exponential stability analysis for genetic regulatory networks with both time varying and continuous distributed delays

Research ArticleExponential Stability Analysis for Genetic Regulatory Networks with Both Time-Varying and Continuous Distributed Delays Lizi Yin1,2and Yungang Liu1 1 School of Control Sc

Research Article

Exponential Stability Analysis for Genetic Regulatory Networks with Both Time-Varying and Continuous Distributed Delays

Lizi Yin1,2and Yungang Liu1

1 School of Control Science and Engineering, Shandong University, Jinan 250061, China

2 School of Mathematical Sciences, University of Jinan, Jinan 250022, China

Correspondence should be addressed to Yungang Liu; lygfr@sdu.edu.cn

Received 23 December 2013; Revised 8 March 2014; Accepted 9 March 2014; Published 6 May 2014

Academic Editor: Sanyi Tang

Copyright © 2014 L Yin and Y Liu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited The global exponential stability is investigated for genetic regulatory networks with time-varying delays and continuous distributed delays By choosing an appropriate Lyapunov-Krasovskii functional, new conditions of delay-dependent stability are obtained in the form of linear matrix inequality (LMI) The lower bound of derivatives of time-varying delay is first taken into account in genetic networks stability analysis, and the main results with less conservatism are established by interactive convex combination method

to estimate the upper bound of derivative function of the Lyapunov-Krasovskii functional In addition, two numerical examples are provided to illustrate the effectiveness of the theoretical results

1 Introduction

From the late 20th century to the early 21st century, more

than a decade’s time, life science, especially molecular biology

science, had great surprising changes The research of genetic

regulatory networks has become an important area in the

molecular biology science and received great attention There

are plenty of results [1, 2] Genetic regulatory networks

can be seen as biochemically dynamical systems, and it

is natural to simulate them by using dynamical system

model There are a variety of models that have been

pro-posed but mainly are the Boolean network model (discrete

model) [3–6] and the differential equation model (continuous

model) [7, 8] In the Boolean models, each gene’s activity

is expressed with ON or OFF, and each gene’s state is

described by the Boolean function of other related genes’

states In differential equation model, the variables

delin-eate the concentrations of gene products, such as mRNAs

and proteins, which are continuous values According to

a large number of biological experiments, we know that

gene expression is usually continuously variable, so it is

more reasonable to describe genetic regulation networks with differential equation model than with the Boolean network model

Gene expression is a complex process regulation by the stimulation and inhibition of protein including transcription, translation, and posttranslation processes, and a large num-ber of reactions and reacting species participate in this pro-cess There are fast reaction and slow reaction in real genetic regulatory systems The fast reaction includes dimerization, binding reaction, and phosphorylation, and the slow reaction contains transcription, translation, and translocation or the finite switching speed of amplifiers Due to the slow reaction, time delays exist in genetic regulatory networks In [9], there

is a biochemistry experiment on mice which has proved that there exists a time lag of about 15 min in the peaks between the mRNA molecules and the proteins of the gene Hes1

The emergence of the time delays will influence the genetic regulatory networks’ dynamic behaviors, which cause the researchers’ interest The stability is one of the very important dynamic characteristics Hence, it is necessary

Abstract and Applied Analysis

Volume 2014, Article ID 897280, 10 pages

http://dx.doi.org/10.1155/2014/897280

to consider the stability of genetic regulatory networks

with time delay; see [10–16] In [11], random time delays

are taken into account, and some stability criteria for the

uncertain delayed genetic networks with SUM regulatory

logic where each transcription factor acts additively to

regulate a gene were obtained; asymptotical stability

cri-teria were proposed for genetic regulatory networks with

interval time-varying delays and nonlinear disturbance in

stability sufficient conditions for the uncertain

stochas-tic genestochas-tic regulatory networks with both mixed

time-varying delays by constructing the Lyapunov functional

and employing stochastic analysis methods In [15,16], the

authors studied genetic regulatory networks with constant

delay

Motivated by the above discussions, we analyze the

exponential stability of genetic regulatory networks with

time-varying delays and continuous distributed delays It is

worth mentioning that the asymptotical stability of genetic

regulatory networks is studied in most literature; see [17,18]

But the exponential stability of our research is of better

sta-bility than asymptotical stasta-bility Literature [11] discusses the

exponential stability of genetic regulatory networks with

ran-dom time delays, which are essentially constant delays Our

paper considers the system with interval time-varying delays

and continuous distributed delays, which is more reasonable

than literature [11] Literature [19] studies robust exponential

stability for stochastic genetic regulatory networks with

time-varying delays, whose derivatives’ upper bound is less than

1 In our models, the derivatives’ upper bound of

inter-val time-varying delays has no limit of less than 1 And

we study the lower bound of derivatives of time-varying

delay to systems stability effect for the first time In the

theorem for evidence, convex combination and interactive

convex combination method were adopted, which have less

conservatism

This paper is organized as follows In Section 2, model

several sufficient results are obtained to check the exponential

stability for genetic regulatory networks with time-varying

delays and continuous distributed delays Some numerical

examples are given to demonstrate the effectiveness of our

Section5

Notations Throughout this paper,R, R𝑛, andR𝑛×𝑚 denote,

respectively, the set of all real numbers, real𝑛-dimensional

inR𝑛.𝐼 and 0 denote, respectively, the identity matrix and

the zero matrix with appropriate dimension For a vector or

used to represent𝐴 + 𝐴T For simplicity, in symmetric block

matrices, we often use∗ to represent the term that is induced

by symmetry

2 Problem Formulation and Some Preliminaries

In this paper, we are devoted to studying the stability to an autoregulatory genetic network with time delays described by the following delay differential equations:

𝑚𝑖(𝑡) = −𝑎𝑖𝑚𝑖(𝑡) + 𝜔𝑖(𝑝1(𝑡 − 𝜎 (𝑡)) 𝑝𝑛(𝑡 − 𝜎 (𝑡))) ,

𝑖 = 1, , 𝑛,

𝑖(𝑡) = −𝑐𝑖𝑝𝑖(𝑡) + 𝑑𝑖𝑚𝑖(𝑡 − 𝜏 (𝑡)) , 𝑖 = 1, , 𝑛,

(1)

where𝑚𝑖(𝑡)’s and 𝑝𝑖(𝑡)’s are the concentrations of mRNAs and proteins, respectively;𝑎𝑖’s and𝑐𝑖’s are the degradation rates

of mRNAs and proteins, respectively;𝑑𝑖’s are the translation rates of proteins; 𝜔𝑖(⋅)’s are the regulatory functions of mRNAs, being generally monotonic to each argument; and 𝜎(𝑡) and 𝜏(𝑡) are time-varying delays

Assumption 1. 𝜎(𝑡) and 𝜏(𝑡) are the time-varying delay satis-fying

0 ≤ 𝜎1≤ 𝜎 (𝑡) ≤ 𝜎2, 𝜎3≤ ̇𝜎 (𝑡) ≤ 𝜎4< ∞,

0 ≤ 𝜏1≤ 𝜏 (𝑡) ≤ 𝜏2, 𝜏3≤ ̇𝜏 (𝑡) ≤ 𝜏4< ∞, (2) where𝜎1, 𝜎2, 𝜎3, 𝜎4, 𝜏1, 𝜏2, 𝜏3, 𝜏4are some constants

Remark 2 In this paper, the lower bound of derivatives of

time-varying delay is considered for the first time in the research of genetic regulatory networks When information

on lower bound of time-varying delay’s derivatives can be measured, our results are better than the previous works

In genetic regulatory networks, some genes can be activated by one of a few different transcription factors (“OR” logic), and others can be activated by two or more transcription factors which must be bounded at the same time (“AND” logic) In this paper, we take a model of genetic regulatory networks where each transcription factor acts additively to regulate the𝑖th gene (“SUM” logic) [20] The regulatory function takes the form 𝜔𝑖(𝑝1(𝑡), , 𝑝𝑛(𝑡)) =

∑𝑛𝑗=1𝜔𝑖𝑗(𝑝𝑗(𝑡)), and 𝜔𝑖𝑗(𝑝𝑗(𝑡)) is a monotonic function with the following Hill form [21]:

𝜔𝑖𝑗(𝑝𝑗(𝑡)) =

{ { { { { { { { { { {

𝛼𝑖𝑗 (𝑝𝑗(𝑡) /𝛽𝑗)

