Research ArticleFinite-Time Stability Analysis of Switched GeneticRegulatory Networks

Lizi Yin12

1 School of Control Science and Engineering Shandong University Jinan 250061 China2 School of Mathematical Sciences University of Jinan Jinan 250022 China

Correspondence should be addressed to Lizi Yin ss yinlzujneducn

Received 11 October 2013 Revised 15 December 2013 Accepted 15 December 2013 Published 19 January 2014

Academic Editor Qiankun Song

Copyright copy 2014 Lizi Yin 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 investigates the finite-time stability problemof switching genetic regulatory networks (GRNs)with interval time-varyingdelays and unbounded continuous distributed delays Based on the piecewise Lyapunov-Krasovskii functional and the average dwelltime method some new finite-time stability criteria are obtained in the form of linear matrix inequalities (LMIs) which are easyto be confirmed by the Matlab toolbox The finite-time stability is taken into account in switching genetic regulatory networks forthe first time and the average dwell time of the switching signal is obtained Two numerical examples are presented to illustrate theeffectiveness of our results

1 Introduction

In the last decade or so the genetic regulatory networks(GRNs) have become an important research area in molec-ular Biology On the one hand how to construct GRNsfrom gene expression data on the other hand what is thedynamic characteristics of gene regulatory networks Thestability is one of vital dynamic characteristics of GRNs andis researched in this paperThere are also many references onthe stability of GRNs [1ndash6]

High throughput biological experiments have proved thattime delays were ubiquitous in GRNs The existence of timedelays influences the stability of GRNs which can give riseto oscillatory or unstable networks Therefore it is necessaryto study the influences of time delays for the stability ofGRNs There are a few theoretical results on GRNs withtime delays [7ndash15] In [2] random time delays are takeninto account and some stability criteria for the uncertaindelayed genetic networks with SUM regulatory logic whereeach transcription factor acts additively to regulate a genewere obtained In [12] some stochastic asymptotic stabilityconditions were established for a class of uncertain stochasticgenetic regulatory networks with mixed time-varying delaysby constructing appropriate Lyapunov-Krasovskii function-als and employing stochastic analysis method In [13 14]

authors studied GRNs with constant delay and asymptoticalstability criteria were proposed for GRNs with interval time-varying delays and nonlinear disturbance in [10]

Cell-cycle regulatory processes can be viewed as finite-state processes So some complex GRNs were describedby continuous time switched system Usually a finite stateMarkov chain was used to simulate this switch process [16ndash19] In [17] authors investigate the global robust stability ofuncertain stochastic GRNs with Markovian switching pro-cess In [19] control theory andmathematical tools were usedto analyze passivity for the stochastic Markovian switchingGRNs with time-varying delays Some other methods arealso used to prove switched GRNsrsquo stability In the literature[20] authors used an average dwell time approach to considerexponential stability of switched GRNs with time delays

Both procedures of these proteins regulate gene expres-sion and gene translated protein that are sometimes accom-plished in a relatively short period of time So it is realisticallysignificant to research the finite-time stability of GRNs andsome articles have researched the finite-time stability [21 22]There are few references about the finite-time stability ofGRNswe research only the literature [23] In [23] the authorsconsidered finite-time robust stability of uncertain stochasticreaction-diffusion GRNs with time delays The finite-timestability of GRNs is the main contents in our paper

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 730292 11 pageshttpdxdoiorg1011552014730292

2 Journal of Applied Mathematics

Motivated by the above discussions we analyze finite-time stability of switching genetic regulatory networks withinterval time-varying delays and continuous distributeddelays Using a novel piecewise Lyapunov-Krasovskii func-tional and finite-time stable definite some new finite-timestability criteria are obtained for the switched GRNs Thefeatures of this paper can be summarized as follows (1)the finite-time stability is researched in the switched GRNsfirstly (2) continuous distributed delays are concerned (3)all sufficient conditions obtained depend on the time delays

This paper is organized as follows In Section 2 modeldescription some assumptions definitions and lemmas aregiven In Section 3 some conditions are obtained to ensurethe finite-time stability of switched GRNs with intervaltime-varying delays and continuous distributed delays Twonumerical examples are given to demonstrate the effective-ness of our analysis in Section 4 Finally conclusions aredrawn in Section 5

Notations Throughout this paper R R119899 and R119899times119898 denoterespectively the set of all real numbers real 119899-dimensionalspace and real 119899 times 119898-dimensional space Z

+denote the set

of all positive integers sdot denote the Euclidean normsin R119899 For a vector or matrix 119860 119860119879 denotes its transposeFor a square matrix 119860 120582max(119860) and 120582min(119860) denote themaximum eigenvalue and minimum eigenvalue of matrix 119860respectively and sym (119860) is used to represent 119860 + 119860

119879 Forsimplicity in symmetric block matrices we often use lowast torepresent the term that is induced by symmetry

2 Problem Formulation andSome Preliminaries

We consider the following genetic regulatory networks

(119905) = minus119860119898 (119905) + 119861119891 (119901 (119905 minus 1205911(119905))) + 119868

(119905) = minus119862119875 (119905) + 119863119898 (119905 minus 1205912(119905))

(1)

where 119898(119905) = [1198981(119905) 119898

119899(119905)]119879 is the concentrations of

mRNAs 119901(119905) = [1199011(119905) 119901

119899(119905)]119879 is the concentrations of

proteins 119891(sdot) = [1198911(sdot) 119891

119899(sdot)]119879 is the regulatory functions

of mRNAs 119860 = diag(1198861 119886

119899) is the degradation rates

of mRNAs 119862 = diag(1198881 119888

119899) is the degradation rates of

proteins 119863 = diag(1198891 119889

119899) is the translation rates of

proteins 119868 = [1198681 119868

119899]119879 is the basal rate and 120591

1(119905) and 120591

2(119905)

are time-varying delaysFor obtaining our conclusions we make the following

assumptions

Assumption 1 119891119894 R rarr R 119894 = 1 119899 are monotonically

increasing functions with saturation and satisfy

0 le119891119894(119886) minus 119891

119894(119887)

119886 minus 119887le 119906119894 forall119886 119887 isin R 119894 = 1 119899 (2)

where 119906119894 119894 = 1 119899 are nonnegative constants

Assumption 2 1205911(119905) and 120591

2(119905) are time-varying delays satisfy-

ing

0 le 12059111le 1205911(119905) le 120591

12 120591

1(119905) le 120591

13lt infin

0 le 12059121le 1205912(119905) le 120591

22 120591

2(119905) le 120591

23lt infin

(3)

Vectors119898lowast 119901lowast are an equilibrium point of the system (1)Let 119909(119905) = 119898(119905) minus 119898lowast 119910(119905) = 119901(119905) minus 119901lowast we get

(119905) = minus119860119909 (119905) + 119861119892 (119910 (119905 minus 1205911(119905)))

119910 (119905) = minus119862119910 (119905) + 119863119909 (119905 minus 1205912(119905))

(4)

where 119909(119905) = [1199091(119905) 119909

119899(119905)]119879 119910(119905) = [119910

1(119905) 119910

119899(119905)]119879

119892(sdot) = [119892(sdot) 119892(sdot)]119879 and 119892

119894(119910119894(119905)) = 119891

119894(119910119894(119905)+119901lowast

119894) minus119891119894(119901lowast119894)

According to Assumption 1 and the definition of 119892119894(sdot) we

know that 119892119894(sdot) is bounded that is exist119866 gt 0 such that |119892

119894(sdot)| le

119866 119894 = 1 119899 and satisfies the following sector condition

0 le119892119894(119886)

119886le 119906119894 forall119886 isin R 0 119894 = 1 119899 (5)

Let 119880 = diag(1199061 119906

119899)

Sometimes GRNs were described by continuous timeswitched system as in [17 24 25] so system (4) can bedescribed as switching system with switching signal

(119905) = minus119860120590(119905)

119909 (119905) + 119861120590(119905)

119892 (119910 (119905 minus 1205911(119905)))

119910 (119905) = minus119862120590(119905)

119910 (119905) + 119863120590(119905)

119909 (119905 minus 1205912(119905))

(6)

where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal which is a piecewise constant function depending ontime 119905 For each 119894 isin N the matrices 119860

119894 119861119894 119862119894 and 119863

119894

are the 119894th subsystem matrices that are constant matrices ofappropriate dimensions

For the switching signal 120590(119905) we have the followingswitching sequence

(1198940 1199050) (119894

119896 119905119896) | 119894

119896isin N 119896 = 0 1 (7)

in otherwords when 119905 isin [119905119896 119905119896+1

) 119894119896th subsystem is activated

The initial condition of system (6) is assumed to be

119909 (119905) = 120593 (119905) 119910 (119905) = 120595 (119905)

minus120588 le 119905 le 0 120588 = max 12059112 12059122

(8)

For proving the theorem we recall the following defini-tion and lemmas

Definition 3 The system (6) is said to be finite-time stablewith respect to positive real numbers (119888

1 1198882 119879) if

1003817100381710038171003817120601 (119905)10038171003817100381710038172

1198621 +

1003817100381710038171003817120593 (119905)10038171003817100381710038172

1198621 le 1198881997904rArr 119909 (119905)

2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882

forall119905 isin [0 119879] (9)

where 120601(119905)1198881 = sup

minus120588le119905le0120601(119905) 120601(119905) 120593(119905)

1198881 =

supminus120588le119905le0

120593(119905) (119905)

Journal of Applied Mathematics 3

Remark 4 The Lyapunov stability implies that by startingsufficiently close to the equilibrium point trajectories canbe guaranteed to stay within any specified ball centered atthe equilibrium point It depicts character of the equilibriumpoint of the system The finite-time stability implies thatby starting from any specified compact set trajectories canbe guaranteed to stay within a large enough compact set Itdepicts character of all solutions of the system and is calledthe finite-time boundedness in some literatures [21 22] Inaddition the Lyapunov stability is considered in infiniteinterval but the finite-time stability is considered in finiteinterval

Remark 5 In [23] the authors considered the finite-timerobust stability of GRNs but in our paper we consideredthe finite-time stability of switching GRNs which are morerealisticGRNs than that of the [23] For the first time switchedGRNsrsquo finite-time stability is researched

Definition 6 For any 119879 ge 119905 ge 0 let 119873120590(119905 119879) denote the

number of switching of 120590(119905) over (119905 119879) If

119873120590(119905 119879) le 119873

0+119879 minus 119905

120591119886

(10)

holds for 1198730ge 0 120591

119886gt 0 then 120591

119886gt 0 is called the average

dwell time and 1198730is the chatter bound As commonly used

in the literature we choose1198730= 0

Lemma 7 (see [26]) For any positive definite matrix 119872 isin

R119899times119899 there exist a scalar 119902 gt 0 and a vector-valued function120596 [0 119902] rarr R119899 such that

(int119902

0

120596 (119904) 119889119904)

119879

119872(int119902

0

120596 (119904) 119889119904) le 119902int119902

0

120596119879

(119904)119872120596 (119904) 119889119904

(11)

3 Main Results

In this section we present a finite time stability theoremfor switching genetic regulatory networks with interval time-varying delays (6)

Theorem 8 For switching system (6) with Assumptions 1 and2 given a scalar 120572 gt 0 if there exist symmetric positive definitematrices119875

1119894119875211989411987611198941198771119894119877211989411987731198941198774119894 for all 119894 isin N the diagonal

matrix Λ119898= diag(120582

1198981 120582

119898119899) ge 0 119898 = 1 2 matrices 119876

2119894

1198763119894 1198721119894 119894 isin 119873 and positive scalars 120583 ge 1 such that the

following inequalities

[1198762119894

1198721119894

lowast 1198763119894

] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2

[1198762119894

1198721119894

lowast 1198763119894

] le 120583 [1198762119895

1198721119895

lowast 1198763119895

] 119877119897119894le 120583119877119897119895

119897 = 1 4 forall119894 119895 isin N

(12)

Ξ = Ω

+ sym 119871119879

111987711198941198712+ 119871119879

1

times [1198751119894119861119894minus 1205912

21119861119879

1198941198771119894119860119894minus (12059122minus 12059121)2

119861119879

1198941198772119894119860119894]

times 11987110+ 119871119879

211987721198941198713+ 119871119879

311987721198941198714+ 119871119879

3

times [1198752119894119863119894minus 1205912

11119863119879

1198941198773119894119862119894minus (12059112minus 12059111)2

119863119879

1198941198774119894119862119894]

times1198715+ 119871119879

511987731198941198716+ 119871119879

5(1198721119894+ Λ1119880) 1198719+119871119879

611987741198941198717

+119871119879

711987741198941198718+ 119871119879

7[1198721119894(1 minus 120591

13) + Λ

2119880] 11987110

lt 0 forall119894 isin N(13)

hold and the average dwell time of the switching signal 120590(119905)satisfies

120591119886gt 120591lowast

119886=

119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879

(14)

where Ω = diag(minus21198751119894119860119894+ 1198761119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 120572119875

1119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591

23) + 120591211119863119879119894

1198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus21198772119894 minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894

1198773119894119862119894+ (12059112minus 12059111)2

1198621198791198941198774119894119862119894minus1198773119894minus 1205721198752119894 minus1198773119894minus1198774119894minus1198762119894(1 minus

12059113) minus 2119877

4119894 minus11987741198941198763119894minus 2Λ

1 minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus2Λ2) 119871119895=[0119899times(119895minus1)119899

119868119899times119899

0119899times(11minus119895)119899

] 119895 =1 10 120582

1= max

119894isinN120582max(1198751119894) + 12059122120582max(1198761119894) + 120582max

(1198752119894) + 120591

12120582max(1198762119894) + 120591

12120582max(1198763119894) 120582max(119880119880) + 2120591

12

120582max(1198721119894) 120582max(119880) + (12) [120591321120582max(1198771119894) + (120591

22minus 12059121)3

120582max(1198772119894) + 1205913

11120582max(1198773119894) + (120591

12minus 12059111)3

120582max(1198774119894)] 1205822 =

min119894isinN[min(120582min(1198751119894) 120582min(1198752119894))] The equilibrium point of

(6) is finite time stable with respect to positive real numbers(1198881 1198882 119879)

Proof Based on the system (6) we construct the followingLyapunov-krasovskii functional

119881120590(119905)

(119905) = 1198811120590(119905)

(119905) + 1198812120590(119905)

(119905) + 1198813120590(119905)

(119905) (15)

where

1198811120590(119905)

(119905) = 119909119879

(119905) 1198751120590(119905)

119909 (119905) + 119910119879

(119905) 1198752120590(119905)

119910 (119905)

1198812120590(119905)

(119905) = int119905

119905minus1205912(119905)

119909119879

(119904) 1198761120590(119905)

119909 (119905) 119889119904

+ int119905

119905minus1205911(119905)

[119910119879

(119904) 119892119879

(119910 (119904))] [1198762120590(119905)

1198721120590(119905)

1198721120590(119905)

1198763120590(119905)

]

times [119910 (119904)

119892 (119910 (119904))] 119889119904

4 Journal of Applied Mathematics

1198813120590(119905)

(119905)

= int0

minus12059121

int119905

119905+120579

12059121119879

(119904) 1198771120590(119905)

(119904) 119889119904 119889120579

+ intminus12059121

minus12059122

int119905

119905+120579

(12059122minus 12059121) 119879

(119904) 1198772120590(119905)

(119904) 119889119904 119889120579

+ int0

minus12059111

int119905

119905+120579

12059111

119910119879

(119904) 1198773120590(119905)

119910 (119904) 119889119904 119889120579

+ intminus12059111

minus12059112

int119905

119905+120579

(12059112minus 12059111) 119910119879

(119904) 1198774120590(119905)

119910 (119904) 119889119904 119889120579

(16)

First when 119905 isin [119905119896 119905119896+1

) taking the derivatives of119881119894 119894 = 1 2 3

along the trajectory of system (6) we have that

1120590(119905119896)(119905) = 2119909

119879

(119905) 1198751120590(119905119896) (119905) + 2119910

119879

(119905) 1198752120590(119905119896)119910 (119905)

2120590(119905119896)(119905) = 119909

119879

(119905) 1198761120590(119905119896)119909 (119905) minus 119909

119879

(119905 minus 1205912(119905)) 119876

1120590(119905119896)119909119879

times (119905 minus 1205912(119905)) (1 minus 120591

2(119905))

+ [119910119879

(119905) 119892119879

(119910 (119905))] [1198762120590(119905119896)1198721120590(119905119896)

lowast 1198763120590(119905119896)

]

times [119910 (119905)

119892 (119910 (119905))]

minus [119910119879

(119905 minus 1205911(119905)) 119892

119879

(119910 (119905 minus 1205911(119905)))]

times [1198762120590(119905119896)1198721120590(119905119896)

lowast 1198763120590(119905119896)

] [119910 (119905 minus 120591

1(119905))

119892 (119910 (119905 minus 1205911(119905)))

]

times (1 minus 1205911(119905))

3120590(119905119896)(119905) = 120591

2

21119879

(119905) 1198771120590(119905119896) (119904)

minus int119905

119905minus12059121

12059121119879

(120579) 1198771120590(119905119896) (120579) 119889120579

+ (12059122minus 12059121)2

119879

(119905) 1198772120590(119905119896) (119904)

minus int119905minus12059121

119905minus12059122

(12059122minus 12059121) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

+ 1205912

11119910119879

(119905) 1198773120590(119905119896)119910 (119904)

minus int119905

119905minus12059111

12059111

119910119879

(120579) 1198773120590(119905119896)119910 (120579) 119889120579

+ (12059112minus 12059111)2

119910119879

(119905) 1198774120590(119905119896)119910 (119904)

minus int119905minus12059111

119905minus12059112

(12059112minus 12059111) 119910119879

(120579) 1198774120590(119905119896)119910 (120579) 119889120579

(17)

By Lemma 7 we have

minus int119905

119905minus12059121

12059121119879

(120579) 1198771120590(119905119896) (120579) 119889120579

le minus[119909 (119905) minus 119909 (119905 minus 12059121)]119879

1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591

21)]

(18)

minus int119905

119905minus12059111

12059111

119910119879

(120579) 1198773120590(119905119896)119910 (120579) 119889120579

le minus[119910 (119905) minus 119910 (119905 minus 12059111)]119879

1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591

11)]

(19)

minus int119905minus12059121

119905minus12059122

(12059122minus 12059121) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

= minus (12059122minus 12059121) [int119905minus1205912(119905)

119905minus12059122

119879

(120579) 1198772120590(119905119896) (120579) 119889120579

+ int119905minus12059121

119905minus1205912(119905)

119879

(120579) 1198772120590(119905119896) (120579) 119889120579]

= minusint119905minus1205912(119905)

119905minus12059122

(12059122minus 1205912(119905)) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus12059121

119905minus1205912(119905)

(1205912(119905) minus 120591

21) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus1205912(119905)

119905minus12059122

(1205912(119905) minus 120591

21) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus12059121

119905minus1205912(119905)

(12059122minus 1205912(119905)) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

le minus[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

minus1205912(119905) minus 120591

21

12059122minus 12059121

[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus12059122minus 1205912(119905)

12059122minus 12059121

[119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

(20)

Journal of Applied Mathematics 5

Similar to (20) we can obtain

minus int119905minus12059111

119905minus12059112

(12059112minus 12059111) 119910119879

(120579) 1198774120590(119905119896)119910 (120579) 119889120579

le minus[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

minus1205911(119905) minus 120591

11

12059112minus 12059111

[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus12059112minus 1205911(119905)

12059112minus 12059111

[119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

(21)

In addition for any Λ119898= diag(120582

1198981 120582

119898119899) ge 0 119898 = 1 2

the following inequality is true from (5)

minus 2

119899

sum119894=1

1205821119894119892119894(119910119894(119905)) [119892

119894(119910119894(119905)) minus 119906

119894119910119894(119905)]

minus 2

119899

sum119894=1

1205822119894119892119894(119910119894(119905 minus 1205911(119905)))

times [119892119894(119910119894(119905 minus 1205911(119905))) minus 119906

119894119910119894(119905 minus 1205911(119905))] ge 0

(22)

It can be written as matrix form

minus 2119892119879

(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)] minus 2119892

119879

(119910 (119905 minus 1205911(119905))) Λ

2

times [119892 (119910 (119905 minus 1205911(119905))) minus 119880119910 (119905 minus 120591

1(119905))] ge 0

(23)

From (15) to (23) we have that

120590(119905119896)(119905)

le 2119909119879

(119905) 1198751120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ 2119910119879

