = 1 2 984858

1= 984858

2= 1 and 120588

1= 120588

2= 12

We take 120591119894119895(119905) = 34 minus (14) sin 3119905 119894 119895 = 1 2 A

straightforward calculation gives 120591 = 1 and 120583119894119895= 34

Then we get 1198642minus 119876 = (

12 minus13

minus13 12) Obviously 119864

2minus 119876 is

an 119872-matrix Thus the conditions required in Theorem 8are satisfied When 119868(119905) is a periodic function in the viewof Theorem 8 this neural network has at least one periodicsolution

For numerical simulations we choose the external input(119868

1(119905) 119868

2(119905))

119879= (065 sin 3119905minus2 065 cos 3119905+2)119879The periodic

0 5 10 15

minus05

minus04

minus03

minus02

minus01

0

01

x1(t)

y1(t)

x1(t)y1(t)

t

Figure 1 Time-domain behavior of the state variables 1199091(119905) and

119910

1(119905) with 119906

119894(119905) = 0

0 5 10 150

01

02

03

04

05

06

07

08

t

x2(t)

y2(t)

x2(t)y2(t)


119910

2(119905) with 119906

119894(119905) = 0

dynamic behavior of the master system (1) and the slavesystem (40) with 119906

119894(119905) = 0 is given in Figures 1 2 and 3 with

the initial states chosen as 119909(119905) = (06 sin 2119905 03 cos 119905)119879 and119910(119905) = (minus02 cos 2119905 minus08 sin 2119905)119879 for 119905 isin [minus1 0]

In order to demonstrate the adaptive controller (42)can realize complete periodic synchronization of memristor-based neural networks some initial parameters are taken as120572

1(119905) = 120572

2(119905) = 02 120573

1(119905) = 120573

2(119905) = 01 for 119905 isin [minus1 0]

120575

119894= 10 120576

119894= 120578

119894= 01 119894 = 1 2 We get the simulation results

shown in Figures 4ndash6 Figure 4 describes the time responsesof synchronization errors 119890

119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 which

turn to zero quickly as time goes Figure 5 shows the timeresponse of 120572(119905) = (120572

1(119905) 120572

2(119905))

119879 Figure 6 depicts the time


minus06 minus05 minus04 minus03 minus02 minus01 0 010

01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)

Figure 3 Phase plane behavior of themaster system (1) and the salvesystem (40) with 119906

119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)

Figure 4 The synchronization errors 1198901(119905) and 119890

2(119905) under the

adaptive controller (42)

response of 120573(119905) = (1205731(119905) 120573

2(119905))

119879 From Figures 5 and 6 onecan see that the control parameters 120572

119894(119905) and 120573

119894(119905) 119894 = 1 2

turn out to be some constants eventually

6 Conclusion

In this paper complete periodic synchronization of a class ofmemristor-based neural networks has been investigated Themaster system synchronizes with the slave system by usingadaptive control The obtained results are novel since thereare few works about complete periodic synchronization issueof memristor-based neural networks via adaptive controlIn addition the easily testable condition which ensures the

0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t

Figure 5 Trajectories of control parameters 1205721(119905) and 120572

2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)

existence of periodic solution of a class of memristor-basedrecurrent neural network is also much different from theexisting work The obtained results are also applicable tothe continuous systems without switching jumps Finallya numerical example has been given to illustrate the validityof the present results

Acknowledgments

This work was supported by the Natural Science Foundationof Hebei Province of China (A2011203103) and the HebeiProvince Education Foundation of China (2009157)


References

[1] L Chua ldquoMemristor-the missing circut elementrdquo IEEE Trans-actions on Circuit Theory vol 18 pp 507ndash519 1971

[2] D Strukov G Snider G Stewart and RWilliams ldquoThemissingmemristor foundrdquo Nature vol 453 pp 80ndash83 2008

[3] J Tour and T He ldquoThe fourth elementrdquo Nature vol 453 pp42ndash43 2008

[4] M Itoh and L O Chua ldquoMemristor oscillatorsrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 18 no 11 pp 3183ndash3206 2008

[5] X Wang Y Chen Y Gu and H Li ldquoSpintronic memristortemperature sensorrdquo IEEE Electron Device Letters vol 31 pp20ndash22 2010

[6] R Riaza ldquoNondegeneracy conditions for active memristivecircuitsrdquo IEEE Transactions on Circuits and Systems II vol 57pp 223ndash227 2010

[7] F Merrikh-Bayat and S Shouraki ldquoMemristor-based circuitsfor performing basic arithmetic operationsrdquo Procedia ComputerScience vol 3 pp 128ndash132 2011

[8] F Merrikh-Bayat and S Shouraki ldquoProgramming of memristorcrossbars by using genetic algorithmrdquo Procedia Computer Sci-ence vol 3 pp 232ndash237 2011

[9] F Corinto A Ascoli and M Gilli ldquoNonlinear dynamicsof memristor oscillatorsrdquo IEEE Transactions on Circuits andSystems I vol 58 no 6 pp 1323ndash1336 2011

[10] A Wu and Z Zeng ldquoExponential stabilization of memristiveneural networks with time delaysrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 23 pp 1919ndash1929 2012

[11] A Wu S Wen Z Zeng X Zhu and J Zhang ldquoExponentialsynchronization of memristor-based recurrent neural networkswith time delaysrdquoNeurocomputing vol 74 pp 3043ndash3050 2011

[12] A Wu and Z Zeng ldquoSynchronization control of a classof memristor-based recurrent neural networksrdquo InformationSciences vol 183 pp 106ndash116 2012

[13] A Wu and Z Zeng ldquoAnti-synchronization control of a classof memristive recurrent neural networksrdquo Communications inNonlinear Science and Numerical Simulation vol 18 no 2 pp373ndash385 2013

[14] S Wen Z Zeng and T Huang ldquoAdaptive synchronization ofmemristor-based Chuarsquos circuitsrdquo Physics Letters A vol 376 pp2775ndash2780 2012

[15] SWen andZ Zeng ldquoDynamics analysis of a class ofmemristor-based recurrent networks with time-varying delays in thepresence of strong external stimulirdquo Neural Processing Lettersvol 35 pp 47ndash59 2012

[16] A Wu and Z Zeng ldquoDynamic behaviors of memristor-basedrecurrent neural networks with time-varying delaysrdquo NeuralNetworks vol 36 pp 1ndash10 2012

[17] SWen Z Zeng andTHuang ldquoExponential stability analysis ofmemristor-based recurrent neural networks with time-varyingdelaysrdquo Neurocomputing vol 97 pp 233ndash240 2012

[18] G Zhang Y Shen and J Sun ldquoGlobal exponential stability of aclass of memristor-based recurrent neural networks with time-varying delaysrdquo Neurocomputing vol 97 pp 149ndash154 2012

[19] A Wu J Zhang and Z Zeng ldquoDynamic behaviors of a classof memristor-based Hopfield networksrdquo Physics Letters A vol375 no 15 pp 1661ndash1665 2011

[20] G Zhang and Y Shen ldquoGlobal exponential periodicity and sta-bility of a class of memristor-based recurrent neural networkswith multiple delaysrdquo Information Sciences vol 232 pp 386ndash396 2013

[21] S Wen Z Zeng and T Huang ldquoDynamic behaviors ofmemristor-based delayed recurrent networksrdquoNeural Comput-ing and Applications 2012

[22] X Liu J Cao and G Huang ldquoComplete periodic synchroniza-tion of delayedneural networkswith discontinuous activationsrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 20 no 7 pp 2151ndash2164 2010

[23] W Zou and M Zhan ldquoComplete synchronization in coupledlimit-cycle systemsrdquo Europhysics Letters vol 81 no 1 p 10062008

[24] W Zou and M Zhan ldquoComplete periodic synchronization incoupled systemsrdquo Chaos vol 18 no 4 Article ID 043115 2008

[25] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990

[26] A Filippov Differential Equations With Discontinuous Right-Hand Side Mathematics and Its Applications Soviet KluwerAcademic Boston Mass USA 1984

[27] M Forti and P Nistri ldquoGlobal convergence of neural networkswith discontinuous neuron activationsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 50 no 11 pp 1421ndash1435 2003

[28] Z Cai L Huang Z Guo and X Chen ldquoOn the periodicdynamics of a class of time-varying delayed neural networksvia differential inclusionsrdquo Neural Networks vol 33 pp 97ndash1132012

[29] H Wu and C Shan ldquoStability analysis for periodic solution ofBAM neural networks with discontinuous neuron activationsand impulsesrdquo Applied Mathematical Modelling vol 33 no 6pp 2564ndash2574 2009

[30] H Wu ldquoGlobal stability analysis of a general class of discontin-uous neural networks with linear growth activation functionsrdquoInformation Sciences vol 179 no 19 pp 3432ndash3441 2009

[31] A Berman and R J Plemmons Nonnegative Matrices in theMathematical Sciences Academic Press New York NY USA1979

[32] Y Li and Z H Lin ldquoPeriodic solutions of differential inclu-sionsrdquo Nonlinear Analysis Theory Methods amp Applications vol24 no 5 pp 631ndash641 1995

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


in which switching jumps 119879119894gt 0

119889

119894gt 0

119889

119894gt 0 119886

119894119895 119886

119894119895

119887

119894119895 119887119894119895 119894 119895 = 1 2 119899 are constants 119891

119895and 119892

119895are feedback

functions 120591119894119895(119905) is the time delay with 0 le 120591

119894119895(119905) le 120591 and

120591

119894119895le 120583

119894119895lt 1 (120591 and 120583

119894119895are negative constants) At first

glance one might intuitively believe that the chaotic motionis more complicated compared with the periodic motionthe synchronization of chaotic oscillators is also complicatedthan those of periodic oscillators [22] However this is notalways true just as indicated in [23 24] where an oppositeresult was given

The rest of this paper is organized as follows In Section 2some preliminaries are introduced In Section 3 the proofof the existence of periodic solutions is presented Com-plete periodic synchronization is discussed in Section 4 InSection 5 a numerical example is presented to demonstratethe validity of the proposed results Some conclusions aredrawn in Section 6

Notation R denotes the set of real numbers R119899 denotes the119899-dimensional Euclidean space and R119898times119899 denotes the set ofall 119898 times 119899 real matrices Given the vectors 119909 = (119909

1 119909

119899)

119879119910 = (119910

1 119910

119899)

119879isin R119899 119909 = (sum

119899

119894=1119909

2

119894)

12 119909119879119910 =

sum

119899

119894=1119909

119894119910

119894 For 119903 gt 0 119862([minus119903 0]R119899) denotes the family of

continuous function 120593 from [minus119903 0] to R119899 with the norm120593 = sup

minus119903le119904le0|120593(119904)| [sdot sdot] represents the interval co(119876)

denotes the closure of the convex hull of 119876 119864119899denotes

the identity matrix of size 119899 A vector or matrix 119860 ge 0

means that all entries of 119860 are greater than or equal to zero119860 gt 0 can be defined similarly For vectors or matrices 119860and 119861 119860 ge 119861 (or 119860 gt 119861) means that 119860 minus 119861 ge 0 (or119860 minus 119861 gt 0) 119870(R119899) denotes the collection of all nonemptycompact subsets of R with the Hausdorff metric 120588 definedby 120588(119860 119861) = max120573(119860 119861) 120573(119861 119860) 119860 119861 isin 119870(R119899) and120588(119860 119861) = supdist(119909 119861) 119909 isin 119860 120588(119861 119860) = supdist(119910 119860) 119910 isin 119861 119870120592(R119899) = 119860 isin 119870(R119899) 119860 is convex

2 Preliminaries

In this section we give some definitions and propertieswhich are needed later

Definition 1 (see [25]) Suppose 119864 sube R119899 then 119909 rarr 119865(119909) iscalled a set-valued map from 119864 rarr R119899 if for each point 119909 isin119864 there exists a nonempty set 119865(119909) sube R119899 A set-valued map119865 with nonempty values is said to be upper semicontinuous(USC) at 119909

0isin 119864 if for any open set119873 containing 119865(119909

0) there

exists a neighborhood119872 of 1199090such that 119865(119872) sube 119873Themap

119865(119909) is said to have a closed (convex compact) image if foreach 119909 isin 119864 119865(119909) is closed (convex compact)

