Research ArticleStability of Uncertain Impulsive Stochastic Genetic RegulatoryNetworks with Time-Varying Delay in the Leakage Term

Jun Li1 Manfeng Hu12 Jinde Cao34 Yongqing Yang1 and Yinghua Jin1

1 School of Science Jiangnan University Wuxi 214122 China2 Key Laboratory of Advanced Process Control for Light Industry Jiangnan University Ministry of Education Wuxi 214122 China3Department of Mathematics Southeast University Nanjing 210096 China4Department of Mathematics Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Manfeng Hu humanfengjiangnaneducn

Received 10 November 2013 Revised 19 January 2014 Accepted 22 January 2014 Published 6 April 2014

Academic Editor Sanyi Tang

Copyright copy 2014 Jun Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with the stability problem for a class of uncertain impulsive stochastic genetic regulatory networks(UISGRNs) with time-varying delays both in the leakage term and in the regulator function By constructing a suitable Lyapunov-Krasovskii functional which uses the information on the lower bound of the delay sufficiently a delay-dependent stability criterionis derived for the proposed UISGRNs model by using the free-weighting matrices method and convex combination techniqueTheconditions obtained here are expressed in terms of LMIs whose feasibility can be checked easily by MATLAB LMI control toolboxIn addition three numerical examples are given to justify the obtained stability results

1 Introduction

Genetic regulatory networks (GRNs) which govern manyessential functions of living cells have received much atten-tion due to their extensive applications in many practicalsystems especially in the biology engineering and otherresearch fields [1ndash6] That is why GRNs have become a hottopic of research recently Several computationalmodels havebeen applied to investigate the behaviours of GRNs Petri netmodels [7ndash9] Bayesian network models [10ndash12] the Booleanmodels [13ndash15] the differential equation models [16ndash18] andso forth In this paper we will use differential equationmodels to encode genetic regulatory networks The rate ofchange in concentration of a particular transcript is given byan influence function of other RNA concentrations

Time delay is an interesting feature of signal transmissionand becomes one of the main sources for causing divergenceinstability and poor performances for networks stabilitySo it is important to consider the delay effects on thedynamical behavior of GRNs Up to now in almost allexisting works on modeling GRNs [5 19ndash21] time delayis included in the regulator function to describe the exist-ing time delays peculiar to transcription translation and

translocation processes in genetic networks Chen andAihara[5] firstly proposed a delay differential equation model forGRNs and studied its stability problem In [19] Ren and Caostudied the asymptotic and robust stability of GRNs withtime-varying delays In [20] Zhang et al investigated thestability analysis for GRNs with random discrete delays anddistributed delays Hu et al [21] proposed a GRNs modelwith hybrid regulatory mechanism and studied its stabilityproblem Recently Gopalsamy [22] put forward a neuralnetwork model with the incorporation of time delays in theleakage terms (ie negative feedback or decay terms whichwidely appeared in themodels of neural networks populationdynamics and GRNs) Along this line a time delay will betaken into consideration in the decay terms of our GRNsmodel and we also call it ldquoleakage delayrdquo

Whenmodeling the GRNs stochastic disturbance shouldbe taken into consideration since molecular noise playsimportant roles in biological functions of GRNs in practiceIn [23 24] the authors studied the model of GRNs withstochastic disturbances Moreover impulsive effects are alsolikely to exist in the genetic networks systems [25] In[26] Li and Sun researched the stability of GRNs underimpulsive control On the other hand it is well known that the

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 706720 15 pageshttpdxdoiorg1011552014706720

2 Abstract and Applied Analysis

stability of well-designed GRNsmay often be destroyed by itsunavoidable uncertainty in practice In [27 28] the authorsinvestigated the stability for uncertain GRNs with intervaltime-varying delays In [29 30] the authors researched thestability problem of GRNs with stochastic disturbance andparameter uncertainties simultaneously In [31] Sakthivel etal dealt with the asymptotic stability of delayed GRNs withboth stochastic disturbance and impulsive effects Howeverso far there has been very little published concerning thestability problem for GRNs with leakage delay impulsiveeffects stochastic disturbances and parameter uncertaintiessimultaneously

Motivated by the above discussion the stability analysisfor UISGRNs with time-varying delays in the leakage termrequires further consideration By constructing a suitableLyapunov-Krasovskii functional which uses the informationon the lower bound of all the delays the derived conditionsare expressed in terms of LMIs whose feasibility can beeasily checked by using numerically efficient MATLAB LMIcontrol toolbox It is believed that the result ismeaningful anduseful for the design and applications of UISGRNs Finallynumerical examples are provided to show the usefulness ofthe derived LMI-based stability conditions

Notations Throughout this paper R119899 and R119899times119899 denoterespectively the 119899-dimensional Euclidean space and the setof all 119899 times 119899 real matrices The superscript 119879 denotes thetransposition and the notation 119883 ge 119884 (resp 119883 gt 119884)where 119883 and 119884 are symmetric matrices and it means that119883minus119884 is positive semidefinite (resp positive definite) Diag(sdot)denotes the diagonal matrix and colsdot means a columnvector In symmetric block matrices we use an asterisk (lowast)to represent a term that is induced by symmetry Matrices iftheir dimensions are not explicitly stated are assumed to becompatible for algebraic operations

2 Problem Formulation and Preliminaries

In this paper we consider the following model

119889119898119894(119905) = ( minus 119886

119894119898119894(119905 minus 119889

1(119905))

+ 119887119894(1199011(119905 minus 120590 (119905)) 119901

2(119905 minus 120590 (119905))

119901119899(119905 minus 120590 (119905)))) 119889119905

+ 120578119894(119905 119898119894(119905 minus 119889

1(119905)) 119901

119894(119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119898119894(119905)1003816100381610038161003816119905=119905119896

= 119898119894(119905119896) minus 119898119894(119905minus

119896) = 119869119896(119898119894(119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119901119894(119905) = (minus119888

119894119901119894(119905 minus 119889

2(119905)) + 119897

119894119898119894(119905 minus 120591 (119905))) 119889119905

+ 120578119894(119905 119898119894(119905 minus 120591 (119905)) 119901

119894(119905 minus 119889

2(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119901119894(119905)1003816100381610038161003816119905=119905119896

= 119901119894(119905119896) minus 119901119894(119905minus

119896) = 119869119896(119901119894(119905minus

119896))

119896 isin 119885+

119905 = 119905119896119894 = 1 2 119899

(1)

where 119898119894(119905) 119901119894(119905) are concentrations of mRNA and protein

of the 119894th node at time 119905 respectively 119886119894and 119888119894are positive

real numbers that are the degradation rates of the mRNA andprotein 119897

119894is a positive constant that represents the translation

rate and 119887119894(sdot) is the regulatory function of the 119894th gene The

first term in the first and third equations of the right side of (1)is called decay term and119889

119894(119905) 119894 = 1 2 is called ldquoleakage delayrdquo

as discussed in the Introduction The regulatory function isof the form 119887

119894(1199011(119905) 1199012(119905) 119901

119899(119905)) = sum

119899

119895=1119887119894119895(119901119895(119905)) which

is called SUM logic [32] The stochastic disturbance 120596(119905) isone-dimensional Brownian motion defined on a completeprobability space (ΩF 119875) with a natural filtrationF

119905ge0and

120578119894(119905 119898119894(119905 minus 120591(119905)) 119901

119894(119905 minus 119889

2(119905))) isin R is the noise inten-

sityThe function 119887

119894119895(119901119895(119905)) is a monotonic function of the Hill

form as follows

119887119894119895(119901119895(119905))

=

120572119894119895

(119901119895(119905) 120573119895)

119867119895

1 + (119901119895(119905) 120573119895)

119867119895

if transcription factor 119895 is an activatorof gene 119894

120572119894119895

1

1 + (119901119895(119905) 120573119895)

119867119895

if transcription factor 119895 is an repressorof gene 119894

(2)

where119867119895is the Hill coefficient 120573

119895is a positive constant and

120572119894119895is the dimensionless transcriptional rate of transcription

factor 119895 to gene 119894 which is a bounded constant Therefore (1)can be rewritten into the following form

119889119898119894(119905) = (minus119886

119894119898119894(119905 minus 119889

1(119905))

+

119899

sum

119895=1

120592119894119895ℎ119895(119901119895(119905 minus 120590 (119905))) + 119906

119894)119889119905

+ 120578119894(119905 119898119894(119905 minus 119889

1(119905)) 119901

119894(119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119898119894(119905)1003816100381610038161003816119905=119905119896

= 119898119894(119905119896) minus 119898119894(119905minus

119896) = 119869119896(119898119894(119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119901119894(119905) = (minus119888

119894119901119894(119905 minus 119889

2(119905)) + 119897

119894119898119894(119905 minus 120591 (119905))) 119889119905

+ 120578119894(119905 119898119894(119905 minus 120591 (119905)) 119901

119894(119905 minus 119889

2(119905))) 119889120596 (119905)

119905 = 119905119896

Abstract and Applied Analysis 3

Δ119901119894(119905)1003816100381610038161003816119905=119905119896

= 119901119894(119905119896) minus 119901119894(119905minus

119896) = 119869119896(119901119894(119905minus

119896))

119896 isin 119885+

119905 = 119905119896 119894 = 1 2 119899

(3)

where ℎ119895(119909) = (119909120573

119895)119867119895(1+(119909120573

119895)119867119895) 119906119894= Σ119895isin119868119894120572119894119895is defined

as a basal rate and 119868119894is the set of all the 119895 which is a repressor

of gene 119894 Thematrix119882 = (120592119894119895) isin R119899times119899 of the genetic network

is defined as follows

120592119894119895=

120572119894119895

if transcription factor 119895 is an activator of gene 1198940

if there is no link from node 119895 to node 119894minus120572119894119895

if transcription factor 119895 is a repressor of gene 119894(4)

Rewriting system (3) into compact matrix form weobtain

119889119898 (119905) = (minus119860119898 (119905 minus 1198891(119905)) + 119882ℎ (119901 (119905 minus 120590 (119905))) + 119906) 119889119905

+ 120578 (119905 119898 (119905 minus 1198891(119905)) 119901 (119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119898 (119905)|119905=119905119896

= 119898 (119905119896) minus 119898 (119905

minus

119896) = 119869119896(119898 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119901 (119905) = (minus119862119901 (119905 minus 1198892(119905)) + 119871119898 (119905 minus 120591 (119905))) 119889119905

+ 120578 (119905 119898 (119905 minus 120591 (119905)) 119901 (119905 minus 1198892(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119901 (119905)1003816100381610038161003816119905=119905119896

= 119901 (119905119896) minus 119901 (119905

minus

119896) = 119869119896(119901 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

(5)

where 119860 = diag(1198861 1198862 119886

119899) 119906 = col119906

1 1199062 119906

119899 119862 =

diag(1198881 1198882 119888

119899) 119871 = diag(119897

1 1198972 119897119899) 119898(119905) = col119898

1(119905)

1198982(119905) 119898

119899(119905) 119901(119905) = col119901

1(119905) 1199012(119905) 119901

119899(119905) ℎ(119901(119905))

= colℎ1(1199011(119905)) ℎ

2(1199012(119905)) ℎ

119899(119901119899(119905)) and 120578(119905 119909 119910) =

col1205781(119905 119909 119910) 120578

2(119905 119909 119910) 120578

119899(119905 119909 119910)

Let (119898lowast 119901lowast) be a nonnegative equilibrium point ofthe system (5) In the following we will always shift theequilibrium point (119898lowast 119901lowast) to the origin by letting 119909(119905) =119898(119905) minus 119898

lowast 119910(119905) = 119901(119905) minus 119901lowast Hence system (5) can be trans-formed into the following form

119889119909 (119905) = (minus119860119909 (119905 minus 1198891(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))) 119889119905

+ 120578 (119905 119909 (119905 minus 1198891(119905)) 119910 (119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119909 (119905)|119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119910 (119905) = (minus119862119910 (119905 minus 1198892(119905)) + 119871119909 (119905 minus 120591 (119905))) 119889119905

+ 120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119910 (119905)1003816100381610038161003816119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

(6)

where 119909(119905minus1198891(119905)) = col119909

1(119905minus1198891(119905)) 119909

2(119905minus1198891(119905)) 119909

119899(119905minus

1198891(119905)) isin R119899 119910(119905 minus 119889

2(119905)) = col119910

1(119905 minus 119889

2(119905)) 119910

2(119905 minus 119889

2(119905))

119910119899(119905 minus 119889

2(119905)) isin R119899 119891(119910(119905)) = col119891

1(1199101(119905)) 119891

2(1199102(119905))

119891119899(119910119899(119905)) isin R119899 the function 119891

119895(119910119895(119905)) = ℎ

119895(119910119895(119905) +119901

lowast

119895) minus

ℎ119895(119901lowast

119895) and obviously fj(0) = 0

Due to the fact that ℎ119895is a monotonically increasing

function with saturation from the relationship of 119891(sdot) andℎ(sdot) we know that for any 119910

119894isin 119877

120574119894le

119891119894(119910119894)

119910119894

le 120572119894 119894 = 1 2 119899 (7)

where 120574119894and 120572

119894are known constant scalars

Taking parameter uncertainties into the GRNsmodel (6)we consider the following UISGRNs model

119889119909 (119905) = (minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905)) + (119882 + Δ119882)

times119891 (119910 (119905 minus 120590 (119905)))) 119889119905

+ 120578 (119905 119909 (119905 minus 1198891(119905)) 119910 (119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119909(119905)|119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119910 (119905) = (minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905))

+ (119871 + Δ119871) 119909 (119905 minus 120591 (119905))) 119889119905

+ 120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119910 (119905)1003816100381610038161003816119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579) forall120579 isin [minus120576 0]

(8)

where 120595(sdot) and 120593(sdot) are the initial function which are contin-uously differentiable on [minus120576 0] with 120576 = maxℎ

2 ℎ4 ℎ6 ℎ8

We extend 984858(120579) on 120579 isin [minus2120576 0] to satisfy 984858120576= 984858

2120576with

984858120576= sup

120579isin[minus1205760]984858(120579) 984858

2120576= sup

120579isin[minus21205760]984858(120579) where

984858 = 120595 120593

4 Abstract and Applied Analysis

Moreover the noise intensity 120578 satisfies

tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times 120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))]

le 119909119879

(119905 minus 120591 (119905)) Σ119879

1Σ1119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) Σ119879

2Σ21199102(119905 minus 119889 (119905))

(9)

where Σ1and Σ

2are constant matrices with appropriate

dimensionsIn order to obtain our main theorem the following

assumptions and lemmas for the system (8) are always madethroughout this paper

Assumption 1 The parametric uncertainties Δ119860(119905) Δ119882(119905)Δ119862(119905) and Δ119871(119905) satisfy

Δ119860 (119905) = 11986611198651(119905)119867119886

Δ119882(119905) = 11986611198651(119905)119867119908

Δ119862 (119905) = 11986621198652(119905)119867119888

Δ119871 (119905) = 11986621198652(119905)119867119897

(10)

where 1198661 1198662 119867119886 119867119908 119867119888 and 119867

119897are some given constant

matrices with appropriate dimensions and 119865119894(119905) satisfies

119865119879

119894(119905)119865119894(119905) le 119868 119894 = 1 2 for any 119905 gt 0

Assumption 2 1198891(119905) 1198892(119905) 120591(119905) and 120590(119905) are the time-varying

delays satisfying

0 le ℎ1le 1198891(119905) le ℎ

2 0 le

1198891(119905) le 119889

1lt infin

0 le ℎ3le 120590 (119905) le ℎ

4 0 le (119905) le 120590 lt infin

0 le ℎ5le 1198892(119905) le ℎ

6 0 le

1198892(119905) le 119889

2lt infin

0 le ℎ7le 120591 (119905) le ℎ

8 0 le 120591 (119905) le 120591 lt infin

ℎ12= ℎ2minus ℎ1 ℎ

34= ℎ4minus ℎ3

ℎ56= ℎ6minus ℎ5 ℎ

78= ℎ8minus ℎ7

(11)

Lemma 3 (Schur complement see [30]) For a given matrix

(

119876 (119909) 119878 (119909)

119878119879

(119909) 119877 (119909)

) gt 0 (12)

where

119876 (119909) = 119876119879

(119909) 119877 (119909) = 119877119879

(119909) (13)

and a vector function 119908(119909) [0 119903] rarr 119877119899 such that the

intergrals concerned as well defined then the following holds

(i) 119876(119909) gt 0 119877(119909) minus 119878119879(119909)119876(119909)minus1119878(119909) gt 0

(ii) 119877(119909) gt 0 119876(119909) minus 119878(119909)119877(119909)minus1119878119879(119909) gt 0

Lemma 4 (see [33]) For any constant symmetric matrix119872 gt

0 scalar 120574 gt 0

[int

120574

0

120603 (119904) 119889119904]

119879

119872[int

120574

0

120603 (119904) 119889119904] le 120574int

120574

0

120603119879

(119904)119872120603 (119904) 119889119904

(14)

Lemma 5 (see [29]) For any vectors 119886 119887 isin R119899 and anypositive matrix 119884 satisfying

plusmn2119886119879

119887 le 119886119879

119884119886 + 119887119879

119884minus1

119887 (15)

3 Main Result

In this section mean square stability result for model (8) issummarized in the following theorem

Theorem 6 If (7) (9) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120588

1gt 0 1205882gt 0 120594im isin [0 1] 119896 = 0 1 119903+

2 and 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (8) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and

1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and any

matrices11987311119873121198732111987322119872111198721211987221119872221198723111987232

11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32to

satisfy the following ten linear matrix inequalities

1198751+ 1198811lt 1205881119868 (16)

1198752+ 1198812lt 1205882119868 (17)

1206011= [

Ξ ℎ121198732

lowast minusℎ121198852

] lt 0 (18)

1206012= [

Ξ ℎ121198733

lowast minusℎ121198852

] lt 0 (19)

1206013= [

Ξ ℎ341198722

lowast minusℎ341198854

] lt 0 (20)

1206014= [

Ξ ℎ341198723

lowast minusℎ341198854

] lt 0 (21)

1206015= [

Ξ ℎ561198782

lowast minusℎ561198856

] lt 0 (22)

1206016= [

Ξ ℎ561198783

lowast minusℎ561198856

] lt 0 (23)

1206017= [

Ξ ℎ781198642

lowast minusℎ781198858

] lt 0 (24)

1206018= [

Ξ ℎ781198643

lowast minusℎ781198858

] lt 0 (25)

Abstract and Applied Analysis 5

where

Ξ =

1

4

[

[

[

[

[

[

[

[

[

[

120601 1198791119883 1198792119884 ℎ11198731ℎ31198721ℎ51198781ℎ71198641

lowast minus119883 0 0 0 0 0

lowast 0 minus119884 0 0 0 0

lowast 0 0 minusℎ11198851

0 0 0

lowast 0 0 0 minusℎ31198853

0 0

lowast 0 0 0 0 ℎ51198855

0

lowast 0 0 0 0 0 minusℎ71198857

]

]

]

]

]

]

]

]

]

]

120601 = [

120601912060110

lowast 12060111

]

1206019=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120601111206011211987321minus1198733112060115

11986421

11986431

11987811+1198721112060119

11987221

lowast 1206012211987322minus1198733212060125

11986422minus1198643211987212+ 1198781212060129

11987222

lowast lowast 12060133

0 0 0 0 0 0 0

lowast lowast lowast minus1198764

0 0 0 0 0 0

lowast lowast lowast lowast 12060155

0 0 12060158

0 0

lowast lowast lowast lowast lowast 12060166

0 0 0 0

lowast lowast lowast lowast lowast lowast minus11987616

0 0 0

lowast lowast lowast lowast lowast lowast lowast 12060188

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 12060199

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1206011010

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12060110=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus11987231120601112

11987812minus11987831120601115

0 120601117

0 0 0

minus11987232120601212

11987822minus11987832120601215

0 0 120601218

0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 120601516

0 0 120601519

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 120601812

0 0 0 120601816

0 0 0 0

0 0 0 0 0 0 120601917

0 0 0

0 0 0 0 0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12060111=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus1198768

0 0 0 0 0 0 0 0 0

lowast 1206011212

0 0 0 1206011216

0 0 0 1206011220

lowast lowast 1206011313

0 0 0 0 0 0 0

lowast lowast lowast minus11987612

0 0 0 0 0 0

lowast lowast lowast lowast 1206011515

0 1206011517

0 0 0

lowast lowast lowast lowast lowast 1206011616

0 0 0 0

lowast lowast lowast lowast lowast lowast 1206011717

0 0 0

lowast lowast lowast lowast lowast lowast lowast 1206011818

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1206011919

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1206012020

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119869119894119898(119909 (119905119898)) = (119869

1119898(1199091(119905119898)) 119869

119899119898(119909119899(119905119898)))119879

12060111= 1198763+ 11987615+ 211987311+ 211986411minus 1198791198846Σ

12060112= minus1198751119860 minus 119875

111986611198651119867119886minus 1198791198812119860 minus 119879119881

211986611198651119867119886minus 11987311+ 11987312+ 11987313minus 11987321+ 11986412

12060115= minus11986411+ 11986431minus 11986421 120601

19= minus119872

11+11987231minus11987221 120601

112= minus11987811+ 11987831minus 11987821

120601115

=

1

2

1198846(119879 + Σ) 120601

117= minus119879119881

2119882minus 119879119881

211986611198651119867119908+ 1198751119882+ 119875

111986611198651119867119908

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232+ 1205881Σ119879

1Σ1minus 1198791198842Σ 120601

25= minus11986412+ 11986432minus 11986422

12060129= minus119872

12+11987232minus11987222 120601

212= minus11987812+ 11987832minus 11987822 120601

215= minus1198812119860 minus 119881

211986611198651119867119886

120601218

=

1

2

1198842(119879 + Σ) 120601

33= 1198761+ 1198764minus 1198763 120601

55= minus (1 minus 120591)119876

13+ 1205882Σ4Σ4minus 1198791198845Σ

6 Abstract and Applied Analysis

12060158= minus119879119881

1119871 minus 119879119881

111986621198652119867119897+ 1198752119871 + 119875211986621198652119867119897 120601

516= 1198811119871 + 119881111986621198652119867119897

120601519

=

1

2

1198845(119879 + Σ) 120601

66= 11987613+ 11987616minus 11987615 120601

88= 1198767+ 11987611minus 1198791198841Σ

120601812

= minus1198752119862 minus 119875211986621198652119867119888+ 1198791198811119862 + 119879119881

111986621198652119867119888 120601

816=

1

2

1198841(119879 + Σ)

12060199= minus (1 minus 120590)119876

5+ 1205881Σ2Σ2minus 1198791198843Σ 120601

917=

1

2

1198843(119879 + Σ) 120601

1010= 1198765minus 1198767+ 1198768

1206011212

= minus (1 minus 1198892) 1198769+ 1205882Σ3Σ3minus 1198791198844Σ 120601

1216= minus1198811119862 minus 119881

111986621198652119867119888

1206011220

=

1

2

1198844(119879 + Σ) 120601

1313= 1198769minus 11987611+ 11987612 120601

1515= 11987614minus 1198846+ 1198762

1206011517

= 1198812119882+119881

211986611198651119867119908 120601

1616= 1198766minus 1198841+ 11987610 120601

1717= minus (1 minus 120590)119876

6minus 1198843

1206011818

= minus (1 minus 1198891) 1198762minus 1198842 120601

1919= minus (1 minus 120591)119876

15minus 1198845 120601

2020= minus (1 minus 119889

2) 11987610minus 1198844

119883 = ℎ11198851+ ℎ121198852+ ℎ71198857+ ℎ781198858 119884 = ℎ

21198853+ ℎ341198854+ ℎ51198855+ ℎ561198856

11987911= [0 minus119860 minus 119866

111986511198671198860 0 0 0 0 0 0 0]

11987912= [0 0 0 0 0 0 119882 + 119866

111986511198671199080 0 0]

11987921= [0 0 0 119871 + 119866

211986521198671198970 0 0 0 0 0]

11987922= [0 minus119862 minus 119866

211986721198671198880 0 0 0 0 0 0 0]

1198791= [1198791111987912]

119879

1198792= [1198792111987922]

119879

1198731= [119873119879

11119873119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198732= [119873119879

21119873119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198733= [119873119879

31119873119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198781= [119878119879

11119878119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198782= [119878119879

21119878119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198783= [119878119879

31119878119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198721= [119872

119879

11119872119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198722= [119872

119879

21119872119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198723= [119872

119879

31119872119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198641= [119864119879

11119864119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198642= [119864119879

21119864119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198643= [119864119879

31119864119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

119879 = diag (1205741 1205742 120574

119899) Σ = diag (120572

1 1205722 120572

119899)

(26)

Abstract and Applied Analysis 7

Proof We consider the following Lyapunov functional can-didate for system (8)

119881 (119905 119909119905 119910119905) = 1198811(119905 119909119905 119910119905) + 1198812(119905 119909119905 119910119905) + 1198813(119905 119909119905 119910119905)

(27)

where

1198811(119905 119909119905 119910119905) = 2

119899

sum

119894=1

1198811119894int

119910119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

1198812119894int

119909119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

(28)

1198812(119905 119909119905 119910119905) = 119909119879

(119905) 1198751119909 (119905) + 119910

119879

(119905) 1198752119910 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119909119879

(119904) 1198761119909 (119904) 119889119904

+ int

119905

119905minus1198891(119905)

119891119879

(119909 (119904)) 1198762119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ1

119909119879

(119904) 1198763119909 (119904) 119889119904

+ int

119905minusℎ1

119905minusℎ2

119909119879

(119904) 1198764119909 (119904) 119889119904

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198765119910 (119904) 119889119904

+ int

119905

119905minus120590(119905)

119891119879

(119910 (119904)) 1198766119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ3

119910119879

(119904) 1198767119910 (119904) 119889119904

+ int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198768119910 (119904) 119889119904

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198769119910 (119904) 119889119904

+ int

119905

119905minus1198892(119905)

119891119879

(119910 (119904)) 11987610119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ5

119910119879

(119904) 11987611119910 (119904) 119889119904

+ int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 11987612119910 (119904) 119889119904

+ int

119905minusℎ7

119905minus120591(119905)

119909119879

(119904) 11987613119909 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119891119879

(119909 (119904)) 11987614119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ7

119909119879

(119904) 11987615119909 (119904) 119889119904

+ int

119905minusℎ7

119905minusℎ8

119909119879

(119904) 11987616119909 (119904) 119889119904

(29)

1198813(119905 119909119905 119910119905) = int

0

minusℎ1

int

119905

119905+120579

119879

(119904) 1198851 (119904) 119889119904 119889120579

+ int

minusℎ1

minusℎ2

int

119905

119905+120579

119879

(119904) 1198852 (119904) 119889119904 119889120579

+ int

0

minusℎ3

int

119905

119905+120579

119910119879

(119904) 1198853119910 (119904) 119889119904 119889120579

+ int

minusℎ3

minusℎ4

int

119905

119905+120579

119910119879

(119904) 1198854119910 (119904) 119889119904 119889120579

+ int

0

minusℎ5

int

119905

119905+120579

119910119879

(119904) 1198855119910 (119904) 119889119904 119889120579

+ int

minusℎ5

minusℎ6

int

119905

119905+120579

119910119879

(119904) 1198856119910 (119904) 119889119904 119889120579

+ int

0

minusℎ7

int

119905

119905+120579

119879

(119904) 1198857 (119904) 119889119904 119889120579

+ int

minusℎ7

minusℎ8

int

119905

119905+120579

119879

(119904) 1198858 (119904) 119889119904 119889120579

(30)

Then by Itorsquos differential formula taking the derivative of119881(119905) along the trajectories of the system (8) we can obtainthe following stochastic differential [29]

119889119881 (119905) = F119881 (119905) 119889119905

+ 2119909119879

(119905) 1198751120578 (119905 119909 (119905 minus 119889

1(119905)) 119910 (119905 minus 120590 (119905)))

+ 2119910119879

1198752120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905)))

(31)

whereF is the diffusion operator and

F119881 (119905 119909119905 119910119905) = F119881

1(119905 119909119905 119910119905)

+F1198812(119905 119909119905 119910119905) +F119881

3(119905 119909119905 119910119905)

(32)

with

F1198811(119905 119909119905 119910119905) = 2 [119891

119879

(119910 (119905)) minus 119910119879

(119905) Γ]

times 1198811[minus119862119910 (119905 minus 119889

2(119905)) + 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

8 Abstract and Applied Analysis

+ tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198811120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 2 [119891119879

(119909 (119905)) minus 119909119879

(119905) Γ]

times 1198812[minus119860119909 (119905 minus 119889

1(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198812120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

F1198812(119905 119909119905 119910119905)

le 2119909119879

(119905) 1198751[minus119860119909 (119905 minus 119889

1(119905))

+ 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198751120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

+ 2119910119879

(119905) 1198752[minus119862119910 (119905 minus 119889

2(119905))

+ 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

+ tr [120578119879 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198752120578 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 119909119879

(119905 minus ℎ1) 1198761119909 (119905 minus ℎ

1)

minus (1 minus 1198891) 119909119879

(119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905))

+ 119891119879

(119909 (119904)) 1198762119891 (119909 (119904))

minus (1 minus 1198891) 119891119879

(119910 (119905 minus 1198891(119905)))

times 1198762119891 (119910 (119905 minus 119889

1(119905)))