(119905) 1198752120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

+ 119909119879

(119905) 1198761120590(119905119896)119909 (119905) minus 119909

119879

(119905 minus 1205912(119905)) 119876

1120590(119905119896)119909119879

times (119905 minus 1205912(119905)) (1 minus 120591

2119889) + 119910119879

(119905) 1198762120590119910 (119905)

+ 2119910119879

(119905)1198721120590119892 (119910 (119905)) + 119892

119879

(119910 (119905)) 1198763120590(119905119896)119892 (119910 (119905))

minus 119910119879

(119905 minus 1205911(119905)) 119876

2120590(119905119896)119910 (119905 minus 120591

1(119905)) (1 minus 120591

1119889)

minus 2119910119879

(119905 minus 1205911(119905))119872

1120590(119905119896)119892 (119910 (119905 minus 120591

1(119905))) (1 minus 120591

1119889)

minus 119892119879

(119910 (119905 minus 1205911(119905))) 119876

3120590(119905119896)119892 (119910 (119905 minus 120591

1(119905))) (1 minus 120591

1119889)

+ 1205912

21[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]119879

times 1198771120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ (12059122minus 12059121)2

[minus119860120590(119905119896)119909(119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]119879

times 1198772120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ 1205912

11[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]119879

times 1198773120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

+ (12059112minus 12059111)2

[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]119879

times 1198774120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

minus [119909 (119905) minus 119909 (119905 minus 12059121)]119879

1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591

21)]

minus [119910 (119905) minus 119910 (119905 minus 12059111)]119879

1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591

11)]

minus [119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

minus [119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

minus 2119892119879

(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)]

minus 2119892119879

(119910 (119905 minus 1205911(119905)))

times Λ2[119892 (119910 (119905 minus 120591

1(119905))) minus 119880119910 (119905 minus 120591

1(119905))]

le 120585119879

Ξ120585 + 120572 (119909119879

(119905) 1198751120590(119905119896)119909 (119905) + 119910

119879

(119905) 1198752120590(119905119896)119910 (119905))

le 120585119879

Ξ120585 + 120572119881120590(119905119896)(119905)

(24)

where

120585119879

= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))]

(25)

6 Journal of Applied Mathematics

By condition (13) we have

120590(119905119896)(119905) le 120572119881

120590(119905119896)(119905) (26)

Integrating (26) from 119905119896to 119905 we obtain that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

119881120590(119905119896)(119905119896) (27)

Using (12) at switching instant 119905119896 we have

119881120590(119905119896)(119905119896) le 120583119881

120590(119905minus

119896)(119905minus

119896) (28)

Therefore it follows from (27) and (28) that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

120583119881120590(119905minus

119896)(119905minus

119896) (29)

For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873

120590(0 119879) le 119879120591

119886 we

have

119881120590(119905)

(119905) le 119890120572119905

120583119879120591119886119881120590(0)

(0) (30)

On the other hand it follows from (15) that

119881120590(0)

(0)

le max119894isinN

[120582max (1198751119894) + 12059122120582max (1198761119894)]

times supminus120588le119905le0

1003817100381710038171003817120593 (119905)10038171003817100381710038172

+ [120582max (1198752119894) + 12059112120582max (1198762119894)

+ 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)]

times supminus120588le119905le0

1003817100381710038171003817120595 (119905)10038171003817100381710038172

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

times 120582max (1198772119894)]

times supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

+1

2[1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

times120582max (1198774119894)] supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

le max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)] 1198881

119881120590(119905)

(119905)

ge min119894isinN

[120582min (1198751119894) 119909 (119905)2

+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)

10038171003817100381710038172

]

ge min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

times (119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

)

(31)

From (30) and (31) it is easy to obtain

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119905

1205831198791205911198861205821

1205822

1198881le 119890120572119879

1205831198791205911198861205821

1205822

1198881 (32)

where

1205821= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894) ]

1205822= min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

(33)

By (14) and (32) for any 119905 isin [0 119879] we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (34)

This completes the proof

Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays

(119905) = minus119860120590(119905)

119909 (119905) + 119861120590(119905)

119892 (119910 (119905 minus 1205911(119905)))

+ 119882120590(119905)

intinfin

0

ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904

119910 (119905) = minus119862120590(119905)

119910 (119905) + 119863120590(119905)

119909 (119905 minus 1205912(119905))

(35)

where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882

119894 119894 isin N is the 119894th subsystem distributively

Journal of Applied Mathematics 7

delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ

1(sdot) ℎ

119899(sdot)) is the delay kernel function and

all the other signs are defined as (6)

Assumption 9 The delay kernel ℎ119895rsquos are some real value

nonnegative continuous functions defined in [0infin) thatsatisfy

intinfin

0

ℎ119895(119904) 119889119904 = 1 int

infin

0

119904ℎ119895(119904) 119889119904 = 119896

119894lt infin

119895 = 1 119899

(36)

Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875

1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite

diagonal matrix 119871119894= diag(119897

1119894 119897119899119894) for all 119894 isin N Λ

119898=

diag(1205821198981 120582

119898119899) 119898 = 1 2 matrices 119876

2119894 1198763119894 1198721119894 119894 isin 119873

and positive scalars 120583 ge 1 such that the following inequalities

[1198762119894

1198721119894

lowast 1198763119894

] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2

[1198762119894

1198721119894

lowast 1198763119894

] le 120583 [1198762119895

1198721119895

lowast 1198763119895

] 119877119897119894le 120583119877119897119895

119897 = 1 4 forall119894 119895 isin N

(37)

Ξ1= Ω1

+ sym 119865119879

111987711198941198652+ 119865119879

1[1198751119894119861119894minus 1205912

21119861119879

1198941198771119894119860119894

minus(12059122minus 12059121)2

119861119879

1198941198772119894119860119894]

times 11986510+ 119865119879

211987721198941198653+ 119865119879

311987721198941198654+ 119865119879

3

times [1198752119894119863119894minus 1205912

11119863119879

1198941198773119894119862119894minus (12059112minus 12059111)2

119863119879

1198941198774119894119862119894]

times 1198655+ 119865119879

511987731198941198656+ 119865119879

5(1198721119894+ Λ1119880)1198659+ 119865119879

611987741198941198657

+119865119879

711987741198941198658+ 119865119879

7[1198721119894(1 minus 120591

13) + Λ

2119880]11986510

lt 0 forall119894 isin N(38)

hold and the average dwell time of the switch signal 120590(119905)satisfies

120591119886gt 120591lowast

119886=

119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879

(39)

where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591

23) + 1205912

11119863119879119894

1198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894

1198773119894119862119894+ (12059112minus 12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894

(1 minus 12059113) minus 2119877

4119894 minus1198774119894 1198762119894minus 2Λ

1+ 119871119894minus1198763119894(1 minus 120591

13) +

1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899

119868119899times119899

0119899times(12minus119895)119899

] 119895 = 1 11 12058210158401= max

119894isinN120582max(1198751119894) +

12059122120582max(1198761119894) + 120582max(1198752119894) + 120591

12120582max(1198762119894) +

12059112120582max(1198763119894) 120582max(119880119880) + 2120591

12120582max(1198721119894)120582max(119880) + (12)

[1205913

21120582max(1198771119894) + (120591

22minus 12059121)3

120582max(1198772119894) + 120591311120582max(1198773119894) +

(12059112minus 12059111)3

120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899

119895=1119897119894119895119896119895 12058210158402= 1205822

The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888

1 1198882 119879)

Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional

119881120590(119905)

(119905) = 1198811120590(119905)

(119905) + 1198812120590(119905)

(119905) + 1198813120590(119905)

(119905) + 1198814120590(119905)

(119905) (40)

where1198811120590(119905)

(119905)1198812120590(119905)

(119905)1198813120590(119905)

(119905) are defined as inTheorem 8

1198814120590(119905)

(119905) =

119899

sum119894=1

119897119894120590(119905)

intinfin

0

ℎ119894(120579) int119905

119905minus120579

1198922

119894(119910119894(119904)) 119889119904 119889120579 (41)

when 119905 isin [119905119896 119905119896+1

) taking the derivatives of 1198814along the

trajectory of system (35) we have that

4120590(119905119896)(119905) =

119899

sum119894=1

119897119894120590(119905119896)1198922

119894(119910119894(119905))

minus

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

(42)

By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2

le (int 1199012(119904)119889119904)

(int 1199022(119904)119889119904) we obtain that

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

=

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 119889120579

times intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

ge

119899

sum119894=1

119897119894120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

2

= [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(43)

Therefore

4120590(119905119896)(119905) le 119892

119879

(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))

minus [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(44)

Combing (17)ndash(23) and (40)ndash(43) we get

120590(119905119896)(119905) le 120585

119879

1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)

8 Journal of Applied Mathematics

where

120585119879

1= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))

[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

]

(46)

Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119879

120583119879120591119886

12058210158401

12058210158402

1198881 (47)

where

1205821015840

1= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)]

+120582max (1198774119894) + 1198662

119899

sum119895=1

119897119894119895119896119895

1205821015840

2= 1205822

(48)

By (39) and (47) we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (49)

This completes the proof

4 Numerical Examples

In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10

Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback

0 1 2 3 4 5

minus01

minus005

0

005

01

015

02

025

03

mRN

A co

ncen

trat

ions

t

Figure 1 mRNA concentrations 119909(119905)

0

0

1 2 3 4 5

Prot

ein

conc

entr

atio

ns

minus02

minus01

t

01

02

03

04

05

06

Figure 2 Protein concentrations 119910(119905)

loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)

119889119898119894

119889119905= minus119898119894+

120572

1 + 119901119899119895

+ 1205720

119889119901119894

119889119905= 120573 (119898

119894minus 119901119894)

(50)

where 119898119894and 119901

119894are the concentrations of two mRNAs and

repressor-protein 1205720is the growth rate of protein in a cell in

the presence of saturating amounts of repressor while 1205720+ 120572

is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient

Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by

Journal of Applied Mathematics 9

adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model

(119905) = minus119860119894119909 (119905) + 119861

119894119892 (119910 (119905 minus 120590 (119905)))

119910 (119905) = minus119862119894119910 (119905) + 119863

119894119909 (119905 minus 120591 (119905)) 119894 = 1 2

(51)

where

1198601= diag (4 4 4) 119862

1= diag (5 5 5)

1198631= diag (2 15 1) 119860

2= diag (3 3 3)

1198622= diag (4 4 4) 119863

2= diag (12 12 12)

1198611= 2 times (

0 0 minus1

minus1 0 0

0 minus1 0

) 1198612= 15 times (

0 0 minus1

minus1 0 0

0 minus1 0

)

(52)

The gene regulation function is taken as 119892(119909) = 1199092(1 +

1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and

1205912(119905) are assumed to be

1205911(119905) = 06 + 04 sin 119905 120591

2(119905) = 03 + 01 cos 119905 (53)

we can get the parameters as following

12059111= 02 120591

12= 1 120591

13= 04

12059121= 04 120591

22= 02 120591

23= 01

(54)

and let 1198881= 2 119888

2= 15 119879 = 5 120572 = 002 120583 = 12 By using

the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI

Ξ =

(((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 119886119894110

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010

)))))))))))))))

)

(55)

where 11988611989411

= minus21198751119894119860119894+ 1198761119894

+ 1205912211198601198791198941198771119894119860119894+ (12059122

minus

12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus

(12059122minus 12059121)2

1198611198791198941198772119894119860119894 11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+

(12059112minus12059111)2

1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus

12059111)2

1198631198791198941198774119894119862119894 11988611989455

= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus

12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877

4119894 119886119894710

=

1198721119894(1 minus 120591

13) + Λ

2119880 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912

21119861119879

1198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

35732 minus02655 minus03039

minus02655 33575 minus02912

minus03039 minus02912 31531

)

11987512= (

46463 minus01506 minus01453

minus01506 47212 minus01477

minus01453 minus01477 47809

)

11987611= (

91959 minus03223 minus02832

minus03223 74980 minus02791

minus02832 minus02791 59270

)

11987612= (

108205 minus04573 minus04597

minus04573 109025 minus04104

minus04597 minus04104 110402

)

(56)

and 120591119886

gt 120591lowast119886

= 04003 The initial condition is 119909(0) =

(02 02 03)119879 119910(0) = (03 02 05)

119879 The simulation resultsof the trajectories are shown in Figures 1 and 2

Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows

1198601= diag (21 12 25) 119862

1= diag (46 4 52)

1198631= diag (05 04 06) 119860

2= diag (18 2 31)

1198622= diag (5 4 5) 119863

2= diag (03 03 04)

1198611= (

0 1 0

0 0 1

1 0 0

) 1198612= (

0 07 0

0 0 07

07 0 0

)

(57)

10 Journal of Applied Mathematics

and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591

1(119905) and 120591

2(119905) are assumed to be

1205911(119905) = 05 + 03 sin 119905 120591

2(119905) = 04 + 01 cos 119905 (58)

we can get the parameters as following

12059111= 02 120591

12= 08 120591

13= 03

12059121= 03 120591

22= 05 120591

23= 01

(59)

and let 1198881= 25 119888

2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ

119895(119904) = 119890minus119904

and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI

Ξ =

(((((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 1198861198940

0

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

0

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499

0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894

10100

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894

)))))))))))))))))

)

(60)

where 11988611989411= minus2119875

1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus

1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2

1198611198791198941198772119894119860119894

11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus

21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2

1198631198791198941198774119894119862119894 11988611989455=

minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894

11988611989477= minus1198762119894(1 minus 120591

13) minus 2119877

4119894 119886119894710

= 1198721119894(1 minus 120591

13) + Λ2119880 11988611989499=

1198762119894minus 2Λ1+ 119871119894 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus

12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

13745 minus00657 minus00145

minus00657 13683 minus00471

minus00145 minus00471 12893

)

11987512= (

16025 minus00233 minus00056

minus00233 15143 minus00073

minus00056 minus00073 12907

)

11987611= (

25617 minus00905 minus00313

minus00905 20794 minus00718

minus00313 minus00718 24995

)

11987612= (

21367 minus00751 minus00258

minus00751 21184 minus00434

minus00258 minus00434 22000

)

(61)

and 120591119886gt 120591lowast119886= 05012

5 Concluding Remarks

This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)

References

[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

Motivated by the above discussions we analyze finite-time stability of switching genetic regulatory networks withinterval time-varying delays and continuous distributeddelays Using a novel piecewise Lyapunov-Krasovskii func-tional and finite-time stable definite some new finite-timestability criteria are obtained for the switched GRNs Thefeatures of this paper can be summarized as follows (1)the finite-time stability is researched in the switched GRNsfirstly (2) continuous distributed delays are concerned (3)all sufficient conditions obtained depend on the time delays

This paper is organized as follows In Section 2 modeldescription some assumptions definitions and lemmas aregiven In Section 3 some conditions are obtained to ensurethe finite-time stability of switched GRNs with intervaltime-varying delays and continuous distributed delays Twonumerical examples are given to demonstrate the effective-ness of our analysis in Section 4 Finally conclusions aredrawn in Section 5

Notations Throughout this paper R R119899 and R119899times119898 denoterespectively the set of all real numbers real 119899-dimensionalspace and real 119899 times 119898-dimensional space Z

+denote the set

of all positive integers sdot denote the Euclidean normsin R119899 For a vector or matrix 119860 119860119879 denotes its transposeFor a square matrix 119860 120582max(119860) and 120582min(119860) denote themaximum eigenvalue and minimum eigenvalue of matrix 119860respectively and sym (119860) is used to represent 119860 + 119860

119879 Forsimplicity in symmetric block matrices we often use lowast torepresent the term that is induced by symmetry

2 Problem Formulation andSome Preliminaries

We consider the following genetic regulatory networks

(119905) = minus119860119898 (119905) + 119861119891 (119901 (119905 minus 1205911(119905))) + 119868

(119905) = minus119862119875 (119905) + 119863119898 (119905 minus 1205912(119905))

(1)

where 119898(119905) = [1198981(119905) 119898

119899(119905)]119879 is the concentrations of

mRNAs 119901(119905) = [1199011(119905) 119901

119899(119905)]119879 is the concentrations of

proteins 119891(sdot) = [1198911(sdot) 119891

119899(sdot)]119879 is the regulatory functions

of mRNAs 119860 = diag(1198861 119886

119899) is the degradation rates

of mRNAs 119862 = diag(1198881 119888

119899) is the degradation rates of

proteins 119863 = diag(1198891 119889

119899) is the translation rates of

proteins 119868 = [1198681 119868

119899]119879 is the basal rate and 120591

1(119905) and 120591

2(119905)

are time-varying delaysFor obtaining our conclusions we make the following

assumptions

Assumption 1 119891119894 R rarr R 119894 = 1 119899 are monotonically

increasing functions with saturation and satisfy

0 le119891119894(119886) minus 119891

119894(119887)

119886 minus 119887le 119906119894 forall119886 119887 isin R 119894 = 1 119899 (2)

where 119906119894 119894 = 1 119899 are nonnegative constants

Assumption 2 1205911(119905) and 120591

2(119905) are time-varying delays satisfy-

ing

0 le 12059111le 1205911(119905) le 120591

12 120591

1(119905) le 120591

13lt infin

0 le 12059121le 1205912(119905) le 120591

22 120591

2(119905) le 120591

23lt infin

(3)

Vectors119898lowast 119901lowast are an equilibrium point of the system (1)Let 119909(119905) = 119898(119905) minus 119898lowast 119910(119905) = 119901(119905) minus 119901lowast we get

(119905) = minus119860119909 (119905) + 119861119892 (119910 (119905 minus 1205911(119905)))

119910 (119905) = minus119862119910 (119905) + 119863119909 (119905 minus 1205912(119905))

(4)

where 119909(119905) = [1199091(119905) 119909

119899(119905)]119879 119910(119905) = [119910

1(119905) 119910

119899(119905)]119879

119892(sdot) = [119892(sdot) 119892(sdot)]119879 and 119892

119894(119910119894(119905)) = 119891

119894(119910119894(119905)+119901lowast

119894) minus119891119894(119901lowast119894)

According to Assumption 1 and the definition of 119892119894(sdot) we

know that 119892119894(sdot) is bounded that is exist119866 gt 0 such that |119892

119894(sdot)| le

119866 119894 = 1 119899 and satisfies the following sector condition

0 le119892119894(119886)

119886le 119906119894 forall119886 isin R 0 119894 = 1 119899 (5)

Let 119880 = diag(1199061 119906

119899)

Sometimes GRNs were described by continuous timeswitched system as in [17 24 25] so system (4) can bedescribed as switching system with switching signal

(119905) = minus119860120590(119905)

119909 (119905) + 119861120590(119905)

119892 (119910 (119905 minus 1205911(119905)))

119910 (119905) = minus119862120590(119905)

119910 (119905) + 119863120590(119905)

119909 (119905 minus 1205912(119905))

(6)

where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal which is a piecewise constant function depending ontime 119905 For each 119894 isin N the matrices 119860

119894 119861119894 119862119894 and 119863

119894

are the 119894th subsystem matrices that are constant matrices ofappropriate dimensions

For the switching signal 120590(119905) we have the followingswitching sequence

(1198940 1199050) (119894

119896 119905119896) | 119894

119896isin N 119896 = 0 1 (7)

in otherwords when 119905 isin [119905119896 119905119896+1

) 119894119896th subsystem is activated

The initial condition of system (6) is assumed to be

119909 (119905) = 120593 (119905) 119910 (119905) = 120595 (119905)

minus120588 le 119905 le 0 120588 = max 12059112 12059122

(8)

For proving the theorem we recall the following defini-tion and lemmas

Definition 3 The system (6) is said to be finite-time stablewith respect to positive real numbers (119888

1 1198882 119879) if

1003817100381710038171003817120601 (119905)10038171003817100381710038172

1198621 +

1003817100381710038171003817120593 (119905)10038171003817100381710038172

1198621 le 1198881997904rArr 119909 (119905)

2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882

forall119905 isin [0 119879] (9)

where 120601(119905)1198881 = sup

minus120588le119905le0120601(119905) 120601(119905) 120593(119905)

1198881 =

supminus120588le119905le0

120593(119905) (119905)