Definition 2 (see [26]) For the system (119905) = 119891(119905 119909) 119909 isin

R119899 with discontinuous right-hand sides a set-valued map isdefined as

119865 (119905 119909) = ⋂

120575gt0

⋂

120583(119873)=0

co [119891 (119861 (119909 120575) 119873)] (3)

where co[119864] is the closure of the convex hull of set119864119861(119909 120575) =119910 119910 minus 119909 le 120575 and 120583(119873) is Lebesgue measure of set

119873 A solution in Filippovrsquos sense of the Cauchy problem forthis system with initial condition 119909(0) = 119909

0is an absolutely

continuous function 119909(119905) 119905 isin [0 119879] which satisfies 119909(0) = 1199090

and differential inclusion

(119905) isin 119865 (119905 119909) for aa 119905 isin [0 119879] (4)

The initial value associated with system (1) is 119909119894(119905) =

120601(119905) isin 119862([minus120591 0]R) 119894 = 1 2 119899 Let 119889119894= min 119889

119894

119889

119894

119889

119894= max 119889

119894

119889

119894 119886119894119895= min 119886

119894119895 119886

119894119895 119886119894119895= max 119886

119894119895 119886

119894119895

119887

119894119895= min 119887

119894119895

119887

119894119895 119887119894119895= max 119887

119894119895

119887

119894119895 119886119894119895= max|119886

119894119895| |119886

119894119895|

and 119887119894119895= max|119887

119894119895| |119887

119894119895| We define

co (119889119894(119909

119894 (119905))) =

119889

119894

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119879

119894

[119889

119894 119889

119894]

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

= 119879

119894

119889

119894

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

co (119886119894119895(119909

119894 (119905))) =

119886

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119879

119894

[119886

119894119895 119886

119894119895]

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

= 119879

119894

119886

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

co (119887119894119895(119909

119894 (119905))) =

119887

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119879

119894

[119887

119894119895 119887

119894119895]

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

= 119879

119894

119887

119894119895

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

(5)

Clearly system (1) is a differential equation with discontin-uous right-hand side its solution in the conventional sensedoes not exist Inspired by [10 20 21 27ndash30] we adopt thefollowing definition of the solution in the sense of Filippovfor system (1)

Definition 3 Suppose that 120601(119904) = (1206011(119904) 120601

2(119904) 120601

119899(119904))

119879isin

119862([minus120591 0]R119899) is a continuous function An absolutely con-tinuous function 119909(119905) is said to be a solution with initial data120601(119904) of system (1) if 119909(119905) satisfies the differential inclusion

119894 (119905) isin minus co (119889119894 (119909119894 (119905))) 119909119894 (119905)

+

119899

sum

119895=1

co (119886119894119895(119909

119894 (119905))) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

co (119887119894119895(119909

119894 (119905))) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(6)

for all 119894 = 1 2 119899 or equivalently there exist 119889119894(119905) isin

co(119889119894(119909

119894(119905))) 119886

119894119895(119905) isin co(119886

119894119895(119909

119894(119905))) and 119887

119894119895(119905) isin co(119887

119894119895(119909

119894(119905)))

such that

119894 (119905) = minus 119889

119894 (119905) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895 (119905) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

119887

119894119895 (119905) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(7)

Remark 4 From the theoretical point of view the aboveparameters 119889

119894(119905) 119886

119894119895(119905) and 119887

119894119895(119905) in (7) are measurable

functions and depend on the state 119909119894(119905) and time 119905



119894119895) isin R119899times119899 is an





2 120585

119899) gt (0 0 0)


1 120578

2 120578

119899)

119879gt (0 0

0)

119879 such that119872120578 gt 0





(119905) isin 120582119865 (119905 119909) (8)


0 isin

1

120596

int

120596

0

119865 (119905 119906) 119889119905 = 120603 (119906) (9)




119894(119905) and 120591


(A2) For 119894 = 1 2 119899 for all 119904

1 119904

2isin R 119904

1= 119904

2 the neural


119894 119892

119894satisfy


119894gt 0

and 119866119894 such that1003816

1003816

1003816

1003816

119891

119894(119904

1) minus 119891

119894(119904

2)

1003816

1003816

1003816

1003816

le 984858

119894

1003816

1003816

1003816

1003816

119904

1minus 119904

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119894(119904

1) minus 119892

119894(119904

2)

1003816

1003816

1003816

1003816

le 120588

119894

1003816

1003816

1003816

1003816

119904

1minus 119904

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119894 (sdot)

1003816

1003816

1003816

1003816

le 119866

119894

(10)




2) if 119864119899minus119876 is an

119872-matrix where 119876 = (119902119894119895)


119902

119894119895=

1

119889

119894

(984858

119895119886

119894119895+

120588

119895

119887

119894119895

radic1 minus 120583

119894119895

) 119894 119895 = 1 2 119899 (11)


Proof Set 119862120596= 119909(119905) isin 119862(RR119899) 119909(119905 + 120596) = 119909(119905)

119909(119905)

119862120596

= sum

119899

119894=1max119905isin[0120596]

|119909

119894(119905)| Then 119862


with the norm sdot

119862120596

Let 119909(119905) = (1199091(119905) 119909

2(119905) 119909

119899(119905))

119879isin

119862

120596 119865(119905 119909) = (119865

1(119905 119909) 119865

2(119905 119909) 119865

119899(119905 119909))

119879 where

119865

119894 (119905 119909) = minus co (119889

119894(119909

119894 (119905))) 119909119894 (

119905)

+

119899

sum

119895=1

co (119886119894119895(119909

119894 (119905))) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

co (119887119894119895(119909

119894 (119905))) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(12)




1(119905) 119909

2(119905) 119909

119899(119905))


119894 (119905) isin 120582

[

[

minus co (119889119894(119909

119894 (119905))) 119909119894 (

119905)

+

119899

sum

119895=1

co (119886119894119895(119909

119894 (119905))) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

co (119887119894119895(119909

119894 (119905))) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

]

]

(13)


co(119889119894(119909

119894(119905))) 119886

119894119895(119905) isin co(119886

119894119895(119909

119894(119905))) and 119887

119894119895(119905) isin co(119887

119894119895(119909

119894(119905)))

such that

119894 (119905) = 120582

[

[

minus119889

119894 (119905) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895 (119905) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

119887

119894119895 (119905) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

]

]

(14)



int

120596

0

119889

119894 (119905) 119909

2

119894(119905) 119889119905

= int

120596

0

119909

119894 (119905)

[

[

119899

sum

119895=1

119886

119894119895 (119905) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

119887

119894119895 (119905) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

]

]

119889119905

(15)


Noting that

int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

= int

120596minus120591119894119895(120596)

minus120591119894119895(0)

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

1 minus 120591

119894119895(120581

minus1

119894119895(119905))

119889119905

= int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

1 minus 120591

119894119895(120581

minus1

119894119895(119905))

119889119905 le

1

1 minus 120583

119894119895

int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

(16)

where 120581minus1119894119895


119894119895(119905) 119894 119895 =


119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905

le

119899

sum

119895=1

119886

119894119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

119886

119894119895984858

119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895120588

119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895

1 minus 120583

119894119895

)int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895

1 minus 120583

119894119895

)(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

times (int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

2

119889119905)

12

+

119868

119894radic120596(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

(17)

where 119868119894= sup

119905isin[0120596]|119868

119894(119905)| This means

(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

le

119899

sum

119895=1

119902

119894119895(int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

2

119889119905)

12

+

radic120596

119868

119894

119889

119894

(18)

Define 119909119894

120596

2= (int

120596

0|119909

119894(119905)|

2119889119905)

12119909119894isin 119862(RR) 119894 = 1 2 119899

and 119868119873= (

119868

1119889

1

119868

2119889

2

119868

119899119889

119899)


(119864

119899minus 119876) (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

119879

leradic120596119868

119873

(19)


(120585

1 120585

2 120585

119899) gt (0 0 0) such that

120585

lowast= (120585

lowast

1 120585

lowast

2 120585

lowast

119899) = 120585 (119864

119899minus 119876) gt (0 0 0) (20)


min 120585lowast1 120585

lowast

2 120585

lowast

119899 (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2+

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2+ sdot sdot sdot +

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

le 120585

lowast

1

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2+ 120585

lowast

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596


lowast

119899

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2

= 120585 (119864

119899minus 119876) (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

119879

le 120585radic120596(

119868

1

119889

1

119868

2

119889

2

119868

119899

119889

119899

)

119879

=radic120596

119899

sum

119894=1

120585

119894

119868

119894

119889

119894

(21)


1003817

1003817

1003817

1003817

119909

119894

1003817

1003817

1003817

1003817

120596

2= (int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

leradic120596119873 119894 = 1 2 119899

(22)

where 119873 = sum

119899

119894=1(120585

119894

119868

119894119889


1 120585

lowast

2 120585

lowast

119899 Then there


1003816

1003816

1003816

1003816

119909

119894(119905

lowast)

1003816

1003816

1003816

1003816

le 119873 119894 = 1 2 119899 (23)


119894(119905

lowast)+int

119905

119905lowast

119894(119904)119889119904 It follows

from (23) that

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

le 119873 + int

120596

0

1003816

1003816

1003816

1003816

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 119894 = 1 2 119899 (24)


int

120596

0

1003816

1003816

1003816

1003816

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 lt int

120596

0

1003816

1003816

1003816

1003816

119889

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

int

120596

0

1003816

1003816

1003816

1003816

1003816

119886

119894119895 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

int

120596

0

1003816

1003816

1003816

1003816

1003816

119887

119894119895 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

119889119905

+ int

120596

0

1003816

1003816

1003816

1003816

119868

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le 119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 +

119899

sum

119895=1

119886

119894119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895(119905)))

1003816

1003816

1003816

1003816

1003816

119889119905 + 120596

119868

119894


le 119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 +

119899

sum

119895=1

119886

119894119895984858

119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895120588

119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905 + 120596

119868

119894

le 119889

119894radic120596

1003817

1003817

1003817

1003817

119909

119894

1003817

1003817

1003817

1003817

120596

2

+

119899

sum

119895=1

(984858

119895119886

119894119895+

120588

119895

119887

119894119895

radic1 minus 120583

119894119895

)radic120596

1003817

1003817

1003817

1003817

1003817

119909

119895

1003817

1003817

1003817

1003817

1003817

120596

2+ 120596

119868

119894

leradic120596119873(119889

119894+ 119889

119894

119899

sum

119895=1

119902

119894119895) + 120596

119868

119894

(25)

Let119872119894= radic120596119873(119889

119894+ 119889

119894sum

119899

119895=1119902

119894119895) + 120596

119868


(25) we can derive

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119873 +119872

119894= 119877

119894 119894 = 1 2 119899 (26)


119899minus 119876 is


1 120578

2 120578

119899)

119879gt

(0 0 0)



lowast

1 120578

lowast

2 120578

lowast

119899)

119879= (120579120578

1 120579120578

2 120579120578

119899)

119879= 120579120578 120578lowast

119894= 120579120578

119894gt 119877

119894

and

(119864

119899minus 119876) 120578

lowastgt 119868

119873 (27)

Taking Ω = 119909(119905) isin 119862

120596| minus120578

lowastlt 119909(119905) lt 120578





2

119906

119899)


0119865(119905 119906)119889119905 =


lowast

119894for


0 isin (120603 (119906))119894

= minus

119906

119894

120596

int

120596

0

co (119889119894(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

1

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119891119895

(119906

119895) 119889119905

+

119899

sum

119895=1

1

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119892119895

(119906

119895) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

(28)


119894(119909

119894(119905))) 119886

119894119895(119905) isin

co(119886119894119895(119909

119894(119905))) and 119887

119894119895(119905) isin co(119887

119894119895(119909

119894(119905))) such that

0 = minus

119906

119894

120596

int

120596

0

119889

119894 (119905) 119889119905 +

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

(29)


0 = minus 119906

119894119889

119894(119905

lowast) +

119899

sum

119895=1

119886

119894119895(119905

lowast) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

lowast) 119892

119895(119906

119895) + 119868

119894(119905

lowast)

(30)