+ 119909119879

(119905) 1198763119909 (119905) minus 119909

119879

(119905 minus ℎ1) 1198763119909 (119905 minus ℎ

1)

+ 119909119879

(119905 minus ℎ1) 1198764119909 (119905 minus ℎ

1)

minus 119909119879

(119905 minus ℎ2) 1198764119909 (119905 minus ℎ

2)

+ 119910119879

(119905 minus ℎ5) 1198765119909 (119905 minus ℎ

5)

minus (1 minus 120590) 119910119879

(119905 minus 120590 (119905)) 1198765119910 (119905 minus 120590 (119905))

+ 119891119879

(119910 (119905)) 1198766119891 (119910 (119905))

minus (1 minus 120590) 119891 (119910119879

(119905 minus 120590 (119905)))

times 1198765119891 (119910 (119905 minus 120590 (119905)))

minus 119910119879

(119905 minus ℎ3) 1198767119910 (119905 minus ℎ

3)

+ 119910119879

(119905) 1198767119910 (119905) + 119910

119879

(119905 minus ℎ3) 1198768119910 (119905 minus ℎ

3)

minus 119910119879

(119905 minus ℎ4) 1198768119910 (119905 minus ℎ

4)

+ 119910119879

(119905 minus ℎ5) 1198769119910 (119905 minus ℎ

5)

minus (1 minus 1198892) 119910119879

(119905 minus 1198892(119905)) 119876

9119910 (119905 minus 119889

2(119905))

+ 119891119879

(119910 (119905)) 11987610119891 (119910 (119905))

minus (1 minus 1198892) 119891119879

(119910 (119905 minus 1198892(119905)))

times 11987610119891 (119910 (119905 minus 119889

2(119905)))

+ 119910119879

11987611119910 (119905) minus 119910

119879

(119905 minus ℎ5) 11987611119910 (119905 minus ℎ

5)

+ 119910119879

(119905 minus ℎ5) 11987612119910 (119905 minus ℎ

5)

minus 119910119879

(119905 minus ℎ6) 11987612119910 (119905 minus ℎ

6)

+ 119909119879

(119905 minus ℎ7) 11987613119909 (119905 minus ℎ

7)

minus (1 minus 120591) 119909119879

(119905 minus 120591 (119905)) 11987613119909 (119905 minus 120591 (119905))

+ 119891119879

(119909 (119905)) 11987614119891 (119909 (119905))

minus (1 minus 120591) 119891119879

(119909 (119905 minus 120591 (119905)))

times 11987614119891 (119909 (119905 minus 120591 (119905))) + 119909

119879

(119905) 11987615119909 (119905)

minus 119909119879

(119905 minus ℎ7) 11987615119909 (119905 minus ℎ

7)

+ 119909119879

(119905 minus ℎ7) 11987616119909 (119905 minus ℎ

7)

minus 119909119879

(119905 minus ℎ8) 11987616119909 (119905 minus ℎ

8)

F1198813(119905 119909119905 119910119905)

= ℎ1119879

(119905) 1198851 (119905)

minus int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

+ ℎ12119879

(119905) 1198852 (119905)

minus int

119905minusℎ1

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

+ ℎ2119910119879

(119905) 1198853119910 (119905)

minus int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

Abstract and Applied Analysis 9

+ ℎ34119910119879

(119905) 1198854119910 (119905)

minus int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

+ ℎ5119910119879

(119905) 1198855119910 (119905)

minus int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

+ ℎ56119910119879

(119905) 1198856119910 (119905)

minus int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

+ ℎ7119879

(119905) 1198857119910 (119905)

minus int

119905

119905minusℎ7

119910119879

(119904) 1198857119910 (119904) 119889119904

+ ℎ78119910119879

(119904) 1198858119910 (119904)

minus int

119905minusℎ7

119905minusℎ8

119910119879

(119905) 1198858119910 (119905) 119889119904

(33)

By Newton-Leibnitz formula we have that

2120576119879

(119905)1198731[119909 (119905) minus 119909 (119905 minus ℎ

1) minus int

119905

119905minusℎ1

(119904) 119889119904] = 0

2120576119879

(119905)1198732[119909 (119905 minus ℎ

1) minus 119909 (119905 minus 119889

1(119905)) minus int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904] = 0

2120576119879

(119905)1198733[119909 (119905 minus 119889

1(119905)) minus 119909 (119905 minus ℎ

2) minus int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904] = 0

2120576119879

(119905)1198721[119910 (119905) minus 119910 (119905 minus 120590 (119905)) minus int

119905

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198722[119910 (119905 minus ℎ

3) minus 119910 (119905 minus 120590 (119905)) minus int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198723[119910 (119905 minus 120590 (119905)) minus 119910 (119905 minus ℎ

4) minus int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198781[119910 (119905) minus 119910 (119905 minus 119889

2(119905)) minus int

119905

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198782[119910 (119905 minus ℎ

5) minus 119910 (119905 minus 119889

2(119905)) minus int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198783[119910 (119905 minus 119889

2(119905)) minus 119910 (119905 minus ℎ

6) minus int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198641[119909 (119905) minus 119909 (119905 minus 120591 (119905)) minus int

119905

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198642[119909 (119905 minus ℎ

7) minus 119909 (119905 minus 120591 (119905)) minus int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198643[119909 (119905 minus ℎ

8) minus 119909 (119905 minus ℎ

7) minus int

119905minusℎ7

119905minusℎ8

(119904) 119889119904] = 0

(34)

where

120576 (119905) = [1205761(119905) 1205762(119905) 1205763(119905) 1205764(119905)]119879

1205761(119905) = [119909

119879

(119905) 119909119879

(119905 minus 1198891(119905)) 119909

119879

(119905 minus ℎ1)

119909119879

(119905 minus ℎ2) 119909119879

(119905 minus 120591 (119905))]

1205762(119905) = [119909

119879

(119905 minus ℎ7) 119909119879

(119905 minus ℎ8) 119910119879

(119905)

119910119879

(119905 minus 120590 (119905)) 119910119879

(119905 minus ℎ3)]

1205763(119905) = [119910

119879

(119905 minus ℎ4) 119910119879

(119905 minus 1198892(119905)) 119910

119879

(119905 minus ℎ5)

119910119879

(119905 minus ℎ6) 119891119879

(119909)]

1205764(119905) = [119891

119879

(119910) 119891119879

(119910 minus 120590 (119905)) 119891119879

(119909 minus 1198891(119905))

119891119879

(119909 minus 120591 (119905)) 119891119879

(119910 minus 1198892(119905))]

(35)

By using Lemmas 4 and 5 we have

minus 2120576119879

(119905)1198731int

119905

119905minusℎ1

(119904) 119889119904

le ℎ1120576119879

(119905)1198731119885minus1

1119873119879

1120576 (119905)

+ int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

minus 2120576119879

(119905)1198732int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904

le (1198891(119905) minus ℎ

1) 120576119879

(119905)1198732119885minus1

2119873119879

2120576 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198733int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904

le (ℎ2minus 1198891(119905)) 120576119879

(119905)1198733119885minus1

2119873119879

3120576 (119905)

+ int

119905minus1198891(119905)

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198721int

119905

119905minusℎ3

119910 (119904) 119889119904

10 Abstract and Applied Analysis

le ℎ3120576119879

(119905)1198721119885minus1

3119872119879

1120576 (119905)

+ int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

minus 2120576119879

(119905)1198722int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904

le (120590 (119905) minus ℎ3) 120576119879

(119905)1198722119885minus1

4119872119879

2120576 (119905)

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905)1198723int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904

le (ℎ4minus 120590 (119905)) 120576

119879

(119905)1198723119885minus1

4119872119879

3120576 (119905)

+ int

119905minus120590(119905)

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905) 1198781int

119905

119905minusℎ5

119910 (119904) 119889119904

le ℎ5120576119879

(119905) 1198781119885minus1

5119878119879

1120576 (119905)

+ int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

minus 2120576119879

(119905) 1198782int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904

le (1198892(119905) minus ℎ

5) 120576119879

(119905) 1198782119885minus1

6119878119879

2120576 (119905)

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198783int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904

le (ℎ6minus 1198892(119905)) 120576119879

(119905) 1198783119885minus1

6119878119879

3120576 (119905)

+ int

119905minus1198892(119905)

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198641int

119905

119905minusℎ7

(119904) 119889119904

le ℎ7120576119879

(119905) 1198641119885minus1

7119864119879

1120576 (119905)

+ int

119905

119905minusℎ7

119879

(119904) 1198857 (119904) 119889119904

minus 2120576119879

(119905) 1198642int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904

le (120591 (119905) minus ℎ7) 120576119879

(119905) 1198642119885minus1

8119864119879

2120576 (119905)

+ int

119905minusℎ7

119905minus120591(119905)

119879

(119904) 1198858 (119904) 119889119904

minus 2120576119879

(119905) 1198643int

119905minus120591(119905)

119905minusℎ8

(119904) 119889119904

le (ℎ8minus 120591 (119905)) 120576

119879

(119905) 1198643119885minus1

8119864119879

3120576 (119905)

+ int

119905minus120591(119905)

119905minusℎ8

119879

(119904) 1198858 (119904) 119889119904

(36)

It follows from (7) that

0 = 119891119879

(119910 (119905)) 1198841119891 (119910 (119905)) minus 119891

119879

(119910 (119905)) 1198841119891 (119910 (119905))

le minus119910119879

(119905) 1198791198841Σ119910 (119905) + 119910

119879

(119905) 1198841(119879 + Σ) 119891 (119910 (119905))

minus 119891119879

(119910 (119905)) 1198841119891 (119910 (119905))

0 = 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

le minus119910119879

(119905 minus 1198891(119905)) 119879119884

2Σ119910 (119905 minus 119889

1(119905))

+ 119910119879

(119905 minus 1198891(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

0 = 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

le minus119910119879

(119905 minus 120590 (119905)) 1198791198843Σ119910 (119905 minus 120590 (119905))

+ 119910119879

(119905 minus 120590 (119905)) 1198843(119879 + Σ) 119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

0 = 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

le minus119910119879

(119905 minus 1198892(119905)) 119879119884

2Σ119910 (119905 minus 119889

2(119905))

+ 119910119879

(119905 minus 1198892(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

2119891 (119910 (119905 minus 119889

2(119905)))

0 = 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

le minus119910119879

(119905 minus 120591 (119905)) 1198791198845Σ119910 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 120591 (119905)) 1198845(119879 + Σ) 119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

0 = 119891119879

(119909 (119905)) 1198846119891 (119909 (119905)) minus 119891

119879

(119909 (119905)) 1198846119891 (119909 (119905))

le minus119909119879

(119905) 1198791198846Σ119909 (119905) + 119909

119879

(119905) 1198846(119879 + Σ) 119891 (119909 (119905))

minus 119891119879

(119909 (119905)) 1198846119891 (119909 (119905))

(37)

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

stability of well-designed GRNsmay often be destroyed by itsunavoidable uncertainty in practice In [27 28] the authorsinvestigated the stability for uncertain GRNs with intervaltime-varying delays In [29 30] the authors researched thestability problem of GRNs with stochastic disturbance andparameter uncertainties simultaneously In [31] Sakthivel etal dealt with the asymptotic stability of delayed GRNs withboth stochastic disturbance and impulsive effects Howeverso far there has been very little published concerning thestability problem for GRNs with leakage delay impulsiveeffects stochastic disturbances and parameter uncertaintiessimultaneously

Motivated by the above discussion the stability analysisfor UISGRNs with time-varying delays in the leakage termrequires further consideration By constructing a suitableLyapunov-Krasovskii functional which uses the informationon the lower bound of all the delays the derived conditionsare expressed in terms of LMIs whose feasibility can beeasily checked by using numerically efficient MATLAB LMIcontrol toolbox It is believed that the result ismeaningful anduseful for the design and applications of UISGRNs Finallynumerical examples are provided to show the usefulness ofthe derived LMI-based stability conditions

Notations Throughout this paper R119899 and R119899times119899 denoterespectively the 119899-dimensional Euclidean space and the setof all 119899 times 119899 real matrices The superscript 119879 denotes thetransposition and the notation 119883 ge 119884 (resp 119883 gt 119884)where 119883 and 119884 are symmetric matrices and it means that119883minus119884 is positive semidefinite (resp positive definite) Diag(sdot)denotes the diagonal matrix and colsdot means a columnvector In symmetric block matrices we use an asterisk (lowast)to represent a term that is induced by symmetry Matrices iftheir dimensions are not explicitly stated are assumed to becompatible for algebraic operations

2 Problem Formulation and Preliminaries

In this paper we consider the following model

119889119898119894(119905) = ( minus 119886

119894119898119894(119905 minus 119889

1(119905))

+ 119887119894(1199011(119905 minus 120590 (119905)) 119901

2(119905 minus 120590 (119905))

119901119899(119905 minus 120590 (119905)))) 119889119905

+ 120578119894(119905 119898119894(119905 minus 119889

1(119905)) 119901

119894(119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119898119894(119905)1003816100381610038161003816119905=119905119896

= 119898119894(119905119896) minus 119898119894(119905minus

119896) = 119869119896(119898119894(119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119901119894(119905) = (minus119888

119894119901119894(119905 minus 119889

2(119905)) + 119897

119894119898119894(119905 minus 120591 (119905))) 119889119905

+ 120578119894(119905 119898119894(119905 minus 120591 (119905)) 119901

119894(119905 minus 119889

2(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119901119894(119905)1003816100381610038161003816119905=119905119896

= 119901119894(119905119896) minus 119901119894(119905minus

119896) = 119869119896(119901119894(119905minus

119896))

119896 isin 119885+

119905 = 119905119896119894 = 1 2 119899

(1)

where 119898119894(119905) 119901119894(119905) are concentrations of mRNA and protein

of the 119894th node at time 119905 respectively 119886119894and 119888119894are positive

real numbers that are the degradation rates of the mRNA andprotein 119897

119894is a positive constant that represents the translation

rate and 119887119894(sdot) is the regulatory function of the 119894th gene The

first term in the first and third equations of the right side of (1)is called decay term and119889

119894(119905) 119894 = 1 2 is called ldquoleakage delayrdquo

as discussed in the Introduction The regulatory function isof the form 119887

119894(1199011(119905) 1199012(119905) 119901

119899(119905)) = sum

119899

119895=1119887119894119895(119901119895(119905)) which

is called SUM logic [32] The stochastic disturbance 120596(119905) isone-dimensional Brownian motion defined on a completeprobability space (ΩF 119875) with a natural filtrationF

119905ge0and

120578119894(119905 119898119894(119905 minus 120591(119905)) 119901

119894(119905 minus 119889

2(119905))) isin R is the noise inten-

sityThe function 119887

119894119895(119901119895(119905)) is a monotonic function of the Hill

form as follows

119887119894119895(119901119895(119905))

=

120572119894119895

(119901119895(119905) 120573119895)

119867119895

1 + (119901119895(119905) 120573119895)

119867119895

if transcription factor 119895 is an activatorof gene 119894

120572119894119895

1

1 + (119901119895(119905) 120573119895)

119867119895

if transcription factor 119895 is an repressorof gene 119894

(2)

where119867119895is the Hill coefficient 120573

119895is a positive constant and

120572119894119895is the dimensionless transcriptional rate of transcription

factor 119895 to gene 119894 which is a bounded constant Therefore (1)can be rewritten into the following form

119889119898119894(119905) = (minus119886

119894119898119894(119905 minus 119889

1(119905))

+

119899

sum

119895=1

120592119894119895ℎ119895(119901119895(119905 minus 120590 (119905))) + 119906

119894)119889119905

+ 120578119894(119905 119898119894(119905 minus 119889

1(119905)) 119901

119894(119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119898119894(119905)1003816100381610038161003816119905=119905119896

= 119898119894(119905119896) minus 119898119894(119905minus

119896) = 119869119896(119898119894(119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119901119894(119905) = (minus119888

119894119901119894(119905 minus 119889

2(119905)) + 119897

119894119898119894(119905 minus 120591 (119905))) 119889119905

+ 120578119894(119905 119898119894(119905 minus 120591 (119905)) 119901

119894(119905 minus 119889

2(119905))) 119889120596 (119905)

119905 = 119905119896

Abstract and Applied Analysis 3

Δ119901119894(119905)1003816100381610038161003816119905=119905119896

= 119901119894(119905119896) minus 119901119894(119905minus

119896) = 119869119896(119901119894(119905minus

119896))

119896 isin 119885+

119905 = 119905119896 119894 = 1 2 119899

(3)

where ℎ119895(119909) = (119909120573

119895)119867119895(1+(119909120573

119895)119867119895) 119906119894= Σ119895isin119868119894120572119894119895is defined

as a basal rate and 119868119894is the set of all the 119895 which is a repressor

of gene 119894 Thematrix119882 = (120592119894119895) isin R119899times119899 of the genetic network

is defined as follows

120592119894119895=

120572119894119895

if transcription factor 119895 is an activator of gene 1198940

if there is no link from node 119895 to node 119894minus120572119894119895

if transcription factor 119895 is a repressor of gene 119894(4)

Rewriting system (3) into compact matrix form weobtain

119889119898 (119905) = (minus119860119898 (119905 minus 1198891(119905)) + 119882ℎ (119901 (119905 minus 120590 (119905))) + 119906) 119889119905

+ 120578 (119905 119898 (119905 minus 1198891(119905)) 119901 (119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119898 (119905)|119905=119905119896

= 119898 (119905119896) minus 119898 (119905

minus

119896) = 119869119896(119898 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119901 (119905) = (minus119862119901 (119905 minus 1198892(119905)) + 119871119898 (119905 minus 120591 (119905))) 119889119905

+ 120578 (119905 119898 (119905 minus 120591 (119905)) 119901 (119905 minus 1198892(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119901 (119905)1003816100381610038161003816119905=119905119896

= 119901 (119905119896) minus 119901 (119905

minus

119896) = 119869119896(119901 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

(5)

where 119860 = diag(1198861 1198862 119886

119899) 119906 = col119906

1 1199062 119906

119899 119862 =

diag(1198881 1198882 119888

119899) 119871 = diag(119897

1 1198972 119897119899) 119898(119905) = col119898

1(119905)

1198982(119905) 119898

119899(119905) 119901(119905) = col119901

1(119905) 1199012(119905) 119901

119899(119905) ℎ(119901(119905))

= colℎ1(1199011(119905)) ℎ

2(1199012(119905)) ℎ

119899(119901119899(119905)) and 120578(119905 119909 119910) =

col1205781(119905 119909 119910) 120578

2(119905 119909 119910) 120578

119899(119905 119909 119910)

Let (119898lowast 119901lowast) be a nonnegative equilibrium point ofthe system (5) In the following we will always shift theequilibrium point (119898lowast 119901lowast) to the origin by letting 119909(119905) =119898(119905) minus 119898

lowast 119910(119905) = 119901(119905) minus 119901lowast Hence system (5) can be trans-formed into the following form

119889119909 (119905) = (minus119860119909 (119905 minus 1198891(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))) 119889119905

+ 120578 (119905 119909 (119905 minus 1198891(119905)) 119910 (119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119909 (119905)|119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119910 (119905) = (minus119862119910 (119905 minus 1198892(119905)) + 119871119909 (119905 minus 120591 (119905))) 119889119905

+ 120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119910 (119905)1003816100381610038161003816119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

(6)

where 119909(119905minus1198891(119905)) = col119909

1(119905minus1198891(119905)) 119909

2(119905minus1198891(119905)) 119909

119899(119905minus

1198891(119905)) isin R119899 119910(119905 minus 119889

2(119905)) = col119910

1(119905 minus 119889

2(119905)) 119910

2(119905 minus 119889

2(119905))

119910119899(119905 minus 119889

2(119905)) isin R119899 119891(119910(119905)) = col119891

1(1199101(119905)) 119891

2(1199102(119905))

119891119899(119910119899(119905)) isin R119899 the function 119891

119895(119910119895(119905)) = ℎ

119895(119910119895(119905) +119901

lowast

119895) minus

ℎ119895(119901lowast

119895) and obviously fj(0) = 0

Due to the fact that ℎ119895is a monotonically increasing

function with saturation from the relationship of 119891(sdot) andℎ(sdot) we know that for any 119910

119894isin 119877

120574119894le

119891119894(119910119894)

119910119894

le 120572119894 119894 = 1 2 119899 (7)

where 120574119894and 120572

119894are known constant scalars

Taking parameter uncertainties into the GRNsmodel (6)we consider the following UISGRNs model

119889119909 (119905) = (minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905)) + (119882 + Δ119882)

times119891 (119910 (119905 minus 120590 (119905)))) 119889119905

+ 120578 (119905 119909 (119905 minus 1198891(119905)) 119910 (119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119909(119905)|119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119910 (119905) = (minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905))

+ (119871 + Δ119871) 119909 (119905 minus 120591 (119905))) 119889119905

+ 120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119910 (119905)1003816100381610038161003816119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579) forall120579 isin [minus120576 0]

(8)

where 120595(sdot) and 120593(sdot) are the initial function which are contin-uously differentiable on [minus120576 0] with 120576 = maxℎ

2 ℎ4 ℎ6 ℎ8

We extend 984858(120579) on 120579 isin [minus2120576 0] to satisfy 984858120576= 984858

2120576with

984858120576= sup

120579isin[minus1205760]984858(120579) 984858

2120576= sup

120579isin[minus21205760]984858(120579) where

984858 = 120595 120593

4 Abstract and Applied Analysis

Moreover the noise intensity 120578 satisfies

tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times 120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))]

le 119909119879

(119905 minus 120591 (119905)) Σ119879

1Σ1119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) Σ119879

2Σ21199102(119905 minus 119889 (119905))

(9)

where Σ1and Σ

2are constant matrices with appropriate

dimensionsIn order to obtain our main theorem the following

assumptions and lemmas for the system (8) are always madethroughout this paper

Assumption 1 The parametric uncertainties Δ119860(119905) Δ119882(119905)Δ119862(119905) and Δ119871(119905) satisfy

Δ119860 (119905) = 11986611198651(119905)119867119886

Δ119882(119905) = 11986611198651(119905)119867119908

Δ119862 (119905) = 11986621198652(119905)119867119888

Δ119871 (119905) = 11986621198652(119905)119867119897

(10)

where 1198661 1198662 119867119886 119867119908 119867119888 and 119867

119897are some given constant

matrices with appropriate dimensions and 119865119894(119905) satisfies

119865119879

119894(119905)119865119894(119905) le 119868 119894 = 1 2 for any 119905 gt 0

Assumption 2 1198891(119905) 1198892(119905) 120591(119905) and 120590(119905) are the time-varying

delays satisfying

0 le ℎ1le 1198891(119905) le ℎ

2 0 le

1198891(119905) le 119889

1lt infin

0 le ℎ3le 120590 (119905) le ℎ

4 0 le (119905) le 120590 lt infin

0 le ℎ5le 1198892(119905) le ℎ

6 0 le

1198892(119905) le 119889

2lt infin

0 le ℎ7le 120591 (119905) le ℎ

8 0 le 120591 (119905) le 120591 lt infin

ℎ12= ℎ2minus ℎ1 ℎ

34= ℎ4minus ℎ3

ℎ56= ℎ6minus ℎ5 ℎ

78= ℎ8minus ℎ7

(11)

Lemma 3 (Schur complement see [30]) For a given matrix

(

119876 (119909) 119878 (119909)

119878119879

(119909) 119877 (119909)

) gt 0 (12)

where

119876 (119909) = 119876119879

(119909) 119877 (119909) = 119877119879

(119909) (13)

and a vector function 119908(119909) [0 119903] rarr 119877119899 such that the

intergrals concerned as well defined then the following holds

(i) 119876(119909) gt 0 119877(119909) minus 119878119879(119909)119876(119909)minus1119878(119909) gt 0

(ii) 119877(119909) gt 0 119876(119909) minus 119878(119909)119877(119909)minus1119878119879(119909) gt 0

Lemma 4 (see [33]) For any constant symmetric matrix119872 gt

0 scalar 120574 gt 0

[int

120574

0

120603 (119904) 119889119904]

119879

119872[int

120574

0

120603 (119904) 119889119904] le 120574int

120574

0

120603119879

(119904)119872120603 (119904) 119889119904

(14)

Lemma 5 (see [29]) For any vectors 119886 119887 isin R119899 and anypositive matrix 119884 satisfying

plusmn2119886119879

119887 le 119886119879

119884119886 + 119887119879

119884minus1

119887 (15)

3 Main Result

In this section mean square stability result for model (8) issummarized in the following theorem

Theorem 6 If (7) (9) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120588

1gt 0 1205882gt 0 120594im isin [0 1] 119896 = 0 1 119903+

2 and 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (8) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and

1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and any

matrices11987311119873121198732111987322119872111198721211987221119872221198723111987232

11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32to

satisfy the following ten linear matrix inequalities

1198751+ 1198811lt 1205881119868 (16)

1198752+ 1198812lt 1205882119868 (17)

1206011= [

Ξ ℎ121198732

lowast minusℎ121198852

] lt 0 (18)

1206012= [

Ξ ℎ121198733

lowast minusℎ121198852

] lt 0 (19)

1206013= [

Ξ ℎ341198722

lowast minusℎ341198854

] lt 0 (20)

1206014= [

Ξ ℎ341198723

lowast minusℎ341198854

] lt 0 (21)

1206015= [

Ξ ℎ561198782

lowast minusℎ561198856

] lt 0 (22)

1206016= [

Ξ ℎ561198783

lowast minusℎ561198856

] lt 0 (23)

1206017= [

Ξ ℎ781198642

lowast minusℎ781198858

] lt 0 (24)

1206018= [

Ξ ℎ781198643

lowast minusℎ781198858

] lt 0 (25)

Abstract and Applied Analysis 5

where

Ξ =

1

4

[

[

[

[

[

[

[

[

[

[

120601 1198791119883 1198792119884 ℎ11198731ℎ31198721ℎ51198781ℎ71198641

lowast minus119883 0 0 0 0 0

lowast 0 minus119884 0 0 0 0

lowast 0 0 minusℎ11198851

0 0 0

lowast 0 0 0 minusℎ31198853

0 0

lowast 0 0 0 0 ℎ51198855

0

lowast 0 0 0 0 0 minusℎ71198857

]

]

]

]

]

]

]

]

]

]

120601 = [

120601912060110

lowast 12060111

]

1206019=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120601111206011211987321minus1198733112060115

11986421

11986431

11987811+1198721112060119

11987221

lowast 1206012211987322minus1198733212060125

11986422minus1198643211987212+ 1198781212060129

11987222

lowast lowast 12060133

0 0 0 0 0 0 0

lowast lowast lowast minus1198764

0 0 0 0 0 0

lowast lowast lowast lowast 12060155

0 0 12060158

0 0

lowast lowast lowast lowast lowast 12060166

0 0 0 0

lowast lowast lowast lowast lowast lowast minus11987616

0 0 0

lowast lowast lowast lowast lowast lowast lowast 12060188

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 12060199

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1206011010

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12060110=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus11987231120601112

11987812minus11987831120601115

0 120601117

0 0 0

minus11987232120601212

11987822minus11987832120601215

0 0 120601218

0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 120601516

0 0 120601519

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 120601812

0 0 0 120601816

0 0 0 0

0 0 0 0 0 0 120601917

0 0 0

0 0 0 0 0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12060111=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus1198768

0 0 0 0 0 0 0 0 0

lowast 1206011212

0 0 0 1206011216

0 0 0 1206011220

lowast lowast 1206011313

0 0 0 0 0 0 0

lowast lowast lowast minus11987612

0 0 0 0 0 0

lowast lowast lowast lowast 1206011515

0 1206011517

0 0 0

lowast lowast lowast lowast lowast 1206011616

0 0 0 0

lowast lowast lowast lowast lowast lowast 1206011717

0 0 0

lowast lowast lowast lowast lowast lowast lowast 1206011818

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1206011919

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1206012020

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119869119894119898(119909 (119905119898)) = (119869

1119898(1199091(119905119898)) 119869

119899119898(119909119899(119905119898)))119879

12060111= 1198763+ 11987615+ 211987311+ 211986411minus 1198791198846Σ

12060112= minus1198751119860 minus 119875

111986611198651119867119886minus 1198791198812119860 minus 119879119881

211986611198651119867119886minus 11987311+ 11987312+ 11987313minus 11987321+ 11986412

12060115= minus11986411+ 11986431minus 11986421 120601

19= minus119872

11+11987231minus11987221 120601

112= minus11987811+ 11987831minus 11987821

120601115

=

1

2

1198846(119879 + Σ) 120601

117= minus119879119881

2119882minus 119879119881

211986611198651119867119908+ 1198751119882+ 119875

111986611198651119867119908

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232+ 1205881Σ119879

1Σ1minus 1198791198842Σ 120601

25= minus11986412+ 11986432minus 11986422

12060129= minus119872

12+11987232minus11987222 120601

212= minus11987812+ 11987832minus 11987822 120601

215= minus1198812119860 minus 119881

211986611198651119867119886

120601218

=

1

2

1198842(119879 + Σ) 120601

33= 1198761+ 1198764minus 1198763 120601

55= minus (1 minus 120591)119876

13+ 1205882Σ4Σ4minus 1198791198845Σ

6 Abstract and Applied Analysis

12060158= minus119879119881

1119871 minus 119879119881

111986621198652119867119897+ 1198752119871 + 119875211986621198652119867119897 120601