𝐻𝑗

1 + (𝑝𝑗(𝑡) /𝛽𝑗)𝐻𝑗

if transcription factor 𝑗 is an activator of gene𝑖,

1 + (𝑝𝑗(𝑡) /𝛽𝑗)𝐻𝑗

if transcription factor 𝑗 is a repressor of gene𝑖,

(3)

where𝐻𝑗is the Hill coefficient,𝛽𝑗is a positive constant, and

𝛼𝑖𝑗is the constant transcriptional rate of𝑗th transcriptional

factor to𝑖th gene

Therefore, (1) can be rewritten as

𝑚𝑖(𝑡) = −𝑎𝑖𝑚𝑖(𝑡) +∑𝑛

𝑗=1

𝑏𝑖𝑗𝑓𝑗(𝑝𝑗(𝑡 − 𝜎 (𝑡))) + 𝑒𝑖,

𝑖 = 1, , 𝑛,

𝑖(𝑡) = −𝑐𝑖𝑝𝑖(𝑡) + 𝑑𝑖𝑚𝑖(𝑡 − 𝜏 (𝑡)) , 𝑖 = 1, , 𝑛,

(4)

where 𝑓𝑗(𝑥) = (𝑥/𝛽𝑗)𝐻𝑗/(1 + (𝑥/𝛽𝑗))𝐻𝑗 is monotonically

increasing function;𝑒𝑖’s are basal rate defined by𝑒𝑖= ∑𝑗∈𝑈𝑘𝛼𝑖𝑗

with𝑈𝑘 = {𝑗 | the 𝑗th transcription factor being a repressor

of the kth gene,𝑗 = 1, , 𝑛}; and matrix 𝐵 = (𝑏𝑖𝑗) ∈ R𝑛×𝑛is

defined as

𝑏𝑖𝑗=

{

{

{

{

{

{

{

{

{

{

{

𝛼𝑖𝑗,

if transcription factor𝑗 is an activator

of gene𝑖,

0,

if there is no link from node𝑗 to node 𝑖,

−𝛼𝑖𝑗,

if transcription factor 𝑗 is a repressor

of gene𝑖

(5)

In the compact matrix form, (4) can be rewritten as

𝑚 (𝑡) = −𝐴𝑚 (𝑡) + 𝐵𝑓 (𝑝 (𝑡 − 𝜎 (𝑡))) + 𝐸,

where

𝑚 (𝑡) = [𝑚1(𝑡) , , 𝑚𝑛(𝑡)]𝑇,

𝑝 (𝑡) = [𝑝1(𝑡) , , 𝑝𝑛(𝑡)]𝑇,

𝑓 (𝑝 (𝑡)) = [𝑓1(𝑝1(𝑡)) , , 𝑓𝑛(𝑝𝑛(𝑡))]𝑇,

𝐴 = diag (𝑎1, , 𝑎𝑛) , 𝐼 = [𝑒1, , 𝑒𝑛]𝑇,

𝐶 = diag (𝑐1, , 𝑐𝑛) , 𝐷 = diag (𝑑1, , 𝑑𝑛)

(7)

In order to get the stability results, the following

assump-tion is necessarily imposed on (6)

Assumption 3. 𝑓𝑖 : R → R, 𝑖 = 1, , 𝑛, are monotonically

increasing functions with saturation and moreover satisfy

𝑚−𝑖 ≤𝑓𝑖(𝑎) − 𝑓𝑎 − 𝑏𝑖(𝑏) ≤ 𝑚𝑖+, ∀𝑎, 𝑏 ∈ R, 𝑖 = 1, , 𝑛, (8)

𝑖 and𝑚+

𝑖 are constants

Remark 4 In the most existing literature, (8) was

strength-ened to0 ≤ ((𝑓𝑖(𝑎) − 𝑓𝑖(𝑏))/(𝑎 − 𝑏)) ≤ 𝑙𝑖, for all 𝑎, 𝑏 ∈ R,

where𝑙𝑖’s are positive constant Therefore, Assumption1 is

somewhat general and, in fact, similar to that of [12]; see the

discussion below

The vectors𝑚∗, 𝑝∗are said to be an equilibrium point of system (6), if they satisfy

0 = −𝐴𝑚∗+ 𝐵𝑓 (𝑝∗(𝑡 − 𝜎 (𝑡))) + 𝐸,

Let𝑥(𝑡) = 𝑚(𝑡) − 𝑚∗and let𝑦(𝑡) = 𝑝(𝑡) − 𝑝∗; we get

̇𝑥 (𝑡) = −𝐴𝑥 (𝑡) + 𝐵𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) ,

where𝑥(𝑡) = [𝑥1(𝑡), , 𝑥𝑛(𝑡)]𝑇,𝑦(𝑡) = [𝑦1(𝑡), , 𝑦𝑛(𝑡)]𝑇, 𝑔(𝑦(𝑡)) = [𝑔(𝑦1(𝑡)), , 𝑔(𝑦𝑛(𝑡))]𝑇, and𝑔𝑖(𝑦𝑖(𝑡)) = 𝑓𝑖(𝑦𝑖(𝑡) +

𝑝∗

𝑖) − 𝑓𝑖(𝑝∗

𝑖)

By the definition of𝑔𝑖(⋅), it satisfies sector condition:

𝑚−𝑖 ≤𝑔𝑖(𝑎)

which implies that

𝑔𝑖(𝑎) − 𝑚−𝑖𝑎

𝑚+

𝑖𝑎 − 𝑔𝑖(𝑎)

1, , 𝑚−

𝑛), 𝑀1 = diag(𝑚+

1, , 𝑚+

𝑛), and

1|, , |𝑚−

𝑛|, |𝑚+

1|, , |𝑚+

𝑛|}

The initial condition of system (10) is assumed to be

𝑥 (𝑡) = 𝜑1(𝑡) , 𝑦 (𝑡) = 𝜓1(𝑡) , −𝜌 ≤ 𝑡 ≤ 0,

regulatory networks with continuous distributed delays:

̇𝑥 (𝑡) = −𝐴𝑥 (𝑡) + 𝐵𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) + 𝑊 ∫𝑡

𝑡−𝜗(𝑡)𝑔 (𝑦 (𝑠)) 𝑑𝑠,

̇𝑦 (𝑡) = −𝐶𝑦 (𝑡) + 𝐷𝑥 (𝑡 − 𝜏 (𝑡)) ,

(14) where0 ≤ 𝜗(𝑡) ≤ 𝜗1

For completeness, we recall the following definition and lemmas

Definition 5 System (10) or (14) is said to be globally exponentially stable, if there exist constants𝜆 > 0 and 𝑀 ≥ 1, such that, for any initial value𝑧𝑡0,

‖𝑧 (𝑡)‖ ≤ 𝑀󵄩󵄩󵄩󵄩󵄩𝑧𝑡0󵄩󵄩󵄩󵄩󵄩𝐶1𝑒−𝜆(𝑡−𝑡0) (15) hold, for all𝑡 ≥ 0, where 𝑧(𝑡) = [𝑥(𝑡), 𝑦(𝑡)]𝑇and‖𝑧(𝑡)‖𝐶1 = sup−𝜌≤𝜃≤0{‖𝑧(𝑡 + 𝜃)‖, ‖ ̇𝑧(𝑡 + 𝜃)‖}

Lemma 6 (see [22]) For any positive definite matrix 𝑀 ∈

R𝑛×𝑛, there exists a scalar 𝑞 > 0 and a vector-valued function

𝜔 : [0, 𝑞] → R𝑛such that

(∫𝑞

0 𝜔(𝑠)𝑑𝑠)𝑇𝑀 (∫𝑞

0 𝜔 (𝑠) 𝑑𝑠) ≤ 𝑞 ∫𝑞

0 𝜔𝑇(𝑠) 𝑀𝜔 (𝑠) 𝑑𝑠

(16)

Lemma 7 (see [23]) Letℎ1, , ℎ𝑁 : R𝑚 → R take

positive values in an open subset D of R𝑚 Then, the reciprocally

convex combination ofℎ𝑖over D satisfies

min

{𝛼𝑖|𝛼𝑖>0,∑𝑖 𝛼𝑖=1}∑𝑖

1

𝛼𝑖ℎ𝑖(𝜂) = ∑𝑖 ℎ𝑖(𝜂) + max𝑘𝑖,𝑗(𝜂)∑

𝑖 ̸= 𝑗𝑘𝑖,𝑗(𝜂) (17)

subject to

{𝑘𝑖,𝑗:R𝑚 󳨃󳨀→ R, 𝑘𝑗,𝑖(𝜂) = 𝑘𝑖,𝑗(𝜂) ,

[ℎ𝑖(𝜂) 𝑘𝑖,𝑗(𝜂)