120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119895+

119868

119894

119889

119894

(31)

This means (119864119899minus 119876)120578

lowastle 119868



[0 1] rarr 119862

120596by 120601(119906 ℎ) = minus diag(ℎ119889

1 ℎ119889

2 ℎ119889

119899)119906 + (1 minus

ℎ)120603(119906)If 119906 = (119906

1 119906

2 119906

119899)



lowast

119894for some 119894 isin 1 2 119899 It

follows that

(120601 (119906 ℎ))

119894

= minusℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(32)



119894 119894 isin 1 2 119899 If 0 isin

(120601(119906 ℎ))

119894 that is


119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(33)

or

0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894 (119905) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(34)


0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894(119905

0) 119906

119894+

119899

sum

119895=1

119886

119894119895(119905

0) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

0) 119892

119895(119906

119895) + 119868

119894(119905

0)

]

]

(35)

that is

ℎ (119889

119894minus 119889

119894(119905

0)) 119906

119894+ 119889

119894(119905

0) 119906

119894

= (1 minus ℎ)

[

[

119899

sum

119895=1

(119886

119894119895(119905

0) 119891

119895(119906

119895) + 119887

119894119895(119905

0) 119892

119895(119906

119895)) + 119868

119894(119905

0)

]

]

(36)

Therefore we have

120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1 minus ℎ

ℎ (119889

119894minus 119889

119894(119905

0)) + 119889

119894(119905

0)

times

[

[

119899

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

119886

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119887

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

)

+

1003816

1003816

1003816

1003816

119868

119894(119905

0)

1003816

1003816

1003816

1003816

]

]

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119894+

119868

119894

119889

119894

(37)


lowastle 119868


(27) Thus 0 notin (120601(119906 ℎ))

119894 119894 = 1 2 119899 It follows that

(0 0 0)

119879notin 120601(119906 ℎ) for any 119906 = (119906

1 119906

2 119906

119899)

119879isin


deg 120603Ω⋂R119899 0 = deg 120601 (119906 0) Ω⋂R

119899 0

= deg 120601 (119906 1) Ω⋂R119899 0

= deg (minus1198891119906

1 minus119889

2119906

2 minus119889

119899119906

119899)

119879

Ω⋂R119899 (0 0 0)

119879

= sign

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus119889

1sdot sdot sdot 0

d

0 minus119889

119899

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= (minus1)

119899= 0

(38)





119894(119905) = 119868

119894

120591

119894119895(119905) = 120591


119894119895)


119902

119894119895=

1

119889

119894

(984858

119895119886

119894119895+ 120588

119895

119887

119894119895) 119894 119895 = 1 2 119899 (39)







119910

119894 (119905) = minus 119889

119894(119910

119894 (119905)) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895(119910

119894 (119905)) 119891119895

(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895(119910

119894 (119905)) 119892119895

(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

119905 ge 0 119894 = 1 2 119899

(40)


119889

119894(119910

119894 (119905)) =

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119886

119894119895(119910

119894 (119905)) =

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119887

119894119895(119910

119894 (119905)) =

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

(41)

Let 119890119894(119905) = 119910

119894(119905) minus 119909


following result


119906

119894 (119905) = minus120572119894 (

119905) 119890119894 (119905) minus 120575119894

120573

119894 (119905) sign (119890119894 (119905))

119894 (119905) = minus120576119894

119890

2

119894(119905) 119894 = 1 2 119899

120573

119894 (119905) = minus120578119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

119894 = 1 2 119899

(42)


119894gt 1


119881 (119905) =

1

2

119899

sum

119894=1

119890

2

119894(119905) +

119899

sum

119894119895=1

1

2 (1 minus 120583

119894119895)

int

119905

119905minus120591119894119895(119905)

119890

2

119894(119904) 119889119904

+

119899

sum

119894=1

1

2120576

119894

(120572

119894 (119905) minus 119870119894

)

2+

119899

sum

119894=1

1

2120578

119894

(119872

119894minus 120573

119894 (119905))

2

(43)

where

119870

119894ge max

minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

minus

119889

119894

+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

119872

119894ge

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895

(44)


Case 1 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| le 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(45)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(t minus 120591119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(46)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(47)

where 119891119895(119890

119895(119905)) = 119891

119895(119910

119895(119905)) minus 119891

119895(119909

119895(119905)) 119892

119895(119890

119895(119905 minus 120591

119894119895(119905))) =

119892

119895(119910

119895(119905 minus 120591

119894119895(119905))) minus 119892

119895(119909

119895(119905 minus 120591


2)


119863

+119881 (119905)

le

119899

sum

119894=1

119890

119894 (119905)

[

[

minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890119895(119905 minus 120591

119894119895 (119905)) minus 120572119894 (

119905) 119890119894 (119905)

minus120575

119894120573

119894 (119905) sign (119890119894 (119905)) ]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))


+ (120572

119894 (119905) minus 119870119894

) 119890

2

119894(119905) minus (119872119894

minus 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

)

le

119899

sum

119894=1

[

[

minus

119889

119894119890

2

119894(119905) +

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895 (119905)

1003816

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

120588

119895

1003816

1003816

1003816

1003816

1003816

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

minus 120575

119894120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119870

119894119890

2

119894(119905) minus 119872119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

+ 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le

119899

sum

119894=1

[

[

minus (119870

119894+

119889

119894) 119890

2

119894(119905) +

119899

sum

119895=1

(

1

2

984858

2

119895119886

2

119894119895119890

119894(119905)

2+

1

2

119890

119895(119905)

2)

+

119899

sum

119895=1

(

1

2

120588

2

119895

119887

2

119894119895119890

119894(119905)

2+

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

=

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(48)


119894gt 1119872

119894gt 0 one has

119863

+119881 (119905) le 0 (49)

Case 2 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| gt 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(50)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(51)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(52)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(53)

Case 3 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| le 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119910

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(54)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119910

119895 (119905))

1003816

1003816

1003816

1003816

1003816


+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119910

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(55)

Note that |119910119894(119905)| le 119879


2) one has

119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(56)


119894gt 1 one has

119863

+119881 (119905) le 0 (57)

Case 4 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| gt 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119909

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119909

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(58)

By using |119909119894(119905)| le 119879


119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(59)



[119889

119894 119889

119894] 119909 minus [119889

119894 119889

119894] 119910 sube [119889

119894 119889

119894] (119909 minus 119910) (60)



5 Numerical Example



119889

1(119909

1 (119905)) =

65

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

6

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4


119889

2(119909

2 (119905)) =

6

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

65

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

11(119909

1 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

22(119909

2 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

11(119909

1 (119905)) =

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

22(119909

2 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

(61)


119891

1 (119904) = 1198912 (

119904) = 119904 119892

1 (119904) = 1198922 (

119904) =

1

2

tanh 119904 (62)


2= 6 119886

11= 119886

22= 2 119886

12= 119886

21=

1 119887119894119895= 1 119894 119895 = 1 2 984858

1= 984858

2= 1 and 120588

1= 120588

2= 12




12 minus13


2minus 119876 is



1(119905) 119868

2(119905))


0 5 10 15

minus05

minus04

minus03

minus02

minus01

0

01

x1(t)

y1(t)

x1(t)y1(t)

t


119910

1(119905) with 119906

119894(119905) = 0

0 5 10 150

01

02

03

04

05

06

07

08

t

x2(t)

y2(t)

x2(t)y2(t)


119910

2(119905) with 119906

119894(119905) = 0





1(119905) = 120572

2(119905) = 02 120573

1(119905) = 120573

2(119905) = 01 for 119905 isin [minus1 0]

120575

119894= 10 120576

119894= 120578



119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 which


1(119905) 120572

2(119905))




01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)


119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)


2(119905) under the


response of 120573(119905) = (1205731(119905) 120573

2(119905))


119894(119905) and 120573

119894(119905) 119894 = 1 2


6 Conclusion


0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t


2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)


Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894119895) isin R119899times119899 is an





2 120585

119899) gt (0 0 0)


1 120578

2 120578

119899)

119879gt (0 0

0)

119879 such that119872120578 gt 0





(119905) isin 120582119865 (119905 119909) (8)


0 isin

1

120596

int

120596

0

119865 (119905 119906) 119889119905 = 120603 (119906) (9)




119894(119905) and 120591


(A2) For 119894 = 1 2 119899 for all 119904

1 119904

2isin R 119904

1= 119904

2 the neural


119894 119892

119894satisfy


119894gt 0

and 119866119894 such that1003816

1003816

1003816

1003816

119891

119894(119904

1) minus 119891

119894(119904

2)

1003816

1003816

1003816

1003816

le 984858

119894

1003816

1003816

1003816

1003816

119904

1minus 119904

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119894(119904

1) minus 119892

119894(119904

2)

1003816

1003816

1003816

1003816

le 120588

119894

1003816

1003816

1003816

1003816

119904

1minus 119904

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119894 (sdot)

1003816

1003816

1003816

1003816

le 119866

119894

(10)




2) if 119864119899minus119876 is an

119872-matrix where 119876 = (119902119894119895)


119902

119894119895=

1

119889

119894

(984858

119895119886

119894119895+

120588

119895

119887

119894119895

radic1 minus 120583

119894119895

) 119894 119895 = 1 2 119899 (11)


Proof Set 119862120596= 119909(119905) isin 119862(RR119899) 119909(119905 + 120596) = 119909(119905)

119909(119905)

119862120596

= sum

119899

119894=1max119905isin[0120596]

|119909

119894(119905)| Then 119862


with the norm sdot

119862120596

Let 119909(119905) = (1199091(119905) 119909

2(119905) 119909

119899(119905))

119879isin

119862

120596 119865(119905 119909) = (119865

1(119905 119909) 119865

2(119905 119909) 119865

119899(119905 119909))

119879 where

119865

119894 (119905 119909) = minus co (119889

119894(119909

119894 (119905))) 119909119894 (

119905)

+

119899

sum

119895=1

co (119886119894119895(119909

119894 (119905))) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

co (119887119894119895(119909

119894 (119905))) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(12)




1(119905) 119909

2(119905) 119909

119899(119905))


119894 (119905) isin 120582

[

[

minus co (119889119894(119909

119894 (119905))) 119909119894 (

119905)

+

119899

sum

119895=1

co (119886119894119895(119909

119894 (119905))) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

co (119887119894119895(119909

119894 (119905))) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

]

]

(13)


co(119889119894(119909

119894(119905))) 119886

119894119895(119905) isin co(119886

119894119895(119909

119894(119905))) and 119887

119894119895(119905) isin co(119887

119894119895(119909

119894(119905)))

such that

119894 (119905) = 120582

[

[

minus119889

119894 (119905) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895 (119905) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

119887

119894119895 (119905) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

]

]

(14)



int

120596

0

119889

119894 (119905) 119909

2

119894(119905) 119889119905

= int

120596

0

119909

119894 (119905)

[

[

119899

sum

119895=1

119886

119894119895 (119905) 119891119895

(119909

119895 (119905))

+

119899

sum

119895=1

119887

119894119895 (119905) 119892119895

(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

]

]

119889119905

(15)


Noting that

int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

= int

120596minus120591119894119895(120596)

minus120591119894119895(0)

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

1 minus 120591

119894119895(120581

minus1

119894119895(119905))

119889119905

= int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

1 minus 120591

119894119895(120581

minus1

119894119895(119905))

119889119905 le

1

1 minus 120583

119894119895

int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

(16)

where 120581minus1119894119895


119894119895(119905) 119894 119895 =


119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905

le

119899

sum

119895=1

119886

119894119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

119886

119894119895984858

119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895120588

119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895

1 minus 120583

119894119895

)int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895

1 minus 120583

119894119895

)(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

times (int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

2

119889119905)

12

+

119868

119894radic120596(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

(17)

where 119868119894= sup

119905isin[0120596]|119868

119894(119905)| This means

(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

le

119899

sum

119895=1

119902

119894119895(int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

2

119889119905)

12

+

radic120596

119868

119894

119889

119894

(18)

Define 119909119894

120596

2= (int

120596

0|119909

119894(119905)|

2119889119905)

12119909119894isin 119862(RR) 119894 = 1 2 119899

and 119868119873= (

119868

1119889

1

119868

2119889

2

119868

119899119889

119899)