516= 1198811119871 + 119881111986621198652119867119897

120601519

=

1

2

1198845(119879 + Σ) 120601

66= 11987613+ 11987616minus 11987615 120601

88= 1198767+ 11987611minus 1198791198841Σ

120601812

= minus1198752119862 minus 119875211986621198652119867119888+ 1198791198811119862 + 119879119881

111986621198652119867119888 120601

816=

1

2

1198841(119879 + Σ)

12060199= minus (1 minus 120590)119876

5+ 1205881Σ2Σ2minus 1198791198843Σ 120601

917=

1

2

1198843(119879 + Σ) 120601

1010= 1198765minus 1198767+ 1198768

1206011212

= minus (1 minus 1198892) 1198769+ 1205882Σ3Σ3minus 1198791198844Σ 120601

1216= minus1198811119862 minus 119881

111986621198652119867119888

1206011220

=

1

2

1198844(119879 + Σ) 120601

1313= 1198769minus 11987611+ 11987612 120601

1515= 11987614minus 1198846+ 1198762

1206011517

= 1198812119882+119881

211986611198651119867119908 120601

1616= 1198766minus 1198841+ 11987610 120601

1717= minus (1 minus 120590)119876

6minus 1198843

1206011818

= minus (1 minus 1198891) 1198762minus 1198842 120601

1919= minus (1 minus 120591)119876

15minus 1198845 120601

2020= minus (1 minus 119889

2) 11987610minus 1198844

119883 = ℎ11198851+ ℎ121198852+ ℎ71198857+ ℎ781198858 119884 = ℎ

21198853+ ℎ341198854+ ℎ51198855+ ℎ561198856

11987911= [0 minus119860 minus 119866

111986511198671198860 0 0 0 0 0 0 0]

11987912= [0 0 0 0 0 0 119882 + 119866

111986511198671199080 0 0]

11987921= [0 0 0 119871 + 119866

211986521198671198970 0 0 0 0 0]

11987922= [0 minus119862 minus 119866

211986721198671198880 0 0 0 0 0 0 0]

1198791= [1198791111987912]

119879

1198792= [1198792111987922]

119879

1198731= [119873119879

11119873119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198732= [119873119879

21119873119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198733= [119873119879

31119873119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198781= [119878119879

11119878119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198782= [119878119879

21119878119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198783= [119878119879

31119878119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198721= [119872

119879

11119872119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198722= [119872

119879

21119872119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198723= [119872

119879

31119872119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198641= [119864119879

11119864119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198642= [119864119879

21119864119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198643= [119864119879

31119864119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

119879 = diag (1205741 1205742 120574

119899) Σ = diag (120572

1 1205722 120572

119899)

(26)

Abstract and Applied Analysis 7

Proof We consider the following Lyapunov functional can-didate for system (8)

119881 (119905 119909119905 119910119905) = 1198811(119905 119909119905 119910119905) + 1198812(119905 119909119905 119910119905) + 1198813(119905 119909119905 119910119905)

(27)

where

1198811(119905 119909119905 119910119905) = 2

119899

sum

119894=1

1198811119894int

119910119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

1198812119894int

119909119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

(28)

1198812(119905 119909119905 119910119905) = 119909119879

(119905) 1198751119909 (119905) + 119910

119879

(119905) 1198752119910 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119909119879

(119904) 1198761119909 (119904) 119889119904

+ int

119905

119905minus1198891(119905)

119891119879

(119909 (119904)) 1198762119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ1

119909119879

(119904) 1198763119909 (119904) 119889119904

+ int

119905minusℎ1

119905minusℎ2

119909119879

(119904) 1198764119909 (119904) 119889119904

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198765119910 (119904) 119889119904

+ int

119905

119905minus120590(119905)

119891119879

(119910 (119904)) 1198766119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ3

119910119879

(119904) 1198767119910 (119904) 119889119904

+ int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198768119910 (119904) 119889119904

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198769119910 (119904) 119889119904

+ int

119905

119905minus1198892(119905)

119891119879

(119910 (119904)) 11987610119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ5

119910119879

(119904) 11987611119910 (119904) 119889119904

+ int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 11987612119910 (119904) 119889119904

+ int

119905minusℎ7

119905minus120591(119905)

119909119879

(119904) 11987613119909 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119891119879

(119909 (119904)) 11987614119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ7

119909119879

(119904) 11987615119909 (119904) 119889119904

+ int

119905minusℎ7

119905minusℎ8

119909119879

(119904) 11987616119909 (119904) 119889119904

(29)

1198813(119905 119909119905 119910119905) = int

0

minusℎ1

int

119905

119905+120579

119879

(119904) 1198851 (119904) 119889119904 119889120579

+ int

minusℎ1

minusℎ2

int

119905

119905+120579

119879

(119904) 1198852 (119904) 119889119904 119889120579

+ int

0

minusℎ3

int

119905

119905+120579

119910119879

(119904) 1198853119910 (119904) 119889119904 119889120579

+ int

minusℎ3

minusℎ4

int

119905

119905+120579

119910119879

(119904) 1198854119910 (119904) 119889119904 119889120579

+ int

0

minusℎ5

int

119905

119905+120579

119910119879

(119904) 1198855119910 (119904) 119889119904 119889120579

+ int

minusℎ5

minusℎ6

int

119905

119905+120579

119910119879

(119904) 1198856119910 (119904) 119889119904 119889120579

+ int

0

minusℎ7

int

119905

119905+120579

119879

(119904) 1198857 (119904) 119889119904 119889120579

+ int

minusℎ7

minusℎ8

int

119905

119905+120579

119879

(119904) 1198858 (119904) 119889119904 119889120579

(30)

Then by Itorsquos differential formula taking the derivative of119881(119905) along the trajectories of the system (8) we can obtainthe following stochastic differential [29]

119889119881 (119905) = F119881 (119905) 119889119905

+ 2119909119879

(119905) 1198751120578 (119905 119909 (119905 minus 119889

1(119905)) 119910 (119905 minus 120590 (119905)))

+ 2119910119879

1198752120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905)))

(31)

whereF is the diffusion operator and

F119881 (119905 119909119905 119910119905) = F119881

1(119905 119909119905 119910119905)

+F1198812(119905 119909119905 119910119905) +F119881

3(119905 119909119905 119910119905)

(32)

with

F1198811(119905 119909119905 119910119905) = 2 [119891

119879

(119910 (119905)) minus 119910119879

(119905) Γ]

times 1198811[minus119862119910 (119905 minus 119889

2(119905)) + 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

8 Abstract and Applied Analysis

+ tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198811120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 2 [119891119879

(119909 (119905)) minus 119909119879

(119905) Γ]

times 1198812[minus119860119909 (119905 minus 119889

1(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198812120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

F1198812(119905 119909119905 119910119905)

le 2119909119879

(119905) 1198751[minus119860119909 (119905 minus 119889

1(119905))

+ 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198751120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

+ 2119910119879

(119905) 1198752[minus119862119910 (119905 minus 119889

2(119905))

+ 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

+ tr [120578119879 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198752120578 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 119909119879

(119905 minus ℎ1) 1198761119909 (119905 minus ℎ

1)

minus (1 minus 1198891) 119909119879

(119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905))

+ 119891119879

(119909 (119904)) 1198762119891 (119909 (119904))

minus (1 minus 1198891) 119891119879

(119910 (119905 minus 1198891(119905)))

times 1198762119891 (119910 (119905 minus 119889

1(119905)))

+ 119909119879

(119905) 1198763119909 (119905) minus 119909

119879

(119905 minus ℎ1) 1198763119909 (119905 minus ℎ

1)

+ 119909119879

(119905 minus ℎ1) 1198764119909 (119905 minus ℎ

1)

minus 119909119879

(119905 minus ℎ2) 1198764119909 (119905 minus ℎ

2)

+ 119910119879

(119905 minus ℎ5) 1198765119909 (119905 minus ℎ

5)

minus (1 minus 120590) 119910119879

(119905 minus 120590 (119905)) 1198765119910 (119905 minus 120590 (119905))

+ 119891119879

(119910 (119905)) 1198766119891 (119910 (119905))

minus (1 minus 120590) 119891 (119910119879

(119905 minus 120590 (119905)))

times 1198765119891 (119910 (119905 minus 120590 (119905)))

minus 119910119879

(119905 minus ℎ3) 1198767119910 (119905 minus ℎ

3)

+ 119910119879

(119905) 1198767119910 (119905) + 119910

119879

(119905 minus ℎ3) 1198768119910 (119905 minus ℎ

3)

minus 119910119879

(119905 minus ℎ4) 1198768119910 (119905 minus ℎ

4)

+ 119910119879

(119905 minus ℎ5) 1198769119910 (119905 minus ℎ

5)

minus (1 minus 1198892) 119910119879

(119905 minus 1198892(119905)) 119876

9119910 (119905 minus 119889

2(119905))

+ 119891119879

(119910 (119905)) 11987610119891 (119910 (119905))

minus (1 minus 1198892) 119891119879

(119910 (119905 minus 1198892(119905)))

times 11987610119891 (119910 (119905 minus 119889

2(119905)))

+ 119910119879

11987611119910 (119905) minus 119910

119879

(119905 minus ℎ5) 11987611119910 (119905 minus ℎ

5)

+ 119910119879

(119905 minus ℎ5) 11987612119910 (119905 minus ℎ

5)

minus 119910119879

(119905 minus ℎ6) 11987612119910 (119905 minus ℎ

6)

+ 119909119879

(119905 minus ℎ7) 11987613119909 (119905 minus ℎ

7)

minus (1 minus 120591) 119909119879

(119905 minus 120591 (119905)) 11987613119909 (119905 minus 120591 (119905))

+ 119891119879

(119909 (119905)) 11987614119891 (119909 (119905))

minus (1 minus 120591) 119891119879

(119909 (119905 minus 120591 (119905)))

times 11987614119891 (119909 (119905 minus 120591 (119905))) + 119909

119879

(119905) 11987615119909 (119905)

minus 119909119879

(119905 minus ℎ7) 11987615119909 (119905 minus ℎ

7)

+ 119909119879

(119905 minus ℎ7) 11987616119909 (119905 minus ℎ

7)

minus 119909119879

(119905 minus ℎ8) 11987616119909 (119905 minus ℎ

8)

F1198813(119905 119909119905 119910119905)

= ℎ1119879

(119905) 1198851 (119905)

minus int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

+ ℎ12119879

(119905) 1198852 (119905)

minus int

119905minusℎ1

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

+ ℎ2119910119879

(119905) 1198853119910 (119905)

minus int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

Abstract and Applied Analysis 9

+ ℎ34119910119879

(119905) 1198854119910 (119905)

minus int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

+ ℎ5119910119879

(119905) 1198855119910 (119905)

minus int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

+ ℎ56119910119879

(119905) 1198856119910 (119905)

minus int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

+ ℎ7119879

(119905) 1198857119910 (119905)

minus int

119905

119905minusℎ7

119910119879

(119904) 1198857119910 (119904) 119889119904

+ ℎ78119910119879

(119904) 1198858119910 (119904)

minus int

119905minusℎ7

119905minusℎ8

119910119879

(119905) 1198858119910 (119905) 119889119904

(33)

By Newton-Leibnitz formula we have that

2120576119879

(119905)1198731[119909 (119905) minus 119909 (119905 minus ℎ

1) minus int

119905

119905minusℎ1

(119904) 119889119904] = 0

2120576119879

(119905)1198732[119909 (119905 minus ℎ

1) minus 119909 (119905 minus 119889

1(119905)) minus int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904] = 0

2120576119879

(119905)1198733[119909 (119905 minus 119889

1(119905)) minus 119909 (119905 minus ℎ

2) minus int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904] = 0

2120576119879

(119905)1198721[119910 (119905) minus 119910 (119905 minus 120590 (119905)) minus int

119905

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198722[119910 (119905 minus ℎ

3) minus 119910 (119905 minus 120590 (119905)) minus int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198723[119910 (119905 minus 120590 (119905)) minus 119910 (119905 minus ℎ

4) minus int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198781[119910 (119905) minus 119910 (119905 minus 119889

2(119905)) minus int

119905

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198782[119910 (119905 minus ℎ

5) minus 119910 (119905 minus 119889

2(119905)) minus int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198783[119910 (119905 minus 119889

2(119905)) minus 119910 (119905 minus ℎ

6) minus int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198641[119909 (119905) minus 119909 (119905 minus 120591 (119905)) minus int

119905

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198642[119909 (119905 minus ℎ

7) minus 119909 (119905 minus 120591 (119905)) minus int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198643[119909 (119905 minus ℎ

8) minus 119909 (119905 minus ℎ

7) minus int

119905minusℎ7

119905minusℎ8

(119904) 119889119904] = 0

(34)

where

120576 (119905) = [1205761(119905) 1205762(119905) 1205763(119905) 1205764(119905)]119879

1205761(119905) = [119909

119879

(119905) 119909119879

(119905 minus 1198891(119905)) 119909

119879

(119905 minus ℎ1)

119909119879

(119905 minus ℎ2) 119909119879

(119905 minus 120591 (119905))]

1205762(119905) = [119909

119879

(119905 minus ℎ7) 119909119879

(119905 minus ℎ8) 119910119879

(119905)

119910119879

(119905 minus 120590 (119905)) 119910119879

(119905 minus ℎ3)]

1205763(119905) = [119910

119879

(119905 minus ℎ4) 119910119879

(119905 minus 1198892(119905)) 119910

119879

(119905 minus ℎ5)

119910119879

(119905 minus ℎ6) 119891119879

(119909)]

1205764(119905) = [119891

119879

(119910) 119891119879

(119910 minus 120590 (119905)) 119891119879

(119909 minus 1198891(119905))

119891119879

(119909 minus 120591 (119905)) 119891119879

(119910 minus 1198892(119905))]

(35)

By using Lemmas 4 and 5 we have

minus 2120576119879

(119905)1198731int

119905

119905minusℎ1

(119904) 119889119904

le ℎ1120576119879

(119905)1198731119885minus1

1119873119879

1120576 (119905)

+ int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

minus 2120576119879

(119905)1198732int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904

le (1198891(119905) minus ℎ

1) 120576119879

(119905)1198732119885minus1

2119873119879

2120576 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198733int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904

le (ℎ2minus 1198891(119905)) 120576119879

(119905)1198733119885minus1

2119873119879

3120576 (119905)

+ int

119905minus1198891(119905)

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198721int

119905

119905minusℎ3

119910 (119904) 119889119904

10 Abstract and Applied Analysis

le ℎ3120576119879

(119905)1198721119885minus1

3119872119879

1120576 (119905)

+ int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

minus 2120576119879

(119905)1198722int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904

le (120590 (119905) minus ℎ3) 120576119879

(119905)1198722119885minus1

4119872119879

2120576 (119905)

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905)1198723int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904

le (ℎ4minus 120590 (119905)) 120576

119879

(119905)1198723119885minus1

4119872119879

3120576 (119905)

+ int

119905minus120590(119905)

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905) 1198781int

119905

119905minusℎ5

119910 (119904) 119889119904

le ℎ5120576119879

(119905) 1198781119885minus1

5119878119879

1120576 (119905)

+ int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

minus 2120576119879

(119905) 1198782int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904

le (1198892(119905) minus ℎ

5) 120576119879

(119905) 1198782119885minus1

6119878119879

2120576 (119905)

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198783int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904

le (ℎ6minus 1198892(119905)) 120576119879

(119905) 1198783119885minus1

6119878119879

3120576 (119905)

+ int

119905minus1198892(119905)

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198641int

119905

119905minusℎ7

(119904) 119889119904

le ℎ7120576119879

(119905) 1198641119885minus1

7119864119879

1120576 (119905)

+ int

119905

119905minusℎ7

119879

(119904) 1198857 (119904) 119889119904

minus 2120576119879

(119905) 1198642int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904

le (120591 (119905) minus ℎ7) 120576119879

(119905) 1198642119885minus1

8119864119879

2120576 (119905)

+ int

119905minusℎ7

119905minus120591(119905)

119879

(119904) 1198858 (119904) 119889119904

minus 2120576119879

(119905) 1198643int

119905minus120591(119905)

119905minusℎ8

(119904) 119889119904

le (ℎ8minus 120591 (119905)) 120576

119879

(119905) 1198643119885minus1

8119864119879

3120576 (119905)

+ int

119905minus120591(119905)

119905minusℎ8

119879

(119904) 1198858 (119904) 119889119904

(36)

It follows from (7) that

0 = 119891119879

(119910 (119905)) 1198841119891 (119910 (119905)) minus 119891

119879

(119910 (119905)) 1198841119891 (119910 (119905))

le minus119910119879

(119905) 1198791198841Σ119910 (119905) + 119910

119879

(119905) 1198841(119879 + Σ) 119891 (119910 (119905))

minus 119891119879

(119910 (119905)) 1198841119891 (119910 (119905))

0 = 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

le minus119910119879

(119905 minus 1198891(119905)) 119879119884

2Σ119910 (119905 minus 119889

1(119905))

+ 119910119879

(119905 minus 1198891(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

0 = 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

le minus119910119879

(119905 minus 120590 (119905)) 1198791198843Σ119910 (119905 minus 120590 (119905))

+ 119910119879

(119905 minus 120590 (119905)) 1198843(119879 + Σ) 119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

0 = 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

le minus119910119879

(119905 minus 1198892(119905)) 119879119884

2Σ119910 (119905 minus 119889

2(119905))

+ 119910119879

(119905 minus 1198892(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

2119891 (119910 (119905 minus 119889

2(119905)))

0 = 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

le minus119910119879

(119905 minus 120591 (119905)) 1198791198845Σ119910 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 120591 (119905)) 1198845(119879 + Σ) 119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

0 = 119891119879

(119909 (119905)) 1198846119891 (119909 (119905)) minus 119891

119879

(119909 (119905)) 1198846119891 (119909 (119905))

le minus119909119879

(119905) 1198791198846Σ119909 (119905) + 119909

119879

(119905) 1198846(119879 + Σ) 119891 (119909 (119905))

minus 119891119879

(119909 (119905)) 1198846119891 (119909 (119905))

(37)

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Δ119901119894(119905)1003816100381610038161003816119905=119905119896

= 119901119894(119905119896) minus 119901119894(119905minus

119896) = 119869119896(119901119894(119905minus

119896))

119896 isin 119885+

119905 = 119905119896 119894 = 1 2 119899

(3)

where ℎ119895(119909) = (119909120573

119895)119867119895(1+(119909120573

119895)119867119895) 119906119894= Σ119895isin119868119894120572119894119895is defined

as a basal rate and 119868119894is the set of all the 119895 which is a repressor

of gene 119894 Thematrix119882 = (120592119894119895) isin R119899times119899 of the genetic network

is defined as follows

120592119894119895=

120572119894119895

if transcription factor 119895 is an activator of gene 1198940

if there is no link from node 119895 to node 119894minus120572119894119895

if transcription factor 119895 is a repressor of gene 119894(4)

Rewriting system (3) into compact matrix form weobtain

119889119898 (119905) = (minus119860119898 (119905 minus 1198891(119905)) + 119882ℎ (119901 (119905 minus 120590 (119905))) + 119906) 119889119905

+ 120578 (119905 119898 (119905 minus 1198891(119905)) 119901 (119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119898 (119905)|119905=119905119896

= 119898 (119905119896) minus 119898 (119905

minus

119896) = 119869119896(119898 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119901 (119905) = (minus119862119901 (119905 minus 1198892(119905)) + 119871119898 (119905 minus 120591 (119905))) 119889119905

+ 120578 (119905 119898 (119905 minus 120591 (119905)) 119901 (119905 minus 1198892(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119901 (119905)1003816100381610038161003816119905=119905119896

= 119901 (119905119896) minus 119901 (119905

minus

119896) = 119869119896(119901 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

(5)

where 119860 = diag(1198861 1198862 119886

119899) 119906 = col119906

1 1199062 119906

119899 119862 =

diag(1198881 1198882 119888

119899) 119871 = diag(119897

1 1198972 119897119899) 119898(119905) = col119898

1(119905)

1198982(119905) 119898

119899(119905) 119901(119905) = col119901

1(119905) 1199012(119905) 119901

119899(119905) ℎ(119901(119905))

= colℎ1(1199011(119905)) ℎ

2(1199012(119905)) ℎ

119899(119901119899(119905)) and 120578(119905 119909 119910) =

col1205781(119905 119909 119910) 120578

2(119905 119909 119910) 120578

119899(119905 119909 119910)

Let (119898lowast 119901lowast) be a nonnegative equilibrium point ofthe system (5) In the following we will always shift theequilibrium point (119898lowast 119901lowast) to the origin by letting 119909(119905) =119898(119905) minus 119898

lowast 119910(119905) = 119901(119905) minus 119901lowast Hence system (5) can be trans-formed into the following form

119889119909 (119905) = (minus119860119909 (119905 minus 1198891(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))) 119889119905

+ 120578 (119905 119909 (119905 minus 1198891(119905)) 119910 (119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119909 (119905)|119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119910 (119905) = (minus119862119910 (119905 minus 1198892(119905)) + 119871119909 (119905 minus 120591 (119905))) 119889119905

+ 120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119910 (119905)1003816100381610038161003816119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

(6)

where 119909(119905minus1198891(119905)) = col119909

1(119905minus1198891(119905)) 119909

2(119905minus1198891(119905)) 119909

119899(119905minus

1198891(119905)) isin R119899 119910(119905 minus 119889

2(119905)) = col119910

1(119905 minus 119889

2(119905)) 119910

2(119905 minus 119889

2(119905))

119910119899(119905 minus 119889

2(119905)) isin R119899 119891(119910(119905)) = col119891

1(1199101(119905)) 119891

2(1199102(119905))

119891119899(119910119899(119905)) isin R119899 the function 119891

119895(119910119895(119905)) = ℎ

119895(119910119895(119905) +119901

lowast

119895) minus

ℎ119895(119901lowast

119895) and obviously fj(0) = 0

Due to the fact that ℎ119895is a monotonically increasing

function with saturation from the relationship of 119891(sdot) andℎ(sdot) we know that for any 119910

119894isin 119877

120574119894le

119891119894(119910119894)

119910119894

le 120572119894 119894 = 1 2 119899 (7)

where 120574119894and 120572

119894are known constant scalars

Taking parameter uncertainties into the GRNsmodel (6)we consider the following UISGRNs model

119889119909 (119905) = (minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905)) + (119882 + Δ119882)

times119891 (119910 (119905 minus 120590 (119905)))) 119889119905

+ 120578 (119905 119909 (119905 minus 1198891(119905)) 119910 (119905 minus 120590 (119905))) 119889120596 (119905)

119905 = 119905119896

Δ119909(119905)|119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119889119910 (119905) = (minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905))

+ (119871 + Δ119871) 119909 (119905 minus 120591 (119905))) 119889119905

+ 120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905))) 119889120596 (119905)

119905 = 119905119896

Δ119910 (119905)1003816100381610038161003816119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579) forall120579 isin [minus120576 0]

(8)

where 120595(sdot) and 120593(sdot) are the initial function which are contin-uously differentiable on [minus120576 0] with 120576 = maxℎ

2 ℎ4 ℎ6 ℎ8

We extend 984858(120579) on 120579 isin [minus2120576 0] to satisfy 984858120576= 984858

2120576with

984858120576= sup

120579isin[minus1205760]984858(120579) 984858

2120576= sup

120579isin[minus21205760]984858(120579) where

984858 = 120595 120593

4 Abstract and Applied Analysis

Moreover the noise intensity 120578 satisfies

tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times 120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))]

le 119909119879

(119905 minus 120591 (119905)) Σ119879

1Σ1119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) Σ119879

2Σ21199102(119905 minus 119889 (119905))

(9)

where Σ1and Σ

2are constant matrices with appropriate

dimensionsIn order to obtain our main theorem the following

assumptions and lemmas for the system (8) are always madethroughout this paper

Assumption 1 The parametric uncertainties Δ119860(119905) Δ119882(119905)Δ119862(119905) and Δ119871(119905) satisfy

Δ119860 (119905) = 11986611198651(119905)119867119886

Δ119882(119905) = 11986611198651(119905)119867119908

Δ119862 (119905) = 11986621198652(119905)119867119888

Δ119871 (119905) = 11986621198652(119905)119867119897

(10)

where 1198661 1198662 119867119886 119867119908 119867119888 and 119867

119897are some given constant

matrices with appropriate dimensions and 119865119894(119905) satisfies

119865119879

119894(119905)119865119894(119905) le 119868 119894 = 1 2 for any 119905 gt 0

Assumption 2 1198891(119905) 1198892(119905) 120591(119905) and 120590(119905) are the time-varying

delays satisfying

0 le ℎ1le 1198891(119905) le ℎ

2 0 le

1198891(119905) le 119889

1lt infin

0 le ℎ3le 120590 (119905) le ℎ

4 0 le (119905) le 120590 lt infin

0 le ℎ5le 1198892(119905) le ℎ

6 0 le

1198892(119905) le 119889

2lt infin

0 le ℎ7le 120591 (119905) le ℎ

8 0 le 120591 (119905) le 120591 lt infin

ℎ12= ℎ2minus ℎ1 ℎ

34= ℎ4minus ℎ3

ℎ56= ℎ6minus ℎ5 ℎ

78= ℎ8minus ℎ7

(11)

Lemma 3 (Schur complement see [30]) For a given matrix

(

119876 (119909) 119878 (119909)

119878119879

(119909) 119877 (119909)

) gt 0 (12)

where

119876 (119909) = 119876119879

(119909) 119877 (119909) = 119877119879

(119909) (13)

and a vector function 119908(119909) [0 119903] rarr 119877119899 such that the

intergrals concerned as well defined then the following holds

(i) 119876(119909) gt 0 119877(119909) minus 119878119879(119909)119876(119909)minus1119878(119909) gt 0

(ii) 119877(119909) gt 0 119876(119909) minus 119878(119909)119877(119909)minus1119878119879(119909) gt 0

Lemma 4 (see [33]) For any constant symmetric matrix119872 gt

0 scalar 120574 gt 0

[int

120574

0

120603 (119904) 119889119904]

119879

119872[int

120574

0

120603 (119904) 119889119904] le 120574int

120574

0

120603119879

(119904)119872120603 (119904) 119889119904

(14)

Lemma 5 (see [29]) For any vectors 119886 119887 isin R119899 and anypositive matrix 119884 satisfying

plusmn2119886119879

119887 le 119886119879

119884119886 + 119887119879

119884minus1

119887 (15)

3 Main Result

In this section mean square stability result for model (8) issummarized in the following theorem

Theorem 6 If (7) (9) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120588

1gt 0 1205882gt 0 120594im isin [0 1] 119896 = 0 1 119903+

2 and 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (8) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and

1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and any

matrices11987311119873121198732111987322119872111198721211987221119872221198723111987232

11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32to

satisfy the following ten linear matrix inequalities

1198751+ 1198811lt 1205881119868 (16)

1198752+ 1198812lt 1205882119868 (17)

1206011= [

Ξ ℎ121198732

lowast minusℎ121198852

] lt 0 (18)

1206012= [

Ξ ℎ121198733

lowast minusℎ121198852

] lt 0 (19)

1206013= [

Ξ ℎ341198722

lowast minusℎ341198854

] lt 0 (20)

1206014= [

Ξ ℎ341198723

lowast minusℎ341198854

] lt 0 (21)

1206015= [

Ξ ℎ561198782

lowast minusℎ561198856

] lt 0 (22)

1206016= [

Ξ ℎ561198783

lowast minusℎ561198856

] lt 0 (23)

1206017= [

Ξ ℎ781198642

lowast minusℎ781198858

] lt 0 (24)

1206018= [

Ξ ℎ781198643

lowast minusℎ781198858

] lt 0 (25)

Abstract and Applied Analysis 5

where

Ξ =

1

4

[

[

[

[

[

[

[

[

[

[

120601 1198791119883 1198792119884 ℎ11198731ℎ31198721ℎ51198781ℎ71198641

lowast minus119883 0 0 0 0 0

lowast 0 minus119884 0 0 0 0

lowast 0 0 minusℎ11198851

0 0 0

lowast 0 0 0 minusℎ31198853

0 0

lowast 0 0 0 0 ℎ51198855

0

lowast 0 0 0 0 0 minusℎ71198857

]

]

]

]

]

]

]

]

]

]

120601 = [

120601912060110

lowast 12060111

]

1206019=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120601111206011211987321minus1198733112060115

11986421

11986431

11987811+1198721112060119

11987221

lowast 1206012211987322minus1198733212060125

11986422minus1198643211987212+ 1198781212060129

11987222

lowast lowast 12060133

0 0 0 0 0 0 0

lowast lowast lowast minus1198764

0 0 0 0 0 0

lowast lowast lowast lowast 12060155

0 0 12060158

0 0

lowast lowast lowast lowast lowast 12060166

0 0 0 0

lowast lowast lowast lowast lowast lowast minus11987616

0 0 0

lowast lowast lowast lowast lowast lowast lowast 12060188

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 12060199

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1206011010

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12060110=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus11987231120601112