Journal of Applied Mathematics 3

Remark 4 The Lyapunov stability implies that by startingsufficiently close to the equilibrium point trajectories canbe guaranteed to stay within any specified ball centered atthe equilibrium point It depicts character of the equilibriumpoint of the system The finite-time stability implies thatby starting from any specified compact set trajectories canbe guaranteed to stay within a large enough compact set Itdepicts character of all solutions of the system and is calledthe finite-time boundedness in some literatures [21 22] Inaddition the Lyapunov stability is considered in infiniteinterval but the finite-time stability is considered in finiteinterval

Remark 5 In [23] the authors considered the finite-timerobust stability of GRNs but in our paper we consideredthe finite-time stability of switching GRNs which are morerealisticGRNs than that of the [23] For the first time switchedGRNsrsquo finite-time stability is researched

Definition 6 For any 119879 ge 119905 ge 0 let 119873120590(119905 119879) denote the

number of switching of 120590(119905) over (119905 119879) If

119873120590(119905 119879) le 119873

0+119879 minus 119905

120591119886

(10)

holds for 1198730ge 0 120591

119886gt 0 then 120591

119886gt 0 is called the average

dwell time and 1198730is the chatter bound As commonly used

in the literature we choose1198730= 0

Lemma 7 (see [26]) For any positive definite matrix 119872 isin

R119899times119899 there exist a scalar 119902 gt 0 and a vector-valued function120596 [0 119902] rarr R119899 such that

(int119902

0

120596 (119904) 119889119904)

119879

119872(int119902

0

120596 (119904) 119889119904) le 119902int119902

0

120596119879

(119904)119872120596 (119904) 119889119904

(11)

3 Main Results

In this section we present a finite time stability theoremfor switching genetic regulatory networks with interval time-varying delays (6)

Theorem 8 For switching system (6) with Assumptions 1 and2 given a scalar 120572 gt 0 if there exist symmetric positive definitematrices119875

1119894119875211989411987611198941198771119894119877211989411987731198941198774119894 for all 119894 isin N the diagonal

matrix Λ119898= diag(120582

1198981 120582

119898119899) ge 0 119898 = 1 2 matrices 119876

2119894

1198763119894 1198721119894 119894 isin 119873 and positive scalars 120583 ge 1 such that the

following inequalities

[1198762119894

1198721119894

lowast 1198763119894

] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2

[1198762119894

1198721119894

lowast 1198763119894

] le 120583 [1198762119895

1198721119895

lowast 1198763119895

] 119877119897119894le 120583119877119897119895

119897 = 1 4 forall119894 119895 isin N

(12)

Ξ = Ω

+ sym 119871119879

111987711198941198712+ 119871119879

1

times [1198751119894119861119894minus 1205912

21119861119879

1198941198771119894119860119894minus (12059122minus 12059121)2

119861119879

1198941198772119894119860119894]

times 11987110+ 119871119879

211987721198941198713+ 119871119879

311987721198941198714+ 119871119879

3

times [1198752119894119863119894minus 1205912

11119863119879

1198941198773119894119862119894minus (12059112minus 12059111)2

119863119879

1198941198774119894119862119894]

times1198715+ 119871119879

511987731198941198716+ 119871119879

5(1198721119894+ Λ1119880) 1198719+119871119879

611987741198941198717

+119871119879

711987741198941198718+ 119871119879

7[1198721119894(1 minus 120591

13) + Λ

2119880] 11987110

lt 0 forall119894 isin N(13)

hold and the average dwell time of the switching signal 120590(119905)satisfies

120591119886gt 120591lowast

119886=

119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879

(14)

where Ω = diag(minus21198751119894119860119894+ 1198761119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 120572119875

1119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591

23) + 120591211119863119879119894

1198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus21198772119894 minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894

1198773119894119862119894+ (12059112minus 12059111)2

1198621198791198941198774119894119862119894minus1198773119894minus 1205721198752119894 minus1198773119894minus1198774119894minus1198762119894(1 minus

12059113) minus 2119877

4119894 minus11987741198941198763119894minus 2Λ

1 minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus2Λ2) 119871119895=[0119899times(119895minus1)119899

119868119899times119899

0119899times(11minus119895)119899

] 119895 =1 10 120582

1= max

119894isinN120582max(1198751119894) + 12059122120582max(1198761119894) + 120582max

(1198752119894) + 120591

12120582max(1198762119894) + 120591

12120582max(1198763119894) 120582max(119880119880) + 2120591

12

120582max(1198721119894) 120582max(119880) + (12) [120591321120582max(1198771119894) + (120591

22minus 12059121)3

120582max(1198772119894) + 1205913

11120582max(1198773119894) + (120591

12minus 12059111)3

120582max(1198774119894)] 1205822 =

min119894isinN[min(120582min(1198751119894) 120582min(1198752119894))] The equilibrium point of

(6) is finite time stable with respect to positive real numbers(1198881 1198882 119879)

Proof Based on the system (6) we construct the followingLyapunov-krasovskii functional

119881120590(119905)

(119905) = 1198811120590(119905)

(119905) + 1198812120590(119905)

(119905) + 1198813120590(119905)

(119905) (15)

where

1198811120590(119905)

(119905) = 119909119879

(119905) 1198751120590(119905)

119909 (119905) + 119910119879

(119905) 1198752120590(119905)

119910 (119905)

1198812120590(119905)

(119905) = int119905

119905minus1205912(119905)

119909119879

(119904) 1198761120590(119905)

119909 (119905) 119889119904

+ int119905

119905minus1205911(119905)

[119910119879

(119904) 119892119879

(119910 (119904))] [1198762120590(119905)

1198721120590(119905)

1198721120590(119905)

1198763120590(119905)

]

times [119910 (119904)

119892 (119910 (119904))] 119889119904

4 Journal of Applied Mathematics

1198813120590(119905)

(119905)

= int0

minus12059121

int119905

119905+120579

12059121119879

(119904) 1198771120590(119905)

(119904) 119889119904 119889120579

+ intminus12059121

minus12059122

int119905

119905+120579

(12059122minus 12059121) 119879

(119904) 1198772120590(119905)

(119904) 119889119904 119889120579

+ int0

minus12059111

int119905

119905+120579

12059111

119910119879

(119904) 1198773120590(119905)

119910 (119904) 119889119904 119889120579

+ intminus12059111

minus12059112

int119905

119905+120579

(12059112minus 12059111) 119910119879

(119904) 1198774120590(119905)

119910 (119904) 119889119904 119889120579

(16)

First when 119905 isin [119905119896 119905119896+1

) taking the derivatives of119881119894 119894 = 1 2 3

along the trajectory of system (6) we have that

1120590(119905119896)(119905) = 2119909

119879

(119905) 1198751120590(119905119896) (119905) + 2119910

119879

(119905) 1198752120590(119905119896)119910 (119905)

2120590(119905119896)(119905) = 119909

119879

(119905) 1198761120590(119905119896)119909 (119905) minus 119909

119879

(119905 minus 1205912(119905)) 119876

1120590(119905119896)119909119879

times (119905 minus 1205912(119905)) (1 minus 120591

2(119905))

+ [119910119879

(119905) 119892119879

(119910 (119905))] [1198762120590(119905119896)1198721120590(119905119896)

lowast 1198763120590(119905119896)

]

times [119910 (119905)

119892 (119910 (119905))]

minus [119910119879

(119905 minus 1205911(119905)) 119892

119879

(119910 (119905 minus 1205911(119905)))]

times [1198762120590(119905119896)1198721120590(119905119896)

lowast 1198763120590(119905119896)

] [119910 (119905 minus 120591

1(119905))

119892 (119910 (119905 minus 1205911(119905)))

]

times (1 minus 1205911(119905))

3120590(119905119896)(119905) = 120591

2

21119879

(119905) 1198771120590(119905119896) (119904)

minus int119905

119905minus12059121

12059121119879

(120579) 1198771120590(119905119896) (120579) 119889120579

+ (12059122minus 12059121)2

119879

(119905) 1198772120590(119905119896) (119904)

minus int119905minus12059121

119905minus12059122

(12059122minus 12059121) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

+ 1205912

11119910119879

(119905) 1198773120590(119905119896)119910 (119904)

minus int119905

119905minus12059111

12059111

119910119879

(120579) 1198773120590(119905119896)119910 (120579) 119889120579

+ (12059112minus 12059111)2

119910119879

(119905) 1198774120590(119905119896)119910 (119904)

minus int119905minus12059111

119905minus12059112

(12059112minus 12059111) 119910119879

(120579) 1198774120590(119905119896)119910 (120579) 119889120579

(17)

By Lemma 7 we have

minus int119905

119905minus12059121

12059121119879

(120579) 1198771120590(119905119896) (120579) 119889120579

le minus[119909 (119905) minus 119909 (119905 minus 12059121)]119879

1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591

21)]

(18)

minus int119905

119905minus12059111

12059111

119910119879

(120579) 1198773120590(119905119896)119910 (120579) 119889120579

le minus[119910 (119905) minus 119910 (119905 minus 12059111)]119879

1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591

11)]

(19)

minus int119905minus12059121

119905minus12059122

(12059122minus 12059121) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

= minus (12059122minus 12059121) [int119905minus1205912(119905)

119905minus12059122

119879

(120579) 1198772120590(119905119896) (120579) 119889120579

+ int119905minus12059121

119905minus1205912(119905)

119879

(120579) 1198772120590(119905119896) (120579) 119889120579]

= minusint119905minus1205912(119905)

119905minus12059122

(12059122minus 1205912(119905)) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus12059121

119905minus1205912(119905)

(1205912(119905) minus 120591

21) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus1205912(119905)

119905minus12059122

(1205912(119905) minus 120591

21) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus12059121

119905minus1205912(119905)

(12059122minus 1205912(119905)) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

le minus[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

minus1205912(119905) minus 120591

21

12059122minus 12059121

[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus12059122minus 1205912(119905)

12059122minus 12059121

[119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

(20)

Journal of Applied Mathematics 5

Similar to (20) we can obtain

minus int119905minus12059111

119905minus12059112

(12059112minus 12059111) 119910119879

(120579) 1198774120590(119905119896)119910 (120579) 119889120579

le minus[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

minus1205911(119905) minus 120591

11

12059112minus 12059111

[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus12059112minus 1205911(119905)

12059112minus 12059111

[119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

(21)

In addition for any Λ119898= diag(120582

1198981 120582

119898119899) ge 0 119898 = 1 2

the following inequality is true from (5)

minus 2

119899

sum119894=1

1205821119894119892119894(119910119894(119905)) [119892

119894(119910119894(119905)) minus 119906

119894119910119894(119905)]

minus 2

119899

sum119894=1

1205822119894119892119894(119910119894(119905 minus 1205911(119905)))

times [119892119894(119910119894(119905 minus 1205911(119905))) minus 119906

119894119910119894(119905 minus 1205911(119905))] ge 0

(22)

It can be written as matrix form

minus 2119892119879

(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)] minus 2119892

119879

(119910 (119905 minus 1205911(119905))) Λ

2

times [119892 (119910 (119905 minus 1205911(119905))) minus 119880119910 (119905 minus 120591

1(119905))] ge 0

(23)

From (15) to (23) we have that

120590(119905119896)(119905)

le 2119909119879

(119905) 1198751120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ 2119910119879

(119905) 1198752120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

+ 119909119879

(119905) 1198761120590(119905119896)119909 (119905) minus 119909

119879

(119905 minus 1205912(119905)) 119876

1120590(119905119896)119909119879

times (119905 minus 1205912(119905)) (1 minus 120591

2119889) + 119910119879

(119905) 1198762120590119910 (119905)

+ 2119910119879

(119905)1198721120590119892 (119910 (119905)) + 119892

119879

(119910 (119905)) 1198763120590(119905119896)119892 (119910 (119905))

minus 119910119879

(119905 minus 1205911(119905)) 119876

2120590(119905119896)119910 (119905 minus 120591

1(119905)) (1 minus 120591

1119889)

minus 2119910119879

(119905 minus 1205911(119905))119872

1120590(119905119896)119892 (119910 (119905 minus 120591

1(119905))) (1 minus 120591

1119889)

minus 119892119879

(119910 (119905 minus 1205911(119905))) 119876

3120590(119905119896)119892 (119910 (119905 minus 120591

1(119905))) (1 minus 120591

1119889)

+ 1205912

21[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]119879

times 1198771120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ (12059122minus 12059121)2

[minus119860120590(119905119896)119909(119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]119879

times 1198772120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ 1205912

11[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]119879

times 1198773120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

+ (12059112minus 12059111)2

[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]119879

times 1198774120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

minus [119909 (119905) minus 119909 (119905 minus 12059121)]119879

1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591

21)]

minus [119910 (119905) minus 119910 (119905 minus 12059111)]119879

1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591

11)]

minus [119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

minus [119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

minus 2119892119879

(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)]

minus 2119892119879

(119910 (119905 minus 1205911(119905)))

times Λ2[119892 (119910 (119905 minus 120591

1(119905))) minus 119880119910 (119905 minus 120591

1(119905))]

le 120585119879

Ξ120585 + 120572 (119909119879

(119905) 1198751120590(119905119896)119909 (119905) + 119910

119879

(119905) 1198752120590(119905119896)119910 (119905))

le 120585119879

Ξ120585 + 120572119881120590(119905119896)(119905)

(24)

where

120585119879

= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))]

(25)

6 Journal of Applied Mathematics

By condition (13) we have

120590(119905119896)(119905) le 120572119881

120590(119905119896)(119905) (26)

Integrating (26) from 119905119896to 119905 we obtain that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

119881120590(119905119896)(119905119896) (27)

Using (12) at switching instant 119905119896 we have

119881120590(119905119896)(119905119896) le 120583119881

120590(119905minus

119896)(119905minus

119896) (28)

Therefore it follows from (27) and (28) that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

120583119881120590(119905minus

119896)(119905minus

119896) (29)

For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873

120590(0 119879) le 119879120591

119886 we

have

119881120590(119905)

(119905) le 119890120572119905

120583119879120591119886119881120590(0)

(0) (30)

On the other hand it follows from (15) that

119881120590(0)

(0)

le max119894isinN

[120582max (1198751119894) + 12059122120582max (1198761119894)]

times supminus120588le119905le0

1003817100381710038171003817120593 (119905)10038171003817100381710038172

+ [120582max (1198752119894) + 12059112120582max (1198762119894)

+ 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)]

times supminus120588le119905le0

1003817100381710038171003817120595 (119905)10038171003817100381710038172

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

times 120582max (1198772119894)]

times supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

+1

2[1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

times120582max (1198774119894)] supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

le max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)] 1198881

119881120590(119905)

(119905)

ge min119894isinN

[120582min (1198751119894) 119909 (119905)2

+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)

10038171003817100381710038172

]

ge min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

times (119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

)

(31)

From (30) and (31) it is easy to obtain

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119905

1205831198791205911198861205821

1205822

1198881le 119890120572119879

1205831198791205911198861205821

1205822

1198881 (32)

where

1205821= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894) ]

1205822= min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

(33)

By (14) and (32) for any 119905 isin [0 119879] we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (34)

This completes the proof

Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays

(119905) = minus119860120590(119905)

119909 (119905) + 119861120590(119905)

119892 (119910 (119905 minus 1205911(119905)))

+ 119882120590(119905)

intinfin

0

ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904

119910 (119905) = minus119862120590(119905)

119910 (119905) + 119863120590(119905)

119909 (119905 minus 1205912(119905))

(35)

where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882

119894 119894 isin N is the 119894th subsystem distributively

Journal of Applied Mathematics 7

delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ

1(sdot) ℎ

119899(sdot)) is the delay kernel function and

all the other signs are defined as (6)

Assumption 9 The delay kernel ℎ119895rsquos are some real value

nonnegative continuous functions defined in [0infin) thatsatisfy

intinfin

0

ℎ119895(119904) 119889119904 = 1 int

infin

0

119904ℎ119895(119904) 119889119904 = 119896

119894lt infin

119895 = 1 119899

(36)

Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875

1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite

diagonal matrix 119871119894= diag(119897

1119894 119897119899119894) for all 119894 isin N Λ

119898=

diag(1205821198981 120582

119898119899) 119898 = 1 2 matrices 119876

2119894 1198763119894 1198721119894 119894 isin 119873

and positive scalars 120583 ge 1 such that the following inequalities

[1198762119894

1198721119894

lowast 1198763119894

] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2

[1198762119894

1198721119894

lowast 1198763119894

] le 120583 [1198762119895

1198721119895

lowast 1198763119895

] 119877119897119894le 120583119877119897119895

119897 = 1 4 forall119894 119895 isin N

(37)

Ξ1= Ω1

+ sym 119865119879

111987711198941198652+ 119865119879

1[1198751119894119861119894minus 1205912

21119861119879

1198941198771119894119860119894

minus(12059122minus 12059121)2

119861119879

1198941198772119894119860119894]

times 11986510+ 119865119879

211987721198941198653+ 119865119879

311987721198941198654+ 119865119879

3

times [1198752119894119863119894minus 1205912

11119863119879

1198941198773119894119862119894minus (12059112minus 12059111)2

119863119879

1198941198774119894119862119894]

times 1198655+ 119865119879

511987731198941198656+ 119865119879

5(1198721119894+ Λ1119880)1198659+ 119865119879

611987741198941198657

+119865119879

711987741198941198658+ 119865119879

7[1198721119894(1 minus 120591

13) + Λ

2119880]11986510

lt 0 forall119894 isin N(38)

hold and the average dwell time of the switch signal 120590(119905)satisfies

120591119886gt 120591lowast

119886=

119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879

(39)

where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591

23) + 1205912

11119863119879119894

1198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894

1198773119894119862119894+ (12059112minus 12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894

(1 minus 12059113) minus 2119877

4119894 minus1198774119894 1198762119894minus 2Λ

1+ 119871119894minus1198763119894(1 minus 120591

13) +

1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899

119868119899times119899

0119899times(12minus119895)119899

] 119895 = 1 11 12058210158401= max

119894isinN120582max(1198751119894) +

12059122120582max(1198761119894) + 120582max(1198752119894) + 120591

12120582max(1198762119894) +

12059112120582max(1198763119894) 120582max(119880119880) + 2120591

12120582max(1198721119894)120582max(119880) + (12)

[1205913

21120582max(1198771119894) + (120591

22minus 12059121)3

120582max(1198772119894) + 120591311120582max(1198773119894) +

(12059112minus 12059111)3

120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899

119895=1119897119894119895119896119895 12058210158402= 1205822

The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888

1 1198882 119879)

Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional

119881120590(119905)

(119905) = 1198811120590(119905)

(119905) + 1198812120590(119905)

(119905) + 1198813120590(119905)

(119905) + 1198814120590(119905)

(119905) (40)

where1198811120590(119905)

(119905)1198812120590(119905)

(119905)1198813120590(119905)

(119905) are defined as inTheorem 8

1198814120590(119905)

(119905) =

119899

sum119894=1

119897119894120590(119905)

intinfin

0

ℎ119894(120579) int119905

119905minus120579

1198922

119894(119910119894(119904)) 119889119904 119889120579 (41)

when 119905 isin [119905119896 119905119896+1

) taking the derivatives of 1198814along the

trajectory of system (35) we have that

4120590(119905119896)(119905) =

119899

sum119894=1

119897119894120590(119905119896)1198922

119894(119910119894(119905))

minus

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

(42)

By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2

le (int 1199012(119904)119889119904)

(int 1199022(119904)119889119904) we obtain that

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

=

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 119889120579

times intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

ge

119899

sum119894=1

119897119894120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

2

= [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(43)

Therefore

4120590(119905119896)(119905) le 119892

119879

(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))

minus [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(44)

Combing (17)ndash(23) and (40)ndash(43) we get

120590(119905119896)(119905) le 120585

119879

1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)

8 Journal of Applied Mathematics

where

120585119879

1= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))

[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

]

(46)

Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119879

120583119879120591119886

12058210158401

12058210158402

1198881 (47)

where

1205821015840

1= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)]

+120582max (1198774119894) + 1198662

119899

sum119895=1

119897119894119895119896119895

1205821015840

2= 1205822

(48)

By (39) and (47) we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (49)

This completes the proof

4 Numerical Examples

In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10

Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback

0 1 2 3 4 5

minus01

minus005

0

005

01

015

02

025

03

mRN

A co

ncen

trat

ions

t

Figure 1 mRNA concentrations 119909(119905)

0

0

1 2 3 4 5

Prot

ein

conc

entr

atio

ns

minus02

minus01

t

01

02

03

04

05

06

Figure 2 Protein concentrations 119910(119905)

loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)

119889119898119894

119889119905= minus119898119894+

120572

1 + 119901119899119895

+ 1205720

119889119901119894

119889119905= 120573 (119898

119894minus 119901119894)