(119864

119899minus 119876) (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

119879

leradic120596119868

119873

(19)


(120585

1 120585

2 120585

119899) gt (0 0 0) such that

120585

lowast= (120585

lowast

1 120585

lowast

2 120585

lowast

119899) = 120585 (119864

119899minus 119876) gt (0 0 0) (20)


min 120585lowast1 120585

lowast

2 120585

lowast

119899 (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2+

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2+ sdot sdot sdot +

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

le 120585

lowast

1

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2+ 120585

lowast

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596


lowast

119899

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2

= 120585 (119864

119899minus 119876) (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

119879

le 120585radic120596(

119868

1

119889

1

119868

2

119889

2

119868

119899

119889

119899

)

119879

=radic120596

119899

sum

119894=1

120585

119894

119868

119894

119889

119894

(21)


1003817

1003817

1003817

1003817

119909

119894

1003817

1003817

1003817

1003817

120596

2= (int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

leradic120596119873 119894 = 1 2 119899

(22)

where 119873 = sum

119899

119894=1(120585

119894

119868

119894119889


1 120585

lowast

2 120585

lowast

119899 Then there


1003816

1003816

1003816

1003816

119909

119894(119905

lowast)

1003816

1003816

1003816

1003816

le 119873 119894 = 1 2 119899 (23)


119894(119905

lowast)+int

119905

119905lowast

119894(119904)119889119904 It follows

from (23) that

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

le 119873 + int

120596

0

1003816

1003816

1003816

1003816

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 119894 = 1 2 119899 (24)


int

120596

0

1003816

1003816

1003816

1003816

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 lt int

120596

0

1003816

1003816

1003816

1003816

119889

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

int

120596

0

1003816

1003816

1003816

1003816

1003816

119886

119894119895 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

int

120596

0

1003816

1003816

1003816

1003816

1003816

119887

119894119895 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

119889119905

+ int

120596

0

1003816

1003816

1003816

1003816

119868

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le 119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 +

119899

sum

119895=1

119886

119894119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895(119905)))

1003816

1003816

1003816

1003816

1003816

119889119905 + 120596

119868

119894


le 119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 +

119899

sum

119895=1

119886

119894119895984858

119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895120588

119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905 + 120596

119868

119894

le 119889

119894radic120596

1003817

1003817

1003817

1003817

119909

119894

1003817

1003817

1003817

1003817

120596

2

+

119899

sum

119895=1

(984858

119895119886

119894119895+

120588

119895

119887

119894119895

radic1 minus 120583

119894119895

)radic120596

1003817

1003817

1003817

1003817

1003817

119909

119895

1003817

1003817

1003817

1003817

1003817

120596

2+ 120596

119868

119894

leradic120596119873(119889

119894+ 119889

119894

119899

sum

119895=1

119902

119894119895) + 120596

119868

119894

(25)

Let119872119894= radic120596119873(119889

119894+ 119889

119894sum

119899

119895=1119902

119894119895) + 120596

119868


(25) we can derive

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119873 +119872

119894= 119877

119894 119894 = 1 2 119899 (26)


119899minus 119876 is


1 120578

2 120578

119899)

119879gt

(0 0 0)



lowast

1 120578

lowast

2 120578

lowast

119899)

119879= (120579120578

1 120579120578

2 120579120578

119899)

119879= 120579120578 120578lowast

119894= 120579120578

119894gt 119877

119894

and

(119864

119899minus 119876) 120578

lowastgt 119868

119873 (27)

Taking Ω = 119909(119905) isin 119862

120596| minus120578

lowastlt 119909(119905) lt 120578





2

119906

119899)


0119865(119905 119906)119889119905 =


lowast

119894for


0 isin (120603 (119906))119894

= minus

119906

119894

120596

int

120596

0

co (119889119894(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

1

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119891119895

(119906

119895) 119889119905

+

119899

sum

119895=1

1

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119892119895

(119906

119895) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

(28)


119894(119909

119894(119905))) 119886

119894119895(119905) isin

co(119886119894119895(119909

119894(119905))) and 119887

119894119895(119905) isin co(119887

119894119895(119909

119894(119905))) such that

0 = minus

119906

119894

120596

int

120596

0

119889

119894 (119905) 119889119905 +

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

(29)


0 = minus 119906

119894119889

119894(119905

lowast) +

119899

sum

119895=1

119886

119894119895(119905

lowast) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

lowast) 119892

119895(119906

119895) + 119868

119894(119905

lowast)

(30)


120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119895+

119868

119894

119889

119894

(31)

This means (119864119899minus 119876)120578

lowastle 119868



[0 1] rarr 119862

120596by 120601(119906 ℎ) = minus diag(ℎ119889

1 ℎ119889

2 ℎ119889

119899)119906 + (1 minus

ℎ)120603(119906)If 119906 = (119906

1 119906

2 119906

119899)



lowast

119894for some 119894 isin 1 2 119899 It

follows that

(120601 (119906 ℎ))

119894

= minusℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(32)



119894 119894 isin 1 2 119899 If 0 isin

(120601(119906 ℎ))

119894 that is


119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(33)

or

0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894 (119905) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(34)


0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894(119905

0) 119906

119894+

119899

sum

119895=1

119886

119894119895(119905

0) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

0) 119892

119895(119906

119895) + 119868

119894(119905

0)

]

]

(35)

that is

ℎ (119889

119894minus 119889

119894(119905

0)) 119906

119894+ 119889

119894(119905

0) 119906

119894

= (1 minus ℎ)

[

[

119899

sum

119895=1

(119886

119894119895(119905

0) 119891

119895(119906

119895) + 119887

119894119895(119905

0) 119892

119895(119906

119895)) + 119868

119894(119905

0)

]

]

(36)

Therefore we have

120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1 minus ℎ

ℎ (119889

119894minus 119889

119894(119905

0)) + 119889

119894(119905

0)

times

[

[

119899

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

119886

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119887

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

)

+

1003816

1003816

1003816

1003816

119868

119894(119905

0)

1003816

1003816

1003816

1003816

]

]

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119894+

119868

119894

119889

119894

(37)


lowastle 119868


(27) Thus 0 notin (120601(119906 ℎ))

119894 119894 = 1 2 119899 It follows that

(0 0 0)

119879notin 120601(119906 ℎ) for any 119906 = (119906

1 119906

2 119906

119899)

119879isin


deg 120603Ω⋂R119899 0 = deg 120601 (119906 0) Ω⋂R

119899 0

= deg 120601 (119906 1) Ω⋂R119899 0

= deg (minus1198891119906

1 minus119889

2119906

2 minus119889

119899119906

119899)

119879

Ω⋂R119899 (0 0 0)

119879

= sign

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus119889

1sdot sdot sdot 0

d

0 minus119889

119899

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= (minus1)

119899= 0

(38)





119894(119905) = 119868

119894

120591

119894119895(119905) = 120591


119894119895)


119902

119894119895=

1

119889

119894

(984858

119895119886

119894119895+ 120588

119895

119887

119894119895) 119894 119895 = 1 2 119899 (39)







119910

119894 (119905) = minus 119889

119894(119910

119894 (119905)) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895(119910

119894 (119905)) 119891119895

(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895(119910

119894 (119905)) 119892119895

(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

119905 ge 0 119894 = 1 2 119899

(40)


119889

119894(119910

119894 (119905)) =

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119886

119894119895(119910

119894 (119905)) =

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119887

119894119895(119910

119894 (119905)) =

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

(41)

Let 119890119894(119905) = 119910

119894(119905) minus 119909


following result


119906

119894 (119905) = minus120572119894 (

119905) 119890119894 (119905) minus 120575119894

120573

119894 (119905) sign (119890119894 (119905))

119894 (119905) = minus120576119894

119890

2

119894(119905) 119894 = 1 2 119899

120573

119894 (119905) = minus120578119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

119894 = 1 2 119899

(42)


119894gt 1


119881 (119905) =

1

2

119899

sum

119894=1

119890

2

119894(119905) +

119899

sum

119894119895=1

1

2 (1 minus 120583

119894119895)

int

119905

119905minus120591119894119895(119905)

119890

2

119894(119904) 119889119904

+

119899

sum

119894=1

1

2120576

119894

(120572

119894 (119905) minus 119870119894

)

2+

119899

sum

119894=1

1

2120578

119894

(119872

119894minus 120573

119894 (119905))

2

(43)

where

119870

119894ge max

minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

minus

119889

119894

+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

119872

119894ge

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895

(44)


Case 1 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| le 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(45)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(t minus 120591119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(46)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(47)

where 119891119895(119890

119895(119905)) = 119891

119895(119910

119895(119905)) minus 119891

119895(119909

119895(119905)) 119892

119895(119890

119895(119905 minus 120591

119894119895(119905))) =

119892

119895(119910

119895(119905 minus 120591

119894119895(119905))) minus 119892

119895(119909

119895(119905 minus 120591


2)


119863

+119881 (119905)

le

119899

sum

119894=1

119890

119894 (119905)

[

[

minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890119895(119905 minus 120591

119894119895 (119905)) minus 120572119894 (

119905) 119890119894 (119905)

minus120575

119894120573

119894 (119905) sign (119890119894 (119905)) ]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))


+ (120572

119894 (119905) minus 119870119894

) 119890

2

119894(119905) minus (119872119894

minus 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

)

le

119899

sum

119894=1

[

[

minus

119889

119894119890

2

119894(119905) +

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895 (119905)

1003816

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

120588

119895

1003816

1003816

1003816

1003816

1003816

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

minus 120575

119894120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119870

119894119890

2

119894(119905) minus 119872119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

+ 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le

119899

sum

119894=1

[

[

minus (119870

119894+

119889

119894) 119890

2

119894(119905) +

119899

sum

119895=1

(

1

2

984858

2

119895119886

2

119894119895119890

119894(119905)

2+

1

2

119890

119895(119905)

2)

+

119899

sum

119895=1

(

1

2

120588

2

119895

119887

2

119894119895119890

119894(119905)

2+

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

=

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(48)


119894gt 1119872

119894gt 0 one has

119863

+119881 (119905) le 0 (49)

Case 2 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| gt 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(50)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(51)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(52)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(53)

Case 3 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| le 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119910

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(54)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119910

119895 (119905))

1003816

1003816

1003816

1003816

1003816


+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119910

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(55)

Note that |119910119894(119905)| le 119879


2) one has

119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(56)


119894gt 1 one has

119863

+119881 (119905) le 0 (57)

Case 4 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| gt 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119909

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119909

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(58)

By using |119909119894(119905)| le 119879


119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(59)



[119889

119894 119889

119894] 119909 minus [119889

119894 119889

119894] 119910 sube [119889

119894 119889

119894] (119909 minus 119910) (60)



5 Numerical Example



119889

1(119909

1 (119905)) =

65

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

6

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4


119889

2(119909

2 (119905)) =

6

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

65

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

11(119909

1 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

22(119909

2 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

11(119909

1 (119905)) =

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

22(119909

2 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

(61)


119891

1 (119904) = 1198912 (

119904) = 119904 119892

1 (119904) = 1198922 (

119904) =

1

2

tanh 119904 (62)


2= 6 119886

11= 119886

22= 2 119886

12= 119886

21=

1 119887119894119895= 1 119894 119895 = 1 2 984858

1= 984858

2= 1 and 120588

1= 120588

2= 12




12 minus13


2minus 119876 is



1(119905) 119868

2(119905))


0 5 10 15

minus05

minus04

minus03

minus02

minus01

0

01

x1(t)

y1(t)

x1(t)y1(t)

t


119910

1(119905) with 119906

119894(119905) = 0

0 5 10 150

01

02

03

04

05

06

07

08

t

x2(t)

y2(t)

x2(t)y2(t)


119910

2(119905) with 119906

119894(119905) = 0





1(119905) = 120572

2(119905) = 02 120573

1(119905) = 120573

2(119905) = 01 for 119905 isin [minus1 0]

120575

119894= 10 120576

119894= 120578



119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 which


1(119905) 120572

2(119905))




01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)


119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)


2(119905) under the


response of 120573(119905) = (1205731(119905) 120573

2(119905))