11987812minus11987831120601115

0 120601117

0 0 0

minus11987232120601212

11987822minus11987832120601215

0 0 120601218

0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 120601516

0 0 120601519

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 120601812

0 0 0 120601816

0 0 0 0

0 0 0 0 0 0 120601917

0 0 0

0 0 0 0 0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12060111=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus1198768

0 0 0 0 0 0 0 0 0

lowast 1206011212

0 0 0 1206011216

0 0 0 1206011220

lowast lowast 1206011313

0 0 0 0 0 0 0

lowast lowast lowast minus11987612

0 0 0 0 0 0

lowast lowast lowast lowast 1206011515

0 1206011517

0 0 0

lowast lowast lowast lowast lowast 1206011616

0 0 0 0

lowast lowast lowast lowast lowast lowast 1206011717

0 0 0

lowast lowast lowast lowast lowast lowast lowast 1206011818

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1206011919

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1206012020

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119869119894119898(119909 (119905119898)) = (119869

1119898(1199091(119905119898)) 119869

119899119898(119909119899(119905119898)))119879

12060111= 1198763+ 11987615+ 211987311+ 211986411minus 1198791198846Σ

12060112= minus1198751119860 minus 119875

111986611198651119867119886minus 1198791198812119860 minus 119879119881

211986611198651119867119886minus 11987311+ 11987312+ 11987313minus 11987321+ 11986412

12060115= minus11986411+ 11986431minus 11986421 120601

19= minus119872

11+11987231minus11987221 120601

112= minus11987811+ 11987831minus 11987821

120601115

=

1

2

1198846(119879 + Σ) 120601

117= minus119879119881

2119882minus 119879119881

211986611198651119867119908+ 1198751119882+ 119875

111986611198651119867119908

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232+ 1205881Σ119879

1Σ1minus 1198791198842Σ 120601

25= minus11986412+ 11986432minus 11986422

12060129= minus119872

12+11987232minus11987222 120601

212= minus11987812+ 11987832minus 11987822 120601

215= minus1198812119860 minus 119881

211986611198651119867119886

120601218

=

1

2

1198842(119879 + Σ) 120601

33= 1198761+ 1198764minus 1198763 120601

55= minus (1 minus 120591)119876

13+ 1205882Σ4Σ4minus 1198791198845Σ

6 Abstract and Applied Analysis

12060158= minus119879119881

1119871 minus 119879119881

111986621198652119867119897+ 1198752119871 + 119875211986621198652119867119897 120601

516= 1198811119871 + 119881111986621198652119867119897

120601519

=

1

2

1198845(119879 + Σ) 120601

66= 11987613+ 11987616minus 11987615 120601

88= 1198767+ 11987611minus 1198791198841Σ

120601812

= minus1198752119862 minus 119875211986621198652119867119888+ 1198791198811119862 + 119879119881

111986621198652119867119888 120601

816=

1

2

1198841(119879 + Σ)

12060199= minus (1 minus 120590)119876

5+ 1205881Σ2Σ2minus 1198791198843Σ 120601

917=

1

2

1198843(119879 + Σ) 120601

1010= 1198765minus 1198767+ 1198768

1206011212

= minus (1 minus 1198892) 1198769+ 1205882Σ3Σ3minus 1198791198844Σ 120601

1216= minus1198811119862 minus 119881

111986621198652119867119888

1206011220

=

1

2

1198844(119879 + Σ) 120601

1313= 1198769minus 11987611+ 11987612 120601

1515= 11987614minus 1198846+ 1198762

1206011517

= 1198812119882+119881

211986611198651119867119908 120601

1616= 1198766minus 1198841+ 11987610 120601

1717= minus (1 minus 120590)119876

6minus 1198843

1206011818

= minus (1 minus 1198891) 1198762minus 1198842 120601

1919= minus (1 minus 120591)119876

15minus 1198845 120601

2020= minus (1 minus 119889

2) 11987610minus 1198844

119883 = ℎ11198851+ ℎ121198852+ ℎ71198857+ ℎ781198858 119884 = ℎ

21198853+ ℎ341198854+ ℎ51198855+ ℎ561198856

11987911= [0 minus119860 minus 119866

111986511198671198860 0 0 0 0 0 0 0]

11987912= [0 0 0 0 0 0 119882 + 119866

111986511198671199080 0 0]

11987921= [0 0 0 119871 + 119866

211986521198671198970 0 0 0 0 0]

11987922= [0 minus119862 minus 119866

211986721198671198880 0 0 0 0 0 0 0]

1198791= [1198791111987912]

119879

1198792= [1198792111987922]

119879

1198731= [119873119879

11119873119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198732= [119873119879

21119873119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198733= [119873119879

31119873119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198781= [119878119879

11119878119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198782= [119878119879

21119878119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198783= [119878119879

31119878119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198721= [119872

119879

11119872119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198722= [119872

119879

21119872119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198723= [119872

119879

31119872119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198641= [119864119879

11119864119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198642= [119864119879

21119864119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198643= [119864119879

31119864119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

119879 = diag (1205741 1205742 120574

119899) Σ = diag (120572

1 1205722 120572

119899)

(26)

Abstract and Applied Analysis 7

Proof We consider the following Lyapunov functional can-didate for system (8)

119881 (119905 119909119905 119910119905) = 1198811(119905 119909119905 119910119905) + 1198812(119905 119909119905 119910119905) + 1198813(119905 119909119905 119910119905)

(27)

where

1198811(119905 119909119905 119910119905) = 2

119899

sum

119894=1

1198811119894int

119910119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

1198812119894int

119909119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

(28)

1198812(119905 119909119905 119910119905) = 119909119879

(119905) 1198751119909 (119905) + 119910

119879

(119905) 1198752119910 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119909119879

(119904) 1198761119909 (119904) 119889119904

+ int

119905

119905minus1198891(119905)

119891119879

(119909 (119904)) 1198762119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ1

119909119879

(119904) 1198763119909 (119904) 119889119904

+ int

119905minusℎ1

119905minusℎ2

119909119879

(119904) 1198764119909 (119904) 119889119904

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198765119910 (119904) 119889119904

+ int

119905

119905minus120590(119905)

119891119879

(119910 (119904)) 1198766119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ3

119910119879

(119904) 1198767119910 (119904) 119889119904

+ int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198768119910 (119904) 119889119904

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198769119910 (119904) 119889119904

+ int

119905

119905minus1198892(119905)

119891119879

(119910 (119904)) 11987610119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ5

119910119879

(119904) 11987611119910 (119904) 119889119904

+ int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 11987612119910 (119904) 119889119904

+ int

119905minusℎ7

119905minus120591(119905)

119909119879

(119904) 11987613119909 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119891119879

(119909 (119904)) 11987614119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ7

119909119879

(119904) 11987615119909 (119904) 119889119904

+ int

119905minusℎ7

119905minusℎ8

119909119879

(119904) 11987616119909 (119904) 119889119904

(29)

1198813(119905 119909119905 119910119905) = int

0

minusℎ1

int

119905

119905+120579

119879

(119904) 1198851 (119904) 119889119904 119889120579

+ int

minusℎ1

minusℎ2

int

119905

119905+120579

119879

(119904) 1198852 (119904) 119889119904 119889120579

+ int

0

minusℎ3

int

119905

119905+120579

119910119879

(119904) 1198853119910 (119904) 119889119904 119889120579

+ int

minusℎ3

minusℎ4

int

119905

119905+120579

119910119879

(119904) 1198854119910 (119904) 119889119904 119889120579

+ int

0

minusℎ5

int

119905

119905+120579

119910119879

(119904) 1198855119910 (119904) 119889119904 119889120579

+ int

minusℎ5

minusℎ6

int

119905

119905+120579

119910119879

(119904) 1198856119910 (119904) 119889119904 119889120579

+ int

0

minusℎ7

int

119905

119905+120579

119879

(119904) 1198857 (119904) 119889119904 119889120579

+ int

minusℎ7

minusℎ8

int

119905

119905+120579

119879

(119904) 1198858 (119904) 119889119904 119889120579

(30)

Then by Itorsquos differential formula taking the derivative of119881(119905) along the trajectories of the system (8) we can obtainthe following stochastic differential [29]

119889119881 (119905) = F119881 (119905) 119889119905

+ 2119909119879

(119905) 1198751120578 (119905 119909 (119905 minus 119889

1(119905)) 119910 (119905 minus 120590 (119905)))

+ 2119910119879

1198752120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905)))

(31)

whereF is the diffusion operator and

F119881 (119905 119909119905 119910119905) = F119881

1(119905 119909119905 119910119905)

+F1198812(119905 119909119905 119910119905) +F119881

3(119905 119909119905 119910119905)

(32)

with

F1198811(119905 119909119905 119910119905) = 2 [119891

119879

(119910 (119905)) minus 119910119879

(119905) Γ]

times 1198811[minus119862119910 (119905 minus 119889

2(119905)) + 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

8 Abstract and Applied Analysis

+ tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198811120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 2 [119891119879

(119909 (119905)) minus 119909119879

(119905) Γ]

times 1198812[minus119860119909 (119905 minus 119889

1(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198812120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

F1198812(119905 119909119905 119910119905)

le 2119909119879

(119905) 1198751[minus119860119909 (119905 minus 119889

1(119905))

+ 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198751120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

+ 2119910119879

(119905) 1198752[minus119862119910 (119905 minus 119889

2(119905))

+ 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

+ tr [120578119879 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198752120578 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 119909119879

(119905 minus ℎ1) 1198761119909 (119905 minus ℎ

1)

minus (1 minus 1198891) 119909119879

(119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905))

+ 119891119879

(119909 (119904)) 1198762119891 (119909 (119904))

minus (1 minus 1198891) 119891119879

(119910 (119905 minus 1198891(119905)))

times 1198762119891 (119910 (119905 minus 119889

1(119905)))

+ 119909119879

(119905) 1198763119909 (119905) minus 119909

119879

(119905 minus ℎ1) 1198763119909 (119905 minus ℎ

1)

+ 119909119879

(119905 minus ℎ1) 1198764119909 (119905 minus ℎ

1)

minus 119909119879

(119905 minus ℎ2) 1198764119909 (119905 minus ℎ

2)

+ 119910119879

(119905 minus ℎ5) 1198765119909 (119905 minus ℎ

5)

minus (1 minus 120590) 119910119879

(119905 minus 120590 (119905)) 1198765119910 (119905 minus 120590 (119905))

+ 119891119879

(119910 (119905)) 1198766119891 (119910 (119905))

minus (1 minus 120590) 119891 (119910119879

(119905 minus 120590 (119905)))

times 1198765119891 (119910 (119905 minus 120590 (119905)))

minus 119910119879

(119905 minus ℎ3) 1198767119910 (119905 minus ℎ

3)

+ 119910119879

(119905) 1198767119910 (119905) + 119910

119879

(119905 minus ℎ3) 1198768119910 (119905 minus ℎ

3)

minus 119910119879

(119905 minus ℎ4) 1198768119910 (119905 minus ℎ

4)

+ 119910119879

(119905 minus ℎ5) 1198769119910 (119905 minus ℎ

5)

minus (1 minus 1198892) 119910119879

(119905 minus 1198892(119905)) 119876

9119910 (119905 minus 119889

2(119905))

+ 119891119879

(119910 (119905)) 11987610119891 (119910 (119905))

minus (1 minus 1198892) 119891119879

(119910 (119905 minus 1198892(119905)))

times 11987610119891 (119910 (119905 minus 119889

2(119905)))

+ 119910119879

11987611119910 (119905) minus 119910

119879

(119905 minus ℎ5) 11987611119910 (119905 minus ℎ

5)

+ 119910119879

(119905 minus ℎ5) 11987612119910 (119905 minus ℎ

5)

minus 119910119879

(119905 minus ℎ6) 11987612119910 (119905 minus ℎ

6)

+ 119909119879

(119905 minus ℎ7) 11987613119909 (119905 minus ℎ

7)

minus (1 minus 120591) 119909119879

(119905 minus 120591 (119905)) 11987613119909 (119905 minus 120591 (119905))

+ 119891119879

(119909 (119905)) 11987614119891 (119909 (119905))

minus (1 minus 120591) 119891119879

(119909 (119905 minus 120591 (119905)))

times 11987614119891 (119909 (119905 minus 120591 (119905))) + 119909

119879

(119905) 11987615119909 (119905)

minus 119909119879

(119905 minus ℎ7) 11987615119909 (119905 minus ℎ

7)

+ 119909119879

(119905 minus ℎ7) 11987616119909 (119905 minus ℎ

7)

minus 119909119879

(119905 minus ℎ8) 11987616119909 (119905 minus ℎ

8)

F1198813(119905 119909119905 119910119905)

= ℎ1119879

(119905) 1198851 (119905)

minus int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

+ ℎ12119879

(119905) 1198852 (119905)

minus int

119905minusℎ1

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

+ ℎ2119910119879

(119905) 1198853119910 (119905)

minus int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

Abstract and Applied Analysis 9

+ ℎ34119910119879

(119905) 1198854119910 (119905)

minus int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

+ ℎ5119910119879

(119905) 1198855119910 (119905)

minus int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

+ ℎ56119910119879

(119905) 1198856119910 (119905)

minus int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

+ ℎ7119879

(119905) 1198857119910 (119905)

minus int

119905

119905minusℎ7

119910119879

(119904) 1198857119910 (119904) 119889119904

+ ℎ78119910119879

(119904) 1198858119910 (119904)

minus int

119905minusℎ7

119905minusℎ8

119910119879

(119905) 1198858119910 (119905) 119889119904

(33)

By Newton-Leibnitz formula we have that

2120576119879

(119905)1198731[119909 (119905) minus 119909 (119905 minus ℎ

1) minus int

119905

119905minusℎ1

(119904) 119889119904] = 0

2120576119879

(119905)1198732[119909 (119905 minus ℎ

1) minus 119909 (119905 minus 119889

1(119905)) minus int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904] = 0

2120576119879

(119905)1198733[119909 (119905 minus 119889

1(119905)) minus 119909 (119905 minus ℎ

2) minus int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904] = 0

2120576119879

(119905)1198721[119910 (119905) minus 119910 (119905 minus 120590 (119905)) minus int

119905

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198722[119910 (119905 minus ℎ

3) minus 119910 (119905 minus 120590 (119905)) minus int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198723[119910 (119905 minus 120590 (119905)) minus 119910 (119905 minus ℎ

4) minus int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198781[119910 (119905) minus 119910 (119905 minus 119889

2(119905)) minus int

119905

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198782[119910 (119905 minus ℎ

5) minus 119910 (119905 minus 119889

2(119905)) minus int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198783[119910 (119905 minus 119889

2(119905)) minus 119910 (119905 minus ℎ

6) minus int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198641[119909 (119905) minus 119909 (119905 minus 120591 (119905)) minus int

119905

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198642[119909 (119905 minus ℎ

7) minus 119909 (119905 minus 120591 (119905)) minus int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198643[119909 (119905 minus ℎ

8) minus 119909 (119905 minus ℎ

7) minus int

119905minusℎ7

119905minusℎ8

(119904) 119889119904] = 0

(34)

where

120576 (119905) = [1205761(119905) 1205762(119905) 1205763(119905) 1205764(119905)]119879

1205761(119905) = [119909

119879

(119905) 119909119879

(119905 minus 1198891(119905)) 119909

119879

(119905 minus ℎ1)

119909119879

(119905 minus ℎ2) 119909119879

(119905 minus 120591 (119905))]

1205762(119905) = [119909

119879

(119905 minus ℎ7) 119909119879

(119905 minus ℎ8) 119910119879

(119905)

119910119879

(119905 minus 120590 (119905)) 119910119879

(119905 minus ℎ3)]

1205763(119905) = [119910

119879

(119905 minus ℎ4) 119910119879

(119905 minus 1198892(119905)) 119910

119879

(119905 minus ℎ5)

119910119879

(119905 minus ℎ6) 119891119879

(119909)]

1205764(119905) = [119891

119879

(119910) 119891119879

(119910 minus 120590 (119905)) 119891119879

(119909 minus 1198891(119905))

119891119879

(119909 minus 120591 (119905)) 119891119879

(119910 minus 1198892(119905))]

(35)

By using Lemmas 4 and 5 we have

minus 2120576119879

(119905)1198731int

119905

119905minusℎ1

(119904) 119889119904

le ℎ1120576119879

(119905)1198731119885minus1

1119873119879

1120576 (119905)

+ int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

minus 2120576119879

(119905)1198732int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904

le (1198891(119905) minus ℎ

1) 120576119879

(119905)1198732119885minus1

2119873119879

2120576 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198733int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904

le (ℎ2minus 1198891(119905)) 120576119879

(119905)1198733119885minus1

2119873119879

3120576 (119905)

+ int

119905minus1198891(119905)

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198721int

119905

119905minusℎ3

119910 (119904) 119889119904

10 Abstract and Applied Analysis

le ℎ3120576119879

(119905)1198721119885minus1

3119872119879

1120576 (119905)

+ int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

minus 2120576119879

(119905)1198722int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904

le (120590 (119905) minus ℎ3) 120576119879

(119905)1198722119885minus1

4119872119879

2120576 (119905)

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905)1198723int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904

le (ℎ4minus 120590 (119905)) 120576

119879

(119905)1198723119885minus1

4119872119879

3120576 (119905)

+ int

119905minus120590(119905)

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905) 1198781int

119905

119905minusℎ5

119910 (119904) 119889119904

le ℎ5120576119879

(119905) 1198781119885minus1

5119878119879

1120576 (119905)

+ int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

minus 2120576119879

(119905) 1198782int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904

le (1198892(119905) minus ℎ

5) 120576119879

(119905) 1198782119885minus1

6119878119879

2120576 (119905)

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198783int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904

le (ℎ6minus 1198892(119905)) 120576119879

(119905) 1198783119885minus1

6119878119879

3120576 (119905)

+ int

119905minus1198892(119905)

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198641int

119905

119905minusℎ7

(119904) 119889119904

le ℎ7120576119879

(119905) 1198641119885minus1

7119864119879

1120576 (119905)

+ int

119905

119905minusℎ7

119879

(119904) 1198857 (119904) 119889119904

minus 2120576119879

(119905) 1198642int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904

le (120591 (119905) minus ℎ7) 120576119879

(119905) 1198642119885minus1

8119864119879

2120576 (119905)

+ int

119905minusℎ7

119905minus120591(119905)

119879

(119904) 1198858 (119904) 119889119904

minus 2120576119879

(119905) 1198643int

119905minus120591(119905)

119905minusℎ8

(119904) 119889119904

le (ℎ8minus 120591 (119905)) 120576

119879

(119905) 1198643119885minus1

8119864119879

3120576 (119905)

+ int

119905minus120591(119905)

119905minusℎ8

119879

(119904) 1198858 (119904) 119889119904

(36)

It follows from (7) that

0 = 119891119879

(119910 (119905)) 1198841119891 (119910 (119905)) minus 119891

119879

(119910 (119905)) 1198841119891 (119910 (119905))

le minus119910119879

(119905) 1198791198841Σ119910 (119905) + 119910

119879

(119905) 1198841(119879 + Σ) 119891 (119910 (119905))

minus 119891119879

(119910 (119905)) 1198841119891 (119910 (119905))

0 = 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

le minus119910119879

(119905 minus 1198891(119905)) 119879119884

2Σ119910 (119905 minus 119889

1(119905))

+ 119910119879

(119905 minus 1198891(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

0 = 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

le minus119910119879

(119905 minus 120590 (119905)) 1198791198843Σ119910 (119905 minus 120590 (119905))

+ 119910119879

(119905 minus 120590 (119905)) 1198843(119879 + Σ) 119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

0 = 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

le minus119910119879

(119905 minus 1198892(119905)) 119879119884

2Σ119910 (119905 minus 119889

2(119905))

+ 119910119879

(119905 minus 1198892(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

2119891 (119910 (119905 minus 119889

2(119905)))

0 = 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

le minus119910119879

(119905 minus 120591 (119905)) 1198791198845Σ119910 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 120591 (119905)) 1198845(119879 + Σ) 119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

0 = 119891119879

(119909 (119905)) 1198846119891 (119909 (119905)) minus 119891

119879

(119909 (119905)) 1198846119891 (119909 (119905))

le minus119909119879

(119905) 1198791198846Σ119909 (119905) + 119909

119879

(119905) 1198846(119879 + Σ) 119891 (119909 (119905))

minus 119891119879

(119909 (119905)) 1198846119891 (119909 (119905))

(37)

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Moreover the noise intensity 120578 satisfies

tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times 120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))]

le 119909119879

(119905 minus 120591 (119905)) Σ119879

1Σ1119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) Σ119879

2Σ21199102(119905 minus 119889 (119905))

(9)

where Σ1and Σ

2are constant matrices with appropriate

dimensionsIn order to obtain our main theorem the following

assumptions and lemmas for the system (8) are always madethroughout this paper

Assumption 1 The parametric uncertainties Δ119860(119905) Δ119882(119905)Δ119862(119905) and Δ119871(119905) satisfy

Δ119860 (119905) = 11986611198651(119905)119867119886

Δ119882(119905) = 11986611198651(119905)119867119908

Δ119862 (119905) = 11986621198652(119905)119867119888

Δ119871 (119905) = 11986621198652(119905)119867119897

(10)

where 1198661 1198662 119867119886 119867119908 119867119888 and 119867

119897are some given constant

matrices with appropriate dimensions and 119865119894(119905) satisfies

119865119879

119894(119905)119865119894(119905) le 119868 119894 = 1 2 for any 119905 gt 0

Assumption 2 1198891(119905) 1198892(119905) 120591(119905) and 120590(119905) are the time-varying

delays satisfying

0 le ℎ1le 1198891(119905) le ℎ

2 0 le

1198891(119905) le 119889

1lt infin

0 le ℎ3le 120590 (119905) le ℎ

4 0 le (119905) le 120590 lt infin

0 le ℎ5le 1198892(119905) le ℎ

6 0 le

1198892(119905) le 119889

2lt infin

0 le ℎ7le 120591 (119905) le ℎ

8 0 le 120591 (119905) le 120591 lt infin

ℎ12= ℎ2minus ℎ1 ℎ

34= ℎ4minus ℎ3

ℎ56= ℎ6minus ℎ5 ℎ

78= ℎ8minus ℎ7

(11)

Lemma 3 (Schur complement see [30]) For a given matrix

(

119876 (119909) 119878 (119909)

119878119879

(119909) 119877 (119909)

) gt 0 (12)

where

119876 (119909) = 119876119879

(119909) 119877 (119909) = 119877119879

(119909) (13)

and a vector function 119908(119909) [0 119903] rarr 119877119899 such that the

intergrals concerned as well defined then the following holds

(i) 119876(119909) gt 0 119877(119909) minus 119878119879(119909)119876(119909)minus1119878(119909) gt 0

(ii) 119877(119909) gt 0 119876(119909) minus 119878(119909)119877(119909)minus1119878119879(119909) gt 0

Lemma 4 (see [33]) For any constant symmetric matrix119872 gt

0 scalar 120574 gt 0

[int

120574

0

120603 (119904) 119889119904]

119879

119872[int

120574

0

120603 (119904) 119889119904] le 120574int

120574

0

120603119879

(119904)119872120603 (119904) 119889119904

(14)

Lemma 5 (see [29]) For any vectors 119886 119887 isin R119899 and anypositive matrix 119884 satisfying

plusmn2119886119879

119887 le 119886119879

119884119886 + 119887119879

119884minus1

119887 (15)

3 Main Result

In this section mean square stability result for model (8) issummarized in the following theorem

Theorem 6 If (7) (9) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120588

1gt 0 1205882gt 0 120594im isin [0 1] 119896 = 0 1 119903+

2 and 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (8) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and

1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and any

matrices11987311119873121198732111987322119872111198721211987221119872221198723111987232

11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32to

satisfy the following ten linear matrix inequalities

1198751+ 1198811lt 1205881119868 (16)

1198752+ 1198812lt 1205882119868 (17)

1206011= [

Ξ ℎ121198732

lowast minusℎ121198852

] lt 0 (18)

1206012= [

Ξ ℎ121198733

lowast minusℎ121198852

] lt 0 (19)

1206013= [

Ξ ℎ341198722

lowast minusℎ341198854

] lt 0 (20)

1206014= [

Ξ ℎ341198723

lowast minusℎ341198854

] lt 0 (21)

1206015= [

Ξ ℎ561198782

lowast minusℎ561198856

] lt 0 (22)

1206016= [

Ξ ℎ561198783

lowast minusℎ561198856

] lt 0 (23)

1206017= [

Ξ ℎ781198642

lowast minusℎ781198858

] lt 0 (24)

1206018= [

Ξ ℎ781198643

lowast minusℎ781198858

] lt 0 (25)

Abstract and Applied Analysis 5

where

Ξ =

1

4

[

[

[

[

[

[

[

[

[

[

120601 1198791119883 1198792119884 ℎ11198731ℎ31198721ℎ51198781ℎ71198641

lowast minus119883 0 0 0 0 0

lowast 0 minus119884 0 0 0 0

lowast 0 0 minusℎ11198851

0 0 0

lowast 0 0 0 minusℎ31198853

0 0

lowast 0 0 0 0 ℎ51198855

0

lowast 0 0 0 0 0 minusℎ71198857

]

]

]

]

]

]

]

]

]

]

120601 = [

120601912060110

lowast 12060111

]

1206019=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120601111206011211987321minus1198733112060115

11986421

11986431

11987811+1198721112060119

11987221

lowast 1206012211987322minus1198733212060125

11986422minus1198643211987212+ 1198781212060129

11987222

lowast lowast 12060133

0 0 0 0 0 0 0

lowast lowast lowast minus1198764

0 0 0 0 0 0

lowast lowast lowast lowast 12060155

0 0 12060158

0 0

lowast lowast lowast lowast lowast 12060166

0 0 0 0

lowast lowast lowast lowast lowast lowast minus11987616

0 0 0

lowast lowast lowast lowast lowast lowast lowast 12060188

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 12060199

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1206011010

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12060110=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus11987231120601112

11987812minus11987831120601115

0 120601117

0 0 0

minus11987232120601212

11987822minus11987832120601215

0 0 120601218

0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 120601516

0 0 120601519

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 120601812

0 0 0 120601816

0 0 0 0

0 0 0 0 0 0 120601917

0 0 0

0 0 0 0 0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12060111=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus1198768

0 0 0 0 0 0 0 0 0

lowast 1206011212

0 0 0 1206011216

0 0 0 1206011220

lowast lowast 1206011313

0 0 0 0 0 0 0

lowast lowast lowast minus11987612

0 0 0 0 0 0

lowast lowast lowast lowast 1206011515

0 1206011517

0 0 0

lowast lowast lowast lowast lowast 1206011616

0 0 0 0

lowast lowast lowast lowast lowast lowast 1206011717

0 0 0

lowast lowast lowast lowast lowast lowast lowast 1206011818

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1206011919

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1206012020

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119869119894119898(119909 (119905119898)) = (119869

1119898(1199091(119905119898)) 119869

119899119898(119909119899(119905119898)))119879

12060111= 1198763+ 11987615+ 211987311+ 211986411minus 1198791198846Σ

12060112= minus1198751119860 minus 119875

111986611198651119867119886minus 1198791198812119860 minus 119879119881

211986611198651119867119886minus 11987311+ 11987312+ 11987313minus 11987321+ 11986412

12060115= minus11986411+ 11986431minus 11986421 120601

19= minus119872

11+11987231minus11987221 120601

112= minus11987811+ 11987831minus 11987821

120601115

=

1

2

1198846(119879 + Σ) 120601

117= minus119879119881

2119882minus 119879119881

211986611198651119867119908+ 1198751119882+ 119875

111986611198651119867119908

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232+ 1205881Σ119879

1Σ1minus 1198791198842Σ 120601

25= minus11986412+ 11986432minus 11986422

12060129= minus119872

12+11987232minus11987222 120601

212= minus11987812+ 11987832minus 11987822 120601

215= minus1198812119860 minus 119881

211986611198651119867119886

120601218

=

1

2

1198842(119879 + Σ) 120601

33= 1198761+ 1198764minus 1198763 120601

55= minus (1 minus 120591)119876

13+ 1205882Σ4Σ4minus 1198791198845Σ

6 Abstract and Applied Analysis

12060158= minus119879119881

1119871 minus 119879119881

111986621198652119867119897+ 1198752119871 + 119875211986621198652119867119897 120601

516= 1198811119871 + 119881111986621198652119867119897

120601519

=

1

2

1198845(119879 + Σ) 120601

66= 11987613+ 11987616minus 11987615 120601

88= 1198767+ 11987611minus 1198791198841Σ

120601812

= minus1198752119862 minus 119875211986621198652119867119888+ 1198791198811119862 + 119879119881

111986621198652119867119888 120601

816=

1

2

1198841(119879 + Σ)