(50)

where 119898119894and 119901

119894are the concentrations of two mRNAs and

repressor-protein 1205720is the growth rate of protein in a cell in

the presence of saturating amounts of repressor while 1205720+ 120572

is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient

Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by

Journal of Applied Mathematics 9

adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model

(119905) = minus119860119894119909 (119905) + 119861

119894119892 (119910 (119905 minus 120590 (119905)))

119910 (119905) = minus119862119894119910 (119905) + 119863

119894119909 (119905 minus 120591 (119905)) 119894 = 1 2

(51)

where

1198601= diag (4 4 4) 119862

1= diag (5 5 5)

1198631= diag (2 15 1) 119860

2= diag (3 3 3)

1198622= diag (4 4 4) 119863

2= diag (12 12 12)

1198611= 2 times (

0 0 minus1

minus1 0 0

0 minus1 0

) 1198612= 15 times (

0 0 minus1

minus1 0 0

0 minus1 0

)

(52)

The gene regulation function is taken as 119892(119909) = 1199092(1 +

1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and

1205912(119905) are assumed to be

1205911(119905) = 06 + 04 sin 119905 120591

2(119905) = 03 + 01 cos 119905 (53)

we can get the parameters as following

12059111= 02 120591

12= 1 120591

13= 04

12059121= 04 120591

22= 02 120591

23= 01

(54)

and let 1198881= 2 119888

2= 15 119879 = 5 120572 = 002 120583 = 12 By using

the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI

Ξ =

(((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 119886119894110

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010

)))))))))))))))

)

(55)

where 11988611989411

= minus21198751119894119860119894+ 1198761119894

+ 1205912211198601198791198941198771119894119860119894+ (12059122

minus

12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus

(12059122minus 12059121)2

1198611198791198941198772119894119860119894 11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+

(12059112minus12059111)2

1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus

12059111)2

1198631198791198941198774119894119862119894 11988611989455

= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus

12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877

4119894 119886119894710

=

1198721119894(1 minus 120591

13) + Λ

2119880 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912

21119861119879

1198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

35732 minus02655 minus03039

minus02655 33575 minus02912

minus03039 minus02912 31531

)

11987512= (

46463 minus01506 minus01453

minus01506 47212 minus01477

minus01453 minus01477 47809

)

11987611= (

91959 minus03223 minus02832

minus03223 74980 minus02791

minus02832 minus02791 59270

)

11987612= (

108205 minus04573 minus04597

minus04573 109025 minus04104

minus04597 minus04104 110402

)

(56)

and 120591119886

gt 120591lowast119886

= 04003 The initial condition is 119909(0) =

(02 02 03)119879 119910(0) = (03 02 05)

119879 The simulation resultsof the trajectories are shown in Figures 1 and 2

Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows

1198601= diag (21 12 25) 119862

1= diag (46 4 52)

1198631= diag (05 04 06) 119860

2= diag (18 2 31)

1198622= diag (5 4 5) 119863

2= diag (03 03 04)

1198611= (

0 1 0

0 0 1

1 0 0

) 1198612= (

0 07 0

0 0 07

07 0 0

)

(57)

10 Journal of Applied Mathematics

and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591

1(119905) and 120591

2(119905) are assumed to be

1205911(119905) = 05 + 03 sin 119905 120591

2(119905) = 04 + 01 cos 119905 (58)

we can get the parameters as following

12059111= 02 120591

12= 08 120591

13= 03

12059121= 03 120591

22= 05 120591

23= 01

(59)

and let 1198881= 25 119888

2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ

119895(119904) = 119890minus119904

and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI

Ξ =

(((((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 1198861198940

0

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

0

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499

0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894

10100

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894

)))))))))))))))))

)

(60)

where 11988611989411= minus2119875

1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus

1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2

1198611198791198941198772119894119860119894

11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus

21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2

1198631198791198941198774119894119862119894 11988611989455=

minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894

11988611989477= minus1198762119894(1 minus 120591

13) minus 2119877

4119894 119886119894710

= 1198721119894(1 minus 120591

13) + Λ2119880 11988611989499=

1198762119894minus 2Λ1+ 119871119894 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus

12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

13745 minus00657 minus00145

minus00657 13683 minus00471

minus00145 minus00471 12893

)

11987512= (

16025 minus00233 minus00056

minus00233 15143 minus00073

minus00056 minus00073 12907

)

11987611= (

25617 minus00905 minus00313

minus00905 20794 minus00718

minus00313 minus00718 24995

)

11987612= (

21367 minus00751 minus00258

minus00751 21184 minus00434

minus00258 minus00434 22000

)

(61)

and 120591119886gt 120591lowast119886= 05012

5 Concluding Remarks

This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)

References

[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

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

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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

Journal of Applied Mathematics 3

Remark 4 The Lyapunov stability implies that by startingsufficiently close to the equilibrium point trajectories canbe guaranteed to stay within any specified ball centered atthe equilibrium point It depicts character of the equilibriumpoint of the system The finite-time stability implies thatby starting from any specified compact set trajectories canbe guaranteed to stay within a large enough compact set Itdepicts character of all solutions of the system and is calledthe finite-time boundedness in some literatures [21 22] Inaddition the Lyapunov stability is considered in infiniteinterval but the finite-time stability is considered in finiteinterval

Remark 5 In [23] the authors considered the finite-timerobust stability of GRNs but in our paper we consideredthe finite-time stability of switching GRNs which are morerealisticGRNs than that of the [23] For the first time switchedGRNsrsquo finite-time stability is researched

Definition 6 For any 119879 ge 119905 ge 0 let 119873120590(119905 119879) denote the

number of switching of 120590(119905) over (119905 119879) If

119873120590(119905 119879) le 119873

0+119879 minus 119905

120591119886

(10)

holds for 1198730ge 0 120591

119886gt 0 then 120591

119886gt 0 is called the average

dwell time and 1198730is the chatter bound As commonly used

in the literature we choose1198730= 0

Lemma 7 (see [26]) For any positive definite matrix 119872 isin

R119899times119899 there exist a scalar 119902 gt 0 and a vector-valued function120596 [0 119902] rarr R119899 such that

(int119902

0

120596 (119904) 119889119904)

119879

119872(int119902

0

120596 (119904) 119889119904) le 119902int119902

0

120596119879

(119904)119872120596 (119904) 119889119904

(11)

3 Main Results

In this section we present a finite time stability theoremfor switching genetic regulatory networks with interval time-varying delays (6)

Theorem 8 For switching system (6) with Assumptions 1 and2 given a scalar 120572 gt 0 if there exist symmetric positive definitematrices119875

1119894119875211989411987611198941198771119894119877211989411987731198941198774119894 for all 119894 isin N the diagonal

matrix Λ119898= diag(120582

1198981 120582

119898119899) ge 0 119898 = 1 2 matrices 119876

2119894

1198763119894 1198721119894 119894 isin 119873 and positive scalars 120583 ge 1 such that the

following inequalities

[1198762119894

1198721119894

lowast 1198763119894

] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2

[1198762119894

1198721119894

lowast 1198763119894

] le 120583 [1198762119895

1198721119895

lowast 1198763119895

] 119877119897119894le 120583119877119897119895

119897 = 1 4 forall119894 119895 isin N

(12)

Ξ = Ω

+ sym 119871119879

111987711198941198712+ 119871119879

1

times [1198751119894119861119894minus 1205912

21119861119879

1198941198771119894119860119894minus (12059122minus 12059121)2

119861119879

1198941198772119894119860119894]

times 11987110+ 119871119879

211987721198941198713+ 119871119879

311987721198941198714+ 119871119879

3

times [1198752119894119863119894minus 1205912

11119863119879

1198941198773119894119862119894minus (12059112minus 12059111)2

119863119879

1198941198774119894119862119894]

times1198715+ 119871119879

511987731198941198716+ 119871119879

5(1198721119894+ Λ1119880) 1198719+119871119879

611987741198941198717

+119871119879

711987741198941198718+ 119871119879

7[1198721119894(1 minus 120591

13) + Λ

2119880] 11987110

lt 0 forall119894 isin N(13)

hold and the average dwell time of the switching signal 120590(119905)satisfies

120591119886gt 120591lowast

119886=

119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879

(14)

where Ω = diag(minus21198751119894119860119894+ 1198761119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 120572119875

1119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591

23) + 120591211119863119879119894

1198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus21198772119894 minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894

1198773119894119862119894+ (12059112minus 12059111)2

1198621198791198941198774119894119862119894minus1198773119894minus 1205721198752119894 minus1198773119894minus1198774119894minus1198762119894(1 minus

12059113) minus 2119877

4119894 minus11987741198941198763119894minus 2Λ

1 minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus2Λ2) 119871119895=[0119899times(119895minus1)119899

119868119899times119899

0119899times(11minus119895)119899

] 119895 =1 10 120582

1= max

119894isinN120582max(1198751119894) + 12059122120582max(1198761119894) + 120582max

(1198752119894) + 120591

12120582max(1198762119894) + 120591

12120582max(1198763119894) 120582max(119880119880) + 2120591

12

120582max(1198721119894) 120582max(119880) + (12) [120591321120582max(1198771119894) + (120591

22minus 12059121)3

120582max(1198772119894) + 1205913

11120582max(1198773119894) + (120591

12minus 12059111)3

120582max(1198774119894)] 1205822 =

min119894isinN[min(120582min(1198751119894) 120582min(1198752119894))] The equilibrium point of

(6) is finite time stable with respect to positive real numbers(1198881 1198882 119879)

Proof Based on the system (6) we construct the followingLyapunov-krasovskii functional

119881120590(119905)

(119905) = 1198811120590(119905)

(119905) + 1198812120590(119905)

(119905) + 1198813120590(119905)

(119905) (15)

where

1198811120590(119905)

(119905) = 119909119879

(119905) 1198751120590(119905)

119909 (119905) + 119910119879

(119905) 1198752120590(119905)

119910 (119905)

1198812120590(119905)

(119905) = int119905

119905minus1205912(119905)

119909119879

(119904) 1198761120590(119905)

119909 (119905) 119889119904

+ int119905

119905minus1205911(119905)

[119910119879

(119904) 119892119879

(119910 (119904))] [1198762120590(119905)

1198721120590(119905)

1198721120590(119905)

1198763120590(119905)

]

times [119910 (119904)

119892 (119910 (119904))] 119889119904

4 Journal of Applied Mathematics

1198813120590(119905)

(119905)

= int0

minus12059121

int119905

119905+120579

12059121119879

(119904) 1198771120590(119905)

(119904) 119889119904 119889120579

+ intminus12059121

minus12059122

int119905

119905+120579

(12059122minus 12059121) 119879

(119904) 1198772120590(119905)

(119904) 119889119904 119889120579

+ int0

minus12059111

int119905

119905+120579

12059111

119910119879

(119904) 1198773120590(119905)

119910 (119904) 119889119904 119889120579

+ intminus12059111

minus12059112

int119905

119905+120579

(12059112minus 12059111) 119910119879

(119904) 1198774120590(119905)

119910 (119904) 119889119904 119889120579

(16)

First when 119905 isin [119905119896 119905119896+1

) taking the derivatives of119881119894 119894 = 1 2 3

along the trajectory of system (6) we have that

1120590(119905119896)(119905) = 2119909

119879

(119905) 1198751120590(119905119896) (119905) + 2119910

119879

(119905) 1198752120590(119905119896)119910 (119905)

2120590(119905119896)(119905) = 119909

119879

(119905) 1198761120590(119905119896)119909 (119905) minus 119909

119879

(119905 minus 1205912(119905)) 119876

1120590(119905119896)119909119879

times (119905 minus 1205912(119905)) (1 minus 120591

2(119905))

+ [119910119879

(119905) 119892119879

(119910 (119905))] [1198762120590(119905119896)1198721120590(119905119896)

lowast 1198763120590(119905119896)

]

times [119910 (119905)

119892 (119910 (119905))]

minus [119910119879

(119905 minus 1205911(119905)) 119892

119879

(119910 (119905 minus 1205911(119905)))]

times [1198762120590(119905119896)1198721120590(119905119896)

lowast 1198763120590(119905119896)

] [119910 (119905 minus 120591

1(119905))

119892 (119910 (119905 minus 1205911(119905)))

]

times (1 minus 1205911(119905))

3120590(119905119896)(119905) = 120591

2

21119879

(119905) 1198771120590(119905119896) (119904)

minus int119905

119905minus12059121

12059121119879

(120579) 1198771120590(119905119896) (120579) 119889120579

+ (12059122minus 12059121)2

119879

(119905) 1198772120590(119905119896) (119904)

minus int119905minus12059121

119905minus12059122

(12059122minus 12059121) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

+ 1205912

11119910119879

(119905) 1198773120590(119905119896)119910 (119904)

minus int119905

119905minus12059111

12059111

119910119879

(120579) 1198773120590(119905119896)119910 (120579) 119889120579

+ (12059112minus 12059111)2

119910119879

(119905) 1198774120590(119905119896)119910 (119904)

minus int119905minus12059111

119905minus12059112

(12059112minus 12059111) 119910119879

(120579) 1198774120590(119905119896)119910 (120579) 119889120579

(17)

By Lemma 7 we have

minus int119905

119905minus12059121

12059121119879

(120579) 1198771120590(119905119896) (120579) 119889120579

le minus[119909 (119905) minus 119909 (119905 minus 12059121)]119879

1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591

21)]

(18)

minus int119905

119905minus12059111

12059111

119910119879

(120579) 1198773120590(119905119896)119910 (120579) 119889120579

le minus[119910 (119905) minus 119910 (119905 minus 12059111)]119879

1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591

11)]

(19)

minus int119905minus12059121

119905minus12059122

(12059122minus 12059121) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

= minus (12059122minus 12059121) [int119905minus1205912(119905)

119905minus12059122

119879

(120579) 1198772120590(119905119896) (120579) 119889120579

+ int119905minus12059121

119905minus1205912(119905)

119879

(120579) 1198772120590(119905119896) (120579) 119889120579]

= minusint119905minus1205912(119905)

119905minus12059122

(12059122minus 1205912(119905)) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus12059121

119905minus1205912(119905)

(1205912(119905) minus 120591

21) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus1205912(119905)

119905minus12059122

(1205912(119905) minus 120591

21) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus12059121

119905minus1205912(119905)

(12059122minus 1205912(119905)) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

le minus[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

minus1205912(119905) minus 120591

21

12059122minus 12059121

[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus12059122minus 1205912(119905)

12059122minus 12059121

[119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

(20)

Journal of Applied Mathematics 5

Similar to (20) we can obtain

minus int119905minus12059111

119905minus12059112

(12059112minus 12059111) 119910119879

(120579) 1198774120590(119905119896)119910 (120579) 119889120579

le minus[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

minus1205911(119905) minus 120591

11

12059112minus 12059111

[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus12059112minus 1205911(119905)

12059112minus 12059111

[119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

(21)

In addition for any Λ119898= diag(120582

1198981 120582

119898119899) ge 0 119898 = 1 2

the following inequality is true from (5)

minus 2

119899

sum119894=1

1205821119894119892119894(119910119894(119905)) [119892

119894(119910119894(119905)) minus 119906

119894119910119894(119905)]

minus 2

119899

sum119894=1

1205822119894119892119894(119910119894(119905 minus 1205911(119905)))

times [119892119894(119910119894(119905 minus 1205911(119905))) minus 119906

119894119910119894(119905 minus 1205911(119905))] ge 0

(22)

It can be written as matrix form

minus 2119892119879

(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)] minus 2119892

119879

(119910 (119905 minus 1205911(119905))) Λ

2

times [119892 (119910 (119905 minus 1205911(119905))) minus 119880119910 (119905 minus 120591

1(119905))] ge 0

(23)

From (15) to (23) we have that

120590(119905119896)(119905)

le 2119909119879

(119905) 1198751120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ 2119910119879

(119905) 1198752120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

+ 119909119879

(119905) 1198761120590(119905119896)119909 (119905) minus 119909

119879

(119905 minus 1205912(119905)) 119876

1120590(119905119896)119909119879

times (119905 minus 1205912(119905)) (1 minus 120591

2119889) + 119910119879

(119905) 1198762120590119910 (119905)

+ 2119910119879

(119905)1198721120590119892 (119910 (119905)) + 119892

119879

(119910 (119905)) 1198763120590(119905119896)119892 (119910 (119905))

minus 119910119879

(119905 minus 1205911(119905)) 119876

2120590(119905119896)119910 (119905 minus 120591

1(119905)) (1 minus 120591

1119889)

minus 2119910119879

(119905 minus 1205911(119905))119872

1120590(119905119896)119892 (119910 (119905 minus 120591

1(119905))) (1 minus 120591

1119889)

minus 119892119879

(119910 (119905 minus 1205911(119905))) 119876

3120590(119905119896)119892 (119910 (119905 minus 120591

1(119905))) (1 minus 120591

1119889)

+ 1205912

21[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]119879

times 1198771120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ (12059122minus 12059121)2

[minus119860120590(119905119896)119909(119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]119879

times 1198772120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ 1205912

11[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]119879

times 1198773120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

+ (12059112minus 12059111)2

[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]119879

times 1198774120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

minus [119909 (119905) minus 119909 (119905 minus 12059121)]119879

1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591

21)]

minus [119910 (119905) minus 119910 (119905 minus 12059111)]119879

1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591

11)]

minus [119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

minus [119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

minus 2119892119879

(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)]

minus 2119892119879

(119910 (119905 minus 1205911(119905)))

times Λ2[119892 (119910 (119905 minus 120591

1(119905))) minus 119880119910 (119905 minus 120591

1(119905))]

le 120585119879

Ξ120585 + 120572 (119909119879

(119905) 1198751120590(119905119896)119909 (119905) + 119910

119879

(119905) 1198752120590(119905119896)119910 (119905))

le 120585119879

Ξ120585 + 120572119881120590(119905119896)(119905)

(24)

where

120585119879

= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))]

(25)

6 Journal of Applied Mathematics

By condition (13) we have

120590(119905119896)(119905) le 120572119881

120590(119905119896)(119905) (26)

Integrating (26) from 119905119896to 119905 we obtain that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

119881120590(119905119896)(119905119896) (27)

Using (12) at switching instant 119905119896 we have

119881120590(119905119896)(119905119896) le 120583119881

120590(119905minus

119896)(119905minus

119896) (28)

Therefore it follows from (27) and (28) that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

120583119881120590(119905minus

119896)(119905minus

119896) (29)

For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873

120590(0 119879) le 119879120591

119886 we

have

119881120590(119905)

(119905) le 119890120572119905

120583119879120591119886119881120590(0)

(0) (30)

On the other hand it follows from (15) that

119881120590(0)

(0)

le max119894isinN

[120582max (1198751119894) + 12059122120582max (1198761119894)]

times supminus120588le119905le0

1003817100381710038171003817120593 (119905)10038171003817100381710038172

+ [120582max (1198752119894) + 12059112120582max (1198762119894)

+ 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)]

times supminus120588le119905le0

1003817100381710038171003817120595 (119905)10038171003817100381710038172

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

times 120582max (1198772119894)]

times supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

+1

2[1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

times120582max (1198774119894)] supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

le max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)] 1198881

119881120590(119905)

(119905)

ge min119894isinN

[120582min (1198751119894) 119909 (119905)2

+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)

10038171003817100381710038172

]

ge min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

times (119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

)

(31)

From (30) and (31) it is easy to obtain

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119905

1205831198791205911198861205821

1205822

1198881le 119890120572119879

1205831198791205911198861205821

1205822

1198881 (32)

where

1205821= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894) ]

1205822= min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

(33)

By (14) and (32) for any 119905 isin [0 119879] we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (34)

This completes the proof

Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays

(119905) = minus119860120590(119905)

119909 (119905) + 119861120590(119905)

119892 (119910 (119905 minus 1205911(119905)))

+ 119882120590(119905)

intinfin

0

ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904

119910 (119905) = minus119862120590(119905)

119910 (119905) + 119863120590(119905)

119909 (119905 minus 1205912(119905))

(35)

where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882

119894 119894 isin N is the 119894th subsystem distributively

Journal of Applied Mathematics 7

delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ

1(sdot) ℎ

119899(sdot)) is the delay kernel function and

all the other signs are defined as (6)