119894(119905) and 120573

119894(119905) 119894 = 1 2


6 Conclusion


0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t


2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)


Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Noting that

int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

= int

120596minus120591119894119895(120596)

minus120591119894119895(0)

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

1 minus 120591

119894119895(120581

minus1

119894119895(119905))

119889119905

= int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

1 minus 120591

119894119895(120581

minus1

119894119895(119905))

119889119905 le

1

1 minus 120583

119894119895

int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

(16)

where 120581minus1119894119895


119894119895(119905) 119894 119895 =


119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905

le

119899

sum

119895=1

119886

119894119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

119886

119894119895984858

119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895120588

119895int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895

1 minus 120583

119894119895

)int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119868

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895

1 minus 120583

119894119895

)(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

times (int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

2

119889119905)

12

+

119868

119894radic120596(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

(17)

where 119868119894= sup

119905isin[0120596]|119868

119894(119905)| This means

(int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

le

119899

sum

119895=1

119902

119894119895(int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

2

119889119905)

12

+

radic120596

119868

119894

119889

119894

(18)

Define 119909119894

120596

2= (int

120596

0|119909

119894(119905)|

2119889119905)

12119909119894isin 119862(RR) 119894 = 1 2 119899

and 119868119873= (

119868

1119889

1

119868

2119889

2

119868

119899119889

119899)


(119864

119899minus 119876) (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

119879

leradic120596119868

119873

(19)


(120585

1 120585

2 120585

119899) gt (0 0 0) such that

120585

lowast= (120585

lowast

1 120585

lowast

2 120585

lowast

119899) = 120585 (119864

119899minus 119876) gt (0 0 0) (20)


min 120585lowast1 120585

lowast

2 120585

lowast

119899 (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2+

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2+ sdot sdot sdot +

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

le 120585

lowast

1

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2+ 120585

lowast

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596


lowast

119899

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2

= 120585 (119864

119899minus 119876) (

1003817

1003817

1003817

1003817

119909

1

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

2

1003817

1003817

1003817

1003817

120596

2

1003817

1003817

1003817

1003817

119909

119899

1003817

1003817

1003817

1003817

120596

2)

119879

le 120585radic120596(

119868

1

119889

1

119868

2

119889

2

119868

119899

119889

119899

)

119879

=radic120596

119899

sum

119894=1

120585

119894

119868

119894

119889

119894

(21)


1003817

1003817

1003817

1003817

119909

119894

1003817

1003817

1003817

1003817

120596

2= (int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

2119889119905)

12

leradic120596119873 119894 = 1 2 119899

(22)

where 119873 = sum

119899

119894=1(120585

119894

119868

119894119889


1 120585

lowast

2 120585

lowast

119899 Then there


1003816

1003816

1003816

1003816

119909

119894(119905

lowast)

1003816

1003816

1003816

1003816

le 119873 119894 = 1 2 119899 (23)


119894(119905

lowast)+int

119905

119905lowast

119894(119904)119889119904 It follows

from (23) that

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

le 119873 + int

120596

0

1003816

1003816

1003816

1003816

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 119894 = 1 2 119899 (24)


int

120596

0

1003816

1003816

1003816

1003816

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 lt int

120596

0

1003816

1003816

1003816

1003816

119889

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

int

120596

0

1003816

1003816

1003816

1003816

1003816

119886

119894119895 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

int

120596

0

1003816

1003816

1003816

1003816

1003816

119887

119894119895 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

119889119905

+ int

120596

0

1003816

1003816

1003816

1003816

119868

119894 (119905)

1003816

1003816

1003816

1003816

119889119905

le 119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 +

119899

sum

119895=1

119886

119894119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119891

119895(119909

119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119892

119895(119909

119895(119905 minus 120591

119894119895(119905)))

1003816

1003816

1003816

1003816

1003816

119889119905 + 120596

119868

119894


le 119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 +

119899

sum

119895=1

119886

119894119895984858

119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895120588

119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905 + 120596

119868

119894

le 119889

119894radic120596

1003817

1003817

1003817

1003817

119909

119894

1003817

1003817

1003817

1003817

120596

2

+

119899

sum

119895=1

(984858

119895119886

119894119895+

120588

119895

119887

119894119895

radic1 minus 120583

119894119895

)radic120596

1003817

1003817

1003817

1003817

1003817

119909

119895

1003817

1003817

1003817

1003817

1003817

120596

2+ 120596

119868

119894

leradic120596119873(119889

119894+ 119889

119894

119899

sum

119895=1

119902

119894119895) + 120596

119868

119894

(25)

Let119872119894= radic120596119873(119889

119894+ 119889

119894sum

119899

119895=1119902

119894119895) + 120596

119868


(25) we can derive

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119873 +119872

119894= 119877

119894 119894 = 1 2 119899 (26)


119899minus 119876 is


1 120578

2 120578

119899)

119879gt

(0 0 0)



lowast

1 120578

lowast

2 120578

lowast

119899)

119879= (120579120578

1 120579120578

2 120579120578

119899)

119879= 120579120578 120578lowast

119894= 120579120578

119894gt 119877

119894

and

(119864

119899minus 119876) 120578

lowastgt 119868

119873 (27)

Taking Ω = 119909(119905) isin 119862

120596| minus120578

lowastlt 119909(119905) lt 120578





2

119906

119899)


0119865(119905 119906)119889119905 =


lowast

119894for


0 isin (120603 (119906))119894

= minus

119906

119894

120596

int

120596

0

co (119889119894(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

1

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119891119895

(119906

119895) 119889119905

+

119899

sum

119895=1

1

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119892119895

(119906

119895) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

(28)


119894(119909

119894(119905))) 119886

119894119895(119905) isin

co(119886119894119895(119909

119894(119905))) and 119887

119894119895(119905) isin co(119887

119894119895(119909

119894(119905))) such that

0 = minus

119906

119894

120596

int

120596

0

119889

119894 (119905) 119889119905 +

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

(29)


0 = minus 119906

119894119889

119894(119905

lowast) +

119899

sum

119895=1

119886

119894119895(119905

lowast) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

lowast) 119892

119895(119906

119895) + 119868

119894(119905

lowast)

(30)


120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119895+

119868

119894

119889

119894

(31)

This means (119864119899minus 119876)120578

lowastle 119868



[0 1] rarr 119862

120596by 120601(119906 ℎ) = minus diag(ℎ119889

1 ℎ119889

2 ℎ119889

119899)119906 + (1 minus

ℎ)120603(119906)If 119906 = (119906

1 119906

2 119906

119899)



lowast

119894for some 119894 isin 1 2 119899 It

follows that

(120601 (119906 ℎ))

119894

= minusℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(32)



119894 119894 isin 1 2 119899 If 0 isin

(120601(119906 ℎ))

119894 that is


119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(33)

or

0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894 (119905) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(34)


0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894(119905

0) 119906

119894+

119899

sum

119895=1

119886

119894119895(119905

0) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

0) 119892

119895(119906

119895) + 119868

119894(119905

0)

]

]

(35)

that is

ℎ (119889

119894minus 119889

119894(119905

0)) 119906

119894+ 119889

119894(119905

0) 119906

119894

= (1 minus ℎ)

[

[

119899

sum

119895=1

(119886

119894119895(119905

0) 119891

119895(119906

119895) + 119887

119894119895(119905

0) 119892

119895(119906

119895)) + 119868

119894(119905

0)

]

]

(36)

Therefore we have

120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1 minus ℎ

ℎ (119889

119894minus 119889

119894(119905

0)) + 119889

119894(119905

0)

times

[

[

119899

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

119886

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119887

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

)

+

1003816

1003816

1003816

1003816

119868

119894(119905

0)

1003816

1003816

1003816

1003816

]

]

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119894+

119868

119894

119889

119894

(37)


lowastle 119868


(27) Thus 0 notin (120601(119906 ℎ))

119894 119894 = 1 2 119899 It follows that

(0 0 0)

119879notin 120601(119906 ℎ) for any 119906 = (119906

1 119906

2 119906

119899)

119879isin


deg 120603Ω⋂R119899 0 = deg 120601 (119906 0) Ω⋂R

119899 0

= deg 120601 (119906 1) Ω⋂R119899 0

= deg (minus1198891119906

1 minus119889

2119906

2 minus119889

119899119906

119899)

119879

Ω⋂R119899 (0 0 0)

119879

= sign

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus119889

1sdot sdot sdot 0

d

0 minus119889

119899

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= (minus1)

119899= 0

(38)





119894(119905) = 119868

119894

120591

119894119895(119905) = 120591


119894119895)


119902

119894119895=

1

119889

119894

(984858

119895119886

119894119895+ 120588

119895

119887

119894119895) 119894 119895 = 1 2 119899 (39)







119910

119894 (119905) = minus 119889

119894(119910

119894 (119905)) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895(119910

119894 (119905)) 119891119895

(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895(119910

119894 (119905)) 119892119895

(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

119905 ge 0 119894 = 1 2 119899

(40)


119889

119894(119910

119894 (119905)) =

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119886

119894119895(119910

119894 (119905)) =

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119887

119894119895(119910

119894 (119905)) =

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

(41)

Let 119890119894(119905) = 119910

119894(119905) minus 119909


following result


119906

119894 (119905) = minus120572119894 (

119905) 119890119894 (119905) minus 120575119894

120573

119894 (119905) sign (119890119894 (119905))

119894 (119905) = minus120576119894

119890

2

119894(119905) 119894 = 1 2 119899

120573

119894 (119905) = minus120578119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

119894 = 1 2 119899

(42)


119894gt 1


119881 (119905) =

1

2

119899

sum

119894=1

119890

2

119894(119905) +

119899

sum

119894119895=1

1

2 (1 minus 120583

119894119895)

int

119905

119905minus120591119894119895(119905)

119890

2

119894(119904) 119889119904

+

119899

sum

119894=1

1

2120576

119894

(120572

119894 (119905) minus 119870119894

)

2+

119899

sum

119894=1

1

2120578

119894

(119872

119894minus 120573

119894 (119905))

2

(43)

where

119870

119894ge max

minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

minus

119889

119894

+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

119872

119894ge

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895

(44)


Case 1 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| le 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(45)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(t minus 120591119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(46)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(47)

where 119891119895(119890

119895(119905)) = 119891

119895(119910

119895(119905)) minus 119891

119895(119909

119895(119905)) 119892

119895(119890

119895(119905 minus 120591

119894119895(119905))) =

119892

119895(119910

119895(119905 minus 120591

119894119895(119905))) minus 119892

119895(119909

119895(119905 minus 120591


2)


119863

+119881 (119905)

le

119899

sum

119894=1

119890

119894 (119905)

[

[

minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890119895(119905 minus 120591

119894119895 (119905)) minus 120572119894 (

119905) 119890119894 (119905)

minus120575

119894120573

119894 (119905) sign (119890119894 (119905)) ]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))


+ (120572

119894 (119905) minus 119870119894

) 119890

2

119894(119905) minus (119872119894

minus 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

)

le

119899

sum

119894=1

[

[

minus

119889

119894119890

2

119894(119905) +

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895 (119905)

1003816

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

120588

119895

1003816

1003816

1003816

1003816

1003816

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

minus 120575

119894120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119870

119894119890

2

119894(119905) minus 119872119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

+ 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le

119899

sum

119894=1

[

[

minus (119870

119894+

119889

119894) 119890

2

119894(119905) +

119899

sum

119895=1

(

1

2

984858

2

119895119886

2

119894119895119890

119894(119905)

2+

1

2

119890

119895(119905)

2)

+

119899

sum

119895=1

(

1

2

120588

2

119895

119887

2

119894119895119890

119894(119905)

2+

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

=

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(48)


119894gt 1119872

119894gt 0 one has

119863

+119881 (119905) le 0 (49)

Case 2 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| gt 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(50)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(51)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(52)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(53)

Case 3 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| le 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119910

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(54)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119910

119895 (119905))

1003816

1003816

1003816

1003816

1003816


+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119910

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(55)

Note that |119910119894(119905)| le 119879


2) one has

119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(56)


119894gt 1 one has

119863

+119881 (119905) le 0 (57)

Case 4 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| gt 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119909

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119909

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(58)

By using |119909119894(119905)| le 119879


119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(59)