12060199= minus (1 minus 120590)119876

5+ 1205881Σ2Σ2minus 1198791198843Σ 120601

917=

1

2

1198843(119879 + Σ) 120601

1010= 1198765minus 1198767+ 1198768

1206011212

= minus (1 minus 1198892) 1198769+ 1205882Σ3Σ3minus 1198791198844Σ 120601

1216= minus1198811119862 minus 119881

111986621198652119867119888

1206011220

=

1

2

1198844(119879 + Σ) 120601

1313= 1198769minus 11987611+ 11987612 120601

1515= 11987614minus 1198846+ 1198762

1206011517

= 1198812119882+119881

211986611198651119867119908 120601

1616= 1198766minus 1198841+ 11987610 120601

1717= minus (1 minus 120590)119876

6minus 1198843

1206011818

= minus (1 minus 1198891) 1198762minus 1198842 120601

1919= minus (1 minus 120591)119876

15minus 1198845 120601

2020= minus (1 minus 119889

2) 11987610minus 1198844

119883 = ℎ11198851+ ℎ121198852+ ℎ71198857+ ℎ781198858 119884 = ℎ

21198853+ ℎ341198854+ ℎ51198855+ ℎ561198856

11987911= [0 minus119860 minus 119866

111986511198671198860 0 0 0 0 0 0 0]

11987912= [0 0 0 0 0 0 119882 + 119866

111986511198671199080 0 0]

11987921= [0 0 0 119871 + 119866

211986521198671198970 0 0 0 0 0]

11987922= [0 minus119862 minus 119866

211986721198671198880 0 0 0 0 0 0 0]

1198791= [1198791111987912]

119879

1198792= [1198792111987922]

119879

1198731= [119873119879

11119873119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198732= [119873119879

21119873119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198733= [119873119879

31119873119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198781= [119878119879

11119878119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198782= [119878119879

21119878119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198783= [119878119879

31119878119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198721= [119872

119879

11119872119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198722= [119872

119879

21119872119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198723= [119872

119879

31119872119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198641= [119864119879

11119864119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198642= [119864119879

21119864119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198643= [119864119879

31119864119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

119879 = diag (1205741 1205742 120574

119899) Σ = diag (120572

1 1205722 120572

119899)

(26)

Abstract and Applied Analysis 7

Proof We consider the following Lyapunov functional can-didate for system (8)

119881 (119905 119909119905 119910119905) = 1198811(119905 119909119905 119910119905) + 1198812(119905 119909119905 119910119905) + 1198813(119905 119909119905 119910119905)

(27)

where

1198811(119905 119909119905 119910119905) = 2

119899

sum

119894=1

1198811119894int

119910119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

1198812119894int

119909119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

(28)

1198812(119905 119909119905 119910119905) = 119909119879

(119905) 1198751119909 (119905) + 119910

119879

(119905) 1198752119910 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119909119879

(119904) 1198761119909 (119904) 119889119904

+ int

119905

119905minus1198891(119905)

119891119879

(119909 (119904)) 1198762119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ1

119909119879

(119904) 1198763119909 (119904) 119889119904

+ int

119905minusℎ1

119905minusℎ2

119909119879

(119904) 1198764119909 (119904) 119889119904

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198765119910 (119904) 119889119904

+ int

119905

119905minus120590(119905)

119891119879

(119910 (119904)) 1198766119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ3

119910119879

(119904) 1198767119910 (119904) 119889119904

+ int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198768119910 (119904) 119889119904

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198769119910 (119904) 119889119904

+ int

119905

119905minus1198892(119905)

119891119879

(119910 (119904)) 11987610119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ5

119910119879

(119904) 11987611119910 (119904) 119889119904

+ int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 11987612119910 (119904) 119889119904

+ int

119905minusℎ7

119905minus120591(119905)

119909119879

(119904) 11987613119909 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119891119879

(119909 (119904)) 11987614119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ7

119909119879

(119904) 11987615119909 (119904) 119889119904

+ int

119905minusℎ7

119905minusℎ8

119909119879

(119904) 11987616119909 (119904) 119889119904

(29)

1198813(119905 119909119905 119910119905) = int

0

minusℎ1

int

119905

119905+120579

119879

(119904) 1198851 (119904) 119889119904 119889120579

+ int

minusℎ1

minusℎ2

int

119905

119905+120579

119879

(119904) 1198852 (119904) 119889119904 119889120579

+ int

0

minusℎ3

int

119905

119905+120579

119910119879

(119904) 1198853119910 (119904) 119889119904 119889120579

+ int

minusℎ3

minusℎ4

int

119905

119905+120579

119910119879

(119904) 1198854119910 (119904) 119889119904 119889120579

+ int

0

minusℎ5

int

119905

119905+120579

119910119879

(119904) 1198855119910 (119904) 119889119904 119889120579

+ int

minusℎ5

minusℎ6

int

119905

119905+120579

119910119879

(119904) 1198856119910 (119904) 119889119904 119889120579

+ int

0

minusℎ7

int

119905

119905+120579

119879

(119904) 1198857 (119904) 119889119904 119889120579

+ int

minusℎ7

minusℎ8

int

119905

119905+120579

119879

(119904) 1198858 (119904) 119889119904 119889120579

(30)

Then by Itorsquos differential formula taking the derivative of119881(119905) along the trajectories of the system (8) we can obtainthe following stochastic differential [29]

119889119881 (119905) = F119881 (119905) 119889119905

+ 2119909119879

(119905) 1198751120578 (119905 119909 (119905 minus 119889

1(119905)) 119910 (119905 minus 120590 (119905)))

+ 2119910119879

1198752120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905)))

(31)

whereF is the diffusion operator and

F119881 (119905 119909119905 119910119905) = F119881

1(119905 119909119905 119910119905)

+F1198812(119905 119909119905 119910119905) +F119881

3(119905 119909119905 119910119905)

(32)

with

F1198811(119905 119909119905 119910119905) = 2 [119891

119879

(119910 (119905)) minus 119910119879

(119905) Γ]

times 1198811[minus119862119910 (119905 minus 119889

2(119905)) + 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

8 Abstract and Applied Analysis

+ tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198811120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 2 [119891119879

(119909 (119905)) minus 119909119879

(119905) Γ]

times 1198812[minus119860119909 (119905 minus 119889

1(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198812120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

F1198812(119905 119909119905 119910119905)

le 2119909119879

(119905) 1198751[minus119860119909 (119905 minus 119889

1(119905))

+ 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198751120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

+ 2119910119879

(119905) 1198752[minus119862119910 (119905 minus 119889

2(119905))

+ 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

+ tr [120578119879 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198752120578 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 119909119879

(119905 minus ℎ1) 1198761119909 (119905 minus ℎ

1)

minus (1 minus 1198891) 119909119879

(119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905))

+ 119891119879

(119909 (119904)) 1198762119891 (119909 (119904))

minus (1 minus 1198891) 119891119879

(119910 (119905 minus 1198891(119905)))

times 1198762119891 (119910 (119905 minus 119889

1(119905)))

+ 119909119879

(119905) 1198763119909 (119905) minus 119909

119879

(119905 minus ℎ1) 1198763119909 (119905 minus ℎ

1)

+ 119909119879

(119905 minus ℎ1) 1198764119909 (119905 minus ℎ

1)

minus 119909119879

(119905 minus ℎ2) 1198764119909 (119905 minus ℎ

2)

+ 119910119879

(119905 minus ℎ5) 1198765119909 (119905 minus ℎ

5)

minus (1 minus 120590) 119910119879

(119905 minus 120590 (119905)) 1198765119910 (119905 minus 120590 (119905))

+ 119891119879

(119910 (119905)) 1198766119891 (119910 (119905))

minus (1 minus 120590) 119891 (119910119879

(119905 minus 120590 (119905)))

times 1198765119891 (119910 (119905 minus 120590 (119905)))

minus 119910119879

(119905 minus ℎ3) 1198767119910 (119905 minus ℎ

3)

+ 119910119879

(119905) 1198767119910 (119905) + 119910

119879

(119905 minus ℎ3) 1198768119910 (119905 minus ℎ

3)

minus 119910119879

(119905 minus ℎ4) 1198768119910 (119905 minus ℎ

4)

+ 119910119879

(119905 minus ℎ5) 1198769119910 (119905 minus ℎ

5)

minus (1 minus 1198892) 119910119879

(119905 minus 1198892(119905)) 119876

9119910 (119905 minus 119889

2(119905))

+ 119891119879

(119910 (119905)) 11987610119891 (119910 (119905))

minus (1 minus 1198892) 119891119879

(119910 (119905 minus 1198892(119905)))

times 11987610119891 (119910 (119905 minus 119889

2(119905)))

+ 119910119879

11987611119910 (119905) minus 119910

119879

(119905 minus ℎ5) 11987611119910 (119905 minus ℎ

5)

+ 119910119879

(119905 minus ℎ5) 11987612119910 (119905 minus ℎ

5)

minus 119910119879

(119905 minus ℎ6) 11987612119910 (119905 minus ℎ

6)

+ 119909119879

(119905 minus ℎ7) 11987613119909 (119905 minus ℎ

7)

minus (1 minus 120591) 119909119879

(119905 minus 120591 (119905)) 11987613119909 (119905 minus 120591 (119905))

+ 119891119879

(119909 (119905)) 11987614119891 (119909 (119905))

minus (1 minus 120591) 119891119879

(119909 (119905 minus 120591 (119905)))

times 11987614119891 (119909 (119905 minus 120591 (119905))) + 119909

119879

(119905) 11987615119909 (119905)

minus 119909119879

(119905 minus ℎ7) 11987615119909 (119905 minus ℎ

7)

+ 119909119879

(119905 minus ℎ7) 11987616119909 (119905 minus ℎ

7)

minus 119909119879

(119905 minus ℎ8) 11987616119909 (119905 minus ℎ

8)

F1198813(119905 119909119905 119910119905)

= ℎ1119879

(119905) 1198851 (119905)

minus int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

+ ℎ12119879

(119905) 1198852 (119905)

minus int

119905minusℎ1

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

+ ℎ2119910119879

(119905) 1198853119910 (119905)

minus int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

Abstract and Applied Analysis 9

+ ℎ34119910119879

(119905) 1198854119910 (119905)

minus int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

+ ℎ5119910119879

(119905) 1198855119910 (119905)

minus int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

+ ℎ56119910119879

(119905) 1198856119910 (119905)

minus int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

+ ℎ7119879

(119905) 1198857119910 (119905)

minus int

119905

119905minusℎ7

119910119879

(119904) 1198857119910 (119904) 119889119904

+ ℎ78119910119879

(119904) 1198858119910 (119904)

minus int

119905minusℎ7

119905minusℎ8

119910119879

(119905) 1198858119910 (119905) 119889119904

(33)

By Newton-Leibnitz formula we have that

2120576119879

(119905)1198731[119909 (119905) minus 119909 (119905 minus ℎ

1) minus int

119905

119905minusℎ1

(119904) 119889119904] = 0

2120576119879

(119905)1198732[119909 (119905 minus ℎ

1) minus 119909 (119905 minus 119889

1(119905)) minus int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904] = 0

2120576119879

(119905)1198733[119909 (119905 minus 119889

1(119905)) minus 119909 (119905 minus ℎ

2) minus int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904] = 0

2120576119879

(119905)1198721[119910 (119905) minus 119910 (119905 minus 120590 (119905)) minus int

119905

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198722[119910 (119905 minus ℎ

3) minus 119910 (119905 minus 120590 (119905)) minus int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198723[119910 (119905 minus 120590 (119905)) minus 119910 (119905 minus ℎ

4) minus int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198781[119910 (119905) minus 119910 (119905 minus 119889

2(119905)) minus int

119905

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198782[119910 (119905 minus ℎ

5) minus 119910 (119905 minus 119889

2(119905)) minus int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198783[119910 (119905 minus 119889

2(119905)) minus 119910 (119905 minus ℎ

6) minus int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198641[119909 (119905) minus 119909 (119905 minus 120591 (119905)) minus int

119905

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198642[119909 (119905 minus ℎ

7) minus 119909 (119905 minus 120591 (119905)) minus int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198643[119909 (119905 minus ℎ

8) minus 119909 (119905 minus ℎ

7) minus int

119905minusℎ7

119905minusℎ8

(119904) 119889119904] = 0

(34)

where

120576 (119905) = [1205761(119905) 1205762(119905) 1205763(119905) 1205764(119905)]119879

1205761(119905) = [119909

119879

(119905) 119909119879

(119905 minus 1198891(119905)) 119909

119879

(119905 minus ℎ1)

119909119879

(119905 minus ℎ2) 119909119879

(119905 minus 120591 (119905))]

1205762(119905) = [119909

119879

(119905 minus ℎ7) 119909119879

(119905 minus ℎ8) 119910119879

(119905)

119910119879

(119905 minus 120590 (119905)) 119910119879

(119905 minus ℎ3)]

1205763(119905) = [119910

119879

(119905 minus ℎ4) 119910119879

(119905 minus 1198892(119905)) 119910

119879

(119905 minus ℎ5)

119910119879

(119905 minus ℎ6) 119891119879

(119909)]

1205764(119905) = [119891

119879

(119910) 119891119879

(119910 minus 120590 (119905)) 119891119879

(119909 minus 1198891(119905))

119891119879

(119909 minus 120591 (119905)) 119891119879

(119910 minus 1198892(119905))]

(35)

By using Lemmas 4 and 5 we have

minus 2120576119879

(119905)1198731int

119905

119905minusℎ1

(119904) 119889119904

le ℎ1120576119879

(119905)1198731119885minus1

1119873119879

1120576 (119905)

+ int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

minus 2120576119879

(119905)1198732int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904

le (1198891(119905) minus ℎ

1) 120576119879

(119905)1198732119885minus1

2119873119879

2120576 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198733int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904

le (ℎ2minus 1198891(119905)) 120576119879

(119905)1198733119885minus1

2119873119879

3120576 (119905)

+ int

119905minus1198891(119905)

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198721int

119905

119905minusℎ3

119910 (119904) 119889119904

10 Abstract and Applied Analysis

le ℎ3120576119879

(119905)1198721119885minus1

3119872119879

1120576 (119905)

+ int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

minus 2120576119879

(119905)1198722int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904

le (120590 (119905) minus ℎ3) 120576119879

(119905)1198722119885minus1

4119872119879

2120576 (119905)

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905)1198723int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904

le (ℎ4minus 120590 (119905)) 120576

119879

(119905)1198723119885minus1

4119872119879

3120576 (119905)

+ int

119905minus120590(119905)

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905) 1198781int

119905

119905minusℎ5

119910 (119904) 119889119904

le ℎ5120576119879

(119905) 1198781119885minus1

5119878119879

1120576 (119905)

+ int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

minus 2120576119879

(119905) 1198782int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904

le (1198892(119905) minus ℎ

5) 120576119879

(119905) 1198782119885minus1

6119878119879

2120576 (119905)

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198783int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904

le (ℎ6minus 1198892(119905)) 120576119879

(119905) 1198783119885minus1

6119878119879

3120576 (119905)

+ int

119905minus1198892(119905)

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198641int

119905

119905minusℎ7

(119904) 119889119904

le ℎ7120576119879

(119905) 1198641119885minus1

7119864119879

1120576 (119905)

+ int

119905

119905minusℎ7

119879

(119904) 1198857 (119904) 119889119904

minus 2120576119879

(119905) 1198642int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904

le (120591 (119905) minus ℎ7) 120576119879

(119905) 1198642119885minus1

8119864119879

2120576 (119905)

+ int

119905minusℎ7

119905minus120591(119905)

119879

(119904) 1198858 (119904) 119889119904

minus 2120576119879

(119905) 1198643int

119905minus120591(119905)

119905minusℎ8

(119904) 119889119904

le (ℎ8minus 120591 (119905)) 120576

119879

(119905) 1198643119885minus1

8119864119879

3120576 (119905)

+ int

119905minus120591(119905)

119905minusℎ8

119879

(119904) 1198858 (119904) 119889119904

(36)

It follows from (7) that

0 = 119891119879

(119910 (119905)) 1198841119891 (119910 (119905)) minus 119891

119879

(119910 (119905)) 1198841119891 (119910 (119905))

le minus119910119879

(119905) 1198791198841Σ119910 (119905) + 119910

119879

(119905) 1198841(119879 + Σ) 119891 (119910 (119905))

minus 119891119879

(119910 (119905)) 1198841119891 (119910 (119905))

0 = 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

le minus119910119879

(119905 minus 1198891(119905)) 119879119884

2Σ119910 (119905 minus 119889

1(119905))

+ 119910119879

(119905 minus 1198891(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

0 = 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

le minus119910119879

(119905 minus 120590 (119905)) 1198791198843Σ119910 (119905 minus 120590 (119905))

+ 119910119879

(119905 minus 120590 (119905)) 1198843(119879 + Σ) 119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

0 = 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

le minus119910119879

(119905 minus 1198892(119905)) 119879119884

2Σ119910 (119905 minus 119889

2(119905))

+ 119910119879

(119905 minus 1198892(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

2119891 (119910 (119905 minus 119889

2(119905)))

0 = 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

le minus119910119879

(119905 minus 120591 (119905)) 1198791198845Σ119910 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 120591 (119905)) 1198845(119879 + Σ) 119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

0 = 119891119879

(119909 (119905)) 1198846119891 (119909 (119905)) minus 119891

119879

(119909 (119905)) 1198846119891 (119909 (119905))

le minus119909119879

(119905) 1198791198846Σ119909 (119905) + 119909

119879

(119905) 1198846(119879 + Σ) 119891 (119909 (119905))

minus 119891119879

(119909 (119905)) 1198846119891 (119909 (119905))

(37)

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

where

Ξ =

1

4

[

[

[

[

[

[

[

[

[

[

120601 1198791119883 1198792119884 ℎ11198731ℎ31198721ℎ51198781ℎ71198641

lowast minus119883 0 0 0 0 0

lowast 0 minus119884 0 0 0 0

lowast 0 0 minusℎ11198851

0 0 0

lowast 0 0 0 minusℎ31198853

0 0

lowast 0 0 0 0 ℎ51198855

0

lowast 0 0 0 0 0 minusℎ71198857

]

]

]

]

]

]

]

]

]

]

120601 = [

120601912060110

lowast 12060111

]

1206019=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120601111206011211987321minus1198733112060115

11986421

11986431

11987811+1198721112060119

11987221

lowast 1206012211987322minus1198733212060125

11986422minus1198643211987212+ 1198781212060129

11987222

lowast lowast 12060133

0 0 0 0 0 0 0

lowast lowast lowast minus1198764

0 0 0 0 0 0

lowast lowast lowast lowast 12060155

0 0 12060158

0 0

lowast lowast lowast lowast lowast 12060166

0 0 0 0

lowast lowast lowast lowast lowast lowast minus11987616

0 0 0

lowast lowast lowast lowast lowast lowast lowast 12060188

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 12060199

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1206011010

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12060110=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus11987231120601112

11987812minus11987831120601115

0 120601117

0 0 0

minus11987232120601212

11987822minus11987832120601215

0 0 120601218

0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 120601516

0 0 120601519

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 120601812

0 0 0 120601816

0 0 0 0

0 0 0 0 0 0 120601917

0 0 0

0 0 0 0 0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12060111=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus1198768

0 0 0 0 0 0 0 0 0

lowast 1206011212

0 0 0 1206011216

0 0 0 1206011220

lowast lowast 1206011313

0 0 0 0 0 0 0

lowast lowast lowast minus11987612

0 0 0 0 0 0

lowast lowast lowast lowast 1206011515

0 1206011517

0 0 0

lowast lowast lowast lowast lowast 1206011616

0 0 0 0

lowast lowast lowast lowast lowast lowast 1206011717

0 0 0

lowast lowast lowast lowast lowast lowast lowast 1206011818

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1206011919

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1206012020

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119869119894119898(119909 (119905119898)) = (119869

1119898(1199091(119905119898)) 119869

119899119898(119909119899(119905119898)))119879

12060111= 1198763+ 11987615+ 211987311+ 211986411minus 1198791198846Σ

12060112= minus1198751119860 minus 119875

111986611198651119867119886minus 1198791198812119860 minus 119879119881

211986611198651119867119886minus 11987311+ 11987312+ 11987313minus 11987321+ 11986412

12060115= minus11986411+ 11986431minus 11986421 120601

19= minus119872

11+11987231minus11987221 120601

112= minus11987811+ 11987831minus 11987821

120601115

=

1

2

1198846(119879 + Σ) 120601

117= minus119879119881

2119882minus 119879119881

211986611198651119867119908+ 1198751119882+ 119875

111986611198651119867119908

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232+ 1205881Σ119879

1Σ1minus 1198791198842Σ 120601

25= minus11986412+ 11986432minus 11986422

12060129= minus119872

12+11987232minus11987222 120601

212= minus11987812+ 11987832minus 11987822 120601

215= minus1198812119860 minus 119881

211986611198651119867119886

120601218

=

1

2

1198842(119879 + Σ) 120601

33= 1198761+ 1198764minus 1198763 120601

55= minus (1 minus 120591)119876

13+ 1205882Σ4Σ4minus 1198791198845Σ

6 Abstract and Applied Analysis

12060158= minus119879119881

1119871 minus 119879119881

111986621198652119867119897+ 1198752119871 + 119875211986621198652119867119897 120601

516= 1198811119871 + 119881111986621198652119867119897

120601519

=

1

2

1198845(119879 + Σ) 120601

66= 11987613+ 11987616minus 11987615 120601

88= 1198767+ 11987611minus 1198791198841Σ

120601812

= minus1198752119862 minus 119875211986621198652119867119888+ 1198791198811119862 + 119879119881

111986621198652119867119888 120601

816=

1

2

1198841(119879 + Σ)

12060199= minus (1 minus 120590)119876

5+ 1205881Σ2Σ2minus 1198791198843Σ 120601

917=

1

2

1198843(119879 + Σ) 120601

1010= 1198765minus 1198767+ 1198768

1206011212

= minus (1 minus 1198892) 1198769+ 1205882Σ3Σ3minus 1198791198844Σ 120601

1216= minus1198811119862 minus 119881

111986621198652119867119888

1206011220

=

1

2

1198844(119879 + Σ) 120601

1313= 1198769minus 11987611+ 11987612 120601

1515= 11987614minus 1198846+ 1198762

1206011517

= 1198812119882+119881

211986611198651119867119908 120601

1616= 1198766minus 1198841+ 11987610 120601

1717= minus (1 minus 120590)119876

6minus 1198843

1206011818

= minus (1 minus 1198891) 1198762minus 1198842 120601

1919= minus (1 minus 120591)119876

15minus 1198845 120601

2020= minus (1 minus 119889

2) 11987610minus 1198844

119883 = ℎ11198851+ ℎ121198852+ ℎ71198857+ ℎ781198858 119884 = ℎ

21198853+ ℎ341198854+ ℎ51198855+ ℎ561198856

11987911= [0 minus119860 minus 119866

111986511198671198860 0 0 0 0 0 0 0]

11987912= [0 0 0 0 0 0 119882 + 119866

111986511198671199080 0 0]

11987921= [0 0 0 119871 + 119866

211986521198671198970 0 0 0 0 0]

11987922= [0 minus119862 minus 119866

211986721198671198880 0 0 0 0 0 0 0]

1198791= [1198791111987912]

119879

1198792= [1198792111987922]

119879

1198731= [119873119879

11119873119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198732= [119873119879

21119873119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198733= [119873119879

31119873119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198781= [119878119879

11119878119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198782= [119878119879

21119878119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198783= [119878119879

31119878119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198721= [119872

119879

11119872119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198722= [119872

119879

21119872119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198723= [119872

119879

31119872119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198641= [119864119879

11119864119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198642= [119864119879

21119864119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198643= [119864119879

31119864119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

119879 = diag (1205741 1205742 120574

119899) Σ = diag (120572

1 1205722 120572

119899)

(26)

Abstract and Applied Analysis 7

Proof We consider the following Lyapunov functional can-didate for system (8)

119881 (119905 119909119905 119910119905) = 1198811(119905 119909119905 119910119905) + 1198812(119905 119909119905 119910119905) + 1198813(119905 119909119905 119910119905)

(27)

where

1198811(119905 119909119905 119910119905) = 2

119899

sum

119894=1

1198811119894int

119910119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

1198812119894int

119909119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

(28)

1198812(119905 119909119905 119910119905) = 119909119879

(119905) 1198751119909 (119905) + 119910

119879

(119905) 1198752119910 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119909119879

(119904) 1198761119909 (119904) 119889119904

+ int

119905

119905minus1198891(119905)

119891119879

(119909 (119904)) 1198762119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ1

119909119879

(119904) 1198763119909 (119904) 119889119904

+ int

119905minusℎ1

119905minusℎ2

119909119879

(119904) 1198764119909 (119904) 119889119904

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198765119910 (119904) 119889119904

+ int

119905

119905minus120590(119905)

119891119879

(119910 (119904)) 1198766119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ3

119910119879

(119904) 1198767119910 (119904) 119889119904

+ int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198768119910 (119904) 119889119904

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198769119910 (119904) 119889119904

+ int

119905

119905minus1198892(119905)

119891119879

(119910 (119904)) 11987610119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ5

119910119879

(119904) 11987611119910 (119904) 119889119904

+ int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 11987612119910 (119904) 119889119904

+ int

119905minusℎ7

119905minus120591(119905)

119909119879

(119904) 11987613119909 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119891119879

(119909 (119904)) 11987614119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ7

119909119879

(119904) 11987615119909 (119904) 119889119904

+ int

119905minusℎ7

119905minusℎ8

119909119879

(119904) 11987616119909 (119904) 119889119904

(29)

1198813(119905 119909119905 119910119905) = int

0

minusℎ1

int

119905

119905+120579

119879

(119904) 1198851 (119904) 119889119904 119889120579

+ int

minusℎ1

minusℎ2

int

119905

119905+120579

119879

(119904) 1198852 (119904) 119889119904 119889120579

+ int

0

minusℎ3

int

119905

119905+120579

119910119879

(119904) 1198853119910 (119904) 119889119904 119889120579

+ int

minusℎ3

minusℎ4

int

119905

119905+120579

119910119879

(119904) 1198854119910 (119904) 119889119904 119889120579

+ int

0

minusℎ5

int

119905

119905+120579

119910119879

(119904) 1198855119910 (119904) 119889119904 119889120579

+ int

minusℎ5

minusℎ6

int

119905

119905+120579

119910119879

(119904) 1198856119910 (119904) 119889119904 119889120579

+ int

0

minusℎ7

int

119905

119905+120579

119879

(119904) 1198857 (119904) 119889119904 119889120579

+ int

minusℎ7

minusℎ8

int

119905

119905+120579

119879

(119904) 1198858 (119904) 119889119904 119889120579

(30)

Then by Itorsquos differential formula taking the derivative of119881(119905) along the trajectories of the system (8) we can obtainthe following stochastic differential [29]

119889119881 (119905) = F119881 (119905) 119889119905

+ 2119909119879

(119905) 1198751120578 (119905 119909 (119905 minus 119889

1(119905)) 119910 (119905 minus 120590 (119905)))

+ 2119910119879

1198752120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905)))

(31)

whereF is the diffusion operator and

F119881 (119905 119909119905 119910119905) = F119881

1(119905 119909119905 119910119905)

+F1198812(119905 119909119905 119910119905) +F119881

3(119905 119909119905 119910119905)

(32)

with

F1198811(119905 119909119905 119910119905) = 2 [119891

119879

(119910 (119905)) minus 119910119879

(119905) Γ]

times 1198811[minus119862119910 (119905 minus 119889

2(119905)) + 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

8 Abstract and Applied Analysis

+ tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198811120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 2 [119891119879

(119909 (119905)) minus 119909119879

(119905) Γ]

times 1198812[minus119860119909 (119905 minus 119889

1(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198812120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

F1198812(119905 119909119905 119910119905)

le 2119909119879

(119905) 1198751[minus119860119909 (119905 minus 119889

1(119905))

+ 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198751120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

+ 2119910119879

(119905) 1198752[minus119862119910 (119905 minus 119889

2(119905))

+ 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

+ tr [120578119879 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198752120578 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 119909119879

(119905 minus ℎ1) 1198761119909 (119905 minus ℎ

1)

minus (1 minus 1198891) 119909119879

(119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905))

+ 119891119879

(119909 (119904)) 1198762119891 (119909 (119904))

minus (1 minus 1198891) 119891119879

(119910 (119905 minus 1198891(119905)))

times 1198762119891 (119910 (119905 minus 119889

1(119905)))

+ 119909119879

(119905) 1198763119909 (119905) minus 119909

119879

(119905 minus ℎ1) 1198763119909 (119905 minus ℎ

1)

+ 119909119879

(119905 minus ℎ1) 1198764119909 (119905 minus ℎ

1)

minus 119909119879

(119905 minus ℎ2) 1198764119909 (119905 minus ℎ

2)

+ 119910119879

(119905 minus ℎ5) 1198765119909 (119905 minus ℎ

5)

minus (1 minus 120590) 119910119879

(119905 minus 120590 (119905)) 1198765119910 (119905 minus 120590 (119905))

+ 119891119879

(119910 (119905)) 1198766119891 (119910 (119905))

minus (1 minus 120590) 119891 (119910119879

(119905 minus 120590 (119905)))

times 1198765119891 (119910 (119905 minus 120590 (119905)))

minus 119910119879

(119905 minus ℎ3) 1198767119910 (119905 minus ℎ

3)

+ 119910119879

(119905) 1198767119910 (119905) + 119910

119879

(119905 minus ℎ3) 1198768119910 (119905 minus ℎ

3)

minus 119910119879

(119905 minus ℎ4) 1198768119910 (119905 minus ℎ

4)

+ 119910119879

(119905 minus ℎ5) 1198769119910 (119905 minus ℎ

5)

minus (1 minus 1198892) 119910119879

(119905 minus 1198892(119905)) 119876

9119910 (119905 minus 119889

2(119905))