Assumption 9 The delay kernel ℎ119895rsquos are some real value

nonnegative continuous functions defined in [0infin) thatsatisfy

intinfin

0

ℎ119895(119904) 119889119904 = 1 int

infin

0

119904ℎ119895(119904) 119889119904 = 119896

119894lt infin

119895 = 1 119899

(36)

Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875

1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite

diagonal matrix 119871119894= diag(119897

1119894 119897119899119894) for all 119894 isin N Λ

119898=

diag(1205821198981 120582

119898119899) 119898 = 1 2 matrices 119876

2119894 1198763119894 1198721119894 119894 isin 119873

and positive scalars 120583 ge 1 such that the following inequalities

[1198762119894

1198721119894

lowast 1198763119894

] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2

[1198762119894

1198721119894

lowast 1198763119894

] le 120583 [1198762119895

1198721119895

lowast 1198763119895

] 119877119897119894le 120583119877119897119895

119897 = 1 4 forall119894 119895 isin N

(37)

Ξ1= Ω1

+ sym 119865119879

111987711198941198652+ 119865119879

1[1198751119894119861119894minus 1205912

21119861119879

1198941198771119894119860119894

minus(12059122minus 12059121)2

119861119879

1198941198772119894119860119894]

times 11986510+ 119865119879

211987721198941198653+ 119865119879

311987721198941198654+ 119865119879

3

times [1198752119894119863119894minus 1205912

11119863119879

1198941198773119894119862119894minus (12059112minus 12059111)2

119863119879

1198941198774119894119862119894]

times 1198655+ 119865119879

511987731198941198656+ 119865119879

5(1198721119894+ Λ1119880)1198659+ 119865119879

611987741198941198657

+119865119879

711987741198941198658+ 119865119879

7[1198721119894(1 minus 120591

13) + Λ

2119880]11986510

lt 0 forall119894 isin N(38)

hold and the average dwell time of the switch signal 120590(119905)satisfies

120591119886gt 120591lowast

119886=

119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879

(39)

where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591

23) + 1205912

11119863119879119894

1198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894

1198773119894119862119894+ (12059112minus 12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894

(1 minus 12059113) minus 2119877

4119894 minus1198774119894 1198762119894minus 2Λ

1+ 119871119894minus1198763119894(1 minus 120591

13) +

1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899

119868119899times119899

0119899times(12minus119895)119899

] 119895 = 1 11 12058210158401= max

119894isinN120582max(1198751119894) +

12059122120582max(1198761119894) + 120582max(1198752119894) + 120591

12120582max(1198762119894) +

12059112120582max(1198763119894) 120582max(119880119880) + 2120591

12120582max(1198721119894)120582max(119880) + (12)

[1205913

21120582max(1198771119894) + (120591

22minus 12059121)3

120582max(1198772119894) + 120591311120582max(1198773119894) +

(12059112minus 12059111)3

120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899

119895=1119897119894119895119896119895 12058210158402= 1205822

The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888

1 1198882 119879)

Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional

119881120590(119905)

(119905) = 1198811120590(119905)

(119905) + 1198812120590(119905)

(119905) + 1198813120590(119905)

(119905) + 1198814120590(119905)

(119905) (40)

where1198811120590(119905)

(119905)1198812120590(119905)

(119905)1198813120590(119905)

(119905) are defined as inTheorem 8

1198814120590(119905)

(119905) =

119899

sum119894=1

119897119894120590(119905)

intinfin

0

ℎ119894(120579) int119905

119905minus120579

1198922

119894(119910119894(119904)) 119889119904 119889120579 (41)

when 119905 isin [119905119896 119905119896+1

) taking the derivatives of 1198814along the

trajectory of system (35) we have that

4120590(119905119896)(119905) =

119899

sum119894=1

119897119894120590(119905119896)1198922

119894(119910119894(119905))

minus

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

(42)

By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2

le (int 1199012(119904)119889119904)

(int 1199022(119904)119889119904) we obtain that

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

=

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 119889120579

times intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

ge

119899

sum119894=1

119897119894120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

2

= [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(43)

Therefore

4120590(119905119896)(119905) le 119892

119879

(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))

minus [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(44)

Combing (17)ndash(23) and (40)ndash(43) we get

120590(119905119896)(119905) le 120585

119879

1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)

8 Journal of Applied Mathematics

where

120585119879

1= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))

[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

]

(46)

Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119879

120583119879120591119886

12058210158401

12058210158402

1198881 (47)

where

1205821015840

1= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)]

+120582max (1198774119894) + 1198662

119899

sum119895=1

119897119894119895119896119895

1205821015840

2= 1205822

(48)

By (39) and (47) we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (49)

This completes the proof

4 Numerical Examples

In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10

Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback

0 1 2 3 4 5

minus01

minus005

0

005

01

015

02

025

03

mRN

A co

ncen

trat

ions

t

Figure 1 mRNA concentrations 119909(119905)

0

0

1 2 3 4 5

Prot

ein

conc

entr

atio

ns

minus02

minus01

t

01

02

03

04

05

06

Figure 2 Protein concentrations 119910(119905)

loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)

119889119898119894

119889119905= minus119898119894+

120572

1 + 119901119899119895

+ 1205720

119889119901119894

119889119905= 120573 (119898

119894minus 119901119894)

(50)

where 119898119894and 119901

119894are the concentrations of two mRNAs and

repressor-protein 1205720is the growth rate of protein in a cell in

the presence of saturating amounts of repressor while 1205720+ 120572

is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient

Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by

Journal of Applied Mathematics 9

adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model

(119905) = minus119860119894119909 (119905) + 119861

119894119892 (119910 (119905 minus 120590 (119905)))

119910 (119905) = minus119862119894119910 (119905) + 119863

119894119909 (119905 minus 120591 (119905)) 119894 = 1 2

(51)

where

1198601= diag (4 4 4) 119862

1= diag (5 5 5)

1198631= diag (2 15 1) 119860

2= diag (3 3 3)

1198622= diag (4 4 4) 119863

2= diag (12 12 12)

1198611= 2 times (

0 0 minus1

minus1 0 0

0 minus1 0

) 1198612= 15 times (

0 0 minus1

minus1 0 0

0 minus1 0

)

(52)

The gene regulation function is taken as 119892(119909) = 1199092(1 +

1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and

1205912(119905) are assumed to be

1205911(119905) = 06 + 04 sin 119905 120591

2(119905) = 03 + 01 cos 119905 (53)

we can get the parameters as following

12059111= 02 120591

12= 1 120591

13= 04

12059121= 04 120591

22= 02 120591

23= 01

(54)

and let 1198881= 2 119888

2= 15 119879 = 5 120572 = 002 120583 = 12 By using

the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI

Ξ =

(((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 119886119894110

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010

)))))))))))))))

)

(55)

where 11988611989411

= minus21198751119894119860119894+ 1198761119894

+ 1205912211198601198791198941198771119894119860119894+ (12059122

minus

12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus

(12059122minus 12059121)2

1198611198791198941198772119894119860119894 11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+

(12059112minus12059111)2

1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus

12059111)2

1198631198791198941198774119894119862119894 11988611989455

= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus

12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877

4119894 119886119894710

=

1198721119894(1 minus 120591

13) + Λ

2119880 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912

21119861119879

1198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

35732 minus02655 minus03039

minus02655 33575 minus02912

minus03039 minus02912 31531

)

11987512= (

46463 minus01506 minus01453

minus01506 47212 minus01477

minus01453 minus01477 47809

)

11987611= (

91959 minus03223 minus02832

minus03223 74980 minus02791

minus02832 minus02791 59270

)

11987612= (

108205 minus04573 minus04597

minus04573 109025 minus04104

minus04597 minus04104 110402

)

(56)

and 120591119886

gt 120591lowast119886

= 04003 The initial condition is 119909(0) =

(02 02 03)119879 119910(0) = (03 02 05)

119879 The simulation resultsof the trajectories are shown in Figures 1 and 2

Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows

1198601= diag (21 12 25) 119862

1= diag (46 4 52)

1198631= diag (05 04 06) 119860

2= diag (18 2 31)

1198622= diag (5 4 5) 119863

2= diag (03 03 04)

1198611= (

0 1 0

0 0 1

1 0 0

) 1198612= (

0 07 0

0 0 07

07 0 0

)

(57)

10 Journal of Applied Mathematics

and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591

1(119905) and 120591

2(119905) are assumed to be

1205911(119905) = 05 + 03 sin 119905 120591

2(119905) = 04 + 01 cos 119905 (58)

we can get the parameters as following

12059111= 02 120591

12= 08 120591

13= 03

12059121= 03 120591

22= 05 120591

23= 01

(59)

and let 1198881= 25 119888

2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ

119895(119904) = 119890minus119904

and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI

Ξ =

(((((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 1198861198940

0

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

0

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499

0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894

10100

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894

)))))))))))))))))

)

(60)

where 11988611989411= minus2119875

1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus

1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2

1198611198791198941198772119894119860119894

11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus

21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2

1198631198791198941198774119894119862119894 11988611989455=

minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894

11988611989477= minus1198762119894(1 minus 120591

13) minus 2119877

4119894 119886119894710

= 1198721119894(1 minus 120591

13) + Λ2119880 11988611989499=

1198762119894minus 2Λ1+ 119871119894 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus

12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

13745 minus00657 minus00145

minus00657 13683 minus00471

minus00145 minus00471 12893

)

11987512= (

16025 minus00233 minus00056

minus00233 15143 minus00073

minus00056 minus00073 12907

)

11987611= (

25617 minus00905 minus00313

minus00905 20794 minus00718

minus00313 minus00718 24995

)

11987612= (

21367 minus00751 minus00258

minus00751 21184 minus00434

minus00258 minus00434 22000

)

(61)

and 120591119886gt 120591lowast119886= 05012

5 Concluding Remarks

This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)

References

[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

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 Journal of Applied Mathematics

1198813120590(119905)

(119905)

= int0

minus12059121

int119905

119905+120579

12059121119879

(119904) 1198771120590(119905)

(119904) 119889119904 119889120579

+ intminus12059121

minus12059122

int119905

119905+120579

(12059122minus 12059121) 119879

(119904) 1198772120590(119905)

(119904) 119889119904 119889120579

+ int0

minus12059111

int119905

119905+120579

12059111

119910119879

(119904) 1198773120590(119905)

119910 (119904) 119889119904 119889120579

+ intminus12059111

minus12059112

int119905

119905+120579

(12059112minus 12059111) 119910119879

(119904) 1198774120590(119905)

119910 (119904) 119889119904 119889120579

(16)

First when 119905 isin [119905119896 119905119896+1

) taking the derivatives of119881119894 119894 = 1 2 3

along the trajectory of system (6) we have that

1120590(119905119896)(119905) = 2119909

119879

(119905) 1198751120590(119905119896) (119905) + 2119910

119879

(119905) 1198752120590(119905119896)119910 (119905)

2120590(119905119896)(119905) = 119909

119879

(119905) 1198761120590(119905119896)119909 (119905) minus 119909

119879

(119905 minus 1205912(119905)) 119876

1120590(119905119896)119909119879

times (119905 minus 1205912(119905)) (1 minus 120591

2(119905))

+ [119910119879

(119905) 119892119879

(119910 (119905))] [1198762120590(119905119896)1198721120590(119905119896)

lowast 1198763120590(119905119896)

]

times [119910 (119905)

119892 (119910 (119905))]

minus [119910119879

(119905 minus 1205911(119905)) 119892

119879

(119910 (119905 minus 1205911(119905)))]

times [1198762120590(119905119896)1198721120590(119905119896)

lowast 1198763120590(119905119896)

] [119910 (119905 minus 120591

1(119905))

119892 (119910 (119905 minus 1205911(119905)))

]

times (1 minus 1205911(119905))

3120590(119905119896)(119905) = 120591

2

21119879

(119905) 1198771120590(119905119896) (119904)

minus int119905

119905minus12059121

12059121119879

(120579) 1198771120590(119905119896) (120579) 119889120579

+ (12059122minus 12059121)2

119879

(119905) 1198772120590(119905119896) (119904)

minus int119905minus12059121

119905minus12059122

(12059122minus 12059121) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

+ 1205912

11119910119879

(119905) 1198773120590(119905119896)119910 (119904)

minus int119905

119905minus12059111

12059111

119910119879

(120579) 1198773120590(119905119896)119910 (120579) 119889120579

+ (12059112minus 12059111)2

119910119879

(119905) 1198774120590(119905119896)119910 (119904)

minus int119905minus12059111

119905minus12059112

(12059112minus 12059111) 119910119879

(120579) 1198774120590(119905119896)119910 (120579) 119889120579

(17)

By Lemma 7 we have

minus int119905

119905minus12059121

12059121119879

(120579) 1198771120590(119905119896) (120579) 119889120579

le minus[119909 (119905) minus 119909 (119905 minus 12059121)]119879

1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591

21)]

(18)

minus int119905

119905minus12059111

12059111

119910119879

(120579) 1198773120590(119905119896)119910 (120579) 119889120579

le minus[119910 (119905) minus 119910 (119905 minus 12059111)]119879

1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591

11)]

(19)

minus int119905minus12059121

119905minus12059122

(12059122minus 12059121) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

= minus (12059122minus 12059121) [int119905minus1205912(119905)

119905minus12059122

119879

(120579) 1198772120590(119905119896) (120579) 119889120579

+ int119905minus12059121

119905minus1205912(119905)

119879

(120579) 1198772120590(119905119896) (120579) 119889120579]

= minusint119905minus1205912(119905)

119905minus12059122

(12059122minus 1205912(119905)) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus12059121

119905minus1205912(119905)

(1205912(119905) minus 120591

21) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus1205912(119905)

119905minus12059122

(1205912(119905) minus 120591

21) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

minus int119905minus12059121

119905minus1205912(119905)

(12059122minus 1205912(119905)) 119879

(120579) 1198772120590(119905119896) (120579) 119889120579

le minus[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

minus1205912(119905) minus 120591

21

12059122minus 12059121

[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus12059122minus 1205912(119905)

12059122minus 12059121

[119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

(20)

Journal of Applied Mathematics 5

Similar to (20) we can obtain

minus int119905minus12059111

119905minus12059112

(12059112minus 12059111) 119910119879

(120579) 1198774120590(119905119896)119910 (120579) 119889120579

le minus[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

minus1205911(119905) minus 120591

11

12059112minus 12059111

[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus12059112minus 1205911(119905)

12059112minus 12059111

[119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

(21)

In addition for any Λ119898= diag(120582

1198981 120582

119898119899) ge 0 119898 = 1 2

the following inequality is true from (5)

minus 2

119899

sum119894=1

1205821119894119892119894(119910119894(119905)) [119892

119894(119910119894(119905)) minus 119906

119894119910119894(119905)]

minus 2

119899

sum119894=1

1205822119894119892119894(119910119894(119905 minus 1205911(119905)))

times [119892119894(119910119894(119905 minus 1205911(119905))) minus 119906

119894119910119894(119905 minus 1205911(119905))] ge 0

(22)

It can be written as matrix form

minus 2119892119879

(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)] minus 2119892

119879

(119910 (119905 minus 1205911(119905))) Λ

2

times [119892 (119910 (119905 minus 1205911(119905))) minus 119880119910 (119905 minus 120591

1(119905))] ge 0

(23)

From (15) to (23) we have that

120590(119905119896)(119905)

le 2119909119879

(119905) 1198751120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ 2119910119879

(119905) 1198752120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

+ 119909119879

(119905) 1198761120590(119905119896)119909 (119905) minus 119909

119879

(119905 minus 1205912(119905)) 119876

1120590(119905119896)119909119879

times (119905 minus 1205912(119905)) (1 minus 120591

2119889) + 119910119879

(119905) 1198762120590119910 (119905)

+ 2119910119879

(119905)1198721120590119892 (119910 (119905)) + 119892

119879

(119910 (119905)) 1198763120590(119905119896)119892 (119910 (119905))

minus 119910119879

(119905 minus 1205911(119905)) 119876

2120590(119905119896)119910 (119905 minus 120591

1(119905)) (1 minus 120591

1119889)

minus 2119910119879

(119905 minus 1205911(119905))119872

1120590(119905119896)119892 (119910 (119905 minus 120591

1(119905))) (1 minus 120591

1119889)

minus 119892119879

(119910 (119905 minus 1205911(119905))) 119876

3120590(119905119896)119892 (119910 (119905 minus 120591

1(119905))) (1 minus 120591

1119889)

+ 1205912

21[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]119879

times 1198771120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ (12059122minus 12059121)2

[minus119860120590(119905119896)119909(119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]119879

times 1198772120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ 1205912

11[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]119879

times 1198773120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

+ (12059112minus 12059111)2

[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]119879

times 1198774120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

minus [119909 (119905) minus 119909 (119905 minus 12059121)]119879

1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591

21)]

minus [119910 (119905) minus 119910 (119905 minus 12059111)]119879

1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591

11)]

minus [119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

minus [119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

minus 2119892119879

(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)]

minus 2119892119879

(119910 (119905 minus 1205911(119905)))

times Λ2[119892 (119910 (119905 minus 120591

1(119905))) minus 119880119910 (119905 minus 120591

1(119905))]

le 120585119879

Ξ120585 + 120572 (119909119879

(119905) 1198751120590(119905119896)119909 (119905) + 119910

119879

(119905) 1198752120590(119905119896)119910 (119905))

le 120585119879

Ξ120585 + 120572119881120590(119905119896)(119905)

(24)

where

120585119879

= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))]

(25)

6 Journal of Applied Mathematics

By condition (13) we have

120590(119905119896)(119905) le 120572119881

120590(119905119896)(119905) (26)

Integrating (26) from 119905119896to 119905 we obtain that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

119881120590(119905119896)(119905119896) (27)

Using (12) at switching instant 119905119896 we have

119881120590(119905119896)(119905119896) le 120583119881

120590(119905minus

119896)(119905minus

119896) (28)

Therefore it follows from (27) and (28) that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

120583119881120590(119905minus

119896)(119905minus

119896) (29)

For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873

120590(0 119879) le 119879120591

119886 we

have

119881120590(119905)

(119905) le 119890120572119905

120583119879120591119886119881120590(0)

(0) (30)

On the other hand it follows from (15) that

119881120590(0)

(0)

le max119894isinN

[120582max (1198751119894) + 12059122120582max (1198761119894)]

times supminus120588le119905le0

1003817100381710038171003817120593 (119905)10038171003817100381710038172

+ [120582max (1198752119894) + 12059112120582max (1198762119894)

+ 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)]

times supminus120588le119905le0

1003817100381710038171003817120595 (119905)10038171003817100381710038172

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

times 120582max (1198772119894)]

times supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

+1

2[1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

times120582max (1198774119894)] supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

le max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)] 1198881

119881120590(119905)

(119905)

ge min119894isinN

[120582min (1198751119894) 119909 (119905)2

+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)

10038171003817100381710038172

]

ge min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

times (119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

)

(31)

From (30) and (31) it is easy to obtain

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119905

1205831198791205911198861205821

1205822

1198881le 119890120572119879

1205831198791205911198861205821

1205822

1198881 (32)

where

1205821= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894) ]

1205822= min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

(33)

By (14) and (32) for any 119905 isin [0 119879] we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (34)

This completes the proof

Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays

(119905) = minus119860120590(119905)

119909 (119905) + 119861120590(119905)

119892 (119910 (119905 minus 1205911(119905)))

+ 119882120590(119905)

intinfin

0

ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904

119910 (119905) = minus119862120590(119905)

119910 (119905) + 119863120590(119905)

119909 (119905 minus 1205912(119905))

(35)

where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882

119894 119894 isin N is the 119894th subsystem distributively

Journal of Applied Mathematics 7

delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ

1(sdot) ℎ

119899(sdot)) is the delay kernel function and

all the other signs are defined as (6)

Assumption 9 The delay kernel ℎ119895rsquos are some real value

nonnegative continuous functions defined in [0infin) thatsatisfy

intinfin

0

ℎ119895(119904) 119889119904 = 1 int

infin

0

119904ℎ119895(119904) 119889119904 = 119896

119894lt infin

119895 = 1 119899

(36)

Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875

1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite

diagonal matrix 119871119894= diag(119897

1119894 119897119899119894) for all 119894 isin N Λ

119898=

diag(1205821198981 120582

119898119899) 119898 = 1 2 matrices 119876

2119894 1198763119894 1198721119894 119894 isin 119873

and positive scalars 120583 ge 1 such that the following inequalities

[1198762119894

1198721119894

lowast 1198763119894

] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2

[1198762119894

1198721119894

lowast 1198763119894

] le 120583 [1198762119895

1198721119895

lowast 1198763119895

] 119877119897119894le 120583119877119897119895

119897 = 1 4 forall119894 119895 isin N

(37)