[119889

119894 119889

119894] 119909 minus [119889

119894 119889

119894] 119910 sube [119889

119894 119889

119894] (119909 minus 119910) (60)



5 Numerical Example



119889

1(119909

1 (119905)) =

65

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

6

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4


119889

2(119909

2 (119905)) =

6

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

65

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

11(119909

1 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

22(119909

2 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

11(119909

1 (119905)) =

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

22(119909

2 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

(61)


119891

1 (119904) = 1198912 (

119904) = 119904 119892

1 (119904) = 1198922 (

119904) =

1

2

tanh 119904 (62)


2= 6 119886

11= 119886

22= 2 119886

12= 119886

21=

1 119887119894119895= 1 119894 119895 = 1 2 984858

1= 984858

2= 1 and 120588

1= 120588

2= 12




12 minus13


2minus 119876 is



1(119905) 119868

2(119905))


0 5 10 15

minus05

minus04

minus03

minus02

minus01

0

01

x1(t)

y1(t)

x1(t)y1(t)

t


119910

1(119905) with 119906

119894(119905) = 0

0 5 10 150

01

02

03

04

05

06

07

08

t

x2(t)

y2(t)

x2(t)y2(t)


119910

2(119905) with 119906

119894(119905) = 0





1(119905) = 120572

2(119905) = 02 120573

1(119905) = 120573

2(119905) = 01 for 119905 isin [minus1 0]

120575

119894= 10 120576

119894= 120578



119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 which


1(119905) 120572

2(119905))




01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)


119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)


2(119905) under the


response of 120573(119905) = (1205731(119905) 120573

2(119905))


119894(119905) and 120573

119894(119905) 119894 = 1 2


6 Conclusion


0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t


2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)


Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 119889

119894int

120596

0

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

119889119905 +

119899

sum

119895=1

119886

119894119895984858

119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

119889119905

+

119899

sum

119895=1

119887

119894119895120588

119895int

120596

0

1003816

1003816

1003816

1003816

1003816

119909

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

119889119905 + 120596

119868

119894

le 119889

119894radic120596

1003817

1003817

1003817

1003817

119909

119894

1003817

1003817

1003817

1003817

120596

2

+

119899

sum

119895=1

(984858

119895119886

119894119895+

120588

119895

119887

119894119895

radic1 minus 120583

119894119895

)radic120596

1003817

1003817

1003817

1003817

1003817

119909

119895

1003817

1003817

1003817

1003817

1003817

120596

2+ 120596

119868

119894

leradic120596119873(119889

119894+ 119889

119894

119899

sum

119895=1

119902

119894119895) + 120596

119868

119894

(25)

Let119872119894= radic120596119873(119889

119894+ 119889

119894sum

119899

119895=1119902

119894119895) + 120596

119868


(25) we can derive

1003816

1003816

1003816

1003816

119909

119894 (119905)

1003816

1003816

1003816

1003816

lt 119873 +119872

119894= 119877

119894 119894 = 1 2 119899 (26)


119899minus 119876 is


1 120578

2 120578

119899)

119879gt

(0 0 0)



lowast

1 120578

lowast

2 120578

lowast

119899)

119879= (120579120578

1 120579120578

2 120579120578

119899)

119879= 120579120578 120578lowast

119894= 120579120578

119894gt 119877

119894

and

(119864

119899minus 119876) 120578

lowastgt 119868

119873 (27)

Taking Ω = 119909(119905) isin 119862

120596| minus120578

lowastlt 119909(119905) lt 120578





2

119906

119899)


0119865(119905 119906)119889119905 =


lowast

119894for


0 isin (120603 (119906))119894

= minus

119906

119894

120596

int

120596

0

co (119889119894(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

1

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119891119895

(119906

119895) 119889119905

+

119899

sum

119895=1

1

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119892119895

(119906

119895) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

(28)


119894(119909

119894(119905))) 119886

119894119895(119905) isin

co(119886119894119895(119909

119894(119905))) and 119887

119894119895(119905) isin co(119887

119894119895(119909

119894(119905))) such that

0 = minus

119906

119894

120596

int

120596

0

119889

119894 (119905) 119889119905 +

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

(29)


0 = minus 119906

119894119889

119894(119905

lowast) +

119899

sum

119895=1

119886

119894119895(119905

lowast) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

lowast) 119892

119895(119906

119895) + 119868

119894(119905

lowast)

(30)


120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119895+

119868

119894

119889

119894

(31)

This means (119864119899minus 119876)120578

lowastle 119868



[0 1] rarr 119862

120596by 120601(119906 ℎ) = minus diag(ℎ119889

1 ℎ119889

2 ℎ119889

119899)119906 + (1 minus

ℎ)120603(119906)If 119906 = (119906

1 119906

2 119906

119899)



lowast

119894for some 119894 isin 1 2 119899 It

follows that

(120601 (119906 ℎ))

119894

= minusℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(32)



119894 119894 isin 1 2 119899 If 0 isin

(120601(119906 ℎ))

119894 that is


119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(33)

or

0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894 (119905) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(34)


0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894(119905

0) 119906

119894+

119899

sum

119895=1

119886

119894119895(119905

0) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

0) 119892

119895(119906

119895) + 119868

119894(119905

0)

]

]

(35)

that is

ℎ (119889

119894minus 119889

119894(119905

0)) 119906

119894+ 119889

119894(119905

0) 119906

119894

= (1 minus ℎ)

[

[

119899

sum

119895=1

(119886

119894119895(119905

0) 119891

119895(119906

119895) + 119887

119894119895(119905

0) 119892

119895(119906

119895)) + 119868

119894(119905

0)

]

]

(36)

Therefore we have

120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1 minus ℎ

ℎ (119889

119894minus 119889

119894(119905

0)) + 119889

119894(119905

0)

times

[

[

119899

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

119886

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119887

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

)

+

1003816

1003816

1003816

1003816

119868

119894(119905

0)

1003816

1003816

1003816

1003816

]

]

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119894+

119868

119894

119889

119894

(37)


lowastle 119868


(27) Thus 0 notin (120601(119906 ℎ))

119894 119894 = 1 2 119899 It follows that

(0 0 0)

119879notin 120601(119906 ℎ) for any 119906 = (119906

1 119906

2 119906

119899)

119879isin


deg 120603Ω⋂R119899 0 = deg 120601 (119906 0) Ω⋂R

119899 0

= deg 120601 (119906 1) Ω⋂R119899 0

= deg (minus1198891119906

1 minus119889

2119906

2 minus119889

119899119906

119899)

119879

Ω⋂R119899 (0 0 0)

119879

= sign

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus119889

1sdot sdot sdot 0

d

0 minus119889

119899

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= (minus1)

119899= 0

(38)





119894(119905) = 119868

119894

120591

119894119895(119905) = 120591


119894119895)


119902

119894119895=

1

119889

119894

(984858

119895119886

119894119895+ 120588

119895

119887

119894119895) 119894 119895 = 1 2 119899 (39)







119910

119894 (119905) = minus 119889

119894(119910

119894 (119905)) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895(119910

119894 (119905)) 119891119895

(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895(119910

119894 (119905)) 119892119895

(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

119905 ge 0 119894 = 1 2 119899

(40)


119889

119894(119910

119894 (119905)) =

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119886

119894119895(119910

119894 (119905)) =

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119887

119894119895(119910

119894 (119905)) =

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

(41)

Let 119890119894(119905) = 119910

119894(119905) minus 119909


following result


119906

119894 (119905) = minus120572119894 (

119905) 119890119894 (119905) minus 120575119894

120573

119894 (119905) sign (119890119894 (119905))

119894 (119905) = minus120576119894

119890

2

119894(119905) 119894 = 1 2 119899

120573

119894 (119905) = minus120578119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

119894 = 1 2 119899

(42)


119894gt 1


119881 (119905) =

1

2

119899

sum

119894=1

119890

2

119894(119905) +

119899

sum

119894119895=1

1

2 (1 minus 120583

119894119895)

int

119905

119905minus120591119894119895(119905)

119890

2

119894(119904) 119889119904

+

119899

sum

119894=1

1

2120576

119894

(120572

119894 (119905) minus 119870119894

)

2+

119899

sum

119894=1

1

2120578

119894

(119872

119894minus 120573

119894 (119905))

2

(43)

where

119870

119894ge max

minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

minus

119889

119894

+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

119872

119894ge

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895

(44)


Case 1 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| le 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(45)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(t minus 120591119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(46)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(47)

where 119891119895(119890

119895(119905)) = 119891

119895(119910

119895(119905)) minus 119891

119895(119909

119895(119905)) 119892

119895(119890

119895(119905 minus 120591

119894119895(119905))) =

119892

119895(119910

119895(119905 minus 120591

119894119895(119905))) minus 119892

119895(119909

119895(119905 minus 120591


2)


119863

+119881 (119905)

le

119899

sum

119894=1

119890

119894 (119905)

[

[

minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890119895(119905 minus 120591

119894119895 (119905)) minus 120572119894 (

119905) 119890119894 (119905)

minus120575

119894120573

119894 (119905) sign (119890119894 (119905)) ]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))


+ (120572

119894 (119905) minus 119870119894

) 119890

2

119894(119905) minus (119872119894

minus 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

)

le

119899

sum

119894=1

[

[

minus

119889

119894119890

2

119894(119905) +

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895 (119905)

1003816

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

120588

119895

1003816

1003816

1003816

1003816

1003816

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

minus 120575

119894120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119870

119894119890

2

119894(119905) minus 119872119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

+ 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le

119899

sum

119894=1

[

[

minus (119870

119894+

119889

119894) 119890

2

119894(119905) +

119899

sum

119895=1

(

1

2

984858

2

119895119886

2

119894119895119890

119894(119905)

2+

1

2

119890

119895(119905)

2)

+

119899

sum

119895=1

(

1

2

120588

2

119895

119887

2

119894119895119890

119894(119905)

2+

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

=

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(48)


119894gt 1119872

119894gt 0 one has

119863

+119881 (119905) le 0 (49)

Case 2 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| gt 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(50)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(51)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(52)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(53)

Case 3 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| le 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119910

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(54)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119910

119895 (119905))

1003816

1003816

1003816

1003816

1003816


+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119910

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(55)

Note that |119910119894(119905)| le 119879


2) one has

119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(56)


119894gt 1 one has

119863

+119881 (119905) le 0 (57)

Case 4 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| gt 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119909

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119909

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(58)

By using |119909119894(119905)| le 119879


119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(59)



[119889

119894 119889

119894] 119909 minus [119889

119894 119889

119894] 119910 sube [119889

119894 119889

119894] (119909 minus 119910) (60)



5 Numerical Example



119889

1(119909

1 (119905)) =

65

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

6

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4


119889

2(119909

2 (119905)) =

6

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

65

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

11(119909

1 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

22(119909

2 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

11(119909

1 (119905)) =

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

22(119909

2 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

(61)


119891

1 (119904) = 1198912 (

119904) = 119904 119892

1 (119904) = 1198922 (

119904) =

1

2

tanh 119904 (62)


2= 6 119886

11= 119886

22= 2 119886

12= 119886

21=

1 119887119894119895= 1 119894 119895 = 1 2 984858

1= 984858

2= 1 and 120588

1= 120588

2= 12




12 minus13


2minus 119876 is



1(119905) 119868

2(119905))


0 5 10 15

minus05

minus04

minus03

minus02

minus01

0

01

x1(t)

y1(t)

x1(t)y1(t)

t


119910

1(119905) with 119906

119894(119905) = 0

0 5 10 150

01

02

03

04

05

06

07

08

t

x2(t)

y2(t)

x2(t)y2(t)


119910

2(119905) with 119906

119894(119905) = 0





1(119905) = 120572

2(119905) = 02 120573

1(119905) = 120573

2(119905) = 01 for 119905 isin [minus1 0]

120575

119894= 10 120576

119894= 120578



119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 which


1(119905) 120572

2(119905))




01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)


119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)


2(119905) under the


response of 120573(119905) = (1205731(119905) 120573

2(119905))


119894(119905) and 120573

119894(119905) 119894 = 1 2


6 Conclusion


0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t


2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)


Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894 119894 isin 1 2 119899 If 0 isin