+ 119891119879

(119910 (119905)) 11987610119891 (119910 (119905))

minus (1 minus 1198892) 119891119879

(119910 (119905 minus 1198892(119905)))

times 11987610119891 (119910 (119905 minus 119889

2(119905)))

+ 119910119879

11987611119910 (119905) minus 119910

119879

(119905 minus ℎ5) 11987611119910 (119905 minus ℎ

5)

+ 119910119879

(119905 minus ℎ5) 11987612119910 (119905 minus ℎ

5)

minus 119910119879

(119905 minus ℎ6) 11987612119910 (119905 minus ℎ

6)

+ 119909119879

(119905 minus ℎ7) 11987613119909 (119905 minus ℎ

7)

minus (1 minus 120591) 119909119879

(119905 minus 120591 (119905)) 11987613119909 (119905 minus 120591 (119905))

+ 119891119879

(119909 (119905)) 11987614119891 (119909 (119905))

minus (1 minus 120591) 119891119879

(119909 (119905 minus 120591 (119905)))

times 11987614119891 (119909 (119905 minus 120591 (119905))) + 119909

119879

(119905) 11987615119909 (119905)

minus 119909119879

(119905 minus ℎ7) 11987615119909 (119905 minus ℎ

7)

+ 119909119879

(119905 minus ℎ7) 11987616119909 (119905 minus ℎ

7)

minus 119909119879

(119905 minus ℎ8) 11987616119909 (119905 minus ℎ

8)

F1198813(119905 119909119905 119910119905)

= ℎ1119879

(119905) 1198851 (119905)

minus int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

+ ℎ12119879

(119905) 1198852 (119905)

minus int

119905minusℎ1

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

+ ℎ2119910119879

(119905) 1198853119910 (119905)

minus int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

Abstract and Applied Analysis 9

+ ℎ34119910119879

(119905) 1198854119910 (119905)

minus int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

+ ℎ5119910119879

(119905) 1198855119910 (119905)

minus int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

+ ℎ56119910119879

(119905) 1198856119910 (119905)

minus int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

+ ℎ7119879

(119905) 1198857119910 (119905)

minus int

119905

119905minusℎ7

119910119879

(119904) 1198857119910 (119904) 119889119904

+ ℎ78119910119879

(119904) 1198858119910 (119904)

minus int

119905minusℎ7

119905minusℎ8

119910119879

(119905) 1198858119910 (119905) 119889119904

(33)

By Newton-Leibnitz formula we have that

2120576119879

(119905)1198731[119909 (119905) minus 119909 (119905 minus ℎ

1) minus int

119905

119905minusℎ1

(119904) 119889119904] = 0

2120576119879

(119905)1198732[119909 (119905 minus ℎ

1) minus 119909 (119905 minus 119889

1(119905)) minus int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904] = 0

2120576119879

(119905)1198733[119909 (119905 minus 119889

1(119905)) minus 119909 (119905 minus ℎ

2) minus int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904] = 0

2120576119879

(119905)1198721[119910 (119905) minus 119910 (119905 minus 120590 (119905)) minus int

119905

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198722[119910 (119905 minus ℎ

3) minus 119910 (119905 minus 120590 (119905)) minus int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198723[119910 (119905 minus 120590 (119905)) minus 119910 (119905 minus ℎ

4) minus int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198781[119910 (119905) minus 119910 (119905 minus 119889

2(119905)) minus int

119905

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198782[119910 (119905 minus ℎ

5) minus 119910 (119905 minus 119889

2(119905)) minus int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198783[119910 (119905 minus 119889

2(119905)) minus 119910 (119905 minus ℎ

6) minus int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198641[119909 (119905) minus 119909 (119905 minus 120591 (119905)) minus int

119905

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198642[119909 (119905 minus ℎ

7) minus 119909 (119905 minus 120591 (119905)) minus int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198643[119909 (119905 minus ℎ

8) minus 119909 (119905 minus ℎ

7) minus int

119905minusℎ7

119905minusℎ8

(119904) 119889119904] = 0

(34)

where

120576 (119905) = [1205761(119905) 1205762(119905) 1205763(119905) 1205764(119905)]119879

1205761(119905) = [119909

119879

(119905) 119909119879

(119905 minus 1198891(119905)) 119909

119879

(119905 minus ℎ1)

119909119879

(119905 minus ℎ2) 119909119879

(119905 minus 120591 (119905))]

1205762(119905) = [119909

119879

(119905 minus ℎ7) 119909119879

(119905 minus ℎ8) 119910119879

(119905)

119910119879

(119905 minus 120590 (119905)) 119910119879

(119905 minus ℎ3)]

1205763(119905) = [119910

119879

(119905 minus ℎ4) 119910119879

(119905 minus 1198892(119905)) 119910

119879

(119905 minus ℎ5)

119910119879

(119905 minus ℎ6) 119891119879

(119909)]

1205764(119905) = [119891

119879

(119910) 119891119879

(119910 minus 120590 (119905)) 119891119879

(119909 minus 1198891(119905))

119891119879

(119909 minus 120591 (119905)) 119891119879

(119910 minus 1198892(119905))]

(35)

By using Lemmas 4 and 5 we have

minus 2120576119879

(119905)1198731int

119905

119905minusℎ1

(119904) 119889119904

le ℎ1120576119879

(119905)1198731119885minus1

1119873119879

1120576 (119905)

+ int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

minus 2120576119879

(119905)1198732int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904

le (1198891(119905) minus ℎ

1) 120576119879

(119905)1198732119885minus1

2119873119879

2120576 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198733int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904

le (ℎ2minus 1198891(119905)) 120576119879

(119905)1198733119885minus1

2119873119879

3120576 (119905)

+ int

119905minus1198891(119905)

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198721int

119905

119905minusℎ3

119910 (119904) 119889119904

10 Abstract and Applied Analysis

le ℎ3120576119879

(119905)1198721119885minus1

3119872119879

1120576 (119905)

+ int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

minus 2120576119879

(119905)1198722int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904

le (120590 (119905) minus ℎ3) 120576119879

(119905)1198722119885minus1

4119872119879

2120576 (119905)

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905)1198723int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904

le (ℎ4minus 120590 (119905)) 120576

119879

(119905)1198723119885minus1

4119872119879

3120576 (119905)

+ int

119905minus120590(119905)

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905) 1198781int

119905

119905minusℎ5

119910 (119904) 119889119904

le ℎ5120576119879

(119905) 1198781119885minus1

5119878119879

1120576 (119905)

+ int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

minus 2120576119879

(119905) 1198782int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904

le (1198892(119905) minus ℎ

5) 120576119879

(119905) 1198782119885minus1

6119878119879

2120576 (119905)

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198783int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904

le (ℎ6minus 1198892(119905)) 120576119879

(119905) 1198783119885minus1

6119878119879

3120576 (119905)

+ int

119905minus1198892(119905)

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198641int

119905

119905minusℎ7

(119904) 119889119904

le ℎ7120576119879

(119905) 1198641119885minus1

7119864119879

1120576 (119905)

+ int

119905

119905minusℎ7

119879

(119904) 1198857 (119904) 119889119904

minus 2120576119879

(119905) 1198642int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904

le (120591 (119905) minus ℎ7) 120576119879

(119905) 1198642119885minus1

8119864119879

2120576 (119905)

+ int

119905minusℎ7

119905minus120591(119905)

119879

(119904) 1198858 (119904) 119889119904

minus 2120576119879

(119905) 1198643int

119905minus120591(119905)

119905minusℎ8

(119904) 119889119904

le (ℎ8minus 120591 (119905)) 120576

119879

(119905) 1198643119885minus1

8119864119879

3120576 (119905)

+ int

119905minus120591(119905)

119905minusℎ8

119879

(119904) 1198858 (119904) 119889119904

(36)

It follows from (7) that

0 = 119891119879

(119910 (119905)) 1198841119891 (119910 (119905)) minus 119891

119879

(119910 (119905)) 1198841119891 (119910 (119905))

le minus119910119879

(119905) 1198791198841Σ119910 (119905) + 119910

119879

(119905) 1198841(119879 + Σ) 119891 (119910 (119905))

minus 119891119879

(119910 (119905)) 1198841119891 (119910 (119905))

0 = 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

le minus119910119879

(119905 minus 1198891(119905)) 119879119884

2Σ119910 (119905 minus 119889

1(119905))

+ 119910119879

(119905 minus 1198891(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

0 = 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

le minus119910119879

(119905 minus 120590 (119905)) 1198791198843Σ119910 (119905 minus 120590 (119905))

+ 119910119879

(119905 minus 120590 (119905)) 1198843(119879 + Σ) 119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

0 = 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

le minus119910119879

(119905 minus 1198892(119905)) 119879119884

2Σ119910 (119905 minus 119889

2(119905))

+ 119910119879

(119905 minus 1198892(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

2119891 (119910 (119905 minus 119889

2(119905)))

0 = 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

le minus119910119879

(119905 minus 120591 (119905)) 1198791198845Σ119910 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 120591 (119905)) 1198845(119879 + Σ) 119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

0 = 119891119879

(119909 (119905)) 1198846119891 (119909 (119905)) minus 119891

119879

(119909 (119905)) 1198846119891 (119909 (119905))

le minus119909119879

(119905) 1198791198846Σ119909 (119905) + 119909

119879

(119905) 1198846(119879 + Σ) 119891 (119909 (119905))

minus 119891119879

(119909 (119905)) 1198846119891 (119909 (119905))

(37)

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

12060158= minus119879119881

1119871 minus 119879119881

111986621198652119867119897+ 1198752119871 + 119875211986621198652119867119897 120601

516= 1198811119871 + 119881111986621198652119867119897

120601519

=

1

2

1198845(119879 + Σ) 120601

66= 11987613+ 11987616minus 11987615 120601

88= 1198767+ 11987611minus 1198791198841Σ

120601812

= minus1198752119862 minus 119875211986621198652119867119888+ 1198791198811119862 + 119879119881

111986621198652119867119888 120601

816=

1

2

1198841(119879 + Σ)

12060199= minus (1 minus 120590)119876

5+ 1205881Σ2Σ2minus 1198791198843Σ 120601

917=

1

2

1198843(119879 + Σ) 120601

1010= 1198765minus 1198767+ 1198768

1206011212

= minus (1 minus 1198892) 1198769+ 1205882Σ3Σ3minus 1198791198844Σ 120601

1216= minus1198811119862 minus 119881

111986621198652119867119888

1206011220

=

1

2

1198844(119879 + Σ) 120601

1313= 1198769minus 11987611+ 11987612 120601

1515= 11987614minus 1198846+ 1198762

1206011517

= 1198812119882+119881

211986611198651119867119908 120601

1616= 1198766minus 1198841+ 11987610 120601

1717= minus (1 minus 120590)119876

6minus 1198843

1206011818

= minus (1 minus 1198891) 1198762minus 1198842 120601

1919= minus (1 minus 120591)119876

15minus 1198845 120601

2020= minus (1 minus 119889

2) 11987610minus 1198844

119883 = ℎ11198851+ ℎ121198852+ ℎ71198857+ ℎ781198858 119884 = ℎ

21198853+ ℎ341198854+ ℎ51198855+ ℎ561198856

11987911= [0 minus119860 minus 119866

111986511198671198860 0 0 0 0 0 0 0]

11987912= [0 0 0 0 0 0 119882 + 119866

111986511198671199080 0 0]

11987921= [0 0 0 119871 + 119866

211986521198671198970 0 0 0 0 0]

11987922= [0 minus119862 minus 119866

211986721198671198880 0 0 0 0 0 0 0]

1198791= [1198791111987912]

119879

1198792= [1198792111987922]

119879

1198731= [119873119879

11119873119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198732= [119873119879

21119873119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198733= [119873119879

31119873119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198781= [119878119879

11119878119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198782= [119878119879

21119878119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198783= [119878119879

31119878119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198721= [119872

119879

11119872119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198722= [119872

119879

21119872119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198723= [119872

119879

31119872119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198641= [119864119879

11119864119879

120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198642= [119864119879

21119864119879

220 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

1198643= [119864119879

31119864119879

320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

119879

119879 = diag (1205741 1205742 120574

119899) Σ = diag (120572

1 1205722 120572

119899)

(26)

Abstract and Applied Analysis 7

Proof We consider the following Lyapunov functional can-didate for system (8)

119881 (119905 119909119905 119910119905) = 1198811(119905 119909119905 119910119905) + 1198812(119905 119909119905 119910119905) + 1198813(119905 119909119905 119910119905)

(27)

where

1198811(119905 119909119905 119910119905) = 2

119899

sum

119894=1

1198811119894int

119910119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

1198812119894int

119909119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

(28)

1198812(119905 119909119905 119910119905) = 119909119879

(119905) 1198751119909 (119905) + 119910

119879

(119905) 1198752119910 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119909119879

(119904) 1198761119909 (119904) 119889119904

+ int

119905

119905minus1198891(119905)

119891119879

(119909 (119904)) 1198762119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ1

119909119879

(119904) 1198763119909 (119904) 119889119904

+ int

119905minusℎ1

119905minusℎ2

119909119879

(119904) 1198764119909 (119904) 119889119904

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198765119910 (119904) 119889119904

+ int

119905

119905minus120590(119905)

119891119879

(119910 (119904)) 1198766119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ3

119910119879

(119904) 1198767119910 (119904) 119889119904

+ int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198768119910 (119904) 119889119904

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198769119910 (119904) 119889119904

+ int

119905

119905minus1198892(119905)

119891119879

(119910 (119904)) 11987610119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ5

119910119879

(119904) 11987611119910 (119904) 119889119904

+ int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 11987612119910 (119904) 119889119904

+ int

119905minusℎ7

119905minus120591(119905)

119909119879

(119904) 11987613119909 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119891119879

(119909 (119904)) 11987614119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ7

119909119879

(119904) 11987615119909 (119904) 119889119904

+ int

119905minusℎ7

119905minusℎ8

119909119879

(119904) 11987616119909 (119904) 119889119904

(29)

1198813(119905 119909119905 119910119905) = int

0

minusℎ1

int

119905

119905+120579

119879

(119904) 1198851 (119904) 119889119904 119889120579

+ int

minusℎ1

minusℎ2

int

119905

119905+120579

119879

(119904) 1198852 (119904) 119889119904 119889120579

+ int

0

minusℎ3

int

119905

119905+120579

119910119879

(119904) 1198853119910 (119904) 119889119904 119889120579

+ int

minusℎ3

minusℎ4

int

119905

119905+120579

119910119879

(119904) 1198854119910 (119904) 119889119904 119889120579

+ int

0

minusℎ5

int

119905

119905+120579

119910119879

(119904) 1198855119910 (119904) 119889119904 119889120579

+ int

minusℎ5

minusℎ6

int

119905

119905+120579

119910119879

(119904) 1198856119910 (119904) 119889119904 119889120579

+ int

0

minusℎ7

int

119905

119905+120579

119879

(119904) 1198857 (119904) 119889119904 119889120579

+ int

minusℎ7

minusℎ8

int

119905

119905+120579

119879

(119904) 1198858 (119904) 119889119904 119889120579

(30)

Then by Itorsquos differential formula taking the derivative of119881(119905) along the trajectories of the system (8) we can obtainthe following stochastic differential [29]

119889119881 (119905) = F119881 (119905) 119889119905

+ 2119909119879

(119905) 1198751120578 (119905 119909 (119905 minus 119889

1(119905)) 119910 (119905 minus 120590 (119905)))

+ 2119910119879

1198752120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905)))

(31)

whereF is the diffusion operator and

F119881 (119905 119909119905 119910119905) = F119881

1(119905 119909119905 119910119905)

+F1198812(119905 119909119905 119910119905) +F119881

3(119905 119909119905 119910119905)

(32)

with

F1198811(119905 119909119905 119910119905) = 2 [119891

119879

(119910 (119905)) minus 119910119879

(119905) Γ]

times 1198811[minus119862119910 (119905 minus 119889

2(119905)) + 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

8 Abstract and Applied Analysis

+ tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198811120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 2 [119891119879

(119909 (119905)) minus 119909119879

(119905) Γ]

times 1198812[minus119860119909 (119905 minus 119889

1(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198812120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

F1198812(119905 119909119905 119910119905)

le 2119909119879

(119905) 1198751[minus119860119909 (119905 minus 119889

1(119905))

+ 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198751120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

+ 2119910119879

(119905) 1198752[minus119862119910 (119905 minus 119889

2(119905))

+ 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

+ tr [120578119879 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198752120578 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 119909119879

(119905 minus ℎ1) 1198761119909 (119905 minus ℎ

1)

minus (1 minus 1198891) 119909119879

(119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905))

+ 119891119879

(119909 (119904)) 1198762119891 (119909 (119904))

minus (1 minus 1198891) 119891119879

(119910 (119905 minus 1198891(119905)))

times 1198762119891 (119910 (119905 minus 119889

1(119905)))

+ 119909119879

(119905) 1198763119909 (119905) minus 119909

119879

(119905 minus ℎ1) 1198763119909 (119905 minus ℎ

1)

+ 119909119879

(119905 minus ℎ1) 1198764119909 (119905 minus ℎ

1)

minus 119909119879

(119905 minus ℎ2) 1198764119909 (119905 minus ℎ

2)

+ 119910119879

(119905 minus ℎ5) 1198765119909 (119905 minus ℎ

5)

minus (1 minus 120590) 119910119879

(119905 minus 120590 (119905)) 1198765119910 (119905 minus 120590 (119905))

+ 119891119879

(119910 (119905)) 1198766119891 (119910 (119905))

minus (1 minus 120590) 119891 (119910119879

(119905 minus 120590 (119905)))

times 1198765119891 (119910 (119905 minus 120590 (119905)))

minus 119910119879

(119905 minus ℎ3) 1198767119910 (119905 minus ℎ

3)

+ 119910119879

(119905) 1198767119910 (119905) + 119910

119879

(119905 minus ℎ3) 1198768119910 (119905 minus ℎ

3)

minus 119910119879

(119905 minus ℎ4) 1198768119910 (119905 minus ℎ

4)

+ 119910119879

(119905 minus ℎ5) 1198769119910 (119905 minus ℎ

5)

minus (1 minus 1198892) 119910119879

(119905 minus 1198892(119905)) 119876

9119910 (119905 minus 119889

2(119905))

+ 119891119879

(119910 (119905)) 11987610119891 (119910 (119905))

minus (1 minus 1198892) 119891119879

(119910 (119905 minus 1198892(119905)))

times 11987610119891 (119910 (119905 minus 119889

2(119905)))

+ 119910119879

11987611119910 (119905) minus 119910

119879

(119905 minus ℎ5) 11987611119910 (119905 minus ℎ

5)

+ 119910119879

(119905 minus ℎ5) 11987612119910 (119905 minus ℎ

5)

minus 119910119879

(119905 minus ℎ6) 11987612119910 (119905 minus ℎ

6)

+ 119909119879

(119905 minus ℎ7) 11987613119909 (119905 minus ℎ

7)

minus (1 minus 120591) 119909119879

(119905 minus 120591 (119905)) 11987613119909 (119905 minus 120591 (119905))

+ 119891119879

(119909 (119905)) 11987614119891 (119909 (119905))

minus (1 minus 120591) 119891119879

(119909 (119905 minus 120591 (119905)))

times 11987614119891 (119909 (119905 minus 120591 (119905))) + 119909

119879

(119905) 11987615119909 (119905)

minus 119909119879

(119905 minus ℎ7) 11987615119909 (119905 minus ℎ

7)

+ 119909119879

(119905 minus ℎ7) 11987616119909 (119905 minus ℎ

7)

minus 119909119879

(119905 minus ℎ8) 11987616119909 (119905 minus ℎ

8)

F1198813(119905 119909119905 119910119905)

= ℎ1119879

(119905) 1198851 (119905)

minus int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

+ ℎ12119879

(119905) 1198852 (119905)

minus int

119905minusℎ1

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

+ ℎ2119910119879

(119905) 1198853119910 (119905)

minus int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

Abstract and Applied Analysis 9

+ ℎ34119910119879

(119905) 1198854119910 (119905)

minus int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

+ ℎ5119910119879

(119905) 1198855119910 (119905)

minus int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

+ ℎ56119910119879

(119905) 1198856119910 (119905)

minus int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

+ ℎ7119879

(119905) 1198857119910 (119905)

minus int

119905

119905minusℎ7

119910119879

(119904) 1198857119910 (119904) 119889119904

+ ℎ78119910119879

(119904) 1198858119910 (119904)

minus int

119905minusℎ7

119905minusℎ8

119910119879

(119905) 1198858119910 (119905) 119889119904

(33)

By Newton-Leibnitz formula we have that

2120576119879

(119905)1198731[119909 (119905) minus 119909 (119905 minus ℎ

1) minus int

119905

119905minusℎ1

(119904) 119889119904] = 0

2120576119879

(119905)1198732[119909 (119905 minus ℎ

1) minus 119909 (119905 minus 119889

1(119905)) minus int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904] = 0

2120576119879

(119905)1198733[119909 (119905 minus 119889

1(119905)) minus 119909 (119905 minus ℎ

2) minus int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904] = 0

2120576119879

(119905)1198721[119910 (119905) minus 119910 (119905 minus 120590 (119905)) minus int

119905

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198722[119910 (119905 minus ℎ

3) minus 119910 (119905 minus 120590 (119905)) minus int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198723[119910 (119905 minus 120590 (119905)) minus 119910 (119905 minus ℎ

4) minus int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198781[119910 (119905) minus 119910 (119905 minus 119889

2(119905)) minus int

119905

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198782[119910 (119905 minus ℎ

5) minus 119910 (119905 minus 119889

2(119905)) minus int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198783[119910 (119905 minus 119889

2(119905)) minus 119910 (119905 minus ℎ

6) minus int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198641[119909 (119905) minus 119909 (119905 minus 120591 (119905)) minus int

119905

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198642[119909 (119905 minus ℎ

7) minus 119909 (119905 minus 120591 (119905)) minus int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198643[119909 (119905 minus ℎ

8) minus 119909 (119905 minus ℎ

7) minus int

119905minusℎ7

119905minusℎ8

(119904) 119889119904] = 0

(34)

where

120576 (119905) = [1205761(119905) 1205762(119905) 1205763(119905) 1205764(119905)]119879

1205761(119905) = [119909

119879

(119905) 119909119879

(119905 minus 1198891(119905)) 119909

119879

(119905 minus ℎ1)

119909119879

(119905 minus ℎ2) 119909119879

(119905 minus 120591 (119905))]

1205762(119905) = [119909

119879

(119905 minus ℎ7) 119909119879

(119905 minus ℎ8) 119910119879

(119905)

119910119879

(119905 minus 120590 (119905)) 119910119879

(119905 minus ℎ3)]

1205763(119905) = [119910

119879

(119905 minus ℎ4) 119910119879

(119905 minus 1198892(119905)) 119910

119879

(119905 minus ℎ5)

119910119879

(119905 minus ℎ6) 119891119879

(119909)]

1205764(119905) = [119891

119879

(119910) 119891119879

(119910 minus 120590 (119905)) 119891119879

(119909 minus 1198891(119905))

119891119879

(119909 minus 120591 (119905)) 119891119879

(119910 minus 1198892(119905))]

(35)

By using Lemmas 4 and 5 we have

minus 2120576119879

(119905)1198731int

119905

119905minusℎ1

(119904) 119889119904

le ℎ1120576119879

(119905)1198731119885minus1

1119873119879

1120576 (119905)

+ int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

minus 2120576119879

(119905)1198732int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904

le (1198891(119905) minus ℎ

1) 120576119879

(119905)1198732119885minus1

2119873119879

2120576 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198733int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904

le (ℎ2minus 1198891(119905)) 120576119879

(119905)1198733119885minus1

2119873119879

3120576 (119905)

+ int

119905minus1198891(119905)

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198721int

119905

119905minusℎ3

119910 (119904) 119889119904

10 Abstract and Applied Analysis

le ℎ3120576119879

(119905)1198721119885minus1

3119872119879

1120576 (119905)

+ int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

minus 2120576119879

(119905)1198722int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904

le (120590 (119905) minus ℎ3) 120576119879

(119905)1198722119885minus1

4119872119879

2120576 (119905)

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905)1198723int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904

le (ℎ4minus 120590 (119905)) 120576

119879

(119905)1198723119885minus1

4119872119879

3120576 (119905)

+ int

119905minus120590(119905)

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905) 1198781int

119905

119905minusℎ5

119910 (119904) 119889119904

le ℎ5120576119879

(119905) 1198781119885minus1

5119878119879

1120576 (119905)

+ int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

minus 2120576119879

(119905) 1198782int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904

le (1198892(119905) minus ℎ

5) 120576119879

(119905) 1198782119885minus1

6119878119879

2120576 (119905)

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198783int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904

le (ℎ6minus 1198892(119905)) 120576119879

(119905) 1198783119885minus1

6119878119879

3120576 (119905)

+ int

119905minus1198892(119905)

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198641int

119905

119905minusℎ7

(119904) 119889119904

le ℎ7120576119879

(119905) 1198641119885minus1

7119864119879

1120576 (119905)

+ int

119905

119905minusℎ7

119879

(119904) 1198857 (119904) 119889119904

minus 2120576119879

(119905) 1198642int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904

le (120591 (119905) minus ℎ7) 120576119879

(119905) 1198642119885minus1

8119864119879

2120576 (119905)

+ int

119905minusℎ7

119905minus120591(119905)

119879

(119904) 1198858 (119904) 119889119904

minus 2120576119879

(119905) 1198643int

119905minus120591(119905)

119905minusℎ8

(119904) 119889119904

le (ℎ8minus 120591 (119905)) 120576

119879

(119905) 1198643119885minus1

8119864119879

3120576 (119905)

+ int

119905minus120591(119905)

119905minusℎ8

119879

(119904) 1198858 (119904) 119889119904

(36)

It follows from (7) that

0 = 119891119879

(119910 (119905)) 1198841119891 (119910 (119905)) minus 119891

119879

(119910 (119905)) 1198841119891 (119910 (119905))

le minus119910119879

(119905) 1198791198841Σ119910 (119905) + 119910

119879

(119905) 1198841(119879 + Σ) 119891 (119910 (119905))

minus 119891119879

(119910 (119905)) 1198841119891 (119910 (119905))

0 = 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

le minus119910119879

(119905 minus 1198891(119905)) 119879119884

2Σ119910 (119905 minus 119889

1(119905))

+ 119910119879

(119905 minus 1198891(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

0 = 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

le minus119910119879

(119905 minus 120590 (119905)) 1198791198843Σ119910 (119905 minus 120590 (119905))

+ 119910119879

(119905 minus 120590 (119905)) 1198843(119879 + Σ) 119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

0 = 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

le minus119910119879

(119905 minus 1198892(119905)) 119879119884

2Σ119910 (119905 minus 119889

2(119905))

+ 119910119879

(119905 minus 1198892(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

2119891 (119910 (119905 minus 119889

2(119905)))

0 = 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

le minus119910119879

(119905 minus 120591 (119905)) 1198791198845Σ119910 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 120591 (119905)) 1198845(119879 + Σ) 119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

0 = 119891119879

(119909 (119905)) 1198846119891 (119909 (119905)) minus 119891

119879

(119909 (119905)) 1198846119891 (119909 (119905))

le minus119909119879

(119905) 1198791198846Σ119909 (119905) + 119909

119879

(119905) 1198846(119879 + Σ) 119891 (119909 (119905))

minus 119891119879

(119909 (119905)) 1198846119891 (119909 (119905))

(37)

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Proof We consider the following Lyapunov functional can-didate for system (8)

119881 (119905 119909119905 119910119905) = 1198811(119905 119909119905 119910119905) + 1198812(119905 119909119905 119910119905) + 1198813(119905 119909119905 119910119905)

(27)

where

1198811(119905 119909119905 119910119905) = 2

119899

sum

119894=1

1198811119894int

119910119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

1198812119894int

119909119894

0

(119891119894(119904) minus 120574

119894119904) 119889119904

(28)

1198812(119905 119909119905 119910119905) = 119909119879

(119905) 1198751119909 (119905) + 119910

119879

(119905) 1198752119910 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119909119879

(119904) 1198761119909 (119904) 119889119904

+ int

119905

119905minus1198891(119905)

119891119879

(119909 (119904)) 1198762119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ1

119909119879

(119904) 1198763119909 (119904) 119889119904

+ int

119905minusℎ1

119905minusℎ2

119909119879

(119904) 1198764119909 (119904) 119889119904

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198765119910 (119904) 119889119904

+ int

119905

119905minus120590(119905)

119891119879

(119910 (119904)) 1198766119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ3

119910119879

(119904) 1198767119910 (119904) 119889119904

+ int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198768119910 (119904) 119889119904

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198769119910 (119904) 119889119904

+ int

119905

119905minus1198892(119905)

119891119879

(119910 (119904)) 11987610119891 (119910 (119904)) 119889119904

+ int

119905

119905minusℎ5

119910119879

(119904) 11987611119910 (119904) 119889119904

+ int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 11987612119910 (119904) 119889119904

+ int

119905minusℎ7

119905minus120591(119905)

119909119879

(119904) 11987613119909 (119904) 119889119904

+ int

119905

119905minus120591(119905)