Ξ1= Ω1

+ sym 119865119879

111987711198941198652+ 119865119879

1[1198751119894119861119894minus 1205912

21119861119879

1198941198771119894119860119894

minus(12059122minus 12059121)2

119861119879

1198941198772119894119860119894]

times 11986510+ 119865119879

211987721198941198653+ 119865119879

311987721198941198654+ 119865119879

3

times [1198752119894119863119894minus 1205912

11119863119879

1198941198773119894119862119894minus (12059112minus 12059111)2

119863119879

1198941198774119894119862119894]

times 1198655+ 119865119879

511987731198941198656+ 119865119879

5(1198721119894+ Λ1119880)1198659+ 119865119879

611987741198941198657

+119865119879

711987741198941198658+ 119865119879

7[1198721119894(1 minus 120591

13) + Λ

2119880]11986510

lt 0 forall119894 isin N(38)

hold and the average dwell time of the switch signal 120590(119905)satisfies

120591119886gt 120591lowast

119886=

119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879

(39)

where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591

23) + 1205912

11119863119879119894

1198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894

1198773119894119862119894+ (12059112minus 12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894

(1 minus 12059113) minus 2119877

4119894 minus1198774119894 1198762119894minus 2Λ

1+ 119871119894minus1198763119894(1 minus 120591

13) +

1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899

119868119899times119899

0119899times(12minus119895)119899

] 119895 = 1 11 12058210158401= max

119894isinN120582max(1198751119894) +

12059122120582max(1198761119894) + 120582max(1198752119894) + 120591

12120582max(1198762119894) +

12059112120582max(1198763119894) 120582max(119880119880) + 2120591

12120582max(1198721119894)120582max(119880) + (12)

[1205913

21120582max(1198771119894) + (120591

22minus 12059121)3

120582max(1198772119894) + 120591311120582max(1198773119894) +

(12059112minus 12059111)3

120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899

119895=1119897119894119895119896119895 12058210158402= 1205822

The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888

1 1198882 119879)

Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional

119881120590(119905)

(119905) = 1198811120590(119905)

(119905) + 1198812120590(119905)

(119905) + 1198813120590(119905)

(119905) + 1198814120590(119905)

(119905) (40)

where1198811120590(119905)

(119905)1198812120590(119905)

(119905)1198813120590(119905)

(119905) are defined as inTheorem 8

1198814120590(119905)

(119905) =

119899

sum119894=1

119897119894120590(119905)

intinfin

0

ℎ119894(120579) int119905

119905minus120579

1198922

119894(119910119894(119904)) 119889119904 119889120579 (41)

when 119905 isin [119905119896 119905119896+1

) taking the derivatives of 1198814along the

trajectory of system (35) we have that

4120590(119905119896)(119905) =

119899

sum119894=1

119897119894120590(119905119896)1198922

119894(119910119894(119905))

minus

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

(42)

By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2

le (int 1199012(119904)119889119904)

(int 1199022(119904)119889119904) we obtain that

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

=

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 119889120579

times intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

ge

119899

sum119894=1

119897119894120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

2

= [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(43)

Therefore

4120590(119905119896)(119905) le 119892

119879

(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))

minus [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(44)

Combing (17)ndash(23) and (40)ndash(43) we get

120590(119905119896)(119905) le 120585

119879

1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)

8 Journal of Applied Mathematics

where

120585119879

1= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))

[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

]

(46)

Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119879

120583119879120591119886

12058210158401

12058210158402

1198881 (47)

where

1205821015840

1= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)]

+120582max (1198774119894) + 1198662

119899

sum119895=1

119897119894119895119896119895

1205821015840

2= 1205822

(48)

By (39) and (47) we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (49)

This completes the proof

4 Numerical Examples

In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10

Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback

0 1 2 3 4 5

minus01

minus005

0

005

01

015

02

025

03

mRN

A co

ncen

trat

ions

t

Figure 1 mRNA concentrations 119909(119905)

0

0

1 2 3 4 5

Prot

ein

conc

entr

atio

ns

minus02

minus01

t

01

02

03

04

05

06

Figure 2 Protein concentrations 119910(119905)

loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)

119889119898119894

119889119905= minus119898119894+

120572

1 + 119901119899119895

+ 1205720

119889119901119894

119889119905= 120573 (119898

119894minus 119901119894)

(50)

where 119898119894and 119901

119894are the concentrations of two mRNAs and

repressor-protein 1205720is the growth rate of protein in a cell in

the presence of saturating amounts of repressor while 1205720+ 120572

is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient

Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by

Journal of Applied Mathematics 9

adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model

(119905) = minus119860119894119909 (119905) + 119861

119894119892 (119910 (119905 minus 120590 (119905)))

119910 (119905) = minus119862119894119910 (119905) + 119863

119894119909 (119905 minus 120591 (119905)) 119894 = 1 2

(51)

where

1198601= diag (4 4 4) 119862

1= diag (5 5 5)

1198631= diag (2 15 1) 119860

2= diag (3 3 3)

1198622= diag (4 4 4) 119863

2= diag (12 12 12)

1198611= 2 times (

0 0 minus1

minus1 0 0

0 minus1 0

) 1198612= 15 times (

0 0 minus1

minus1 0 0

0 minus1 0

)

(52)

The gene regulation function is taken as 119892(119909) = 1199092(1 +

1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and

1205912(119905) are assumed to be

1205911(119905) = 06 + 04 sin 119905 120591

2(119905) = 03 + 01 cos 119905 (53)

we can get the parameters as following

12059111= 02 120591

12= 1 120591

13= 04

12059121= 04 120591

22= 02 120591

23= 01

(54)

and let 1198881= 2 119888

2= 15 119879 = 5 120572 = 002 120583 = 12 By using

the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI

Ξ =

(((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 119886119894110

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010

)))))))))))))))

)

(55)

where 11988611989411

= minus21198751119894119860119894+ 1198761119894

+ 1205912211198601198791198941198771119894119860119894+ (12059122

minus

12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus

(12059122minus 12059121)2

1198611198791198941198772119894119860119894 11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+

(12059112minus12059111)2

1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus

12059111)2

1198631198791198941198774119894119862119894 11988611989455

= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus

12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877

4119894 119886119894710

=

1198721119894(1 minus 120591

13) + Λ

2119880 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912

21119861119879

1198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

35732 minus02655 minus03039

minus02655 33575 minus02912

minus03039 minus02912 31531

)

11987512= (

46463 minus01506 minus01453

minus01506 47212 minus01477

minus01453 minus01477 47809

)

11987611= (

91959 minus03223 minus02832

minus03223 74980 minus02791

minus02832 minus02791 59270

)

11987612= (

108205 minus04573 minus04597

minus04573 109025 minus04104

minus04597 minus04104 110402

)

(56)

and 120591119886

gt 120591lowast119886

= 04003 The initial condition is 119909(0) =

(02 02 03)119879 119910(0) = (03 02 05)

119879 The simulation resultsof the trajectories are shown in Figures 1 and 2

Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows

1198601= diag (21 12 25) 119862

1= diag (46 4 52)

1198631= diag (05 04 06) 119860

2= diag (18 2 31)

1198622= diag (5 4 5) 119863

2= diag (03 03 04)

1198611= (

0 1 0

0 0 1

1 0 0

) 1198612= (

0 07 0

0 0 07

07 0 0

)

(57)

10 Journal of Applied Mathematics

and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591

1(119905) and 120591

2(119905) are assumed to be

1205911(119905) = 05 + 03 sin 119905 120591

2(119905) = 04 + 01 cos 119905 (58)

we can get the parameters as following

12059111= 02 120591

12= 08 120591

13= 03

12059121= 03 120591

22= 05 120591

23= 01

(59)

and let 1198881= 25 119888

2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ

119895(119904) = 119890minus119904

and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI

Ξ =

(((((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 1198861198940

0

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

0

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499

0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894

10100

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894

)))))))))))))))))

)

(60)

where 11988611989411= minus2119875

1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus

1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2

1198611198791198941198772119894119860119894

11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus

21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2

1198631198791198941198774119894119862119894 11988611989455=

minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894

11988611989477= minus1198762119894(1 minus 120591

13) minus 2119877

4119894 119886119894710

= 1198721119894(1 minus 120591

13) + Λ2119880 11988611989499=

1198762119894minus 2Λ1+ 119871119894 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus

12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

13745 minus00657 minus00145

minus00657 13683 minus00471

minus00145 minus00471 12893

)

11987512= (

16025 minus00233 minus00056

minus00233 15143 minus00073

minus00056 minus00073 12907

)

11987611= (

25617 minus00905 minus00313

minus00905 20794 minus00718

minus00313 minus00718 24995

)

11987612= (

21367 minus00751 minus00258

minus00751 21184 minus00434

minus00258 minus00434 22000

)

(61)

and 120591119886gt 120591lowast119886= 05012

5 Concluding Remarks

This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)

References

[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

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

Journal of Applied Mathematics 5

Similar to (20) we can obtain

minus int119905minus12059111

119905minus12059112

(12059112minus 12059111) 119910119879

(120579) 1198774120590(119905119896)119910 (120579) 119889120579

le minus[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

minus1205911(119905) minus 120591

11

12059112minus 12059111

[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus12059112minus 1205911(119905)

12059112minus 12059111

[119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

(21)

In addition for any Λ119898= diag(120582

1198981 120582

119898119899) ge 0 119898 = 1 2

the following inequality is true from (5)

minus 2

119899

sum119894=1

1205821119894119892119894(119910119894(119905)) [119892

119894(119910119894(119905)) minus 119906

119894119910119894(119905)]

minus 2

119899

sum119894=1

1205822119894119892119894(119910119894(119905 minus 1205911(119905)))

times [119892119894(119910119894(119905 minus 1205911(119905))) minus 119906

119894119910119894(119905 minus 1205911(119905))] ge 0

(22)

It can be written as matrix form

minus 2119892119879

(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)] minus 2119892

119879

(119910 (119905 minus 1205911(119905))) Λ

2

times [119892 (119910 (119905 minus 1205911(119905))) minus 119880119910 (119905 minus 120591

1(119905))] ge 0

(23)

From (15) to (23) we have that

120590(119905119896)(119905)

le 2119909119879

(119905) 1198751120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ 2119910119879

(119905) 1198752120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

+ 119909119879

(119905) 1198761120590(119905119896)119909 (119905) minus 119909

119879

(119905 minus 1205912(119905)) 119876

1120590(119905119896)119909119879

times (119905 minus 1205912(119905)) (1 minus 120591

2119889) + 119910119879

(119905) 1198762120590119910 (119905)

+ 2119910119879

(119905)1198721120590119892 (119910 (119905)) + 119892

119879

(119910 (119905)) 1198763120590(119905119896)119892 (119910 (119905))

minus 119910119879

(119905 minus 1205911(119905)) 119876

2120590(119905119896)119910 (119905 minus 120591

1(119905)) (1 minus 120591

1119889)

minus 2119910119879

(119905 minus 1205911(119905))119872

1120590(119905119896)119892 (119910 (119905 minus 120591

1(119905))) (1 minus 120591

1119889)

minus 119892119879

(119910 (119905 minus 1205911(119905))) 119876

3120590(119905119896)119892 (119910 (119905 minus 120591

1(119905))) (1 minus 120591

1119889)

+ 1205912

21[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]119879

times 1198771120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ (12059122minus 12059121)2

[minus119860120590(119905119896)119909(119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]119879

times 1198772120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861

120590(119905119896)119892 (119910 (119905 minus 120591

1(119905)))]

+ 1205912

11[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]119879

times 1198773120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

+ (12059112minus 12059111)2

[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]119879

times 1198774120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863

120590(119905119896)119909 (119905 minus 120591

2(119905))]

minus [119909 (119905) minus 119909 (119905 minus 12059121)]119879

1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591

21)]

minus [119910 (119905) minus 119910 (119905 minus 12059111)]119879

1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591

11)]

minus [119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591

22)]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

2(119905)) minus 119909 (119905 minus 120591

22)]

minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591

2(119905))]119879

times 1198772120590(119905119896)[119909 (119905 minus 120591

21) minus 119909 (119905 minus 120591

2(119905))]

minus [119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591

12)]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

1(119905)) minus 119910 (119905 minus 120591

12)]

minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591

1(119905))]119879

times 1198774120590(119905119896)[119910 (119905 minus 120591

11) minus 119910 (119905 minus 120591

1(119905))]

minus 2119892119879

(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)]

minus 2119892119879

(119910 (119905 minus 1205911(119905)))

times Λ2[119892 (119910 (119905 minus 120591

1(119905))) minus 119880119910 (119905 minus 120591

1(119905))]

le 120585119879

Ξ120585 + 120572 (119909119879

(119905) 1198751120590(119905119896)119909 (119905) + 119910

119879

(119905) 1198752120590(119905119896)119910 (119905))

le 120585119879

Ξ120585 + 120572119881120590(119905119896)(119905)

(24)

where

120585119879

= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))]

(25)

6 Journal of Applied Mathematics

By condition (13) we have

120590(119905119896)(119905) le 120572119881

120590(119905119896)(119905) (26)

Integrating (26) from 119905119896to 119905 we obtain that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

119881120590(119905119896)(119905119896) (27)

Using (12) at switching instant 119905119896 we have

119881120590(119905119896)(119905119896) le 120583119881

120590(119905minus

119896)(119905minus

119896) (28)

Therefore it follows from (27) and (28) that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

120583119881120590(119905minus

119896)(119905minus

119896) (29)

For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873

120590(0 119879) le 119879120591

119886 we

have

119881120590(119905)

(119905) le 119890120572119905

120583119879120591119886119881120590(0)

(0) (30)

On the other hand it follows from (15) that

119881120590(0)

(0)

le max119894isinN

[120582max (1198751119894) + 12059122120582max (1198761119894)]

times supminus120588le119905le0

1003817100381710038171003817120593 (119905)10038171003817100381710038172

+ [120582max (1198752119894) + 12059112120582max (1198762119894)

+ 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)]

times supminus120588le119905le0

1003817100381710038171003817120595 (119905)10038171003817100381710038172

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

times 120582max (1198772119894)]

times supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

+1

2[1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

times120582max (1198774119894)] supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

le max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)] 1198881

119881120590(119905)

(119905)

ge min119894isinN

[120582min (1198751119894) 119909 (119905)2

+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)

10038171003817100381710038172

]

ge min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

times (119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

)

(31)

From (30) and (31) it is easy to obtain

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119905

1205831198791205911198861205821

1205822

1198881le 119890120572119879

1205831198791205911198861205821

1205822

1198881 (32)

where

1205821= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894) ]

1205822= min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

(33)

By (14) and (32) for any 119905 isin [0 119879] we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (34)

This completes the proof

Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays

(119905) = minus119860120590(119905)

119909 (119905) + 119861120590(119905)

119892 (119910 (119905 minus 1205911(119905)))

+ 119882120590(119905)

intinfin

0

ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904

119910 (119905) = minus119862120590(119905)

119910 (119905) + 119863120590(119905)

119909 (119905 minus 1205912(119905))

(35)

where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882

119894 119894 isin N is the 119894th subsystem distributively

Journal of Applied Mathematics 7

delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ

1(sdot) ℎ

119899(sdot)) is the delay kernel function and

all the other signs are defined as (6)

Assumption 9 The delay kernel ℎ119895rsquos are some real value

nonnegative continuous functions defined in [0infin) thatsatisfy

intinfin

0

ℎ119895(119904) 119889119904 = 1 int

infin

0

119904ℎ119895(119904) 119889119904 = 119896

119894lt infin

119895 = 1 119899

(36)

Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875

1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite

diagonal matrix 119871119894= diag(119897

1119894 119897119899119894) for all 119894 isin N Λ

119898=

diag(1205821198981 120582

119898119899) 119898 = 1 2 matrices 119876

2119894 1198763119894 1198721119894 119894 isin 119873

and positive scalars 120583 ge 1 such that the following inequalities

[1198762119894

1198721119894

lowast 1198763119894

] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2

[1198762119894

1198721119894

lowast 1198763119894

] le 120583 [1198762119895

1198721119895

lowast 1198763119895

] 119877119897119894le 120583119877119897119895

119897 = 1 4 forall119894 119895 isin N

(37)

Ξ1= Ω1

+ sym 119865119879

111987711198941198652+ 119865119879

1[1198751119894119861119894minus 1205912

21119861119879

1198941198771119894119860119894

minus(12059122minus 12059121)2

119861119879

1198941198772119894119860119894]

times 11986510+ 119865119879

211987721198941198653+ 119865119879

311987721198941198654+ 119865119879

3

times [1198752119894119863119894minus 1205912

11119863119879

1198941198773119894119862119894minus (12059112minus 12059111)2

119863119879

1198941198774119894119862119894]

times 1198655+ 119865119879

511987731198941198656+ 119865119879

5(1198721119894+ Λ1119880)1198659+ 119865119879

611987741198941198657

+119865119879

711987741198941198658+ 119865119879

7[1198721119894(1 minus 120591

13) + Λ

2119880]11986510

lt 0 forall119894 isin N(38)

hold and the average dwell time of the switch signal 120590(119905)satisfies

120591119886gt 120591lowast

119886=

119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879

(39)

where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591

23) + 1205912

11119863119879119894

1198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894

1198773119894119862119894+ (12059112minus 12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894

(1 minus 12059113) minus 2119877

4119894 minus1198774119894 1198762119894minus 2Λ

1+ 119871119894minus1198763119894(1 minus 120591

13) +

1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899

119868119899times119899

0119899times(12minus119895)119899

] 119895 = 1 11 12058210158401= max

119894isinN120582max(1198751119894) +

12059122120582max(1198761119894) + 120582max(1198752119894) + 120591

12120582max(1198762119894) +

12059112120582max(1198763119894) 120582max(119880119880) + 2120591

12120582max(1198721119894)120582max(119880) + (12)

[1205913

21120582max(1198771119894) + (120591

22minus 12059121)3

120582max(1198772119894) + 120591311120582max(1198773119894) +

(12059112minus 12059111)3

120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899

119895=1119897119894119895119896119895 12058210158402= 1205822

The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888

1 1198882 119879)

Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional

119881120590(119905)

(119905) = 1198811120590(119905)

(119905) + 1198812120590(119905)

(119905) + 1198813120590(119905)

(119905) + 1198814120590(119905)

(119905) (40)

where1198811120590(119905)

(119905)1198812120590(119905)

(119905)1198813120590(119905)

(119905) are defined as inTheorem 8

1198814120590(119905)

(119905) =

119899

sum119894=1

119897119894120590(119905)

intinfin

0

ℎ119894(120579) int119905

119905minus120579

1198922

119894(119910119894(119904)) 119889119904 119889120579 (41)

when 119905 isin [119905119896 119905119896+1

) taking the derivatives of 1198814along the

trajectory of system (35) we have that

4120590(119905119896)(119905) =

119899

sum119894=1

119897119894120590(119905119896)1198922

119894(119910119894(119905))

minus

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

(42)

By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2

le (int 1199012(119904)119889119904)

(int 1199022(119904)119889119904) we obtain that

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

=

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 119889120579

times intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

ge

119899

sum119894=1

119897119894120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

2

= [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(43)

Therefore

4120590(119905119896)(119905) le 119892

119879

(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))

minus [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(44)

Combing (17)ndash(23) and (40)ndash(43) we get

120590(119905119896)(119905) le 120585

119879

1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)

8 Journal of Applied Mathematics

where

120585119879

1= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))

[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

]

(46)

Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119879

120583119879120591119886

12058210158401

12058210158402

1198881 (47)

where

1205821015840

1= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)]

+120582max (1198774119894) + 1198662

119899

sum119895=1

119897119894119895119896119895

1205821015840

2= 1205822

(48)

By (39) and (47) we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (49)

This completes the proof

4 Numerical Examples

In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10

Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback

0 1 2 3 4 5

minus01

minus005

0

005

01

015

02

025

03

mRN

A co

ncen

trat

ions

t

Figure 1 mRNA concentrations 119909(119905)

0

0

1 2 3 4 5

Prot

ein

conc

entr

atio

ns

minus02

minus01

t

01

02

03

04

05

06

Figure 2 Protein concentrations 119910(119905)

loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)

119889119898119894

119889119905= minus119898119894+

120572

1 + 119901119899119895

+ 1205720

119889119901119894

119889119905= 120573 (119898

119894minus 119901119894)