(120601(119906 ℎ))

119894 that is


119894119906

119894

+ (1 minus ℎ)

[

[

minus co (119889119894(119909

119894 (119905))) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

co (119886119894119895(119909

119894 (119905))) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

co (119887119894119895(119909

119894 (119905))) 119889119905

+

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(33)

or

0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894 (119905) 119906119894

+

119899

sum

119895=1

119891

119895(119906

119895)

120596

int

120596

0

119886

119894119895 (119905) 119889119905

+

119899

sum

119895=1

119892

119895(119906

119895)

120596

int

120596

0

119887

119894119895 (119905) 119889119905 +

1

120596

int

120596

0

119868

119894 (119905) 119889119905

]

]

(34)


0 = minus ℎ119889

119894119906

119894

+ (1 minus ℎ)

[

[

minus119889

119894(119905

0) 119906

119894+

119899

sum

119895=1

119886

119894119895(119905

0) 119891

119895(119906

119895)

+

119899

sum

119895=1

119887

119894119895(119905

0) 119892

119895(119906

119895) + 119868

119894(119905

0)

]

]

(35)

that is

ℎ (119889

119894minus 119889

119894(119905

0)) 119906

119894+ 119889

119894(119905

0) 119906

119894

= (1 minus ℎ)

[

[

119899

sum

119895=1

(119886

119894119895(119905

0) 119891

119895(119906

119895) + 119887

119894119895(119905

0) 119892

119895(119906

119895)) + 119868

119894(119905

0)

]

]

(36)

Therefore we have

120578

lowast

119894=

1003816

1003816

1003816

1003816

119906

119894

1003816

1003816

1003816

1003816

le

1 minus ℎ

ℎ (119889

119894minus 119889

119894(119905

0)) + 119889

119894(119905

0)

times

[

[

119899

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

119886

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119887

119894119895(119905

0)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119906

119895)

1003816

1003816

1003816

1003816

1003816

)

+

1003816

1003816

1003816

1003816

119868

119894(119905

0)

1003816

1003816

1003816

1003816

]

]

le

1

119889

119894

[

[

119899

sum

119895=1

(119886

119894119895984858

119895+

119887

119894119895120588

119895)

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894]

]

le

119899

sum

119895=1

119902

119894119895

1003816

1003816

1003816

1003816

1003816

119906

119895

1003816

1003816

1003816

1003816

1003816

+

119868

119894

119889

119894

=

119899

sum

119895=1

119902

119894119895120578

lowast

119894+

119868

119894

119889

119894

(37)


lowastle 119868


(27) Thus 0 notin (120601(119906 ℎ))

119894 119894 = 1 2 119899 It follows that

(0 0 0)

119879notin 120601(119906 ℎ) for any 119906 = (119906

1 119906

2 119906

119899)

119879isin


deg 120603Ω⋂R119899 0 = deg 120601 (119906 0) Ω⋂R

119899 0

= deg 120601 (119906 1) Ω⋂R119899 0

= deg (minus1198891119906

1 minus119889

2119906

2 minus119889

119899119906

119899)

119879

Ω⋂R119899 (0 0 0)

119879

= sign

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus119889

1sdot sdot sdot 0

d

0 minus119889

119899

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= (minus1)

119899= 0

(38)





119894(119905) = 119868

119894

120591

119894119895(119905) = 120591


119894119895)


119902

119894119895=

1

119889

119894

(984858

119895119886

119894119895+ 120588

119895

119887

119894119895) 119894 119895 = 1 2 119899 (39)







119910

119894 (119905) = minus 119889

119894(119910

119894 (119905)) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895(119910

119894 (119905)) 119891119895

(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895(119910

119894 (119905)) 119892119895

(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

119905 ge 0 119894 = 1 2 119899

(40)


119889

119894(119910

119894 (119905)) =

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119886

119894119895(119910

119894 (119905)) =

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119887

119894119895(119910

119894 (119905)) =

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

(41)

Let 119890119894(119905) = 119910

119894(119905) minus 119909


following result


119906

119894 (119905) = minus120572119894 (

119905) 119890119894 (119905) minus 120575119894

120573

119894 (119905) sign (119890119894 (119905))

119894 (119905) = minus120576119894

119890

2

119894(119905) 119894 = 1 2 119899

120573

119894 (119905) = minus120578119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

119894 = 1 2 119899

(42)


119894gt 1


119881 (119905) =

1

2

119899

sum

119894=1

119890

2

119894(119905) +

119899

sum

119894119895=1

1

2 (1 minus 120583

119894119895)

int

119905

119905minus120591119894119895(119905)

119890

2

119894(119904) 119889119904

+

119899

sum

119894=1

1

2120576

119894

(120572

119894 (119905) minus 119870119894

)

2+

119899

sum

119894=1

1

2120578

119894

(119872

119894minus 120573

119894 (119905))

2

(43)

where

119870

119894ge max

minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

minus

119889

119894

+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

119872

119894ge

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895

(44)


Case 1 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| le 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(45)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(t minus 120591119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(46)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(47)

where 119891119895(119890

119895(119905)) = 119891

119895(119910

119895(119905)) minus 119891

119895(119909

119895(119905)) 119892

119895(119890

119895(119905 minus 120591

119894119895(119905))) =

119892

119895(119910

119895(119905 minus 120591

119894119895(119905))) minus 119892

119895(119909

119895(119905 minus 120591


2)


119863

+119881 (119905)

le

119899

sum

119894=1

119890

119894 (119905)

[

[

minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890119895(119905 minus 120591

119894119895 (119905)) minus 120572119894 (

119905) 119890119894 (119905)

minus120575

119894120573

119894 (119905) sign (119890119894 (119905)) ]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))


+ (120572

119894 (119905) minus 119870119894

) 119890

2

119894(119905) minus (119872119894

minus 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

)

le

119899

sum

119894=1

[

[

minus

119889

119894119890

2

119894(119905) +

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895 (119905)

1003816

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

120588

119895

1003816

1003816

1003816

1003816

1003816

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

minus 120575

119894120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119870

119894119890

2

119894(119905) minus 119872119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

+ 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le

119899

sum

119894=1

[

[

minus (119870

119894+

119889

119894) 119890

2

119894(119905) +

119899

sum

119895=1

(

1

2

984858

2

119895119886

2

119894119895119890

119894(119905)

2+

1

2

119890

119895(119905)

2)

+

119899

sum

119895=1

(

1

2

120588

2

119895

119887

2

119894119895119890

119894(119905)

2+

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

=

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(48)


119894gt 1119872

119894gt 0 one has

119863

+119881 (119905) le 0 (49)

Case 2 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| gt 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(50)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(51)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(52)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(53)

Case 3 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| le 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119910

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(54)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119910

119895 (119905))

1003816

1003816

1003816

1003816

1003816


+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119910

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(55)

Note that |119910119894(119905)| le 119879


2) one has

119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(56)


119894gt 1 one has

119863

+119881 (119905) le 0 (57)

Case 4 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| gt 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119909

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119909

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(58)

By using |119909119894(119905)| le 119879


119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(59)



[119889

119894 119889

119894] 119909 minus [119889

119894 119889

119894] 119910 sube [119889

119894 119889

119894] (119909 minus 119910) (60)



5 Numerical Example



119889

1(119909

1 (119905)) =

65

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

6

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4


119889

2(119909

2 (119905)) =

6

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

65

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

11(119909

1 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

22(119909

2 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

11(119909

1 (119905)) =

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

22(119909

2 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

(61)


119891

1 (119904) = 1198912 (

119904) = 119904 119892

1 (119904) = 1198922 (

119904) =

1

2

tanh 119904 (62)


2= 6 119886

11= 119886

22= 2 119886

12= 119886

21=

1 119887119894119895= 1 119894 119895 = 1 2 984858

1= 984858

2= 1 and 120588

1= 120588

2= 12




12 minus13


2minus 119876 is



1(119905) 119868

2(119905))


0 5 10 15

minus05

minus04

minus03

minus02

minus01

0

01

x1(t)

y1(t)

x1(t)y1(t)

t


119910

1(119905) with 119906

119894(119905) = 0

0 5 10 150

01

02

03

04

05

06

07

08

t

x2(t)

y2(t)

x2(t)y2(t)


119910

2(119905) with 119906

119894(119905) = 0





1(119905) = 120572

2(119905) = 02 120573

1(119905) = 120573

2(119905) = 01 for 119905 isin [minus1 0]

120575

119894= 10 120576

119894= 120578



119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 which


1(119905) 120572

2(119905))




01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)


119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)


2(119905) under the


response of 120573(119905) = (1205731(119905) 120573

2(119905))


119894(119905) and 120573

119894(119905) 119894 = 1 2


6 Conclusion


0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t


2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)


Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119910

119894 (119905) = minus 119889

119894(119910

119894 (119905)) 119909119894 (

119905) +

119899

sum

119895=1

119886

119894119895(119910

119894 (119905)) 119891119895

(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895(119910

119894 (119905)) 119892119895

(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

119905 ge 0 119894 = 1 2 119899

(40)


119889

119894(119910

119894 (119905)) =

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119889

119894

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119886

119894119895(119910

119894 (119905)) =

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119886

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

119887

119894119895(119910

119894 (119905)) =

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

le 119879

119894

119887

119894119895

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

gt 119879

119894

(41)

Let 119890119894(119905) = 119910

119894(119905) minus 119909


following result


119906

119894 (119905) = minus120572119894 (

119905) 119890119894 (119905) minus 120575119894

120573

119894 (119905) sign (119890119894 (119905))

119894 (119905) = minus120576119894

119890

2

119894(119905) 119894 = 1 2 119899

120573

119894 (119905) = minus120578119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

119894 = 1 2 119899

(42)


119894gt 1


119881 (119905) =

1

2

119899

sum

119894=1

119890

2

119894(119905) +

119899

sum

119894119895=1

1

2 (1 minus 120583

119894119895)

int

119905

119905minus120591119894119895(119905)

119890

2

119894(119904) 119889119904

+

119899

sum

119894=1

1

2120576

119894

(120572

119894 (119905) minus 119870119894

)

2+

119899

sum

119894=1

1

2120578

119894

(119872

119894minus 120573

119894 (119905))

2

(43)

where

119870

119894ge max

minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

minus

119889

119894

+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

119872

119894ge

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895

(44)


Case 1 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| le 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(45)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(t minus 120591119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(46)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(47)

where 119891119895(119890

119895(119905)) = 119891

119895(119910

119895(119905)) minus 119891

119895(119909

119895(119905)) 119892

119895(119890

119895(119905 minus 120591

119894119895(119905))) =

119892

119895(119910

119895(119905 minus 120591

119894119895(119905))) minus 119892

119895(119909

119895(119905 minus 120591


2)


119863

+119881 (119905)

le

119899

sum

119894=1

119890

119894 (119905)

[

[

minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890119895(119905 minus 120591

119894119895 (119905)) minus 120572119894 (

119905) 119890119894 (119905)

minus120575

119894120573

119894 (119905) sign (119890119894 (119905)) ]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))


+ (120572

119894 (119905) minus 119870119894

) 119890

2

119894(119905) minus (119872119894

minus 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

)

le

119899

sum

119894=1

[

[

minus

119889

119894119890

2

119894(119905) +

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895 (119905)

1003816

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

120588

119895

1003816

1003816

1003816

1003816

1003816

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

minus 120575

119894120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119870

119894119890

2

119894(119905) minus 119872119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

+ 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le

119899

sum

119894=1

[

[

minus (119870

119894+

119889

119894) 119890

2

119894(119905) +

119899

sum

119895=1

(

1

2

984858

2

119895119886

2

119894119895119890

119894(119905)

2+

1

2

119890

119895(119905)

2)

+

119899

sum

119895=1

(

1

2

120588

2

119895

119887

2

119894119895119890

119894(119905)

2+

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

=

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(48)


119894gt 1119872

119894gt 0 one has

119863

+119881 (119905) le 0 (49)

Case 2 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| gt 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(50)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(51)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(52)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(53)

Case 3 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| le 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119910

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(54)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119910

119895 (119905))

1003816

1003816

1003816

1003816

1003816


+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119910

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(55)

Note that |119910119894(119905)| le 119879


2) one has

119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(56)