119891119879

(119909 (119904)) 11987614119891 (119909 (119904)) 119889119904

+ int

119905

119905minusℎ7

119909119879

(119904) 11987615119909 (119904) 119889119904

+ int

119905minusℎ7

119905minusℎ8

119909119879

(119904) 11987616119909 (119904) 119889119904

(29)

1198813(119905 119909119905 119910119905) = int

0

minusℎ1

int

119905

119905+120579

119879

(119904) 1198851 (119904) 119889119904 119889120579

+ int

minusℎ1

minusℎ2

int

119905

119905+120579

119879

(119904) 1198852 (119904) 119889119904 119889120579

+ int

0

minusℎ3

int

119905

119905+120579

119910119879

(119904) 1198853119910 (119904) 119889119904 119889120579

+ int

minusℎ3

minusℎ4

int

119905

119905+120579

119910119879

(119904) 1198854119910 (119904) 119889119904 119889120579

+ int

0

minusℎ5

int

119905

119905+120579

119910119879

(119904) 1198855119910 (119904) 119889119904 119889120579

+ int

minusℎ5

minusℎ6

int

119905

119905+120579

119910119879

(119904) 1198856119910 (119904) 119889119904 119889120579

+ int

0

minusℎ7

int

119905

119905+120579

119879

(119904) 1198857 (119904) 119889119904 119889120579

+ int

minusℎ7

minusℎ8

int

119905

119905+120579

119879

(119904) 1198858 (119904) 119889119904 119889120579

(30)

Then by Itorsquos differential formula taking the derivative of119881(119905) along the trajectories of the system (8) we can obtainthe following stochastic differential [29]

119889119881 (119905) = F119881 (119905) 119889119905

+ 2119909119879

(119905) 1198751120578 (119905 119909 (119905 minus 119889

1(119905)) 119910 (119905 minus 120590 (119905)))

+ 2119910119879

1198752120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905)))

(31)

whereF is the diffusion operator and

F119881 (119905 119909119905 119910119905) = F119881

1(119905 119909119905 119910119905)

+F1198812(119905 119909119905 119910119905) +F119881

3(119905 119909119905 119910119905)

(32)

with

F1198811(119905 119909119905 119910119905) = 2 [119891

119879

(119910 (119905)) minus 119910119879

(119905) Γ]

times 1198811[minus119862119910 (119905 minus 119889

2(119905)) + 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

8 Abstract and Applied Analysis

+ tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198811120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 2 [119891119879

(119909 (119905)) minus 119909119879

(119905) Γ]

times 1198812[minus119860119909 (119905 minus 119889

1(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198812120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

F1198812(119905 119909119905 119910119905)

le 2119909119879

(119905) 1198751[minus119860119909 (119905 minus 119889

1(119905))

+ 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198751120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

+ 2119910119879

(119905) 1198752[minus119862119910 (119905 minus 119889

2(119905))

+ 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

+ tr [120578119879 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198752120578 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 119909119879

(119905 minus ℎ1) 1198761119909 (119905 minus ℎ

1)

minus (1 minus 1198891) 119909119879

(119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905))

+ 119891119879

(119909 (119904)) 1198762119891 (119909 (119904))

minus (1 minus 1198891) 119891119879

(119910 (119905 minus 1198891(119905)))

times 1198762119891 (119910 (119905 minus 119889

1(119905)))

+ 119909119879

(119905) 1198763119909 (119905) minus 119909

119879

(119905 minus ℎ1) 1198763119909 (119905 minus ℎ

1)

+ 119909119879

(119905 minus ℎ1) 1198764119909 (119905 minus ℎ

1)

minus 119909119879

(119905 minus ℎ2) 1198764119909 (119905 minus ℎ

2)

+ 119910119879

(119905 minus ℎ5) 1198765119909 (119905 minus ℎ

5)

minus (1 minus 120590) 119910119879

(119905 minus 120590 (119905)) 1198765119910 (119905 minus 120590 (119905))

+ 119891119879

(119910 (119905)) 1198766119891 (119910 (119905))

minus (1 minus 120590) 119891 (119910119879

(119905 minus 120590 (119905)))

times 1198765119891 (119910 (119905 minus 120590 (119905)))

minus 119910119879

(119905 minus ℎ3) 1198767119910 (119905 minus ℎ

3)

+ 119910119879

(119905) 1198767119910 (119905) + 119910

119879

(119905 minus ℎ3) 1198768119910 (119905 minus ℎ

3)

minus 119910119879

(119905 minus ℎ4) 1198768119910 (119905 minus ℎ

4)

+ 119910119879

(119905 minus ℎ5) 1198769119910 (119905 minus ℎ

5)

minus (1 minus 1198892) 119910119879

(119905 minus 1198892(119905)) 119876

9119910 (119905 minus 119889

2(119905))

+ 119891119879

(119910 (119905)) 11987610119891 (119910 (119905))

minus (1 minus 1198892) 119891119879

(119910 (119905 minus 1198892(119905)))

times 11987610119891 (119910 (119905 minus 119889

2(119905)))

+ 119910119879

11987611119910 (119905) minus 119910

119879

(119905 minus ℎ5) 11987611119910 (119905 minus ℎ

5)

+ 119910119879

(119905 minus ℎ5) 11987612119910 (119905 minus ℎ

5)

minus 119910119879

(119905 minus ℎ6) 11987612119910 (119905 minus ℎ

6)

+ 119909119879

(119905 minus ℎ7) 11987613119909 (119905 minus ℎ

7)

minus (1 minus 120591) 119909119879

(119905 minus 120591 (119905)) 11987613119909 (119905 minus 120591 (119905))

+ 119891119879

(119909 (119905)) 11987614119891 (119909 (119905))

minus (1 minus 120591) 119891119879

(119909 (119905 minus 120591 (119905)))

times 11987614119891 (119909 (119905 minus 120591 (119905))) + 119909

119879

(119905) 11987615119909 (119905)

minus 119909119879

(119905 minus ℎ7) 11987615119909 (119905 minus ℎ

7)

+ 119909119879

(119905 minus ℎ7) 11987616119909 (119905 minus ℎ

7)

minus 119909119879

(119905 minus ℎ8) 11987616119909 (119905 minus ℎ

8)

F1198813(119905 119909119905 119910119905)

= ℎ1119879

(119905) 1198851 (119905)

minus int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

+ ℎ12119879

(119905) 1198852 (119905)

minus int

119905minusℎ1

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

+ ℎ2119910119879

(119905) 1198853119910 (119905)

minus int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

Abstract and Applied Analysis 9

+ ℎ34119910119879

(119905) 1198854119910 (119905)

minus int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

+ ℎ5119910119879

(119905) 1198855119910 (119905)

minus int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

+ ℎ56119910119879

(119905) 1198856119910 (119905)

minus int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

+ ℎ7119879

(119905) 1198857119910 (119905)

minus int

119905

119905minusℎ7

119910119879

(119904) 1198857119910 (119904) 119889119904

+ ℎ78119910119879

(119904) 1198858119910 (119904)

minus int

119905minusℎ7

119905minusℎ8

119910119879

(119905) 1198858119910 (119905) 119889119904

(33)

By Newton-Leibnitz formula we have that

2120576119879

(119905)1198731[119909 (119905) minus 119909 (119905 minus ℎ

1) minus int

119905

119905minusℎ1

(119904) 119889119904] = 0

2120576119879

(119905)1198732[119909 (119905 minus ℎ

1) minus 119909 (119905 minus 119889

1(119905)) minus int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904] = 0

2120576119879

(119905)1198733[119909 (119905 minus 119889

1(119905)) minus 119909 (119905 minus ℎ

2) minus int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904] = 0

2120576119879

(119905)1198721[119910 (119905) minus 119910 (119905 minus 120590 (119905)) minus int

119905

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198722[119910 (119905 minus ℎ

3) minus 119910 (119905 minus 120590 (119905)) minus int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198723[119910 (119905 minus 120590 (119905)) minus 119910 (119905 minus ℎ

4) minus int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198781[119910 (119905) minus 119910 (119905 minus 119889

2(119905)) minus int

119905

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198782[119910 (119905 minus ℎ

5) minus 119910 (119905 minus 119889

2(119905)) minus int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198783[119910 (119905 minus 119889

2(119905)) minus 119910 (119905 minus ℎ

6) minus int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198641[119909 (119905) minus 119909 (119905 minus 120591 (119905)) minus int

119905

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198642[119909 (119905 minus ℎ

7) minus 119909 (119905 minus 120591 (119905)) minus int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198643[119909 (119905 minus ℎ

8) minus 119909 (119905 minus ℎ

7) minus int

119905minusℎ7

119905minusℎ8

(119904) 119889119904] = 0

(34)

where

120576 (119905) = [1205761(119905) 1205762(119905) 1205763(119905) 1205764(119905)]119879

1205761(119905) = [119909

119879

(119905) 119909119879

(119905 minus 1198891(119905)) 119909

119879

(119905 minus ℎ1)

119909119879

(119905 minus ℎ2) 119909119879

(119905 minus 120591 (119905))]

1205762(119905) = [119909

119879

(119905 minus ℎ7) 119909119879

(119905 minus ℎ8) 119910119879

(119905)

119910119879

(119905 minus 120590 (119905)) 119910119879

(119905 minus ℎ3)]

1205763(119905) = [119910

119879

(119905 minus ℎ4) 119910119879

(119905 minus 1198892(119905)) 119910

119879

(119905 minus ℎ5)

119910119879

(119905 minus ℎ6) 119891119879

(119909)]

1205764(119905) = [119891

119879

(119910) 119891119879

(119910 minus 120590 (119905)) 119891119879

(119909 minus 1198891(119905))

119891119879

(119909 minus 120591 (119905)) 119891119879

(119910 minus 1198892(119905))]

(35)

By using Lemmas 4 and 5 we have

minus 2120576119879

(119905)1198731int

119905

119905minusℎ1

(119904) 119889119904

le ℎ1120576119879

(119905)1198731119885minus1

1119873119879

1120576 (119905)

+ int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

minus 2120576119879

(119905)1198732int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904

le (1198891(119905) minus ℎ

1) 120576119879

(119905)1198732119885minus1

2119873119879

2120576 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198733int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904

le (ℎ2minus 1198891(119905)) 120576119879

(119905)1198733119885minus1

2119873119879

3120576 (119905)

+ int

119905minus1198891(119905)

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198721int

119905

119905minusℎ3

119910 (119904) 119889119904

10 Abstract and Applied Analysis

le ℎ3120576119879

(119905)1198721119885minus1

3119872119879

1120576 (119905)

+ int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

minus 2120576119879

(119905)1198722int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904

le (120590 (119905) minus ℎ3) 120576119879

(119905)1198722119885minus1

4119872119879

2120576 (119905)

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905)1198723int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904

le (ℎ4minus 120590 (119905)) 120576

119879

(119905)1198723119885minus1

4119872119879

3120576 (119905)

+ int

119905minus120590(119905)

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905) 1198781int

119905

119905minusℎ5

119910 (119904) 119889119904

le ℎ5120576119879

(119905) 1198781119885minus1

5119878119879

1120576 (119905)

+ int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

minus 2120576119879

(119905) 1198782int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904

le (1198892(119905) minus ℎ

5) 120576119879

(119905) 1198782119885minus1

6119878119879

2120576 (119905)

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198783int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904

le (ℎ6minus 1198892(119905)) 120576119879

(119905) 1198783119885minus1

6119878119879

3120576 (119905)

+ int

119905minus1198892(119905)

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198641int

119905

119905minusℎ7

(119904) 119889119904

le ℎ7120576119879

(119905) 1198641119885minus1

7119864119879

1120576 (119905)

+ int

119905

119905minusℎ7

119879

(119904) 1198857 (119904) 119889119904

minus 2120576119879

(119905) 1198642int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904

le (120591 (119905) minus ℎ7) 120576119879

(119905) 1198642119885minus1

8119864119879

2120576 (119905)

+ int

119905minusℎ7

119905minus120591(119905)

119879

(119904) 1198858 (119904) 119889119904

minus 2120576119879

(119905) 1198643int

119905minus120591(119905)

119905minusℎ8

(119904) 119889119904

le (ℎ8minus 120591 (119905)) 120576

119879

(119905) 1198643119885minus1

8119864119879

3120576 (119905)

+ int

119905minus120591(119905)

119905minusℎ8

119879

(119904) 1198858 (119904) 119889119904

(36)

It follows from (7) that

0 = 119891119879

(119910 (119905)) 1198841119891 (119910 (119905)) minus 119891

119879

(119910 (119905)) 1198841119891 (119910 (119905))

le minus119910119879

(119905) 1198791198841Σ119910 (119905) + 119910

119879

(119905) 1198841(119879 + Σ) 119891 (119910 (119905))

minus 119891119879

(119910 (119905)) 1198841119891 (119910 (119905))

0 = 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

le minus119910119879

(119905 minus 1198891(119905)) 119879119884

2Σ119910 (119905 minus 119889

1(119905))

+ 119910119879

(119905 minus 1198891(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

0 = 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

le minus119910119879

(119905 minus 120590 (119905)) 1198791198843Σ119910 (119905 minus 120590 (119905))

+ 119910119879

(119905 minus 120590 (119905)) 1198843(119879 + Σ) 119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

0 = 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

le minus119910119879

(119905 minus 1198892(119905)) 119879119884

2Σ119910 (119905 minus 119889

2(119905))

+ 119910119879

(119905 minus 1198892(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

2119891 (119910 (119905 minus 119889

2(119905)))

0 = 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

le minus119910119879

(119905 minus 120591 (119905)) 1198791198845Σ119910 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 120591 (119905)) 1198845(119879 + Σ) 119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

0 = 119891119879

(119909 (119905)) 1198846119891 (119909 (119905)) minus 119891

119879

(119909 (119905)) 1198846119891 (119909 (119905))

le minus119909119879

(119905) 1198791198846Σ119909 (119905) + 119909

119879

(119905) 1198846(119879 + Σ) 119891 (119909 (119905))

minus 119891119879

(119909 (119905)) 1198846119891 (119909 (119905))

(37)

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

+ tr [120578119879 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198811120578 (119905 119909 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 2 [119891119879

(119909 (119905)) minus 119909119879

(119905) Γ]

times 1198812[minus119860119909 (119905 minus 119889

1(119905)) + 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198812120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

F1198812(119905 119909119905 119910119905)

le 2119909119879

(119905) 1198751[minus119860119909 (119905 minus 119889

1(119905))

+ 119882119891 (119910 (119905 minus 120590 (119905)))

minus 11986611198651119867119886119909 (119905 minus 119889

1(119905))

+ 11986611198651119867119908119891 (119910 (119905 minus 120590 (119905)))]

+ tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 1198891(119905)))

times1198751120578 (119905 119910 (119905 minus 120590 (119905)) 119910 (119905 minus 119889

1(119905))) ]

+ 2119910119879

(119905) 1198752[minus119862119910 (119905 minus 119889

2(119905))

+ 119871119909 (119905 minus 120591 (119905))

minus 11986621198652119867119888119910 (119905 minus 119889

2(119905))

+ 11986621198652119867119897119909 (119905 minus 120591 (119905))]

+ tr [120578119879 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 1198892(119905)))

times1198752120578 (119905 119910 (119905 minus 120591 (119905)) 119910 (119905 minus 119889

2(119905))) ]

+ 119909119879

(119905 minus ℎ1) 1198761119909 (119905 minus ℎ

1)

minus (1 minus 1198891) 119909119879

(119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905))

+ 119891119879

(119909 (119904)) 1198762119891 (119909 (119904))

minus (1 minus 1198891) 119891119879

(119910 (119905 minus 1198891(119905)))

times 1198762119891 (119910 (119905 minus 119889

1(119905)))

+ 119909119879

(119905) 1198763119909 (119905) minus 119909

119879

(119905 minus ℎ1) 1198763119909 (119905 minus ℎ

1)

+ 119909119879

(119905 minus ℎ1) 1198764119909 (119905 minus ℎ

1)

minus 119909119879

(119905 minus ℎ2) 1198764119909 (119905 minus ℎ

2)

+ 119910119879

(119905 minus ℎ5) 1198765119909 (119905 minus ℎ

5)

minus (1 minus 120590) 119910119879

(119905 minus 120590 (119905)) 1198765119910 (119905 minus 120590 (119905))

+ 119891119879

(119910 (119905)) 1198766119891 (119910 (119905))

minus (1 minus 120590) 119891 (119910119879

(119905 minus 120590 (119905)))

times 1198765119891 (119910 (119905 minus 120590 (119905)))

minus 119910119879

(119905 minus ℎ3) 1198767119910 (119905 minus ℎ

3)

+ 119910119879

(119905) 1198767119910 (119905) + 119910

119879

(119905 minus ℎ3) 1198768119910 (119905 minus ℎ

3)

minus 119910119879

(119905 minus ℎ4) 1198768119910 (119905 minus ℎ

4)

+ 119910119879

(119905 minus ℎ5) 1198769119910 (119905 minus ℎ

5)

minus (1 minus 1198892) 119910119879

(119905 minus 1198892(119905)) 119876

9119910 (119905 minus 119889

2(119905))

+ 119891119879

(119910 (119905)) 11987610119891 (119910 (119905))

minus (1 minus 1198892) 119891119879

(119910 (119905 minus 1198892(119905)))

times 11987610119891 (119910 (119905 minus 119889

2(119905)))

+ 119910119879

11987611119910 (119905) minus 119910

119879

(119905 minus ℎ5) 11987611119910 (119905 minus ℎ

5)

+ 119910119879

(119905 minus ℎ5) 11987612119910 (119905 minus ℎ

5)

minus 119910119879

(119905 minus ℎ6) 11987612119910 (119905 minus ℎ

6)

+ 119909119879

(119905 minus ℎ7) 11987613119909 (119905 minus ℎ

7)

minus (1 minus 120591) 119909119879

(119905 minus 120591 (119905)) 11987613119909 (119905 minus 120591 (119905))

+ 119891119879

(119909 (119905)) 11987614119891 (119909 (119905))

minus (1 minus 120591) 119891119879

(119909 (119905 minus 120591 (119905)))

times 11987614119891 (119909 (119905 minus 120591 (119905))) + 119909

119879

(119905) 11987615119909 (119905)

minus 119909119879

(119905 minus ℎ7) 11987615119909 (119905 minus ℎ

7)

+ 119909119879

(119905 minus ℎ7) 11987616119909 (119905 minus ℎ

7)

minus 119909119879

(119905 minus ℎ8) 11987616119909 (119905 minus ℎ

8)

F1198813(119905 119909119905 119910119905)

= ℎ1119879

(119905) 1198851 (119905)

minus int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

+ ℎ12119879

(119905) 1198852 (119905)

minus int

119905minusℎ1

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

+ ℎ2119910119879

(119905) 1198853119910 (119905)

minus int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

Abstract and Applied Analysis 9

+ ℎ34119910119879

(119905) 1198854119910 (119905)

minus int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

+ ℎ5119910119879

(119905) 1198855119910 (119905)

minus int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

+ ℎ56119910119879

(119905) 1198856119910 (119905)

minus int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

+ ℎ7119879

(119905) 1198857119910 (119905)

minus int

119905

119905minusℎ7

119910119879

(119904) 1198857119910 (119904) 119889119904

+ ℎ78119910119879

(119904) 1198858119910 (119904)

minus int

119905minusℎ7

119905minusℎ8

119910119879

(119905) 1198858119910 (119905) 119889119904

(33)

By Newton-Leibnitz formula we have that

2120576119879

(119905)1198731[119909 (119905) minus 119909 (119905 minus ℎ

1) minus int

119905

119905minusℎ1

(119904) 119889119904] = 0

2120576119879

(119905)1198732[119909 (119905 minus ℎ

1) minus 119909 (119905 minus 119889

1(119905)) minus int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904] = 0

2120576119879

(119905)1198733[119909 (119905 minus 119889

1(119905)) minus 119909 (119905 minus ℎ

2) minus int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904] = 0

2120576119879

(119905)1198721[119910 (119905) minus 119910 (119905 minus 120590 (119905)) minus int

119905

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198722[119910 (119905 minus ℎ

3) minus 119910 (119905 minus 120590 (119905)) minus int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198723[119910 (119905 minus 120590 (119905)) minus 119910 (119905 minus ℎ

4) minus int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198781[119910 (119905) minus 119910 (119905 minus 119889

2(119905)) minus int

119905

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198782[119910 (119905 minus ℎ

5) minus 119910 (119905 minus 119889

2(119905)) minus int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198783[119910 (119905 minus 119889

2(119905)) minus 119910 (119905 minus ℎ

6) minus int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198641[119909 (119905) minus 119909 (119905 minus 120591 (119905)) minus int

119905

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198642[119909 (119905 minus ℎ

7) minus 119909 (119905 minus 120591 (119905)) minus int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198643[119909 (119905 minus ℎ

8) minus 119909 (119905 minus ℎ

7) minus int

119905minusℎ7

119905minusℎ8

(119904) 119889119904] = 0

(34)

where

120576 (119905) = [1205761(119905) 1205762(119905) 1205763(119905) 1205764(119905)]119879

1205761(119905) = [119909

119879

(119905) 119909119879

(119905 minus 1198891(119905)) 119909

119879

(119905 minus ℎ1)

119909119879

(119905 minus ℎ2) 119909119879

(119905 minus 120591 (119905))]

1205762(119905) = [119909

119879

(119905 minus ℎ7) 119909119879

(119905 minus ℎ8) 119910119879

(119905)

119910119879

(119905 minus 120590 (119905)) 119910119879

(119905 minus ℎ3)]

1205763(119905) = [119910

119879

(119905 minus ℎ4) 119910119879

(119905 minus 1198892(119905)) 119910

119879

(119905 minus ℎ5)

119910119879

(119905 minus ℎ6) 119891119879

(119909)]

1205764(119905) = [119891

119879

(119910) 119891119879

(119910 minus 120590 (119905)) 119891119879

(119909 minus 1198891(119905))

119891119879

(119909 minus 120591 (119905)) 119891119879

(119910 minus 1198892(119905))]

(35)

By using Lemmas 4 and 5 we have

minus 2120576119879

(119905)1198731int

119905

119905minusℎ1

(119904) 119889119904

le ℎ1120576119879

(119905)1198731119885minus1

1119873119879

1120576 (119905)

+ int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

minus 2120576119879

(119905)1198732int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904

le (1198891(119905) minus ℎ

1) 120576119879

(119905)1198732119885minus1

2119873119879

2120576 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198733int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904

le (ℎ2minus 1198891(119905)) 120576119879

(119905)1198733119885minus1

2119873119879

3120576 (119905)

+ int

119905minus1198891(119905)

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198721int

119905

119905minusℎ3

119910 (119904) 119889119904

10 Abstract and Applied Analysis

le ℎ3120576119879

(119905)1198721119885minus1

3119872119879

1120576 (119905)

+ int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

minus 2120576119879

(119905)1198722int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904

le (120590 (119905) minus ℎ3) 120576119879

(119905)1198722119885minus1

4119872119879

2120576 (119905)

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905)1198723int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904

le (ℎ4minus 120590 (119905)) 120576

119879

(119905)1198723119885minus1

4119872119879

3120576 (119905)

+ int

119905minus120590(119905)

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905) 1198781int

119905

119905minusℎ5

119910 (119904) 119889119904

le ℎ5120576119879

(119905) 1198781119885minus1

5119878119879

1120576 (119905)

+ int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

minus 2120576119879

(119905) 1198782int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904

le (1198892(119905) minus ℎ

5) 120576119879

(119905) 1198782119885minus1

6119878119879

2120576 (119905)

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198783int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904

le (ℎ6minus 1198892(119905)) 120576119879

(119905) 1198783119885minus1

6119878119879

3120576 (119905)

+ int

119905minus1198892(119905)

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198641int

119905

119905minusℎ7

(119904) 119889119904

le ℎ7120576119879

(119905) 1198641119885minus1

7119864119879

1120576 (119905)

+ int

119905

119905minusℎ7

119879

(119904) 1198857 (119904) 119889119904

minus 2120576119879

(119905) 1198642int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904

le (120591 (119905) minus ℎ7) 120576119879

(119905) 1198642119885minus1

8119864119879

2120576 (119905)

+ int

119905minusℎ7

119905minus120591(119905)

119879

(119904) 1198858 (119904) 119889119904

minus 2120576119879

(119905) 1198643int

119905minus120591(119905)

119905minusℎ8

(119904) 119889119904

le (ℎ8minus 120591 (119905)) 120576

119879

(119905) 1198643119885minus1

8119864119879

3120576 (119905)

+ int

119905minus120591(119905)

119905minusℎ8

119879

(119904) 1198858 (119904) 119889119904

(36)

It follows from (7) that

0 = 119891119879

(119910 (119905)) 1198841119891 (119910 (119905)) minus 119891

119879

(119910 (119905)) 1198841119891 (119910 (119905))

le minus119910119879

(119905) 1198791198841Σ119910 (119905) + 119910

119879

(119905) 1198841(119879 + Σ) 119891 (119910 (119905))

minus 119891119879

(119910 (119905)) 1198841119891 (119910 (119905))

0 = 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

le minus119910119879

(119905 minus 1198891(119905)) 119879119884

2Σ119910 (119905 minus 119889

1(119905))

+ 119910119879

(119905 minus 1198891(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

0 = 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

le minus119910119879

(119905 minus 120590 (119905)) 1198791198843Σ119910 (119905 minus 120590 (119905))

+ 119910119879

(119905 minus 120590 (119905)) 1198843(119879 + Σ) 119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

0 = 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

le minus119910119879

(119905 minus 1198892(119905)) 119879119884

2Σ119910 (119905 minus 119889

2(119905))

+ 119910119879

(119905 minus 1198892(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

2119891 (119910 (119905 minus 119889

2(119905)))

0 = 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

le minus119910119879

(119905 minus 120591 (119905)) 1198791198845Σ119910 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 120591 (119905)) 1198845(119879 + Σ) 119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

0 = 119891119879

(119909 (119905)) 1198846119891 (119909 (119905)) minus 119891

119879

(119909 (119905)) 1198846119891 (119909 (119905))

le minus119909119879

(119905) 1198791198846Σ119909 (119905) + 119909

119879

(119905) 1198846(119879 + Σ) 119891 (119909 (119905))

minus 119891119879

(119909 (119905)) 1198846119891 (119909 (119905))

(37)

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

+ ℎ34119910119879

(119905) 1198854119910 (119905)

minus int

119905minusℎ3

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

+ ℎ5119910119879

(119905) 1198855119910 (119905)

minus int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

+ ℎ56119910119879

(119905) 1198856119910 (119905)

minus int

119905minusℎ5

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

+ ℎ7119879

(119905) 1198857119910 (119905)

minus int

119905

119905minusℎ7

119910119879

(119904) 1198857119910 (119904) 119889119904

+ ℎ78119910119879

(119904) 1198858119910 (119904)

minus int

119905minusℎ7

119905minusℎ8

119910119879

(119905) 1198858119910 (119905) 119889119904

(33)

By Newton-Leibnitz formula we have that

2120576119879

(119905)1198731[119909 (119905) minus 119909 (119905 minus ℎ

1) minus int

119905

119905minusℎ1

(119904) 119889119904] = 0

2120576119879

(119905)1198732[119909 (119905 minus ℎ

1) minus 119909 (119905 minus 119889

1(119905)) minus int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904] = 0

2120576119879

(119905)1198733[119909 (119905 minus 119889

1(119905)) minus 119909 (119905 minus ℎ

2) minus int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904] = 0

2120576119879

(119905)1198721[119910 (119905) minus 119910 (119905 minus 120590 (119905)) minus int

119905

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198722[119910 (119905 minus ℎ

3) minus 119910 (119905 minus 120590 (119905)) minus int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905)1198723[119910 (119905 minus 120590 (119905)) minus 119910 (119905 minus ℎ

4) minus int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198781[119910 (119905) minus 119910 (119905 minus 119889

2(119905)) minus int

119905

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198782[119910 (119905 minus ℎ

5) minus 119910 (119905 minus 119889

2(119905)) minus int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198783[119910 (119905 minus 119889

2(119905)) minus 119910 (119905 minus ℎ

6) minus int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904] = 0

2120576119879

(119905) 1198641[119909 (119905) minus 119909 (119905 minus 120591 (119905)) minus int

119905

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198642[119909 (119905 minus ℎ

7) minus 119909 (119905 minus 120591 (119905)) minus int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904] = 0

2120576119879

(119905) 1198643[119909 (119905 minus ℎ

8) minus 119909 (119905 minus ℎ

7) minus int

119905minusℎ7

119905minusℎ8

(119904) 119889119904] = 0

(34)

where

120576 (119905) = [1205761(119905) 1205762(119905) 1205763(119905) 1205764(119905)]119879

1205761(119905) = [119909

119879

(119905) 119909119879

(119905 minus 1198891(119905)) 119909

119879

(119905 minus ℎ1)

119909119879

(119905 minus ℎ2) 119909119879

(119905 minus 120591 (119905))]

1205762(119905) = [119909

119879

(119905 minus ℎ7) 119909119879

(119905 minus ℎ8) 119910119879

(119905)

119910119879

(119905 minus 120590 (119905)) 119910119879

(119905 minus ℎ3)]

1205763(119905) = [119910

119879

(119905 minus ℎ4) 119910119879

(119905 minus 1198892(119905)) 119910

119879

(119905 minus ℎ5)