𝑘𝑖,𝑗(𝜂) ℎ𝑗(𝜂) ] ≥ 0}

(18)

3 Main Results

In this section, several theorems are presented of genetic

regulatory networks with both time-varying delays and

continuous distributed delays Firstly, a globally exponential

stability result is developed for the genetic regulatory network

with time-varying delays

Theorem 8 For system (10) with Assumptions 1 and 3, the

equilibrium point is globally exponentially stable (that is,

there are two positive constants 𝛼 and 𝜆 such that ‖𝑧(𝑡)‖ ≤

𝛼𝑒−𝜆𝑡‖𝑧(𝑡0)‖𝐶1, for all 𝑡 ≥ 𝑡0) if there exist positive definite

matrices𝐻2, 𝐻4, 𝑃1, 𝑃2, 𝑄𝑖, 𝑖 = 1, , 7, 𝑊𝑖, 𝑖 = 1, , 4, Γ1 =

diag(𝛾11, , 𝛾1𝑛), and Γ2 = diag(𝛾21, 𝛾22, , 𝛾2𝑛), such that,

for any appropriate dimensions constant matrices𝐻1, 𝐻3, 𝑋, 𝑌,

the following LMIs hold:

Ω1+ Ω2+ 𝑂𝑇3[𝑒−𝜆𝜏2𝑄2+ 𝑒−𝜆𝜏1𝑄3] 𝑂3

+ 𝑂8𝑇[𝑒−𝜆𝜎2𝑄5+ 𝑒−𝜆𝜎1𝑄6] 𝑂8< 0, (20)

where Ω1 = diag(𝜆𝑃1 + 𝑄1 − 𝑒−𝜆𝜏 1𝑊1 − 𝐻𝑇

1𝐴 − 𝐴𝑇𝐻1,

𝑒−𝜆𝜏1(𝑄1−𝑄2)−𝑒−𝜆𝜏1𝑊1−𝑒−𝜆𝜏2𝑊2,𝑒−𝜆𝜏1(1−𝜏4)𝑄3−𝑒−𝜆𝜏2[(1−

𝜏3)𝑄2+2𝑊2−𝑋𝑇−𝑋], −𝑒−𝜆𝜏2𝑊2, 𝜏2

1𝑊1+(𝜏2−𝜏1)2𝑊2−𝐻𝑇

2−𝐻2,

𝜆𝑃2+𝑄4−𝑒−𝜆𝜎1𝑊3−2𝑀𝑇

1Γ1𝑀0−𝐻𝑇

3𝐶−𝐶𝑇𝐻3,−𝑒−𝜆𝜎1(𝑄4−𝑄5+

𝑊3)−𝑒−𝜆𝜎2𝑊4,𝑒−𝜆𝜎1(1−𝜎4)𝑄6−𝑒−𝜆𝜎2[(1−𝜎3)𝑄5+2𝑊4−𝑌𝑇−

𝑌] − 𝑀𝑇

1Γ2𝑀0,−𝑒−𝜆𝜎2(𝑄4+ 𝑊4), 𝜎2

1𝑊3+ (𝜎2− 𝜎1)2𝑊4− 𝐻𝑇

4 −

𝐻4, 𝑄7−2Γ1,−𝑒−𝜆𝜎2𝑄7(1−𝜎𝑑)−2Γ2), Ω2= sym(𝑂𝑇

1𝑒−𝜆𝜏1𝑊1𝑂2

+ 𝑂𝑇

1[𝑃1 − 𝐻𝑇

1 − 𝐻𝑇

2𝐴]𝑂5 + 𝑂𝑇

1𝐻𝑇

1𝐵𝑂12 + 𝑂𝑇

2𝑒−𝜆𝜏2(𝑊2 − 𝑋)𝑂3+ 𝑂𝑇2𝑒−𝜆𝜏2𝑋𝑂4 +𝑂𝑇3𝑒−𝜆𝜏2(𝑊2 − 𝑋)𝑂4+ 𝑂𝑇3𝐻3𝑇𝐷𝑂6+

𝑂3𝑇𝐻4𝑇𝐷𝑂10+ 𝑂5𝑇𝐻2𝑇𝐵𝑂12+ 𝑂6𝑇𝑒−𝜆𝜎1𝑊3𝑂7+𝑂𝑇6[𝑃2− 𝐻3𝑇−

𝐻4𝑇𝐶]𝑂10 + 𝑂𝑇6[Γ1𝑀0+ 𝑀𝑇1Γ1]𝑂11+𝑂𝑇7𝑒−𝜆𝜎2(𝑊4− 𝑌)𝑂8+

𝑂7𝑇𝑒−𝜆𝜎2𝑌𝑂9+𝑂𝑇8𝑒−𝜆𝜎2(𝑊4− 𝑌)𝑂9+ 𝑂𝑇8[Γ2𝑀0+ 𝑀𝑇1Γ2]𝑂12),

and𝑂𝑖= [0𝑛×(𝑖−1)𝑛, 𝐼𝑛×𝑛, 0𝑛×(13−𝑖)𝑛], 𝑖 = 1, , 12.

Proof Based on system (10), we construct the following Lyapunov-Krasovskii functional:

𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) = 𝑉1(𝑥 (𝑡) , 𝑦 (𝑡)) + 𝑉2(𝑥 (𝑡) , 𝑦 (𝑡))

where

𝑉1(𝑥 (𝑡) , 𝑦 (𝑡)) = 𝑥𝑇(𝑡) 𝑃1𝑥 (𝑡) + 𝑦𝑇(𝑡) 𝑃2𝑦 (𝑡) ,

𝑉2(𝑥 (𝑡) , 𝑦 (𝑡)) = ∫𝑡

𝑡−𝜏1𝑒𝜆(𝑠−𝑡)𝑥𝑇(𝑠) 𝑄1𝑥 (𝑠) 𝑑𝑠 + ∫𝑡−𝜏1

𝑡−𝜏(𝑡)𝑒𝜆(𝑠−𝑡)𝑥𝑇(𝑠) 𝑄2𝑥 (𝑠) 𝑑𝑠 + ∫𝑡−𝜏(𝑡)

𝑡−𝜏2 𝑒𝜆(𝑠−𝑡)𝑥𝑇(𝑠) 𝑄3𝑥 (𝑠) 𝑑𝑠 + ∫𝑡

𝑡−𝜎1𝑒𝜆(𝑠−𝑡)𝑦𝑇(𝑠) 𝑄4𝑦 (𝑠) 𝑑𝑠 + ∫𝑡−𝜎1

𝑡−𝜎(𝑡)𝑒𝜆(𝑠−𝑡)𝑦𝑇(𝑠) 𝑄5𝑦 (𝑠) 𝑑𝑠 + ∫𝑡−𝜎(𝑡)

𝑡−𝜎2 𝑒𝜆(𝑠−𝑡)𝑦𝑇(𝑠) 𝑄6𝑦 (𝑠) 𝑑𝑠 + ∫𝑡

𝑡−𝜎(𝑡)𝑒𝜆(𝑠−𝑡)𝑔𝑇(𝑦 (𝑠)) 𝑄7𝑔 (𝑦 (𝑠)) 𝑑𝑠,

𝑉3(𝑥 (𝑡) , 𝑦 (𝑡))