(50)

where 119898119894and 119901

119894are the concentrations of two mRNAs and

repressor-protein 1205720is the growth rate of protein in a cell in

the presence of saturating amounts of repressor while 1205720+ 120572

is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient

Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by

Journal of Applied Mathematics 9

adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model

(119905) = minus119860119894119909 (119905) + 119861

119894119892 (119910 (119905 minus 120590 (119905)))

119910 (119905) = minus119862119894119910 (119905) + 119863

119894119909 (119905 minus 120591 (119905)) 119894 = 1 2

(51)

where

1198601= diag (4 4 4) 119862

1= diag (5 5 5)

1198631= diag (2 15 1) 119860

2= diag (3 3 3)

1198622= diag (4 4 4) 119863

2= diag (12 12 12)

1198611= 2 times (

0 0 minus1

minus1 0 0

0 minus1 0

) 1198612= 15 times (

0 0 minus1

minus1 0 0

0 minus1 0

)

(52)

The gene regulation function is taken as 119892(119909) = 1199092(1 +

1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and

1205912(119905) are assumed to be

1205911(119905) = 06 + 04 sin 119905 120591

2(119905) = 03 + 01 cos 119905 (53)

we can get the parameters as following

12059111= 02 120591

12= 1 120591

13= 04

12059121= 04 120591

22= 02 120591

23= 01

(54)

and let 1198881= 2 119888

2= 15 119879 = 5 120572 = 002 120583 = 12 By using

the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI

Ξ =

(((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 119886119894110

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010

)))))))))))))))

)

(55)

where 11988611989411

= minus21198751119894119860119894+ 1198761119894

+ 1205912211198601198791198941198771119894119860119894+ (12059122

minus

12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus

(12059122minus 12059121)2

1198611198791198941198772119894119860119894 11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+

(12059112minus12059111)2

1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus

12059111)2

1198631198791198941198774119894119862119894 11988611989455

= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus

12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877

4119894 119886119894710

=

1198721119894(1 minus 120591

13) + Λ

2119880 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912

21119861119879

1198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

35732 minus02655 minus03039

minus02655 33575 minus02912

minus03039 minus02912 31531

)

11987512= (

46463 minus01506 minus01453

minus01506 47212 minus01477

minus01453 minus01477 47809

)

11987611= (

91959 minus03223 minus02832

minus03223 74980 minus02791

minus02832 minus02791 59270

)

11987612= (

108205 minus04573 minus04597

minus04573 109025 minus04104

minus04597 minus04104 110402

)

(56)

and 120591119886

gt 120591lowast119886

= 04003 The initial condition is 119909(0) =

(02 02 03)119879 119910(0) = (03 02 05)

119879 The simulation resultsof the trajectories are shown in Figures 1 and 2

Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows

1198601= diag (21 12 25) 119862

1= diag (46 4 52)

1198631= diag (05 04 06) 119860

2= diag (18 2 31)

1198622= diag (5 4 5) 119863

2= diag (03 03 04)

1198611= (

0 1 0

0 0 1

1 0 0

) 1198612= (

0 07 0

0 0 07

07 0 0

)

(57)

10 Journal of Applied Mathematics

and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591

1(119905) and 120591

2(119905) are assumed to be

1205911(119905) = 05 + 03 sin 119905 120591

2(119905) = 04 + 01 cos 119905 (58)

we can get the parameters as following

12059111= 02 120591

12= 08 120591

13= 03

12059121= 03 120591

22= 05 120591

23= 01

(59)

and let 1198881= 25 119888

2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ

119895(119904) = 119890minus119904

and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI

Ξ =

(((((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 1198861198940

0

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

0

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499

0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894

10100

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894

)))))))))))))))))

)

(60)

where 11988611989411= minus2119875

1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus

1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2

1198611198791198941198772119894119860119894

11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus

21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2

1198631198791198941198774119894119862119894 11988611989455=

minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894

11988611989477= minus1198762119894(1 minus 120591

13) minus 2119877

4119894 119886119894710

= 1198721119894(1 minus 120591

13) + Λ2119880 11988611989499=

1198762119894minus 2Λ1+ 119871119894 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus

12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

13745 minus00657 minus00145

minus00657 13683 minus00471

minus00145 minus00471 12893

)

11987512= (

16025 minus00233 minus00056

minus00233 15143 minus00073

minus00056 minus00073 12907

)

11987611= (

25617 minus00905 minus00313

minus00905 20794 minus00718

minus00313 minus00718 24995

)

11987612= (

21367 minus00751 minus00258

minus00751 21184 minus00434

minus00258 minus00434 22000

)

(61)

and 120591119886gt 120591lowast119886= 05012

5 Concluding Remarks

This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)

References

[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

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 Journal of Applied Mathematics

By condition (13) we have

120590(119905119896)(119905) le 120572119881

120590(119905119896)(119905) (26)

Integrating (26) from 119905119896to 119905 we obtain that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

119881120590(119905119896)(119905119896) (27)

Using (12) at switching instant 119905119896 we have

119881120590(119905119896)(119905119896) le 120583119881

120590(119905minus

119896)(119905minus

119896) (28)

Therefore it follows from (27) and (28) that

119881120590(119905119896)(119905) le 119890

120572(119905minus119905119896)

120583119881120590(119905minus

119896)(119905minus

119896) (29)

For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873

120590(0 119879) le 119879120591

119886 we

have

119881120590(119905)

(119905) le 119890120572119905

120583119879120591119886119881120590(0)

(0) (30)

On the other hand it follows from (15) that

119881120590(0)

(0)

le max119894isinN

[120582max (1198751119894) + 12059122120582max (1198761119894)]

times supminus120588le119905le0

1003817100381710038171003817120593 (119905)10038171003817100381710038172

+ [120582max (1198752119894) + 12059112120582max (1198762119894)

+ 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)]

times supminus120588le119905le0

1003817100381710038171003817120595 (119905)10038171003817100381710038172

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

times 120582max (1198772119894)]

times supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

+1

2[1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

times120582max (1198774119894)] supminus120588le119905le0

1003817100381710038171003817 (119905)10038171003817100381710038172

le max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)] 1198881

119881120590(119905)

(119905)

ge min119894isinN

[120582min (1198751119894) 119909 (119905)2

+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)

10038171003817100381710038172

]

ge min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

times (119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

)

(31)

From (30) and (31) it is easy to obtain

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119905

1205831198791205911198861205821

1205822

1198881le 119890120572119879

1205831198791205911198861205821

1205822

1198881 (32)

where

1205821= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894) ]

1205822= min119894isinN

[min (120582min (1198751119894) 120582min (1198752119894))]

(33)

By (14) and (32) for any 119905 isin [0 119879] we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (34)

This completes the proof

Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays

(119905) = minus119860120590(119905)

119909 (119905) + 119861120590(119905)

119892 (119910 (119905 minus 1205911(119905)))

+ 119882120590(119905)

intinfin

0

ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904

119910 (119905) = minus119862120590(119905)

119910 (119905) + 119863120590(119905)

119909 (119905 minus 1205912(119905))

(35)

where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882

119894 119894 isin N is the 119894th subsystem distributively

Journal of Applied Mathematics 7

delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ

1(sdot) ℎ

119899(sdot)) is the delay kernel function and

all the other signs are defined as (6)

Assumption 9 The delay kernel ℎ119895rsquos are some real value

nonnegative continuous functions defined in [0infin) thatsatisfy

intinfin

0

ℎ119895(119904) 119889119904 = 1 int

infin

0

119904ℎ119895(119904) 119889119904 = 119896

119894lt infin

119895 = 1 119899

(36)

Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875

1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite

diagonal matrix 119871119894= diag(119897

1119894 119897119899119894) for all 119894 isin N Λ

119898=

diag(1205821198981 120582

119898119899) 119898 = 1 2 matrices 119876

2119894 1198763119894 1198721119894 119894 isin 119873

and positive scalars 120583 ge 1 such that the following inequalities

[1198762119894

1198721119894

lowast 1198763119894

] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2

[1198762119894

1198721119894

lowast 1198763119894

] le 120583 [1198762119895

1198721119895

lowast 1198763119895

] 119877119897119894le 120583119877119897119895

119897 = 1 4 forall119894 119895 isin N

(37)

Ξ1= Ω1

+ sym 119865119879

111987711198941198652+ 119865119879

1[1198751119894119861119894minus 1205912

21119861119879

1198941198771119894119860119894

minus(12059122minus 12059121)2

119861119879

1198941198772119894119860119894]

times 11986510+ 119865119879

211987721198941198653+ 119865119879

311987721198941198654+ 119865119879

3

times [1198752119894119863119894minus 1205912

11119863119879

1198941198773119894119862119894minus (12059112minus 12059111)2

119863119879

1198941198774119894119862119894]

times 1198655+ 119865119879

511987731198941198656+ 119865119879

5(1198721119894+ Λ1119880)1198659+ 119865119879

611987741198941198657

+119865119879

711987741198941198658+ 119865119879

7[1198721119894(1 minus 120591

13) + Λ

2119880]11986510

lt 0 forall119894 isin N(38)

hold and the average dwell time of the switch signal 120590(119905)satisfies

120591119886gt 120591lowast

119886=

119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879

(39)

where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591

23) + 1205912

11119863119879119894

1198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894

1198773119894119862119894+ (12059112minus 12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894

(1 minus 12059113) minus 2119877

4119894 minus1198774119894 1198762119894minus 2Λ

1+ 119871119894minus1198763119894(1 minus 120591

13) +

1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899

119868119899times119899

0119899times(12minus119895)119899

] 119895 = 1 11 12058210158401= max

119894isinN120582max(1198751119894) +

12059122120582max(1198761119894) + 120582max(1198752119894) + 120591

12120582max(1198762119894) +

12059112120582max(1198763119894) 120582max(119880119880) + 2120591

12120582max(1198721119894)120582max(119880) + (12)

[1205913

21120582max(1198771119894) + (120591

22minus 12059121)3

120582max(1198772119894) + 120591311120582max(1198773119894) +

(12059112minus 12059111)3

120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899

119895=1119897119894119895119896119895 12058210158402= 1205822

The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888

1 1198882 119879)

Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional

119881120590(119905)

(119905) = 1198811120590(119905)

(119905) + 1198812120590(119905)

(119905) + 1198813120590(119905)

(119905) + 1198814120590(119905)

(119905) (40)

where1198811120590(119905)

(119905)1198812120590(119905)

(119905)1198813120590(119905)

(119905) are defined as inTheorem 8

1198814120590(119905)

(119905) =

119899

sum119894=1

119897119894120590(119905)

intinfin

0

ℎ119894(120579) int119905

119905minus120579

1198922

119894(119910119894(119904)) 119889119904 119889120579 (41)

when 119905 isin [119905119896 119905119896+1

) taking the derivatives of 1198814along the

trajectory of system (35) we have that

4120590(119905119896)(119905) =

119899

sum119894=1

119897119894120590(119905119896)1198922

119894(119910119894(119905))

minus

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

(42)

By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2

le (int 1199012(119904)119889119904)

(int 1199022(119904)119889119904) we obtain that

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

=

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 119889120579

times intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

ge

119899

sum119894=1

119897119894120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

2

= [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(43)

Therefore

4120590(119905119896)(119905) le 119892

119879

(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))

minus [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(44)

Combing (17)ndash(23) and (40)ndash(43) we get

120590(119905119896)(119905) le 120585

119879

1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)

8 Journal of Applied Mathematics

where

120585119879

1= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))

[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

]

(46)

Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119879

120583119879120591119886

12058210158401

12058210158402

1198881 (47)

where

1205821015840

1= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)]

+120582max (1198774119894) + 1198662

119899

sum119895=1

119897119894119895119896119895

1205821015840

2= 1205822

(48)

By (39) and (47) we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (49)

This completes the proof

4 Numerical Examples

In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10

Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback

0 1 2 3 4 5

minus01

minus005

0

005

01

015

02

025

03

mRN

A co

ncen

trat

ions

t

Figure 1 mRNA concentrations 119909(119905)

0

0

1 2 3 4 5

Prot

ein

conc

entr

atio

ns

minus02

minus01

t

01

02

03

04

05

06

Figure 2 Protein concentrations 119910(119905)

loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)

119889119898119894

119889119905= minus119898119894+

120572

1 + 119901119899119895

+ 1205720

119889119901119894

119889119905= 120573 (119898

119894minus 119901119894)

(50)

where 119898119894and 119901

119894are the concentrations of two mRNAs and

repressor-protein 1205720is the growth rate of protein in a cell in

the presence of saturating amounts of repressor while 1205720+ 120572

is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient

Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by

Journal of Applied Mathematics 9

adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model

(119905) = minus119860119894119909 (119905) + 119861

119894119892 (119910 (119905 minus 120590 (119905)))

119910 (119905) = minus119862119894119910 (119905) + 119863

119894119909 (119905 minus 120591 (119905)) 119894 = 1 2

(51)

where

1198601= diag (4 4 4) 119862

1= diag (5 5 5)

1198631= diag (2 15 1) 119860

2= diag (3 3 3)

1198622= diag (4 4 4) 119863

2= diag (12 12 12)

1198611= 2 times (

0 0 minus1

minus1 0 0

0 minus1 0

) 1198612= 15 times (

0 0 minus1

minus1 0 0

0 minus1 0

)

(52)

The gene regulation function is taken as 119892(119909) = 1199092(1 +

1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and

1205912(119905) are assumed to be

1205911(119905) = 06 + 04 sin 119905 120591

2(119905) = 03 + 01 cos 119905 (53)

we can get the parameters as following

12059111= 02 120591

12= 1 120591

13= 04

12059121= 04 120591

22= 02 120591

23= 01

(54)

and let 1198881= 2 119888

2= 15 119879 = 5 120572 = 002 120583 = 12 By using

the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI

Ξ =

(((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 119886119894110

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010

)))))))))))))))

)

(55)

where 11988611989411

= minus21198751119894119860119894+ 1198761119894

+ 1205912211198601198791198941198771119894119860119894+ (12059122

minus

12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus

(12059122minus 12059121)2

1198611198791198941198772119894119860119894 11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+

(12059112minus12059111)2

1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus

12059111)2

1198631198791198941198774119894119862119894 11988611989455

= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus

12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877

4119894 119886119894710

=

1198721119894(1 minus 120591

13) + Λ

2119880 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912

21119861119879

1198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

35732 minus02655 minus03039

minus02655 33575 minus02912

minus03039 minus02912 31531

)

11987512= (

46463 minus01506 minus01453

minus01506 47212 minus01477

minus01453 minus01477 47809

)

11987611= (

91959 minus03223 minus02832

minus03223 74980 minus02791

minus02832 minus02791 59270

)

11987612= (

108205 minus04573 minus04597

minus04573 109025 minus04104

minus04597 minus04104 110402

)

(56)

and 120591119886

gt 120591lowast119886

= 04003 The initial condition is 119909(0) =

(02 02 03)119879 119910(0) = (03 02 05)

119879 The simulation resultsof the trajectories are shown in Figures 1 and 2

Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows

1198601= diag (21 12 25) 119862

1= diag (46 4 52)

1198631= diag (05 04 06) 119860

2= diag (18 2 31)

1198622= diag (5 4 5) 119863

2= diag (03 03 04)

1198611= (

0 1 0

0 0 1

1 0 0

) 1198612= (

0 07 0

0 0 07

07 0 0

)

(57)

10 Journal of Applied Mathematics

and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591

1(119905) and 120591

2(119905) are assumed to be

1205911(119905) = 05 + 03 sin 119905 120591

2(119905) = 04 + 01 cos 119905 (58)

we can get the parameters as following

12059111= 02 120591

12= 08 120591

13= 03

12059121= 03 120591

22= 05 120591

23= 01

(59)

and let 1198881= 25 119888

2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ

119895(119904) = 119890minus119904

and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI

Ξ =

(((((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 1198861198940

0

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

0

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499

0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894

10100

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894

)))))))))))))))))

)

(60)

where 11988611989411= minus2119875

1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus

1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2

1198611198791198941198772119894119860119894

11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus

21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2

1198631198791198941198774119894119862119894 11988611989455=

minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894

11988611989477= minus1198762119894(1 minus 120591

13) minus 2119877

4119894 119886119894710

= 1198721119894(1 minus 120591

13) + Λ2119880 11988611989499=

1198762119894minus 2Λ1+ 119871119894 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus

12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

13745 minus00657 minus00145

minus00657 13683 minus00471

minus00145 minus00471 12893

)

11987512= (

16025 minus00233 minus00056

minus00233 15143 minus00073

minus00056 minus00073 12907

)

11987611= (

25617 minus00905 minus00313

minus00905 20794 minus00718

minus00313 minus00718 24995

)

11987612= (

21367 minus00751 minus00258

minus00751 21184 minus00434

minus00258 minus00434 22000

)

(61)

and 120591119886gt 120591lowast119886= 05012

5 Concluding Remarks

This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)

References

[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

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

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

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

Journal of Applied Mathematics 7

delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ

1(sdot) ℎ

119899(sdot)) is the delay kernel function and

all the other signs are defined as (6)

Assumption 9 The delay kernel ℎ119895rsquos are some real value

nonnegative continuous functions defined in [0infin) thatsatisfy

intinfin

0

ℎ119895(119904) 119889119904 = 1 int

infin

0

119904ℎ119895(119904) 119889119904 = 119896

119894lt infin

119895 = 1 119899

(36)

Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875

1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite

diagonal matrix 119871119894= diag(119897

1119894 119897119899119894) for all 119894 isin N Λ

119898=

diag(1205821198981 120582

119898119899) 119898 = 1 2 matrices 119876

2119894 1198763119894 1198721119894 119894 isin 119873

and positive scalars 120583 ge 1 such that the following inequalities

[1198762119894

1198721119894

lowast 1198763119894

] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2

[1198762119894

1198721119894

lowast 1198763119894

] le 120583 [1198762119895

1198721119895

lowast 1198763119895

] 119877119897119894le 120583119877119897119895

119897 = 1 4 forall119894 119895 isin N

(37)

Ξ1= Ω1

+ sym 119865119879

111987711198941198652+ 119865119879

1[1198751119894119861119894minus 1205912

21119861119879

1198941198771119894119860119894

minus(12059122minus 12059121)2

119861119879

1198941198772119894119860119894]

times 11986510+ 119865119879

211987721198941198653+ 119865119879

311987721198941198654+ 119865119879

3

times [1198752119894119863119894minus 1205912

11119863119879

1198941198773119894119862119894minus (12059112minus 12059111)2

119863119879

1198941198774119894119862119894]

times 1198655+ 119865119879

511987731198941198656+ 119865119879

5(1198721119894+ Λ1119880)1198659+ 119865119879

611987741198941198657

+119865119879

711987741198941198658+ 119865119879

7[1198721119894(1 minus 120591

13) + Λ

2119880]11986510

lt 0 forall119894 isin N(38)

hold and the average dwell time of the switch signal 120590(119905)satisfies

120591119886gt 120591lowast

119886=

119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879

(39)

where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591

23) + 1205912

11119863119879119894

1198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894

1198773119894119862119894+ (12059112minus 12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894

(1 minus 12059113) minus 2119877

4119894 minus1198774119894 1198762119894minus 2Λ

1+ 119871119894minus1198763119894(1 minus 120591

13) +

1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899

119868119899times119899

0119899times(12minus119895)119899

] 119895 = 1 11 12058210158401= max

119894isinN120582max(1198751119894) +

12059122120582max(1198761119894) + 120582max(1198752119894) + 120591

12120582max(1198762119894) +

12059112120582max(1198763119894) 120582max(119880119880) + 2120591

12120582max(1198721119894)120582max(119880) + (12)

[1205913

21120582max(1198771119894) + (120591

22minus 12059121)3

120582max(1198772119894) + 120591311120582max(1198773119894) +

(12059112minus 12059111)3

120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899

119895=1119897119894119895119896119895 12058210158402= 1205822

The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888

1 1198882 119879)

Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional

119881120590(119905)

(119905) = 1198811120590(119905)

(119905) + 1198812120590(119905)

(119905) + 1198813120590(119905)

(119905) + 1198814120590(119905)

(119905) (40)

where1198811120590(119905)

(119905)1198812120590(119905)

(119905)1198813120590(119905)

(119905) are defined as inTheorem 8

1198814120590(119905)