119910119879

(119905 minus ℎ6) 119891119879

(119909)]

1205764(119905) = [119891

119879

(119910) 119891119879

(119910 minus 120590 (119905)) 119891119879

(119909 minus 1198891(119905))

119891119879

(119909 minus 120591 (119905)) 119891119879

(119910 minus 1198892(119905))]

(35)

By using Lemmas 4 and 5 we have

minus 2120576119879

(119905)1198731int

119905

119905minusℎ1

(119904) 119889119904

le ℎ1120576119879

(119905)1198731119885minus1

1119873119879

1120576 (119905)

+ int

119905

119905minusℎ1

119879

(119904) 1198851 (119904) 119889119904

minus 2120576119879

(119905)1198732int

119905minusℎ1

119905minus1198891(119905)

(119904) 119889119904

le (1198891(119905) minus ℎ

1) 120576119879

(119905)1198732119885minus1

2119873119879

2120576 (119905)

+ int

119905minusℎ1

119905minus1198891(119905)

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198733int

119905minus1198891(119905)

119905minusℎ2

(119904) 119889119904

le (ℎ2minus 1198891(119905)) 120576119879

(119905)1198733119885minus1

2119873119879

3120576 (119905)

+ int

119905minus1198891(119905)

119905minusℎ2

119879

(119904) 1198852 (119904) 119889119904

minus 2120576119879

(119905)1198721int

119905

119905minusℎ3

119910 (119904) 119889119904

10 Abstract and Applied Analysis

le ℎ3120576119879

(119905)1198721119885minus1

3119872119879

1120576 (119905)

+ int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

minus 2120576119879

(119905)1198722int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904

le (120590 (119905) minus ℎ3) 120576119879

(119905)1198722119885minus1

4119872119879

2120576 (119905)

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905)1198723int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904

le (ℎ4minus 120590 (119905)) 120576

119879

(119905)1198723119885minus1

4119872119879

3120576 (119905)

+ int

119905minus120590(119905)

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905) 1198781int

119905

119905minusℎ5

119910 (119904) 119889119904

le ℎ5120576119879

(119905) 1198781119885minus1

5119878119879

1120576 (119905)

+ int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

minus 2120576119879

(119905) 1198782int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904

le (1198892(119905) minus ℎ

5) 120576119879

(119905) 1198782119885minus1

6119878119879

2120576 (119905)

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198783int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904

le (ℎ6minus 1198892(119905)) 120576119879

(119905) 1198783119885minus1

6119878119879

3120576 (119905)

+ int

119905minus1198892(119905)

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198641int

119905

119905minusℎ7

(119904) 119889119904

le ℎ7120576119879

(119905) 1198641119885minus1

7119864119879

1120576 (119905)

+ int

119905

119905minusℎ7

119879

(119904) 1198857 (119904) 119889119904

minus 2120576119879

(119905) 1198642int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904

le (120591 (119905) minus ℎ7) 120576119879

(119905) 1198642119885minus1

8119864119879

2120576 (119905)

+ int

119905minusℎ7

119905minus120591(119905)

119879

(119904) 1198858 (119904) 119889119904

minus 2120576119879

(119905) 1198643int

119905minus120591(119905)

119905minusℎ8

(119904) 119889119904

le (ℎ8minus 120591 (119905)) 120576

119879

(119905) 1198643119885minus1

8119864119879

3120576 (119905)

+ int

119905minus120591(119905)

119905minusℎ8

119879

(119904) 1198858 (119904) 119889119904

(36)

It follows from (7) that

0 = 119891119879

(119910 (119905)) 1198841119891 (119910 (119905)) minus 119891

119879

(119910 (119905)) 1198841119891 (119910 (119905))

le minus119910119879

(119905) 1198791198841Σ119910 (119905) + 119910

119879

(119905) 1198841(119879 + Σ) 119891 (119910 (119905))

minus 119891119879

(119910 (119905)) 1198841119891 (119910 (119905))

0 = 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

le minus119910119879

(119905 minus 1198891(119905)) 119879119884

2Σ119910 (119905 minus 119889

1(119905))

+ 119910119879

(119905 minus 1198891(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

0 = 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

le minus119910119879

(119905 minus 120590 (119905)) 1198791198843Σ119910 (119905 minus 120590 (119905))

+ 119910119879

(119905 minus 120590 (119905)) 1198843(119879 + Σ) 119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

0 = 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

le minus119910119879

(119905 minus 1198892(119905)) 119879119884

2Σ119910 (119905 minus 119889

2(119905))

+ 119910119879

(119905 minus 1198892(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

2119891 (119910 (119905 minus 119889

2(119905)))

0 = 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

le minus119910119879

(119905 minus 120591 (119905)) 1198791198845Σ119910 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 120591 (119905)) 1198845(119879 + Σ) 119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

0 = 119891119879

(119909 (119905)) 1198846119891 (119909 (119905)) minus 119891

119879

(119909 (119905)) 1198846119891 (119909 (119905))

le minus119909119879

(119905) 1198791198846Σ119909 (119905) + 119909

119879

(119905) 1198846(119879 + Σ) 119891 (119909 (119905))

minus 119891119879

(119909 (119905)) 1198846119891 (119909 (119905))

(37)

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

le ℎ3120576119879

(119905)1198721119885minus1

3119872119879

1120576 (119905)

+ int

119905

119905minusℎ3

119910119879

(119904) 1198853119910 (119904) 119889119904

minus 2120576119879

(119905)1198722int

119905minusℎ3

119905minus120590(119905)

119910 (119904) 119889119904

le (120590 (119905) minus ℎ3) 120576119879

(119905)1198722119885minus1

4119872119879

2120576 (119905)

+ int

119905minusℎ3

119905minus120590(119905)

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905)1198723int

119905minus120590(119905)

119905minusℎ4

119910 (119904) 119889119904

le (ℎ4minus 120590 (119905)) 120576

119879

(119905)1198723119885minus1

4119872119879

3120576 (119905)

+ int

119905minus120590(119905)

119905minusℎ4

119910119879

(119904) 1198854119910 (119904) 119889119904

minus 2120576119879

(119905) 1198781int

119905

119905minusℎ5

119910 (119904) 119889119904

le ℎ5120576119879

(119905) 1198781119885minus1

5119878119879

1120576 (119905)

+ int

119905

119905minusℎ5

119910119879

(119904) 1198855119910 (119904) 119889119904

minus 2120576119879

(119905) 1198782int

119905minusℎ5

119905minus1198892(119905)

119910 (119904) 119889119904

le (1198892(119905) minus ℎ

5) 120576119879

(119905) 1198782119885minus1

6119878119879

2120576 (119905)

+ int

119905minusℎ5

119905minus1198892(119905)

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198783int

119905minus1198892(119905)

119905minusℎ6

119910 (119904) 119889119904

le (ℎ6minus 1198892(119905)) 120576119879

(119905) 1198783119885minus1

6119878119879

3120576 (119905)

+ int

119905minus1198892(119905)

119905minusℎ6

119910119879

(119904) 1198856119910 (119904) 119889119904

minus 2120576119879

(119905) 1198641int

119905

119905minusℎ7

(119904) 119889119904

le ℎ7120576119879

(119905) 1198641119885minus1

7119864119879

1120576 (119905)

+ int

119905

119905minusℎ7

119879

(119904) 1198857 (119904) 119889119904

minus 2120576119879

(119905) 1198642int

119905minusℎ7

119905minus120591(119905)

(119904) 119889119904

le (120591 (119905) minus ℎ7) 120576119879

(119905) 1198642119885minus1

8119864119879

2120576 (119905)

+ int

119905minusℎ7

119905minus120591(119905)

119879

(119904) 1198858 (119904) 119889119904

minus 2120576119879

(119905) 1198643int

119905minus120591(119905)

119905minusℎ8

(119904) 119889119904

le (ℎ8minus 120591 (119905)) 120576

119879

(119905) 1198643119885minus1

8119864119879

3120576 (119905)

+ int

119905minus120591(119905)

119905minusℎ8

119879

(119904) 1198858 (119904) 119889119904

(36)

It follows from (7) that

0 = 119891119879

(119910 (119905)) 1198841119891 (119910 (119905)) minus 119891

119879

(119910 (119905)) 1198841119891 (119910 (119905))

le minus119910119879

(119905) 1198791198841Σ119910 (119905) + 119910

119879

(119905) 1198841(119879 + Σ) 119891 (119910 (119905))

minus 119891119879

(119910 (119905)) 1198841119891 (119910 (119905))

0 = 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

le minus119910119879

(119905 minus 1198891(119905)) 119879119884

2Σ119910 (119905 minus 119889

1(119905))

+ 119910119879

(119905 minus 1198891(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

1(119905)))

minus 119891119879

(119910 (119905 minus 1198891(119905))) 119884

2119891 (119910 (119905 minus 119889

1(119905)))

0 = 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

le minus119910119879

(119905 minus 120590 (119905)) 1198791198843Σ119910 (119905 minus 120590 (119905))

+ 119910119879

(119905 minus 120590 (119905)) 1198843(119879 + Σ) 119891 (119910 (119905 minus 120590 (119905)))

minus 119891119879

(119910 (119905 minus 120590 (119905))) 1198843119891 (119910 (119905 minus 120590 (119905)))

0 = 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

4119891 (119910 (119905 minus 119889

2(119905)))

le minus119910119879

(119905 minus 1198892(119905)) 119879119884

2Σ119910 (119905 minus 119889

2(119905))

+ 119910119879

(119905 minus 1198892(119905)) 1198842(119879 + Σ) 119891 (119910 (119905 minus 119889

2(119905)))

minus 119891119879

(119910 (119905 minus 1198892(119905))) 119884

2119891 (119910 (119905 minus 119889

2(119905)))

0 = 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

le minus119910119879

(119905 minus 120591 (119905)) 1198791198845Σ119910 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 120591 (119905)) 1198845(119879 + Σ) 119891 (119910 (119905 minus 120591 (119905)))

minus 119891119879

(119910 (119905 minus 120591 (119905))) 1198845119891 (119910 (119905 minus 120591 (119905)))

0 = 119891119879

(119909 (119905)) 1198846119891 (119909 (119905)) minus 119891

119879

(119909 (119905)) 1198846119891 (119909 (119905))

le minus119909119879

(119905) 1198791198846Σ119909 (119905) + 119909

119879

(119905) 1198846(119879 + Σ) 119891 (119909 (119905))

minus 119891119879

(119909 (119905)) 1198846119891 (119909 (119905))

(37)

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

Next it follows from (9) and (27) that

tr [120578119879 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times (1198751+ 1198811) 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 119889

1(119905))) ]

le 120582max (1198751 + 1198811) [120578119879

(119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905)))

times 120578 (119905 119910 (119905 minus 120590 (119905)) 119909 (119905 minus 1198891(119905))) ]

le 119910119879

(119905 minus 120590 (119905)) 1205881Σ119879

1Σ1119910 (119905 minus 120590 (119905))

+ 119909119879

(119905 minus 1198891(119905)) 1205881Σ119879

2Σ2119909 (119905 minus 119889

1(119905))

tr [120578119879 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times (1198752+ 1198812) 120578 (119905 119910 (119905 minus 119889

2(119905)) 119909 (119905 minus 120591 (119905))) ]

le 120582max (1198752 + 1198812) [120578119879

(119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905)))

times 120578 (119905 119910 (119905 minus 1198892(119905)) 119909 (119905 minus 120591 (119905))) ]

le 119909119879

(119905 minus 120591 (119905)) 1205882Σ119879

4Σ4119909 (119905 minus 120591 (119905))

+ 119910119879

(119905 minus 1198892(119905)) 1205882Σ119879

3Σ3119910 (119905 minus 119889

2(119905))

(38)

Add both sides of (34) and (38) to both sides of (32) andapply (36)ndash(37) one can obtain that

F119881 (119905 119909119905 119910119905) le 120576119879

(119905) Υ120576 (119905) (39)

whereΥ = 120601+11987911198831198791+11987921198841198792+ℎ11198731119885minus1

1119873119879

1+ℎ31198721119885minus1

3119872119879

1+

ℎ51198781119885minus1

5119878119879

1+ℎ71198641119885minus1

7119864119879

1+Θ withΘ = (119889

1(119905)minusℎ

1)1198732119885minus1

2119873119879

2+

(ℎ2minus 1198891(119905))1198733119885minus1

2119873119879

3+ (1198892(119905) minus ℎ

5)1198782119885minus1

6119878119879

2+ (ℎ6minus

1198892(119905))1198783119885minus1

6119878119879

3+(120590(119905)minusℎ

3)1198722119885minus1

4119872119879

2+(ℎ4minus120590(119905))119872

3119885minus1

4119872119879

3+

(120591(119905) minus ℎ7)1198642119885minus1

8119864119879

2+ (ℎ8minus 120591(119905))119864

3119885minus1

8119864119879

3

Noting Assumption 2 Θ can be seen as the convexcombination of119873

2119885minus1

2119873119879

2and119873

3119885minus1

2119873119879

3on1198891(119905)119872

2119885minus1

4119872119879

2

and1198723119885minus1

4119872119879

3on 120590(119905) 119878

2119885minus1

6119878119879

2and 119878

3119885minus1

6119878119879

3on 1198892(119905) and

1198642119885minus1

8119864119879

2and 119864

3119885minus1

8119864119879

3on 120591(119905) Therefore Υ lt 0 holds if

Λ + ℎ121198732119885minus1

2119873119879

2lt 0

Λ + ℎ121198733119885minus1

2119873119879

3lt 0

Λ + ℎ341198722119885minus1

4119872119879

2lt 0

Λ + ℎ341198723119885minus1

4119872119879

3lt 0

Λ + ℎ561198782119885minus1

6119878119879

2lt 0

Λ + ℎ561198783119885minus1

6119878119879

3lt 0

Λ + ℎ781198642119885minus1

8119864119879

2lt 0

Λ + ℎ781198643119885minus1

8119864119879

3lt 0

(40)

where

Λ =

1

4

120601 + 11987911198831198791+ 11987921198841198792+ ℎ11198731119885minus1

1119873119879

1

+ ℎ31198721119885minus1

3119872119879

1+ ℎ51198781119885minus1

5119878119879

1+ ℎ71198641119885minus1

7119864119879

1

(41)

By Schur complements (40) is equivalent to 120601119894lt 0 119894 =

1 2 3 8 respectively Then

(119905 119909119905 119910119905) lt 0 (42)

On the other hand from (27) andTheorem 6 conditionswe note that

V1 (tk xt yt) = 2119899

sum

119894=1

120582119894int

119910119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

1minus120594119894119896119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

le 2

119899

sum

119894=1

120582119894int

119910119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

+ 2

119899

sum

119894=1

120582119894int

119909119894(119905minus

119896)

0

(119891119894(119904) minus 120574

119894119904) 119889119904

= 1198811(119905minus

119896 119909119905 119910119905)

(43)

Moreover it is obvious that 1198812(119905119896 119909119905 119910119905) = 1198812(119905minus

119896 119909119905 119910119905)

1198813(119905119896 119909119905 119910119905) = 119881

3(119905minus

119896 119909119905 119910119905) Hence we get 119881(119905

119896 119909119905 119910119905) le

119881(119905minus

119896 119909119905 119910119905) By Lyapunov-Krasovskii stability theorem the

equilibrium point of (8) is stable in the mean square Theproof is completed

Remark 7 For the UISGRNs (8) without stochastic distur-bances leakage delay and impulsive effects it reduces to themodel of [27] And when themodel of (8) without impulsiveit reduces to themodel of [30] In addition it is easy to see thatthe main theorem obtained above covers the sparse resultsavailable in the literature in the concern of only one or two ofthe complex dynamics generally being involved with GRNsleakage delays parameter uncertainties impulsive effectsand stochastic disturbances

We give a couple of corollaries below in order to showfurther that our main result is general enough to cover twocases that have not been investigated in the literature Hencethey are new and significant Firstly for model (6) or theUISGRNs (8) without parameter uncertainties (ie Δ119860(119905) =Δ119882(119905) = Δ119862(119905) = Δ119871(119905) = 0) we have the followingcorollary

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Abstract and Applied Analysis

Corollary 8 If (7) (9) and Assumption 1 hold there exist120583 ge 0 120582 ge 0 120588

1gt 0 120588

2gt 0 120594im isin [0 1] 119896 = 0 1

119903 + 2 119894 = 1 119899 119898 isin 119885+ such that the impulsive operator

119869119898(sdot) satisfies 119869

119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898)The system (6) is stable

in the mean square if there exist real matrices 1198751gt 0 119875

2gt 0

119876119894gt 0 (119894 = 1 2 16) 119885

119894gt 0 (119894 = 1 2 8) 119881

1gt 0

and 1198812gt 0 diagonal matrices 119884

119894gt 0 (119894 = 1 2 6) and

any matrices 11987311 11987312 11987321 119873221198721111987212119872211198722211987231

11987232 11987811 11987812 11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and

11986432 to satisfy conditions (16)ndash(25) replaced accordingly by the

following

12060112= minus1198751119860 minus 119879119881

2119860 minus 119873

11+ 11987312+ 11987313minus 11987321+ 11986412

120601117

= minus1198791198812119882+ 119875

1119882 120601

215= minus1198812119860

12060158= minus119879119881

1119871 + 1198752119871 120601

516= 1198811119871

120601812

= minus1198752119862 + 119879119881

1119862 120601

1216= minus1198811119862

1206011517

= 1198812119882

1198791= [0 minus119860 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119882 0 0 0]

119879

1198792= [0 0 0 119871 0 0 0 0 0 0 0 minus119862 0 0 0 0 0 0 0 0]

119879

(44)

In the second case we suppose that there are no stochasticdisturbances in the UISGRNs model (8) Hence model (8)can be reduced to

(119905) = minus (119860 + Δ119860) 119909 (119905 minus 1198891(119905))

+ (119882 + Δ119882)119891 (119910 (119905 minus 120590 (119905)))

Δ119909 (119905) |119905=119905119896

= 119909 (119905119896) minus 119909 (119905

minus

119896) = 119869119896(119909 (119905minus

119896))

119896 isin 119885+

119905 = 119905119896

119910 (119905) = minus (119862 + Δ119862) 119910 (119905 minus 1198892(119905)) + (119871 + Δ119871) 119909 (119905 minus 120591 (119905))

Δ119910 (119905) |119905=119905119896

= 119910 (119905119896) minus 119910 (119905

minus

119896) = 119869119896(119910 (119905minus

119896))

119905 = 119905119896 119896 isin 119885

+

1199090= 119909 (120579) = 120593 (120579) 119910

0= 119910 (120579) = 120595 (120579)

forall120579 isin [minus120596 0]

(45)

Then we have the following new result

Corollary 9 If (7) and Assumptions 1 and 2 hold thereexist 120583 ge 0 120582 ge 0 120594im isin [0 1] 119896 = 0 1 119903 + 2119894 = 1 119899 119898 isin 119885

+ such that the impulsive operator 119869119898(sdot)

satisfies 119869119894119898(119909119894(119905119898)) = minus120594

119894119898119909119894(119905119898) The system (45) is stable

if there exist real matrices 1198751gt 0 119875

2gt 0 119876

119894gt 0 (119894 =

1 2 16) 119885119894gt 0 (119894 = 1 2 8) 119881

1gt 0 and 119881

2gt 0

diagonal matrices 119884119894gt 0 (119894 = 1 2 6) and any matrices

11987311 11987312 11987321 11987322119872111198721211987221119872221198723111987232 11987811 11987812

11987821 11987822 11987831 11987832 11986411 11986412 11986421 11986422 11986431 and 119864

32 to satisfy

conditions (18)ndash(25) replaced accordingly by the following

12060122= minus (1 minus 119889

1) 1198761minus 211987312minus 211987322+ 211987232

12060155= minus (1 minus 120591)119876

13minus 1198791198845Σ

12060199= minus (1 minus 120590)119876

5minus 1198791198843Σ

1206011212

= minus (1 minus 1198892) 1198769minus 1198791198844Σ

(46)

4 Numerical Examples

In this section we present three numerical examples so as toillustrate the usefulness of our results derived in this paper

Example 1 Let us firstly consider the system (8) with param-eters as follows

119862 = [

1 0

0 1] A = [11 0

0 11]

L = [08 0

0 08] W = [

08 0

0 08]

Σ1= Σ2= [

01667 0

0 01067]

119868 = [

1 0

0 1] G1 = 02 lowast 119868 G2 = 02 lowast 119868

Ha = Hw = Hc = Hl = 002 lowast 119868

1198651= 05 lowast 119868 119865

2= 06 lowast 119868

(47)

For convenience we assume that 120594119894119898= 13 119889

1(119905) = 01 +

005 sin(119905) 120590(119905) = 02 + 003 cos(119905) 1198892(119905) = 03 + 001 cos(119905)

120591(119905) = 04 + 005 cos(119905) and f1(x) = f2(x) = 1(1 + x2) thenwe can obtain

ℎ1= 005 ℎ

2= 015 ℎ

3= 017

ℎ4= 023 ℎ

5= 029 ℎ

6= 031

ℎ7= 035 ℎ

8= 045

1198891= 05 119889

2= 06 120591 = 07

120590 = 08 Σ = [

1 0

0 1] 119879 = 0

(48)

Using MATLAB LMI control toolbox and by solving theLMIs (16) inTheorem 6 we can obtain the feasible solutionsDue to space limitations we do not list them here We findthat the delayed UISGRNs (8) are stable in the mean squarewhich is shown in Figure 1

Example 2 Consider the system (6) with the parameters thatare the same as in Example 1 other than

119867119886= 119867119888= 119867119908= 119867119897= 0 (49)

We can find feasible solutions for the LMIs in Corollary8 so the impulsive stochastic GRNs (6) are stable in themeansquare which can be shown in Figure 2

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 13

minus5 0 5 10 15 20 25 30 35 40

02

03

04

05

06

07

08

09

1

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

0

02

04

06

08

1

12

14

16

18

t

y

(b)

Figure 1 Trajectories of 119909(119905) and 119910(119905) of the genetic network (8) with randomly chosen initial values

minus5 0 5 10 15 20 25 30 35 40minus05

0

05

1

15

2

25

3

t

x

(a)

minus5 0 5 10 15 20 25 30 35 40

t

minus05

0

05

1

15

2

25

3

35

y

(b)

Figure 2 Trajectories of 119909(119905) and 119910(119905) of the genetic network (6) with randomly chosen initial values

Example 3 When the parameters in (45) are the same as inExample 1 in addition to

120578 = 0 (50)

It is easy to see that the uncertain impulsiveGRNs (45) arestable in themean square by checking Corollary 9 conditionsFigure 3 shows that the result is valid

5 Conclusions

In this paper we have investigated the stability problem for anew UISGRNs model with the introduction of leakage delayBy employing the Lyapunov stability theory free-weightingmatrix and convex combination technique combining withstochastic stability approach and the LMI frameworkwe have

obtained a sufficient condition to justify the stability of theproposed UISGRNs model The obtained stability conditionis expressed in terms of LMIswhich can be easily solved by theefficient MATALB LMI toolbox Finally numerical exampleshave been provided to illustrate the usefulness of the derivedstability results

Conflict of Interests

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

Acknowledgments

This work was jointly supported by the FundamentalResearch Funds for the Central Universities (JUSRP51317B

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Abstract and Applied Analysis

minus5 0 5 10 15 20 25 30 35 40

04

05

06

07

08

09

1

t

x

(a)

04

05

06

07

08

09

1

minus5 0 5 10 15 20 25 30 35 40

t

y

(b)

Figure 3 Trajectories of 119909(119905) and 119910(119905) of the genetic network (45) with randomly chosen initial values

JUSRP211A21) the National Natural Science Foundation ofChina under Grants 61272530 and 11072059 the NaturalScience Foundation of Jiangsu Province of China underGrant BK2012741 and the Specialized Research Fund forthe Doctoral Program of Higher Education under Grants20110092110017 and 20130092110017

References

[1] A Becskel and L Serrano ldquoEngineering stability in gene net-works by autoregulationrdquo Nature vol 405 no 6786 pp 590ndash593 2000

[2] M B Elowitz and S Leibier ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 6767 pp 335ndash338 2000

[3] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[4] H H McAdams and L Shapiro ldquoCircuit simulation of geneticnetworksrdquo Science vol 269 no 5224 pp 650ndash656 1995

[5] L Chen and K Aihara ldquoStability of genetic regulatory networkswith time delayrdquo IEEE Transactions on Circuits and Systems vol49 no 5 pp 602ndash608 2002

[6] DW Austin M S Allen J MMcCollum et al ldquoGene networkshaping of inherent noise spectrardquoNature vol 439 no 7076 pp608ndash611 2006

[7] C Chaouiya E Remy and D Thieffry ldquoPetri net modelling ofbiological regulatory networksrdquo Journal of Discrete Algorithmsvol 6 no 2 pp 165ndash177 2008

[8] SHardy andPN Robillard ldquoModeling and simulation ofmole-cular biology systems using petri nets modeling goals of vari-ous approachesrdquo Journal of Bioinformatics and ComputationalBiology vol 2 no 4 pp 595ndash613 2004

[9] W H Hsu ldquoGenetic wrappers for feature selection in decisiontree induction and variable ordering in Bayesian networkstructure learningrdquo Information Sciences vol 163 no 1ndash3 pp103ndash122 2004

[10] N Friedman M Linial I Nachman and D Persquoer ldquoUsingBayesian networks to analyze expression datardquo Journal ofComputational Biology vol 7 no 3-4 pp 601ndash620 2000

[11] A J Hartemink D K Gifford T S Jaakkola and R AYoung ldquoBayesian methods for elucidating genetic regulatorynetworksrdquo IEEE Intelligent Systems and Their Applications vol17 no 2 pp 37ndash43 2002

[12] B G Marcot ldquoMetrics for evaluating performance and uncer-tainty of Bayesian network modelsrdquo Ecological Modelling vol230 pp 50ndash62 2012

[13] R Somogyi and C A Sniegoski ldquoModeling the complexityof genetic networks understanding multigenic and pleiotropicregulationrdquo Complexity vol 1 no 6 pp 45ndash63 1996

[14] D Weaver C Workman and G Stormo ldquoModeling regulatorynetworks with weight matricesrdquo in Proceedings of the PacificSymposium on Biocomputing (PSB rsquo99) vol 4 pp 112ndash123 BigIsland of Hawaii Hawaii USA 1999

[15] M Chaves R Albert and E D Sontag ldquoRobustness andfragility of Boolean models for genetic regulatory networksrdquoJournal of Theoretical Biology vol 235 no 3 pp 431ndash449 2005

[16] H Bolouri and EHDavidson ldquoModeling transcriptional regu-latory networksrdquo BioEssays vol 24 no 12 pp 1118ndash1129 2002

[17] H de Jong ldquoModeling and simulation of genetic regulatorysystems a literature reviewrdquo Journal of Computational Biologyvol 9 no 1 pp 67ndash103 2002

[18] P Smolen D A Baxter and J H Byrne ldquoMathematical model-ing of gene networksrdquoNeuron vol 26 no 3 pp 567ndash580 2000

[19] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[20] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011

[21] M Hu J Cao and Y Yang ldquoStability of genetic networks withhybrid regulatorymechanismrdquoArabian Journal ofMathematicsvol 1 no 3 pp 319ndash328 2012

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 15

[22] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[23] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011

[24] T Tian K Burrage P M Burrage and M Carletti ldquoStochasticdelay differential equations for genetic regulatory networksrdquoJournal of Computational andAppliedMathematics vol 205 no2 pp 696ndash707 2007

[25] K A Pavlov D A Chistiakov and V P Chekhonin ldquoGeneticdeterminants of aggression and impulsivity in humansrdquo Journalof Applied Genetics vol 53 no 1 pp 61ndash82 2012

[26] F Li and J Sun ldquoAsymptotic stability of a genetic network underimpulsive controlrdquoPhysics Letters A vol 374 no 31-32 pp 3177ndash3184 2010

[27] HWuX LiaoW Feng S Guo andWZhang ldquoRobust stabilityfor uncertain genetic regulatory networks with interval time-varying delaysrdquo Information Sciences vol 180 no 18 pp 3532ndash3545 2010

[28] W Wang S Zhong S Nguang and F Liu ldquoNovel delay-dependent stability criterion for uncertain genetic regulatorynetworks with interval time-varying delaysrdquo Neurocomputingvol 121 pp 170ndash178 2013

[29] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stability ana-lysis of Markovian switching uncertain stochastic genetic reg-ulatory networks with unbounded time-varying delaysrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 10 pp 3894ndash3905 2012

[30] G Wang and J Cao ldquoRobust exponential stability analysis forstochastic genetic networks with uncertain parametersrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 8 pp 3369ndash3378 2009

[31] R Sakthivel R Raja and SM Anthoni ldquoAsymptotic stability ofdelayed stochastic genetic regulatory networks with impulsesrdquoPhysica Scripta vol 82 no 5 Article ID 055009 2010

[32] C Li L Chen andKAihara ldquoStability of genetic networkswithSUM regulatory logic lurrsquoe system and LMI approachrdquo IEEETransactions on Circuits and Systems vol 53 no 11 pp 2451ndash2458 2006

[33] S Blythe X Mao and X Liao ldquoStability of stochastic delayneural networksrdquo Journal of the Franklin Institute vol 338 no4 pp 481ndash495 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of