= ∫0

−𝜏1∫𝑡

𝑡+𝜃𝜏1𝑒𝜆(𝑠−𝑡) ̇𝑥𝑇(𝑠) 𝑊1 ̇𝑥 (𝑠) 𝑑𝑠 𝑑𝜃 + ∫−𝜏1

−𝜏2 ∫𝑡

𝑡+𝜃(𝜏2− 𝜏1) 𝑒𝜆(𝑠−𝑡) ̇𝑥𝑇(𝑠) 𝑊2 ̇𝑥 (𝑠) 𝑑𝑠 𝑑𝜃 + ∫0

−𝜎1∫𝑡

𝑡+𝜃𝜎1𝑒𝜆(𝑠−𝑡) ̇𝑦𝑇(𝑠) 𝑊3 ̇𝑦 (𝑠) 𝑑𝑠 𝑑𝜃 + ∫−𝜎1

−𝜎2 ∫𝑡

𝑡+𝜃(𝜎2− 𝜎1) 𝑒𝜆(𝑠−𝑡) ̇𝑦𝑇(𝑠) 𝑊4 ̇𝑦 (𝑠) 𝑑𝑠 𝑑𝜃

(22) Taking the derivatives of𝑉𝑖, 𝑖 = 1, 2, 3, we have

̇𝑉

1(𝑥 (𝑡) , 𝑦 (𝑡)) = 2𝑥𝑇(𝑡) 𝑃1 ̇𝑥 (𝑡) + 2𝑦𝑇(𝑡) 𝑃2 ̇𝑦 (𝑡) ,

̇𝑉

2(𝑥 (𝑡) , 𝑦 (𝑡))

= −𝜆 ∫𝑡

𝑡−𝜏1𝑒𝜆(𝑠−𝑡)𝑥𝑇(𝑠) 𝑄1𝑥 (𝑠) 𝑑𝑠

− 𝜆 ∫𝑡−𝜏1

𝑡−𝜏(𝑡)𝑒𝜆(𝑠−𝑡)𝑥𝑇(𝑠) 𝑄2𝑥 (𝑠) 𝑑𝑠

− 𝜆 ∫𝑡−𝜏(𝑡)

𝑡−𝜏 𝑒𝜆(𝑠−𝑡)𝑥𝑇(𝑠) 𝑄3𝑥 (𝑠) 𝑑𝑠

− 𝜆 ∫𝑡

𝑡−𝜎1𝑒𝜆(𝑠−𝑡)𝑦𝑇(𝑠) 𝑄4𝑦 (𝑠) 𝑑𝑠

− 𝜆 ∫𝑡−𝜎1

𝑡−𝜎(𝑡)𝑒𝜆(𝑠−𝑡)𝑦𝑇(𝑠) 𝑄5𝑦 (𝑠) 𝑑𝑠

− 𝜆 ∫𝑡−𝜎(𝑡)

𝑡−𝜎2 𝑒𝜆(𝑠−𝑡)𝑦𝑇(𝑠) 𝑄6𝑦 (𝑠) 𝑑𝑠

− 𝜆 ∫𝑡

𝑡−𝜎(𝑡)𝑒𝜆(𝑠−𝑡)𝑔𝑇(𝑦 (𝑠)) 𝑄7𝑔 (𝑦 (𝑠)) 𝑑𝑠

+ 𝑥𝑇(𝑡) 𝑄1𝑥 (𝑡) − 𝑒−𝜆𝜏1𝑥𝑇(𝑡 − 𝜏1) 𝑄1𝑥 (𝑡 − 𝜏1)

+ 𝑒−𝜆𝜏1𝑥𝑇(𝑡 − 𝜏1) 𝑄2𝑥 (𝑡 − 𝜏1)

− 𝑒−𝜆𝜏2𝑥𝑇(𝑡 − 𝜏 (𝑡)) 𝑄2𝑥 (𝑡 − 𝜏 (𝑡)) (1 − ̇𝜏 (𝑡))

− 𝑒−𝜆𝜏2𝑥𝑇(𝑡 − 𝜏2) 𝑄3𝑥 (𝑡 − 𝜏2)

+ 𝑒−𝜆𝜏1𝑥𝑇(𝑡 − 𝜏 (𝑡)) 𝑄3𝑥 (𝑡 − 𝜏 (𝑡)) (1 − ̇𝜏 (𝑡))

+ 𝑦𝑇(𝑡) 𝑄4𝑦 (𝑡) − 𝑒−𝜆𝜎1𝑦𝑇(𝑡 − 𝜎1) 𝑄4𝑦 (𝑡 − 𝜎1)

+ 𝑒−𝜆𝜎1𝑦𝑇(𝑡 − 𝜎1) 𝑄5𝑦 (𝑡 − 𝜎1)

− 𝑒−𝜆𝜎2𝑦𝑇(𝑡 − 𝜎 (𝑡)) 𝑄5𝑦 (𝑡 − 𝜎 (𝑡)) (1 − ̇𝜎 (𝑡))

− 𝑒−𝜆𝜎2𝑦𝑇(𝑡 − 𝜎2) 𝑄6𝑦 (𝑡 − 𝜎2)

+ 𝑒−𝜆𝜎1𝑦𝑇(𝑡 − 𝜎 (𝑡)) 𝑄6𝑦 (𝑡 − 𝜎 (𝑡)) (1 − ̇𝜎 (𝑡))

+ 𝑔𝑇(𝑦 (𝑡)) 𝑄7𝑔 (𝑦 (𝑡)) − 𝑒−𝜆𝜎2𝑔𝑇

× (𝑦 (𝑡 − 𝜎 (𝑡))) 𝑄7𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) (1 − ̇𝜎 (𝑡)) ,

̇𝑉

3(𝑥 (𝑡) , 𝑦 (𝑡))

= −𝜆 ∫0

−𝜏1∫𝑡

𝑡+𝜃𝜏1𝑒𝜆(𝑠−𝑡) ̇𝑥𝑇(𝑠) 𝑊1 ̇𝑥 (𝑠) 𝑑𝑠 𝑑𝜃

− 𝜆 ∫−𝜏1

−𝜏2 ∫𝑡

𝑡+𝜃(𝜏2− 𝜏1) 𝑒𝜆(𝑠−𝑡) ̇𝑥𝑇(𝑠) 𝑊2 ̇𝑥 (𝑠) 𝑑𝑠 𝑑𝜃

− 𝜆 ∫0

−𝜎1∫𝑡

𝑡+𝜃𝜎1𝑒𝜆(𝑠−𝑡) ̇𝑦𝑇(𝑠) 𝑊3 ̇𝑦 (𝑠) 𝑑𝑠𝑑𝜃

− 𝜆 ∫−𝜎1

−𝜎2 ∫𝑡

𝑡+𝜃(𝜎2− 𝜎1) 𝑒𝜆(𝑠−𝑡) ̇𝑦𝑇(𝑠) 𝑊4 ̇𝑦 (𝑠) 𝑑𝑠 𝑑𝜃 + 𝜏12 ̇𝑥𝑇(𝑡) 𝑊1 ̇𝑥 (𝑡) − ∫𝑡

𝑡−𝜏1𝜏1𝑒𝜆(𝜃−𝑡) ̇𝑥𝑇(𝜃) 𝑊1 ̇𝑥 (𝜃) 𝑑𝜃 + (𝜏2− 𝜏1)2 ̇𝑥𝑇(𝑡) 𝑊2 ̇𝑥 (𝑡)

− ∫𝑡−𝜏1

𝑡−𝜏2 (𝜏2− 𝜏1) 𝑒𝜆(𝜃−𝑡) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃

+ 𝜎12 𝑇̇𝑦 (𝑡) 𝑊3 ̇𝑦 (𝑡) − ∫𝑡

𝑡−𝜎 𝜎1𝑒𝜆(𝜃−𝑡) ̇𝑦𝑇(𝜃) 𝑊3 ̇𝑦 (𝜃) 𝑑𝜃

+ (𝜎2− 𝜎1)2 ̇𝑦𝑇(𝑡) 𝑊4 ̇𝑦 (𝑡)

− ∫𝑡−𝜎1

𝑡−𝜎2 (𝜎2− 𝜎1) 𝑒𝜆(𝜃−𝑡) ̇𝑦𝑇(𝜃) 𝑊4 ̇𝑦 (𝜃) 𝑑𝜃

(23)

− ∫𝑡

𝑡−𝜏1𝜏1𝑒𝜆(𝜃−𝑡) ̇𝑥𝑇(𝜃) 𝑊1 ̇𝑥 (𝜃) 𝑑𝜃

≤ −𝑒−𝜆𝜏1[𝑥 (𝑡) − 𝑥 (𝑡 − 𝜏1)]𝑇𝑊1[𝑥 (𝑡) − 𝑥 (𝑡 − 𝜏1)] ,

− ∫𝑡

𝑡−𝜎1𝜎1𝑒𝜆(𝜃−𝑡) ̇𝑦𝑇(𝜃) 𝑊3 ̇𝑦 (𝜃) 𝑑𝜃

≤ −𝑒−𝜆𝜎1[𝑦 (𝑡) − 𝑦 (𝑡 − 𝜎1)]𝑇𝑊3[𝑦 (𝑡) − 𝑦 (𝑡 − 𝜎1)]