(119905) =

119899

sum119894=1

119897119894120590(119905)

intinfin

0

ℎ119894(120579) int119905

119905minus120579

1198922

119894(119910119894(119904)) 119889119904 119889120579 (41)

when 119905 isin [119905119896 119905119896+1

) taking the derivatives of 1198814along the

trajectory of system (35) we have that

4120590(119905119896)(119905) =

119899

sum119894=1

119897119894120590(119905119896)1198922

119894(119910119894(119905))

minus

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

(42)

By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2

le (int 1199012(119904)119889119904)

(int 1199022(119904)119889119904) we obtain that

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

=

119899

sum119894=1

119897119894120590(119905119896)intinfin

0

ℎ119894(120579) 119889120579

times intinfin

0

ℎ119894(120579) 1198922

119894(119910119894(119905 minus 120579)) 119889120579

ge

119899

sum119894=1

119897119894120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

2

= [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(43)

Therefore

4120590(119905119896)(119905) le 119892

119879

(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))

minus [intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

times 119871120590(119905119896)[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

(44)

Combing (17)ndash(23) and (40)ndash(43) we get

120590(119905119896)(119905) le 120585

119879

1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)

8 Journal of Applied Mathematics

where

120585119879

1= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))

[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

]

(46)

Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119879

120583119879120591119886

12058210158401

12058210158402

1198881 (47)

where

1205821015840

1= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)]

+120582max (1198774119894) + 1198662

119899

sum119895=1

119897119894119895119896119895

1205821015840

2= 1205822

(48)

By (39) and (47) we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (49)

This completes the proof

4 Numerical Examples

In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10

Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback

0 1 2 3 4 5

minus01

minus005

0

005

01

015

02

025

03

mRN

A co

ncen

trat

ions

t

Figure 1 mRNA concentrations 119909(119905)

0

0

1 2 3 4 5

Prot

ein

conc

entr

atio

ns

minus02

minus01

t

01

02

03

04

05

06

Figure 2 Protein concentrations 119910(119905)

loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)

119889119898119894

119889119905= minus119898119894+

120572

1 + 119901119899119895

+ 1205720

119889119901119894

119889119905= 120573 (119898

119894minus 119901119894)

(50)

where 119898119894and 119901

119894are the concentrations of two mRNAs and

repressor-protein 1205720is the growth rate of protein in a cell in

the presence of saturating amounts of repressor while 1205720+ 120572

is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient

Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by

Journal of Applied Mathematics 9

adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model

(119905) = minus119860119894119909 (119905) + 119861

119894119892 (119910 (119905 minus 120590 (119905)))

119910 (119905) = minus119862119894119910 (119905) + 119863

119894119909 (119905 minus 120591 (119905)) 119894 = 1 2

(51)

where

1198601= diag (4 4 4) 119862

1= diag (5 5 5)

1198631= diag (2 15 1) 119860

2= diag (3 3 3)

1198622= diag (4 4 4) 119863

2= diag (12 12 12)

1198611= 2 times (

0 0 minus1

minus1 0 0

0 minus1 0

) 1198612= 15 times (

0 0 minus1

minus1 0 0

0 minus1 0

)

(52)

The gene regulation function is taken as 119892(119909) = 1199092(1 +

1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and

1205912(119905) are assumed to be

1205911(119905) = 06 + 04 sin 119905 120591

2(119905) = 03 + 01 cos 119905 (53)

we can get the parameters as following

12059111= 02 120591

12= 1 120591

13= 04

12059121= 04 120591

22= 02 120591

23= 01

(54)

and let 1198881= 2 119888

2= 15 119879 = 5 120572 = 002 120583 = 12 By using

the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI

Ξ =

(((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 119886119894110

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010

)))))))))))))))

)

(55)

where 11988611989411

= minus21198751119894119860119894+ 1198761119894

+ 1205912211198601198791198941198771119894119860119894+ (12059122

minus

12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus

(12059122minus 12059121)2

1198611198791198941198772119894119860119894 11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+

(12059112minus12059111)2

1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus

12059111)2

1198631198791198941198774119894119862119894 11988611989455

= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus

12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877

4119894 119886119894710

=

1198721119894(1 minus 120591

13) + Λ

2119880 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912

21119861119879

1198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

35732 minus02655 minus03039

minus02655 33575 minus02912

minus03039 minus02912 31531

)

11987512= (

46463 minus01506 minus01453

minus01506 47212 minus01477

minus01453 minus01477 47809

)

11987611= (

91959 minus03223 minus02832

minus03223 74980 minus02791

minus02832 minus02791 59270

)

11987612= (

108205 minus04573 minus04597

minus04573 109025 minus04104

minus04597 minus04104 110402

)

(56)

and 120591119886

gt 120591lowast119886

= 04003 The initial condition is 119909(0) =

(02 02 03)119879 119910(0) = (03 02 05)

119879 The simulation resultsof the trajectories are shown in Figures 1 and 2

Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows

1198601= diag (21 12 25) 119862

1= diag (46 4 52)

1198631= diag (05 04 06) 119860

2= diag (18 2 31)

1198622= diag (5 4 5) 119863

2= diag (03 03 04)

1198611= (

0 1 0

0 0 1

1 0 0

) 1198612= (

0 07 0

0 0 07

07 0 0

)

(57)

10 Journal of Applied Mathematics

and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591

1(119905) and 120591

2(119905) are assumed to be

1205911(119905) = 05 + 03 sin 119905 120591

2(119905) = 04 + 01 cos 119905 (58)

we can get the parameters as following

12059111= 02 120591

12= 08 120591

13= 03

12059121= 03 120591

22= 05 120591

23= 01

(59)

and let 1198881= 25 119888

2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ

119895(119904) = 119890minus119904

and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI

Ξ =

(((((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 1198861198940

0

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

0

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499

0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894

10100

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894

)))))))))))))))))

)

(60)

where 11988611989411= minus2119875

1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus

1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2

1198611198791198941198772119894119860119894

11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus

21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2

1198631198791198941198774119894119862119894 11988611989455=

minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894

11988611989477= minus1198762119894(1 minus 120591

13) minus 2119877

4119894 119886119894710

= 1198721119894(1 minus 120591

13) + Λ2119880 11988611989499=

1198762119894minus 2Λ1+ 119871119894 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus

12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

13745 minus00657 minus00145

minus00657 13683 minus00471

minus00145 minus00471 12893

)

11987512= (

16025 minus00233 minus00056

minus00233 15143 minus00073

minus00056 minus00073 12907

)

11987611= (

25617 minus00905 minus00313

minus00905 20794 minus00718

minus00313 minus00718 24995

)

11987612= (

21367 minus00751 minus00258

minus00751 21184 minus00434

minus00258 minus00434 22000

)

(61)

and 120591119886gt 120591lowast119886= 05012

5 Concluding Remarks

This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)

References

[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

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

8 Journal of Applied Mathematics

where

120585119879

1= [119909119879

(119905) 119909119879

(119905 minus 12059121) 119909119879

(119905 minus 1205912(119905)) 119909

119879

(119905 minus 12059122)

119910119879

(119905) 119910119879

(119905 minus 12059111) 119910119879

(119905 minus 1205911(119905)) 119910

119879

(119905 minus 12059112)

119892119879

(119910 (119905)) 119892119879

(119910 (119905 minus 1205911(119905)))

[intinfin

0

ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]

119879

]

(46)

Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 119890120572119879

120583119879120591119886

12058210158401

12058210158402

1198881 (47)

where

1205821015840

1= max119894isinN

120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)

+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)

+ 212059112120582max (1198721119894) 120582max (119880)

+1

2[1205913

21120582max (1198771119894) + (12059122 minus 12059121)

3

120582max (1198772119894)

+ 1205913

11120582max (1198773119894) + (12059112 minus 12059111)

3

120582max (1198774119894)]

+120582max (1198774119894) + 1198662

119899

sum119895=1

119897119894119895119896119895

1205821015840

2= 1205822

(48)

By (39) and (47) we obtain that

119909 (119905)2

+1003817100381710038171003817119910 (119905)

10038171003817100381710038172

le 1198882 (49)

This completes the proof

4 Numerical Examples

In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10

Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback

0 1 2 3 4 5

minus01

minus005

0

005

01

015

02

025

03

mRN

A co

ncen

trat

ions

t

Figure 1 mRNA concentrations 119909(119905)

0

0

1 2 3 4 5

Prot

ein

conc

entr

atio

ns

minus02

minus01

t

01

02

03

04

05

06

Figure 2 Protein concentrations 119910(119905)

loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)

119889119898119894

119889119905= minus119898119894+

120572

1 + 119901119899119895

+ 1205720

119889119901119894

119889119905= 120573 (119898

119894minus 119901119894)

(50)

where 119898119894and 119901

119894are the concentrations of two mRNAs and

repressor-protein 1205720is the growth rate of protein in a cell in

the presence of saturating amounts of repressor while 1205720+ 120572

is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient

Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by

Journal of Applied Mathematics 9

adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model

(119905) = minus119860119894119909 (119905) + 119861

119894119892 (119910 (119905 minus 120590 (119905)))

119910 (119905) = minus119862119894119910 (119905) + 119863

119894119909 (119905 minus 120591 (119905)) 119894 = 1 2

(51)

where

1198601= diag (4 4 4) 119862

1= diag (5 5 5)

1198631= diag (2 15 1) 119860

2= diag (3 3 3)

1198622= diag (4 4 4) 119863

2= diag (12 12 12)

1198611= 2 times (

0 0 minus1

minus1 0 0

0 minus1 0

) 1198612= 15 times (

0 0 minus1

minus1 0 0

0 minus1 0

)

(52)

The gene regulation function is taken as 119892(119909) = 1199092(1 +

1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and

1205912(119905) are assumed to be

1205911(119905) = 06 + 04 sin 119905 120591

2(119905) = 03 + 01 cos 119905 (53)

we can get the parameters as following

12059111= 02 120591

12= 1 120591

13= 04

12059121= 04 120591

22= 02 120591

23= 01

(54)

and let 1198881= 2 119888

2= 15 119879 = 5 120572 = 002 120583 = 12 By using

the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI

Ξ =

(((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 119886119894110

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010

)))))))))))))))

)

(55)

where 11988611989411

= minus21198751119894119860119894+ 1198761119894

+ 1205912211198601198791198941198771119894119860119894+ (12059122

minus

12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus

(12059122minus 12059121)2

1198611198791198941198772119894119860119894 11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+

(12059112minus12059111)2

1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus

12059111)2

1198631198791198941198774119894119862119894 11988611989455

= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus

12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877

4119894 119886119894710

=

1198721119894(1 minus 120591

13) + Λ

2119880 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912

21119861119879

1198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

35732 minus02655 minus03039

minus02655 33575 minus02912

minus03039 minus02912 31531

)

11987512= (

46463 minus01506 minus01453

minus01506 47212 minus01477

minus01453 minus01477 47809

)

11987611= (

91959 minus03223 minus02832

minus03223 74980 minus02791

minus02832 minus02791 59270

)

11987612= (

108205 minus04573 minus04597

minus04573 109025 minus04104

minus04597 minus04104 110402

)

(56)

and 120591119886

gt 120591lowast119886

= 04003 The initial condition is 119909(0) =

(02 02 03)119879 119910(0) = (03 02 05)

119879 The simulation resultsof the trajectories are shown in Figures 1 and 2

Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows

1198601= diag (21 12 25) 119862

1= diag (46 4 52)

1198631= diag (05 04 06) 119860

2= diag (18 2 31)

1198622= diag (5 4 5) 119863

2= diag (03 03 04)

1198611= (

0 1 0

0 0 1

1 0 0

) 1198612= (

0 07 0

0 0 07

07 0 0

)

(57)

10 Journal of Applied Mathematics

and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591

1(119905) and 120591

2(119905) are assumed to be

1205911(119905) = 05 + 03 sin 119905 120591

2(119905) = 04 + 01 cos 119905 (58)

we can get the parameters as following

12059111= 02 120591

12= 08 120591

13= 03

12059121= 03 120591

22= 05 120591

23= 01

(59)

and let 1198881= 25 119888

2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ

119895(119904) = 119890minus119904

and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI

Ξ =

(((((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 1198861198940

0

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

0

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499

0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894

10100

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894

)))))))))))))))))

)

(60)

where 11988611989411= minus2119875

1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus

1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2

1198611198791198941198772119894119860119894

11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus

21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2

1198631198791198941198774119894119862119894 11988611989455=

minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894

11988611989477= minus1198762119894(1 minus 120591

13) minus 2119877

4119894 119886119894710

= 1198721119894(1 minus 120591

13) + Λ2119880 11988611989499=

1198762119894minus 2Λ1+ 119871119894 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus

12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

13745 minus00657 minus00145

minus00657 13683 minus00471

minus00145 minus00471 12893

)

11987512= (

16025 minus00233 minus00056

minus00233 15143 minus00073

minus00056 minus00073 12907

)

11987611= (

25617 minus00905 minus00313

minus00905 20794 minus00718

minus00313 minus00718 24995

)

11987612= (

21367 minus00751 minus00258

minus00751 21184 minus00434

minus00258 minus00434 22000

)

(61)

and 120591119886gt 120591lowast119886= 05012

5 Concluding Remarks

This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)

References

[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

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

Journal of Applied Mathematics 9

adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model

(119905) = minus119860119894119909 (119905) + 119861

119894119892 (119910 (119905 minus 120590 (119905)))

119910 (119905) = minus119862119894119910 (119905) + 119863

119894119909 (119905 minus 120591 (119905)) 119894 = 1 2

(51)

where

1198601= diag (4 4 4) 119862

1= diag (5 5 5)

1198631= diag (2 15 1) 119860

2= diag (3 3 3)

1198622= diag (4 4 4) 119863

2= diag (12 12 12)

1198611= 2 times (

0 0 minus1

minus1 0 0

0 minus1 0

) 1198612= 15 times (

0 0 minus1

minus1 0 0

0 minus1 0

)

(52)

The gene regulation function is taken as 119892(119909) = 1199092(1 +

1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and

1205912(119905) are assumed to be

1205911(119905) = 06 + 04 sin 119905 120591

2(119905) = 03 + 01 cos 119905 (53)

we can get the parameters as following

12059111= 02 120591

12= 1 120591

13= 04

12059121= 04 120591

22= 02 120591

23= 01

(54)

and let 1198881= 2 119888

2= 15 119879 = 5 120572 = 002 120583 = 12 By using

the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI

Ξ =

(((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 119886119894110

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0

lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1

0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010

)))))))))))))))

)

(55)

where 11988611989411

= minus21198751119894119860119894+ 1198761119894

+ 1205912211198601198791198941198771119894119860119894+ (12059122

minus

12059121)2

1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus

(12059122minus 12059121)2

1198611198791198941198772119894119860119894 11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+

(12059112minus12059111)2

1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus

12059111)2

1198631198791198941198774119894119862119894 11988611989455

= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus

12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877

4119894 119886119894710

=

1198721119894(1 minus 120591

13) + Λ

2119880 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912

21119861119879

1198941198771119894119861119894+

(12059122minus 12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

35732 minus02655 minus03039

minus02655 33575 minus02912

minus03039 minus02912 31531

)

11987512= (

46463 minus01506 minus01453

minus01506 47212 minus01477

minus01453 minus01477 47809

)

11987611= (

91959 minus03223 minus02832

minus03223 74980 minus02791

minus02832 minus02791 59270

)

11987612= (

108205 minus04573 minus04597

minus04573 109025 minus04104

minus04597 minus04104 110402

)

(56)

and 120591119886

gt 120591lowast119886

= 04003 The initial condition is 119909(0) =

(02 02 03)119879 119910(0) = (03 02 05)

119879 The simulation resultsof the trajectories are shown in Figures 1 and 2

Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows

1198601= diag (21 12 25) 119862

1= diag (46 4 52)

1198631= diag (05 04 06) 119860

2= diag (18 2 31)

1198622= diag (5 4 5) 119863

2= diag (03 03 04)

1198611= (

0 1 0

0 0 1

1 0 0

) 1198612= (

0 07 0

0 0 07

07 0 0

)

(57)

10 Journal of Applied Mathematics

and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591

1(119905) and 120591

2(119905) are assumed to be

1205911(119905) = 05 + 03 sin 119905 120591

2(119905) = 04 + 01 cos 119905 (58)

we can get the parameters as following

12059111= 02 120591

12= 08 120591

13= 03

12059121= 03 120591

22= 05 120591

23= 01

(59)

and let 1198881= 25 119888

2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ

119895(119904) = 119890minus119904

and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI

Ξ =

(((((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 1198861198940

0

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

0

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499

0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894

10100

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894

)))))))))))))))))

)

(60)

where 11988611989411= minus2119875

1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus

1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2

1198611198791198941198772119894119860119894

11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus

21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2

1198631198791198941198774119894119862119894 11988611989455=

minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894

11988611989477= minus1198762119894(1 minus 120591

13) minus 2119877

4119894 119886119894710

= 1198721119894(1 minus 120591

13) + Λ2119880 11988611989499=

1198762119894minus 2Λ1+ 119871119894 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus

12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

13745 minus00657 minus00145

minus00657 13683 minus00471

minus00145 minus00471 12893

)

11987512= (

16025 minus00233 minus00056

minus00233 15143 minus00073

minus00056 minus00073 12907

)

11987611= (

25617 minus00905 minus00313

minus00905 20794 minus00718

minus00313 minus00718 24995

)

11987612= (

21367 minus00751 minus00258

minus00751 21184 minus00434

minus00258 minus00434 22000

)

(61)

and 120591119886gt 120591lowast119886= 05012

5 Concluding Remarks

This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)

References

[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

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 Journal of Applied Mathematics

and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591

1(119905) and 120591

2(119905) are assumed to be

1205911(119905) = 05 + 03 sin 119905 120591

2(119905) = 04 + 01 cos 119905 (58)

we can get the parameters as following

12059111= 02 120591

12= 08 120591

13= 03

12059121= 03 120591

22= 05 120591

23= 01

(59)

and let 1198881= 25 119888

2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ

119895(119904) = 119890minus119904

and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI

Ξ =

(((((((((((((((((

(

119886119894

111198771119894

0 0 0 0 0 0 0 1198861198940

0

lowast minus1198771119894minus 1198772119894

1198772119894

0 0 0 0 0 0 0 0

lowast lowast 11988611989433

1198772119894

11988611989435

0 0 0 0 0 0

lowast lowast lowast minus1198772119894

0 0 0 0 0 0 0

lowast lowast lowast lowast 11988611989455

1198773119894

0 0 1198721119894+ Λ1119880 0 0

lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894

1198774119894

0 0 0 0

lowast lowast lowast lowast lowast lowast 11988611989477

1198774119894

0 119886119894710

0

lowast lowast lowast lowast lowast lowast lowast minus1198774119894

0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499

0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894

10100

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894

)))))))))))))))))

)

(60)

where 11988611989411= minus2119875

1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2

1198601198791198941198772119894119860119894minus

1198771119894minus 1205721198751119894 119886119894110

= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2

1198611198791198941198772119894119860119894

11988611989433

= minus1198761119894(1 minus 120591

23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2

1198631198791198941198774119894119863119894minus

21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2

1198631198791198941198774119894119862119894 11988611989455=

minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2

1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894

11988611989477= minus1198762119894(1 minus 120591

13) minus 2119877

4119894 119886119894710

= 1198721119894(1 minus 120591

13) + Λ2119880 11988611989499=

1198762119894minus 2Λ1+ 119871119894 1198861198941010

= minus1198763119894(1 minus 120591

13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus

12059121)2

1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2

Due to the space limitation we only list matrices 11987511 11987512

11987611 and 119876

12here

11987511= (

13745 minus00657 minus00145

minus00657 13683 minus00471

minus00145 minus00471 12893

)

11987512= (

16025 minus00233 minus00056

minus00233 15143 minus00073

minus00056 minus00073 12907

)

11987611= (

25617 minus00905 minus00313

minus00905 20794 minus00718

minus00313 minus00718 24995

)

11987612= (

21367 minus00751 minus00258

minus00751 21184 minus00434

minus00258 minus00434 22000

)

(61)

and 120591119886gt 120591lowast119886= 05012

5 Concluding Remarks

This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)

References

[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

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

Journal of Applied Mathematics 11

no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic

regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011

[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010

[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011

[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012

[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010

[8] 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

[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010

[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012

[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010

[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012

[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007

[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011

[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002

[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012

[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011

[19] 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

[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011

[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014

[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013

[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011

[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009

[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009

[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009

[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000

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

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