119894gt 1 one has

119863

+119881 (119905) le 0 (57)

Case 4 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| gt 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119909

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119909

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(58)

By using |119909119894(119905)| le 119879


119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(59)



[119889

119894 119889

119894] 119909 minus [119889

119894 119889

119894] 119910 sube [119889

119894 119889

119894] (119909 minus 119910) (60)



5 Numerical Example



119889

1(119909

1 (119905)) =

65

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

6

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4


119889

2(119909

2 (119905)) =

6

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

65

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

11(119909

1 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

22(119909

2 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

11(119909

1 (119905)) =

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

22(119909

2 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

(61)


119891

1 (119904) = 1198912 (

119904) = 119904 119892

1 (119904) = 1198922 (

119904) =

1

2

tanh 119904 (62)


2= 6 119886

11= 119886

22= 2 119886

12= 119886

21=

1 119887119894119895= 1 119894 119895 = 1 2 984858

1= 984858

2= 1 and 120588

1= 120588

2= 12




12 minus13


2minus 119876 is



1(119905) 119868

2(119905))


0 5 10 15

minus05

minus04

minus03

minus02

minus01

0

01

x1(t)

y1(t)

x1(t)y1(t)

t


119910

1(119905) with 119906

119894(119905) = 0

0 5 10 150

01

02

03

04

05

06

07

08

t

x2(t)

y2(t)

x2(t)y2(t)


119910

2(119905) with 119906

119894(119905) = 0





1(119905) = 120572

2(119905) = 02 120573

1(119905) = 120573

2(119905) = 01 for 119905 isin [minus1 0]

120575

119894= 10 120576

119894= 120578



119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 which


1(119905) 120572

2(119905))




01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)


119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)


2(119905) under the


response of 120573(119905) = (1205731(119905) 120573

2(119905))


119894(119905) and 120573

119894(119905) 119894 = 1 2


6 Conclusion


0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t


2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)


Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ (120572

119894 (119905) minus 119870119894

) 119890

2

119894(119905) minus (119872119894

minus 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

)

le

119899

sum

119894=1

[

[

minus

119889

119894119890

2

119894(119905) +

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895 (119905)

1003816

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

120588

119895

1003816

1003816

1003816

1003816

1003816

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119890

119895(119905 minus 120591

119894119895 (119905))

1003816

1003816

1003816

1003816

1003816

minus 120575

119894120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119870

119894119890

2

119894(119905) minus 119872119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

+ 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le

119899

sum

119894=1

[

[

minus (119870

119894+

119889

119894) 119890

2

119894(119905) +

119899

sum

119895=1

(

1

2

984858

2

119895119886

2

119894119895119890

119894(119905)

2+

1

2

119890

119895(119905)

2)

+

119899

sum

119895=1

(

1

2

120588

2

119895

119887

2

119894119895119890

119894(119905)

2+

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

]

]

+

119899

sum

119895=1

(

1

2 (1 minus 120583

119894119895)

119890

2

119895(119905) minus

1

2

119890

2

119895(119905 minus 120591

119894119895 (119905)))

minus119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

=

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(48)


119894gt 1119872

119894gt 0 one has

119863

+119881 (119905) le 0 (49)

Case 2 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| gt 119879



119894 (119905) = minus

119889

119894119909

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119909

119895 (119905))

+

119899

sum

119894=119895

119887

119894119895119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905)

(50)

119910

119894 (119905) = minus

119889

119894119910

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119868119894 (

119905) + 119906119894 (119905)

(51)


119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905))

+

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(52)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(53)

Case 3 If |119909119894(119905)| gt 119879

119894 |119910119894(119905)| le 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119910

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119910

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119910

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(54)


119863

+119881 (119905)

le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119910

119894 (119905)

1003816

1003816

1003816

1003816

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119891

119895(119910

119895 (119905))

1003816

1003816

1003816

1003816

1003816


+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119910

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(55)

Note that |119910119894(119905)| le 119879


2) one has

119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(56)


119894gt 1 one has

119863

+119881 (119905) le 0 (57)

Case 4 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| gt 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119909

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119909

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(58)

By using |119909119894(119905)| le 119879


119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(59)



[119889

119894 119889

119894] 119909 minus [119889

119894 119889

119894] 119910 sube [119889

119894 119889

119894] (119909 minus 119910) (60)



5 Numerical Example



119889

1(119909

1 (119905)) =

65

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

6

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4


119889

2(119909

2 (119905)) =

6

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

65

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

11(119909

1 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

22(119909

2 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

11(119909

1 (119905)) =

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

22(119909

2 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

(61)


119891

1 (119904) = 1198912 (

119904) = 119904 119892

1 (119904) = 1198922 (

119904) =

1

2

tanh 119904 (62)


2= 6 119886

11= 119886

22= 2 119886

12= 119886

21=

1 119887119894119895= 1 119894 119895 = 1 2 984858

1= 984858

2= 1 and 120588

1= 120588

2= 12




12 minus13


2minus 119876 is



1(119905) 119868

2(119905))


0 5 10 15

minus05

minus04

minus03

minus02

minus01

0

01

x1(t)

y1(t)

x1(t)y1(t)

t


119910

1(119905) with 119906

119894(119905) = 0

0 5 10 150

01

02

03

04

05

06

07

08

t

x2(t)

y2(t)

x2(t)y2(t)


119910

2(119905) with 119906

119894(119905) = 0





1(119905) = 120572

2(119905) = 02 120573

1(119905) = 120573

2(119905) = 01 for 119905 isin [minus1 0]

120575

119894= 10 120576

119894= 120578



119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 which


1(119905) 120572

2(119905))




01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)


119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)


2(119905) under the


response of 120573(119905) = (1205731(119905) 120573

2(119905))


119894(119905) and 120573

119894(119905) 119894 = 1 2


6 Conclusion


0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t


2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)


Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119892

119895(119910

119895(119905 minus 120591

119894119895 (119905)))

1003816

1003816

1003816

1003816

1003816

]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(55)

Note that |119910119894(119905)| le 119879


2) one has

119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

(56)


119894gt 1 one has

119863

+119881 (119905) le 0 (57)

Case 4 If |119909119894(119905)| le 119879

119894 |119910119894(119905)| gt 119879



119890

119894 (119905) = minus

119889

119894119890

119894 (119905) +

119899

sum

119895=1

119886

119894119895119891

119895(119890

119895 (119905)) +

119899

sum

119895=1

119887

119894119895119892

119895(119890

119895(119905 minus 120591

119894119895 (119905)))

+ (

119889

119894minus

119889

119894) 119909

119894 (119905) +

119899

sum

119895=1

( 119886

119894119895minus 119886

119894119895) 119891

119895(119909

119895 (119905))

+

119899

sum

119895=1

(

119887

119894119895minus

119887

119894119895) 119892

119895(119909

119895(119905 minus 120591

119894119895 (119905))) + 119906119894 (

119905)

(58)

By using |119909119894(119905)| le 119879


119863

+119881 (119905) le

119899

sum

119894=1

[

[

minus119870

119894minus

119889

119894+

119899

sum

119895=1

984858

2

119895119886

2

119894119895+ 120588

2

119895

119887

2

119894119895+ 1

2

+

119899

sum

119895=1

1

2 (1 minus 120583

119895119894)

]

]

119890

2

119894(119905)

+

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

[

[

1003816

1003816

1003816

1003816

1003816

119889

119894minus

119889

119894

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

984858

119895

1003816

1003816

1003816

1003816

1003816

119886

119894119895minus 119886

119894119895

1003816

1003816

1003816

1003816

1003816

119879

119894

+

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119887

119894119895minus

119887

119894119895

1003816

1003816

1003816

1003816

1003816

119866

119895]

]

minus (120575

119894minus 1) 120573

119894 (119905)

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

minus 119872

119894

1003816

1003816

1003816

1003816

119890

119894 (119905)

1003816

1003816

1003816

1003816

le 0

(59)



[119889

119894 119889

119894] 119909 minus [119889

119894 119889

119894] 119910 sube [119889

119894 119889

119894] (119909 minus 119910) (60)



5 Numerical Example



119889

1(119909

1 (119905)) =

65

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

6

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4


119889

2(119909

2 (119905)) =

6

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

65

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

11(119909

1 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

22(119909

2 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

11(119909

1 (119905)) =

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

22(119909

2 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

(61)


119891

1 (119904) = 1198912 (

119904) = 119904 119892

1 (119904) = 1198922 (

119904) =

1

2

tanh 119904 (62)


2= 6 119886

11= 119886

22= 2 119886

12= 119886

21=

1 119887119894119895= 1 119894 119895 = 1 2 984858

1= 984858

2= 1 and 120588

1= 120588

2= 12




12 minus13


2minus 119876 is



1(119905) 119868

2(119905))


0 5 10 15

minus05

minus04

minus03

minus02

minus01

0

01

x1(t)

y1(t)

x1(t)y1(t)

t


119910

1(119905) with 119906

119894(119905) = 0

0 5 10 150

01

02

03

04

05

06

07

08

t

x2(t)

y2(t)

x2(t)y2(t)


119910

2(119905) with 119906

119894(119905) = 0





1(119905) = 120572

2(119905) = 02 120573

1(119905) = 120573

2(119905) = 01 for 119905 isin [minus1 0]

120575

119894= 10 120576

119894= 120578



119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 which


1(119905) 120572

2(119905))




01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)


119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)


2(119905) under the


response of 120573(119905) = (1205731(119905) 120573

2(119905))


119894(119905) and 120573

119894(119905) 119894 = 1 2


6 Conclusion


0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t


2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)


Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119889

2(119909

2 (119905)) =

6

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

65

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

11(119909

1 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119886

22(119909

2 (119905)) =

minus2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

2

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

11(119909

1 (119905)) =

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

12(119909

1 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

1 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

21(119909

2 (119905)) =

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

119887

22(119909

2 (119905)) =

minus1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

le

1

4

1

1003816

1003816

1003816

1003816

119909

2 (119905)

1003816

1003816

1003816

1003816

gt

1

4

(61)


119891

1 (119904) = 1198912 (

119904) = 119904 119892

1 (119904) = 1198922 (

119904) =

1

2

tanh 119904 (62)


2= 6 119886

11= 119886

22= 2 119886

12= 119886

21=

1 119887119894119895= 1 119894 119895 = 1 2 984858

1= 984858

2= 1 and 120588

1= 120588

2= 12




12 minus13


2minus 119876 is



1(119905) 119868

2(119905))


0 5 10 15

minus05

minus04

minus03

minus02

minus01

0

01

x1(t)

y1(t)

x1(t)y1(t)

t


119910

1(119905) with 119906

119894(119905) = 0

0 5 10 150

01

02

03

04

05

06

07

08

t

x2(t)

y2(t)

x2(t)y2(t)


119910

2(119905) with 119906

119894(119905) = 0





1(119905) = 120572

2(119905) = 02 120573

1(119905) = 120573

2(119905) = 01 for 119905 isin [minus1 0]

120575

119894= 10 120576

119894= 120578



119894(119905) = 119910

119894(119905) minus 119909

119894(119905) 119894 = 1 2 which


1(119905) 120572

2(119905))




01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)


119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)


2(119905) under the


response of 120573(119905) = (1205731(119905) 120573

2(119905))


119894(119905) and 120573

119894(119905) 119894 = 1 2


6 Conclusion


0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t


2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)


Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











01

02

03

04

05

06

07

08

x2(t)y2(t)

x1(t) y1(t)

x(t)

y(t)


119894= 0

0 05 1 15

minus025

minus02

minus015

minus01

minus005

0

005

e1(t)

e2(t)

t

e 1(t)e 2(t)


2(119905) under the


response of 120573(119905) = (1205731(119905) 120573

2(119905))


119894(119905) and 120573

119894(119905) 119894 = 1 2


6 Conclusion


0 05 1 150199

02

0201

0202

0203

0204

0205

0206

1205721(t)

1205722(t)

1205721(t)1205722(t)

t


2(119905)

1205731(t)1205732(t)

t

1205731(t)

1205732(t)

0 05 1 150095

01

0105

011

0115

012

0125

013


2(119905)


Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









complete periodic synchronization of memristor-based neural

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