(24) Meanwhile,

− ∫𝑡−𝜏1

𝑡−𝜏2 (𝜏2− 𝜏1) 𝑒𝜆(𝜃−𝑡) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃

≤ −𝑒−𝜆𝜏2[∫𝑡−𝜏(𝑡)

𝑡−𝜏2 (𝜏2− 𝜏1) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃 + ∫𝑡−𝜏1

𝑡−𝜏(𝑡)(𝜏2− 𝜏1) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃]

= −𝑒−𝜆𝜏2[∫𝑡−𝜏(𝑡)

𝑡−𝜏2 (𝜏2− 𝜏 (𝑡)) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃 + ∫𝑡−𝜏1

𝑡−𝜏(𝑡)(𝜏 (𝑡) − 𝜏1) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃 + ∫𝑡−𝜏(𝑡)

𝑡−𝜏2 (𝜏 (𝑡) − 𝜏1) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃 + ∫𝑡−𝜏1

𝑡−𝜏(𝑡)(𝜏2− 𝜏 (𝑡)) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃]

= −𝑒−𝜆𝜏2[∫𝑡−𝜏(𝑡)

𝑡−𝜏2 (𝜏2− 𝜏 (𝑡)) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃 + ∫𝑡−𝜏1

𝑡−𝜏(𝑡)(𝜏 (𝑡) − 𝜏1) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃 +𝜏 (𝑡) − 𝜏1

𝜏2− 𝜏 (𝑡)

× ∫𝑡−𝜏(𝑡)

𝑡−𝜏2 (𝜏2− 𝜏 (𝑡)) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃 +𝜏2− 𝜏 (𝑡)

𝜏 (𝑡) − 𝜏1

× ∫𝑡−𝜏1

𝑡−𝜏(𝑡)(𝜏 (𝑡) − 𝜏1) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃]

(25)

By Lemma6, we obtain that

− ∫𝑡−𝜏1

𝑡−𝜏2 (𝜏2− 𝜏1) 𝑒𝜆(𝜃−𝑡) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃

≤ −𝑒−𝜆𝜏2{ 𝜏2− 𝜏1

𝜏2− 𝜏 (𝑡)[𝑥 (𝑡 − 𝜏 (𝑡)) − 𝑥 (𝑡 − 𝜏2)]

𝑇

× 𝑊2[𝑥 (𝑡 − 𝜏 (𝑡)) − 𝑥 (𝑡 − 𝜏2)]

+ 𝜏2− 𝜏1

𝜏 (𝑡) − 𝜏1[𝑥 (𝑡 − 𝜏1) − 𝑥 (𝑡 − 𝜏 (𝑡))]

𝑇

× 𝑊2[𝑥 (𝑡 − 𝜏1) − 𝑥 (𝑡 − 𝜏 (𝑡))] }

(26)

− ∫𝑡−𝜏1

𝑡−𝜏2 (𝜏2− 𝜏1) 𝑒𝜆(𝜃−𝑡) ̇𝑥𝑇(𝜃) 𝑊2 ̇𝑥 (𝜃) 𝑑𝜃

≤ −𝑒−𝜆𝜏2{[𝑥 (𝑡 − 𝜏1) − 𝑥 (𝑡 − 𝜏 (𝑡))]𝑇

× 𝑊2[𝑥 (𝑡 − 𝜏1) − 𝑥 (𝑡 − 𝜏 (𝑡))]

+ [𝑥 (𝑡 − 𝜏 (𝑡)) − 𝑥 (𝑡 − 𝜏2)]𝑇

× 𝑊2[𝑥 (𝑡 − 𝜏 (𝑡)) − 𝑥 (𝑡 − 𝜏2)]

+ 2[𝑥 (𝑡 − 𝜏1) − 𝑥 (𝑡 − 𝜏 (𝑡))]𝑇

×𝑋 [𝑥 (𝑡 − 𝜏 (𝑡)) − 𝑥 (𝑡 − 𝜏2)] }

(27)

Similar to (27),

− ∫𝑡−𝜎1

𝑡−𝜎2 𝑒𝜆(𝜃−𝑡) ̇𝑦𝑇(𝜃) 𝑊4 ̇𝑦 (𝜃) 𝑑𝜃

≤ −𝑒−𝜆𝜎2{[𝑦 (𝑡 − 𝜎1) − 𝑦 (𝑡 − 𝜎 (𝑡))]𝑇

× 𝑊4[𝑦 (𝑡 − 𝜎1) − 𝑦 (𝑡 − 𝜎 (𝑡))]

+ [𝑦 (𝑡 − 𝜎 (𝑡)) − 𝑦 (𝑡 − 𝜎2)]𝑇

× 𝑊4[𝑦 (𝑡 − 𝜎 (𝑡)) − 𝑦 (𝑡 − 𝜎2)]

+ 2[𝑦 (𝑡 − 𝜎1) − 𝑦 (𝑡 − 𝜎 (𝑡))]𝑇

× 𝑌 [𝑦 (𝑡 − 𝜎 (𝑡)) − 𝑦 (𝑡 − 𝜎2)] }

(28)

By Assumption3, for anyΓ𝑖= diag(𝛾𝑖1, , 𝛾𝑖𝑛) ≥ 0, 𝑖 = 1, 2,

the following inequality is true:

− 2∑𝑛

𝑖=1

𝛾𝑖1[𝑔𝑖(𝑦𝑖(𝑡)) − 𝑚+𝑖𝑦𝑖(𝑡)] [𝑔𝑖(𝑦𝑖(𝑡)) − 𝑚−𝑖𝑦𝑖(𝑡)]

− 2∑𝑛

𝑖=1

𝛾𝑖2[𝑔𝑖(𝑦𝑖(𝑡 − 𝜏 (𝑡))) − 𝑚+𝑖𝑦𝑖(𝑡 − 𝜏 (𝑡))]

× [𝑔𝑖(𝑦𝑖(𝑡 − 𝜏 (𝑡))) − 𝑚−𝑖𝑦𝑖(𝑡 − 𝜏 (𝑡))] ≥ 0

(29)

It can be rewritten as

− 2[𝑔 (𝑦 (𝑡)) − 𝑀1𝑦 (𝑡)]𝑇Γ1[𝑔 (𝑦 (𝑡)) − 𝑀0𝑦 (𝑡)]

− 2[𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) − 𝑀1𝑦 (𝑡 − 𝜎 (𝑡))]𝑇

× Γ2[𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) − 𝑀0𝑦 (𝑡 − 𝜎 (𝑡))] ≥ 0

(30)

For any constant matrices of appropriate dimensions𝐻𝑖, 𝑖 =

1, , 4, and from (10), we can obtain that

0 = 2 [𝑥𝑇(𝑡) 𝐻1𝑇+ ̇𝑥𝑇(𝑡) 𝐻2𝑇]

× [− ̇𝑥 (𝑡) − 𝐴𝑥 (𝑡) + 𝐵𝑔 (𝑦 (𝑡 − 𝜎 (𝑡)))] ,

0 = 2 [𝑦𝑇(𝑡) 𝐻3𝑇+ ̇𝑦𝑇(𝑡) 𝐻4𝑇]

× [− ̇𝑦 (𝑡) − 𝐶𝑦 (𝑡) + 𝐷𝑥 (𝑡 − 𝜏 (𝑡))]

(31)

Combining (21)–(31), we have

̇𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) + 𝜆𝑉 (𝑥 (𝑡) , 𝑦 (𝑡))

≤ 𝜉T(𝑡) {Ω + 𝑂3𝑇[𝑒−𝜆𝜏2( ̇𝜏 (𝑡) − 𝜏3) 𝑄2

+𝑒−𝜆𝜏1(𝜏4− ̇𝜏 (𝑡)) 𝑄3] 𝑂3 + 𝑂𝑇8[𝑒−𝜆𝜎2( ̇𝜎 (𝑡) − 𝜎3) 𝑄5

+𝑒−𝜆𝜎1(𝜎4− ̇𝜎 (𝑡)) 𝑄6] 𝑂8} 𝜉 (𝑡) ,

(32) where

𝜉𝑇(𝑡) = [𝑥𝑇(𝑡) , 𝑥𝑇(𝑡 − 𝜏1) , 𝑥𝑇(𝑡 − 𝜏 (𝑡)) , 𝑥𝑇(𝑡 − 𝜏2) ,

𝑇(𝑡) , 𝑦𝑇(𝑡) , 𝑦𝑇(𝑡 − 𝜎1) , 𝑦𝑇(𝑡 − 𝜎 (𝑡)) ,

𝑦𝑇(𝑡 − 𝜎2) , ̇𝑦𝑇(𝑡) , 𝑔𝑇(𝑦 (𝑡)) , 𝑔𝑇(𝑦 (𝑡 − 𝜎 (𝑡)))]

(33) From (19) and (20), we can see that

̇𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) + 𝜆𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) < 0, (34)

for all nonzero 𝜉(𝑡) Integrating the above inequality (34) from𝑡0to𝑡 gives

𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) ≤ 𝑒−𝜆(𝑡−𝑡0)𝑉 (𝑥 (𝑡0) , 𝑦 (𝑡0)) (35) From (21), we know that

𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) ≥ 𝑉1(𝑥 (𝑡) , 𝑦 (𝑡))

≥ min {𝜆min(𝑃1) , 𝜆min(𝑃2)} ‖𝑧 (𝑡)‖2,

𝑉 (𝑥 (𝑡0) , 𝑦 (𝑡0))

≤ [𝜆max(𝑃1) + 𝜆max(𝑃2) + 𝜏1𝜆max(𝑄1)

+ (𝜏2− 𝜏1) [𝜆max(𝑄2) + 𝜆max(𝑄3)]

+ (𝜎2− 𝜎1) [𝜆max(𝑄5) + 𝜆max(𝑄6)]

+ 𝜎1𝜆max(𝑄4) + 𝜎2𝑚2𝜆max(𝑄7)

+12𝜏13𝜆max(𝑊1) +12(𝜏2− 𝜏1)3𝜆max(𝑊2)

+12𝜎13𝜆max(𝑊3) +12(𝜎2− 𝜎1)3𝜆max(𝑊4)] 󵄩󵄩󵄩󵄩𝑧(𝑡0)󵄩󵄩󵄩󵄩2𝐶1

(36) Let

𝜆1= min {𝜆min(𝑃1) , 𝜆min(𝑃2)} ,

𝜆2= 𝜆max(𝑃1) + 𝜆max(𝑃2) + 𝜏1𝜆max(𝑄1)

+ (𝜏2− 𝜏1) [𝜆max(𝑄2) + 𝜆max(𝑄3)]

+ (𝜎2− 𝜎1) [𝜆max(𝑄5) + 𝜆max(𝑄6)]

+ 𝜎1𝜆max(𝑄4) + 𝜎2𝑚2𝜆max(𝑄7)

2𝜏13𝜆max(𝑊1) +1

2(𝜏2− 𝜏1)

3𝜆max(𝑊2)

+12𝜎31𝜆max(𝑊3) +12(𝜎2− 𝜎1)3𝜆max(𝑊4)

(37)

Then, by (35) and (36), we have

𝜆1‖𝑧 (𝑡)‖2≤ 𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) ≤ 𝜆2𝑒−𝜆(𝑡−𝑡0)󵄩󵄩󵄩󵄩𝑧(𝑡0)󵄩󵄩󵄩󵄩2𝐶1 (38)

By (38), we get that

‖𝑧 (𝑡)‖2≤𝜆2

𝜆1𝑒−𝜆(𝑡−𝑡0)󵄩󵄩󵄩󵄩𝑧(𝑡0)󵄩󵄩󵄩󵄩2𝐶1 (39)

Let 𝛼 = (𝜆2/𝜆1)1/2, and, by Definition 5, the genetic

regulatory networks in (10) are exponentially stable

The proof is completed

In the following, we consider the globally exponential

stability of the genetic regulatory networks with time-varying

delays and continuous distributed delays

Theorem 9 For system (14) with Assumptions 1 and 3, the

equilibrium point is globally exponentially stable (that is, there

are two positive constants 𝛼󸀠 and 𝜆 such that ‖𝑧(𝑡)‖ ≤

𝛼𝑒−𝜆𝑡‖𝑧(𝑡0)‖𝐶1, for all 𝑡 ≥ 𝑡0) if there exist positive definite

matrices 𝐻2, 𝐻4, 𝑃1, 𝑃2, 𝑆, 𝑄𝑖, 𝑖 = 1, , 7, 𝑊𝑖, 𝑖 = 1, , 4,

Γ1 = diag(𝛾11, , 𝛾1𝑛), and Γ2 = diag(𝛾21, , 𝛾2𝑛), such that,

for any appropriate dimensions constant matrices𝐻1, 𝐻3, 𝑋, 𝑌,

the following LMIs hold:

Ω󸀠1+ Ω󸀠2+ 𝐿𝑇3[𝑒−𝜆𝜏2𝑄2+ 𝑒−𝜆𝜏1𝑄3] 𝐿3 + 𝐿𝑇8[𝑒−𝜆𝜎2𝑄5+ 𝑒−𝜆𝜎1𝑄6] 𝐿8< 0, (41)

whereΩ󸀠1= diag(𝜆𝑃1+𝑄1−𝑒−𝜆𝜏1𝑊1−𝐻1𝑇𝐴−𝐴𝑇𝐻1,𝑒−𝜆𝜏1(𝑄1−

𝑄2) − 𝑒−𝜆𝜏1𝑊1− 𝑒−𝜆𝜏2𝑊2,𝑒−𝜆𝜏1(1 − 𝜏4)𝑄3− 𝑒−𝜆𝜏2[(1 − 𝜏3)𝑄2+ 2𝑊2− 𝑋𝑇− 𝑋], −𝑒−𝜆𝜏2𝑊2, 𝜏12𝑊1+ (𝜏2− 𝜏1)2𝑊2− 𝐻2𝑇− 𝐻2,

𝜆𝑃2+ 𝑄4− 𝑒−𝜆𝜎1𝑊3− 2𝑀1𝑇Γ1𝑀0− 𝐻3𝑇𝐶 − 𝐶𝑇𝐻3,−𝑒−𝜆𝜎1(𝑄4−

𝑄5+ 𝑊3) − 𝑒−𝜆𝜎2𝑊4,𝑒−𝜆𝜎1(1 − 𝜎4)𝑄6 − 𝑒−𝜆𝜎2[(1 − 𝜎3)𝑄5+ 2𝑊4− 𝑌𝑇− 𝑌] − 𝑀𝑇1Γ2𝑀0,−𝑒−𝜆𝜎2(𝑄4+ 𝑊4), 𝜎21𝑊3+ (𝜎2−

𝜎1)2𝑊4− 𝐻4𝑇− 𝐻4,𝑄7− 2Γ1+ 𝜗21𝑆, −𝑒−𝜆𝜎2𝑄7(1 − 𝜎4) − 2Γ2,

−𝑒−𝜆𝜗1𝑆, −𝑒−𝜆𝜗1𝑆), Ω󸀠

1𝑒−𝜆𝜏1𝑊1𝐿2+ 𝐿𝑇

1[𝑃1− 𝐻𝑇

1 −

𝐻𝑇

2𝐴]𝐿5+𝐼𝑇

1𝐻𝑇

1𝐵𝐿12+ 𝐿𝑇

2𝑒−𝜆𝜏2(𝑊2− 𝑋)𝐿3+ 𝐿𝑇

2𝑒−𝜆𝜏2𝑋𝐿4+

𝐿𝑇

3𝑒−𝜆𝜏2(𝑊2− 𝑋)𝐿4+ 𝐿𝑇

3𝐻𝑇

3𝐷𝐿6+ 𝐿𝑇

3𝐻𝑇

4𝐷𝐿10+𝐼𝑇

5𝐻𝑇

2𝐵𝐿12+

𝐿𝑇

6𝑒−𝜆𝜎1𝑊3𝐿7+𝐿𝑇

6[𝑃2−𝐻𝑇

3−𝐻𝑇

4𝐶]𝐿10+𝐿𝑇

6[Γ1𝑀0+𝑀𝑇

1Γ1]𝐿11

+𝐿𝑇

7𝑒−𝜆𝜎2(𝑊4− 𝑌)𝐿8+𝐿𝑇

7𝑒−𝜆𝜎2𝑌𝐿9+ 𝐿𝑇

8𝑒−𝜆𝜎2(𝑊4− 𝑌)𝐿9+

𝐿𝑇

8[Γ2𝑀0+𝑀𝑇

1Γ2]𝐿12), and 𝐿𝑖= [0𝑛×(𝑖−1)𝑛, 𝐼𝑛×𝑛, 0𝑛×(15−𝑖)𝑛], 𝑖 =

1, , 12.

Proof Based on system (14), we construct the following Lyapunov-Krasovskii functional:

𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) = 𝑉1(𝑥 (𝑡) , 𝑦 (𝑡)) + 𝑉2(𝑥 (𝑡) , 𝑦 (𝑡))

+ 𝑉3(𝑥 (𝑡) , 𝑦 (𝑡)) + 𝑉4(𝑥 (𝑡) , 𝑦 (𝑡)) , (42)

where 𝑉1(𝑥(𝑡), 𝑦(𝑡)), 𝑉2(𝑥(𝑡), 𝑦(𝑡)), and 𝑉3(𝑥(𝑡), 𝑦(𝑡)) are

𝑉4(𝑥 (𝑡) , 𝑦 (𝑡))

= ∫0

−𝜗1∫𝑡

𝑡+𝜃𝜗1𝑒𝜆(𝑠−𝑡)𝑔𝑇(𝑦 (𝑠)) 𝑆𝑔 (𝑦 (𝑠)) 𝑑𝑠 𝑑𝜃 (43)

Taking the derivative of𝑉4,

̇𝑉

4(𝑥 (𝑡) , 𝑦 (𝑡))

≤ −𝜆 ∫0

−𝜗1∫𝑡

𝑡+𝜃𝜗1𝑒𝜆(𝑠−𝑡)𝑔𝑇(𝑦 (𝑠)) 𝑆𝑔 (𝑦 (𝑠)) 𝑑𝑠 𝑑𝜃 + 𝜗21𝑔T

(𝑦 (𝑡)) 𝑆𝑔 (𝑦 (𝑡))

− 𝑒−𝜆𝜗1∫𝑡

𝑡−𝜗 𝜗1𝑔𝑇(𝑦 (𝜃)) 𝑆𝑔 (𝑦 (𝜃)) 𝑑𝜃

(44)

By Lemma6, we get that

̇𝑉

4(𝑥 (𝑡) , 𝑦 (𝑡))

≤ −𝜆 ∫0

−𝜗1∫𝑡

𝑡+𝜃𝜗1𝑒𝜆(𝑠−𝑡)𝑔𝑇(𝑦 (𝑠)) 𝑆𝑔 (𝑦 (𝑠)) 𝑑𝑠 𝑑𝜃 + 𝜗12𝑔𝑇(𝑦 (𝑡)) 𝑆𝑔 (𝑦 (𝑡)) − 𝑒−𝜆𝜗1 𝜗1

𝜗1− 𝜗 (𝑡)

× (∫𝑡−𝜗(𝑡)

𝑇

𝑆 (∫𝑡−𝜗(𝑡)

− 𝑒−𝜆𝜗1 𝜗1

𝜗 (𝑡)

× (∫𝑡

𝑡−𝜗(𝑡)𝑔(𝑦(𝜃))𝑑𝜃)𝑇𝑆 (∫𝑡

𝑡−𝜗(𝑡)𝑔 (𝑦 (𝜃)) 𝑑𝜃)

≤ −𝜆 ∫0

−𝜗1∫𝑡

𝑡+𝜃𝜗1𝑒𝜆(𝑠−𝑡)𝑔𝑇(𝑦 (𝑠)) 𝑆𝑔 (𝑦 (𝑠)) 𝑑𝑠 𝑑𝜃 + 𝜗12𝑔𝑇(𝑦 (𝑡)) 𝑆𝑔 (𝑦 (𝑡)) − 𝑒−𝜆𝜗1(1 +𝜗 (𝑡)

𝜗1 )

× (∫𝑡−𝜗(𝑡)

𝑇

𝑆 (∫𝑡−𝜗(𝑡)

− 𝑒−𝜆𝜗1(1 +𝜗1− 𝜗 (𝑡)

× (∫𝑡

𝑡−𝜗(𝑡)𝑔(𝑦(𝜃))𝑑𝜃)𝑇𝑆 (∫𝑡

𝑡−𝜗(𝑡)𝑔 (𝑦 (𝜃)) 𝑑𝜃)

(45) Combining (23)–(31), (42), and (45), we get

̇𝑉 (𝑥 (𝑡) , 𝑦 (𝑡)) + 𝜆𝑉 (𝑥 (𝑡) , 𝑦 (𝑡))

≤ 𝜉𝑇1(𝑡) {Ω + 𝐿𝑇3[𝑒−𝜆𝜏2( ̇𝜏 (𝑡) − 𝜏3) 𝑄2

+ 𝑒−𝜆𝜏1(𝜏4− ̇𝜏 (𝑡)) 𝑄3] 𝐿3 + 𝐿𝑇8[𝑒−𝜆𝜎2( ̇𝜎 (𝑡) − 𝜎3) 𝑄5 +𝑒−𝜆𝜎1(𝜎4− ̇𝜎 (𝑡)) 𝑄6] 𝐿8} 𝜉1(𝑡) ,

(46) where

𝜉1𝑇(𝑡) = [𝑥𝑇(𝑡) , 𝑥𝑇(𝑡 − 𝜏1) , 𝑥𝑇(𝑡 − 𝜏 (𝑡)) ,

𝑥𝑇(𝑡 − 𝜏2) , ̇𝑥𝑇(𝑡) , 𝑦𝑇(𝑡) , 𝑦𝑇(𝑡 − 𝜎1) ,

𝑦𝑇(𝑡 − 𝜎 (𝑡)) , 𝑦𝑇(𝑡 − 𝜎2) , ̇𝑦𝑇(𝑡) , 𝑔𝑇(𝑦 (𝑡)) ,

𝑔𝑇(𝑦 (𝑡 − 𝜎 (𝑡))) , (∫𝑡−𝜗(𝑡)

𝑇

,

(∫𝑡

𝑡−𝜗(𝑡)𝑔(𝑦(𝜃))𝑑𝜃)𝑇]

(47)

By (40) and (41), we get that ̇𝑉(𝑥(𝑡), 𝑦(𝑡))+𝜆𝑉(𝑥(𝑡), 𝑦(𝑡)) < 0

‖𝑧 (𝑡)‖2≤𝜆󸀠2

𝜆1𝑒−𝜆(𝑡−𝑡0)󵄩󵄩󵄩󵄩𝑧(𝑡0)󵄩󵄩󵄩󵄩2𝐶1 (48) where𝜆󸀠

2 = 𝜆2+ (1/2)𝜗3

1𝑚2𝜆max(𝑆) and 𝜆1,𝜆2 are defined

as Theorem8 Let𝛼󸀠 = (𝜆󸀠

2/𝜆1)1/2, and, by Definition5, the genetic regulatory network (14) is exponentially stable The proof is completed

Remark 10 In the proof of Theorems 8 and 9, we use convex combination and interactive convex combination definition to estimate the upper bound of derivative function

of the Lyapunov-Krasovskii functional and obtain some new conservative weaker sufficient conditions

Remark 11 When the lower bound of derivatives

∫𝑡−𝜏2𝑡−𝜏(𝑡)𝑒𝜆(𝑠−𝑡)𝑥𝑇(𝑠)𝑄3𝑥(𝑠)𝑑𝑠 = 0 and ∫𝑡−𝜎2𝑡−𝜎(𝑡)𝑒𝜆(𝑠−𝑡)𝑦𝑇

our results are true still

4 Numerical Examples

In this section, two examples are given to illustrate the effectiveness of our theoretical results

Example 1 Consider a genetic regulatory network model

dynamics of repressilator which is cyclic negative-feedback

loop comprising three repressor genes (lacl, tetR, and cl) and their promoters (cl, lacl, and tetR):

𝑑𝑥𝑖

𝛼

1 + 𝑦𝑛

𝑗 + 𝛼0,

𝑑𝑦𝑖

𝑑𝑡 = 𝛽 (𝑥𝑖− 𝑦𝑖)

(49)

Taking time-varying delays into account and shifting the equilibrium point to the origin, one gets the following model:

̇𝑥 (𝑡) = −𝐴𝑥 (𝑡) + 𝐵𝑔 (𝑦 (𝑡 − 𝜎 (𝑡))) ,

where𝐴 = diag(2, 2, 2), 𝐶 = diag(3, 3, 3), 𝐷 = diag(1, 1, 1), and the coupling matrix

𝐵 = 1.5 × (−1 0 00 0 −1

0 5 10 15 20 25 30

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

t

(a)

t

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

(b)

Figure 1: (a) mRNA concentrations𝑥(𝑡) (b) Protein concentrations 𝑦(𝑡)

The gene regulation function is taken as𝑔(𝑥) = 𝑥2/(1 +

𝑥2), 𝑀0 = diag(0, 0, 0), and 𝑀1 = diag(0.65, 0.65, 0.65) The

time delays𝜎(𝑡) and 𝜏(𝑡) are assumed to be

We can get the parameters as follows:

Theorem8, system (50) is exponentially stable By using the

MATLAB LMI toolbox, we can get the feasible solutions Due

to the space limitation, we only list matrices𝑃1and𝑃2here as

follows:

𝑃1= (17.1041 7.9997 7.99977.9997 17.1041 7.9997

𝑃2= (13.8943 0.5573 0.55730.5573 13.8943 0.5573

(54)

The initial condition is𝑥(0) = (0.3, 0.5, 0.4)𝑇 and 𝑦(0) =

(0.2, 0.4, 0.6)𝑇 The simulation results of the trajectories are

shown in Figure1

Example 2 In this example, we consider the genetic

distributed delays, in which the parameters are listed as

follows:

𝐴 = diag (1, 2, 3) , 𝐶 = diag (5, 4, 5) ,

𝐷 = diag (0.3, 0.2, 0.4) , 𝐵 = (0 0.8 00 0 0.8

(55)

diag(0.65, 0.65, 0.65) The time delays 𝜎(𝑡), 𝜏(𝑡), and 𝜗(𝑡) are assumed to be

𝜎 (𝑡) = 0.5 + 0.3sin2𝑡, 𝜏 (𝑡) = 0.4 + 0.1cos2𝑡,

We can get the parameters as follows:

can get the feasible solutions Due to the space limitation, we only list matrices𝑃1and𝑃2here as follows:

𝑃1= (−0.4713 7.2143 −0.30717.1735 −0.4713 −0.2152

𝑃2= (7.93630 8.01990 00

(59)

5 Concluding Remarks

This paper has investigated the exponential stability of genetic regulatory networks with time-varying delays and continuous distributed delays By using the novel Lyapunov-Krasovskii functions and employing the Jensen inequality and the interactive convex combination method, some suffi-cient criteria are given to ensure the exponential stability with less conservative All the obtained conditions are dependent

on the delays and on linear matrix inequalities Two examples are provided to illustrate the effectiveness of our results

Conflict of Interests

The authors declare that there is no conflict of interests

regarding the publication of this paper

Acknowledgments

This work was supported by the National Natural Science

Foundations of China (61273084, 61233014, and 61174217),

the Natural Science Foundation for Distinguished Young

Scholar of Shandong Province of China (JQ200919), the

Independent Innovation Foundation of Shandong University

(2012JC014), the Natural Science Foundation of Shandong

Province of China (ZR2010AL016, ZR2011AL007), and the

Doctoral Foundation of University of Jinan (XBS1244)

References

[1] H Li and X Yang, “Asymptotic stability analysis of genetic

regulatory networks with time-varying delay,” in Proceedings of

the Chinese Control and Decision Conference (CCDC ’10), pp.

566–571, May 2010

[2] Y Tang, Z Wang, and J.-A Fang, “Parameters identification of

unknown delayed genetic regulatory networks by a switching

particle swarm optimization algorithm,” Expert Systems with

Applications, vol 38, no 3, pp 2523–2535, 2011.

[3] S Huang, “Gene expression profiling, genetic networks, and

cellular states: an integrating concept for tumorigenesis and

drug discovery,” Journal of Molecular Medicine, vol 77, no 6,

pp 469–480, 1999

[4] S A Kauffman, “Metabolic stability and epigenesis in randomly

constructed genetic nets,” Journal of Theoretical Biology, vol 22,

no 3, pp 437–467, 1969

[5] R Somogyi and C A Sniegoski, “Modeling the complexity

of genetic networks: understanding multigenic and pleiotropic

regulation,” Complexity, vol 1, no 6, pp 45–63, 1996.

[6] R Thomas, “Boolean formalization of genetic control circuits,”

Journal of Theoretical Biology, vol 42, no 3, pp 563–585, 1973.

[7] F Ren and J Cao, “Asymptotic and robust stability of genetic

regulatory networks with time-varying delays,”

Neurocomput-ing, vol 71, no 4–6, pp 834–842, 2008.

[8] M de Hoon, S Imoto, K Kobayashi, N Ogasawara, and S

Miyano, “Inferring gene regulatory networks fromtime-ordered

gene expression data of bacillus subtilis using differential

equations,” in Proceedings Pacific Symposiumon Biocomputing,

vol 8, pp 17–28, 2003

[9] L Chen and K Aihara, “Stability of genetic regulatory networks

with time delay,” IEEE Transactions on Circuits and Systems I:

Fundamental Theory and Applications, vol 49, no 5, pp 602–

608, 2002

[10] W Zhang, J.-A Fang, and Y Tang, “New robust stability analysis

for genetic regulatory networks with random discrete delays

and distributed delays,” Neurocomputing, vol 74, no 14-15, pp.

2344–2360, 2011

[11] X Lou, Q Ye, and B Cui, “Exponential stability of genetic

regulatory networks with random delays,” Neurocomputing, vol.

73, no 4–6, pp 759–769, 2010

[12] W Wang and S Zhong, “Delay-dependent stability criteria

for genetic regulatory networks with time-varying delays and

nonlinear disturbance,” Communications in Nonlinear Science

and Numerical Simulation, vol 17, no 9, pp 3597–3611, 2012.

[13] R Rakkiyappan and P Balasubramaniam, “Delay-probability-distribution-dependent stability of uncertain stochastic genetic regulatory networks with mixed time-varying delays: an LMI

approach,” Nonlinear Analysis: Hybrid Systems, vol 4, no 3, pp.

600–607, 2010

[14] W Wang and S Zhong, “Stochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varying

delays,” Neurocomputing, vol 82, no 1, pp 143–156, 2012.

[15] J Cao and F Ren, “Exponential stability of discrete-time genetic

regulatory networks with delays,” IEEE Transactions on Neural Networks, vol 19, no 3, pp 520–523, 2008.

[16] F Wu, “Stability analysis of genetic regulatory networks with

multiple time delays,” in Proceedings of the 29th Annual Inter-national Conference of the IEEE EMBS Cite InterInter-nationale, Lyon,

France, 2007

[17] W Zhang, J.-A Fang, and Y Tang, “Stochastic stability of Markovian jumping genetic regulatory networks with mixed

time delays,” Applied Mathematics and Computation, vol 217,

no 17, pp 7210–7225, 2011

[18] W Zhang, Y Tang, J Fang, and X Wu, “Stochastic stability of genetic regulatory networks with a finite set delay

characteriza-tion,” Chaos, vol 2, no 22, Article ID 023106, 2012.

[19] G Wang and J Cao, “Robust exponential stability analysis for

stochastic genetic networks with uncertain parameters,” Com-munications in Nonlinear Science and Numerical Simulation, vol.

14, no 8, pp 3369–3378, 2009

[20] C.-H Yuh, H Bolouri, and E H Davidson, “Genomic cis-regulatory logic: experimental and computational analysis of a

sea urchin gene,” Science, vol 279, no 5358, pp 1896–1902, 1998.

[21] C Li, L Chen, and K Aihara, “Stability of genetic networks with

SUM regulatory logic: Lur’e system and LMI approach,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol 53,

no 11, pp 2451–2458, 2006

[22] W Su and Y Chen, “Global robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and

dis-tributed time-varying delays,” Communications in Nonlinear Science and Numerical Simulation, vol 14, no 2, pp 520–528,

2009

[23] P Park, J W Ko, and C Jeong, “Reciprocally convex approach to

stability of systems with time-varying delays,” Automatica, vol.

47, no 1, pp 235–238, 2011

[24] M B Elowitz and S Leibier, “A synthetic oscillatory network of

transcriptional regulators,” Nature, vol 403, no 1, pp 335–338,

2000