Stability of genetic regulatory networks with interval time-varying delays and stochastic perturbation

Asian Journal of Control, Vol. 13, No. 5, pp. 625 634, September 2011Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.336

STABILITY OF GENETIC REGULATORY NETWORKS WITH INTERVAL

TIME-VARYING DELAYS AND STOCHASTIC PERTURBATION

Yong He, Ling-Yun Fu, Jin Zeng, and Min Wu

ABSTRACT

This paper studies the delay-dependent stability for genetic regulatorynetworks with both interval time-varying delays and stochastic perturbation.By considering the relationship among the time-varying delays, its upperbound and their difference, and introducing an improved-free-weightingmatrix (IFWM) approach, some less conservative stability criteria areproposed without ignoring any useful terms in the weak infinitesimal operatorof the Lyapunov-Krasovskii functional. Two numerical examples are given toillustrate the effectiveness and the reduced conservativeness of the obtainedresults.

Key Words: Genetic regulatory networks (GRNs), linear matrix inequality(LMI), time-varying delay, stochastic perturbation, stability,improved-free-weighting matrix (IFWM).

I. INTRODUCTION

Recently, genetic regulatory networks (GRNs)have received considerable attention due to theirgreat effect on human survival and development. Asis known, a genetic regulatory network consists ofsome genes interacting with each other indirectly andgenes regulate their expressions by their own produc-tion, namely protein. Since the encoded informationcontained in genes determines the characters of theorganism, the identification of the steady-state valuesof the practice networks states, i.e., the concentrationsof mRNA and protein, have significant theoreticalmeaning and practical value.

Manuscript received June 28, 2010; revised September 29,2010; accepted October 28, 2010.The authors are with the School of Information Science

and Engineering, Central South University, Changsha 410083,China (Min Wu is the corresponding author: e-mail:[email protected]).This work was supported in part by the National Natural

Science Foundation of China under Grant No. 60974026,the Doctor Subject Foundation of China under Grant No.200805330004, the Hunan Provincial Natural Science Foun-dation of China under Grant No. 08JJ1010 and the Graduatedegree thesis Innovation Foundation of Central South Univer-sity No. 2009ssxt187.

The research on GRNs dates from the 1960s. Withthe increasing of experimental data, modeling GRNshas become more and more attractive over the past fewdecades [1–3]. Up to now, five typical types of GRNshave been identified, directed graphs model, stochasticequation model [4, 5], Boolean model, Bayesian modeland differential equation model [6–10], among whichthe last two are the most widespread. In the stochasticdifferential equation model adapted in this paper, theconcentrations of mRNA and protein are used as thestate variables for modeling.

In GRNs, time delays are encountered inevitablydue to the slow process of transcription, translation,translocation and diffusion, and are often a source ofinstability and oscillations. Besides, stochastic noisefrom both internal and external perturbations as wellas environment fluctuations can not be neglected in thereal word. Hence, it is necessary to take time delaysand stochastic noise into consideration when modelingGRNs.

Based on the identified model, studies on thestability of GRNs with time delays become fruitfuland many important results on this subject have beenreported in the literature [9–21]. Meanwhile, thediscussion on the filtering problem of GRNs is stilla changing area [22, 23]. In [9, 10], some sufficientdelay-independent conditions were proposed for GRNs

q 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

626 Asian Journal of Control, Vol. 13, No. 5, pp. 625 634, September 2011

with SUM regulatory. It is well known that delay-independent criteria tend to be conservative, especiallywhen the size of a delay is small. Moreover, thetime-varying delays should be differentiable and thederivative of the two time-varying delays therein shouldbe less than 1. In [11], in order to eliminate the quadraticintegral terms in the weak infinitesimal operator ofV (t) by bounding the cross terms, the basic inequalitywhich inevitably leads to conservativeness was usedeight times. In [12], Zhou et al. analyzed the stabilityof a class of GRNs with both interval time-varyingdelay and stochastic disturbance by introducing a free-weighting matrix (FWM) approach [24, 25]. However,the lower bounds of two time delays were enlarged astime delays themselves, which may lead to a conser-vative result. In [15], delay-dependent asymptotic androbust stability conditions of GRNs with time-varyingdelays were presented, but they were rate-independent.Above all, although great achievements have beenmade in GRNs, further improvement is needed.

This paper concerns the stability problem ofGRNs with interval time-varying delays and state-dependent stochastic disturbances. First, a new criterionthat considers the relationship between the time-varyingdelay and its lower and upper bounds is proposed whenestimating the upper bound of the weak infinitesimaloperator of the Lyapunov-Krasovskii functional. Thenthe criterion is further extended to the case where therate of time delay is independent. Finally, two numer-ical examples are given to demonstrate the benefits ofour method.

Notation. For any matrix P , P>0 means thatP is symmetric positive definite; The superscript “T ”represents the transpose of a matrix; I and 0 representidentity matrix and zero matrix with appropriate dimen-sions, respectively; ∗ represents a term that is inducedby symmetry.

II. NOTATION AND PRELIMINARIES

Consider the following GRNs with time-varyingdelays [10]:

m(t) = −Am(t)+Bg(p(t−d2(t)))+l

p(t) = −Cp(t)+Wm(t−d1(t))(1)

where m(t)=[m1(t),m2(t), . . . ,mn(t)]T , p(t)=[p1(t), p2(t), . . . , pn(t)]T , mi (t)∈ R and pi (t)∈ R, i =1,2, . . . ,n are the concentrations of mRNA andprotein of the i th node at time t , respectively; A=diag(a1,a2, . . . ,an), C=diag(c1,c2, . . . ,cn), ai andci , i=1,2, . . . ,n are the degradation rates of the mRNA

and protein, respectively; W =diag(w1,w2, . . . ,wn),wi , i=1,2, . . . ,n is the translation rate of the mRNA;d1(t) and d2(t) represent the translation delay andfeedback regulation delay respectively, and satisfy thefollowing conditions:

0≤ds1≤d1(t)≤db1, d1(t)≤�1<∞0≤ds2≤d2(t)≤db2, d2(t)≤�2<∞ (2)

g(p(t))=[g1(p1(t)),g2(p2(t)), . . . ,gn(pn(t))]T , g(p(t)) represents the feedback regulation of the protein onthe transcription, where g j (p j (t))=(p j (t)/�)H/[1+(p j (t)/�)H], j=1,2,. . .,n is amonotonically increasingfunction with H being the Hill coefficient and � beinga positive constant; B=(Bij)∈ Rn×n is the couplingmatrix of the GRNs, which is defined as follows:

Bij=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

�ij, if transcription factor j is an activator

of gene i

0, if there is no link from node j to node i

−�ij, if transcription factor j is an repressor

of gene i

and l=diag(l1, . . . , ln), li =∑j∈Si �ij where Si is a set

of all j nodes which are repressors of gene i .Assuming that vectors m∗ and p∗ are the equilib-

rium point of system (1), and letting e1(t)=m(t)−m∗,e2(t)= p(t)− p∗, then we have

e1(t) = −Ae1(t)+B f (e2(t−d2(t)))

e2(t) = −Ce2(t)+We1(t−d1(t))(3)

where f (e2(t))=g(e2(t)+ p∗)−g(p∗), with f (e2(t))=[ f1(e21(t)), f2(e22(t)), . . . , fn(e2n(t))]T . Since g j (·) isa monotonically increasing function with saturation,f j (·) satisfies the sector condition

0≤ f j (x)/x≤k j , j=1,2, . . . ,n (4)

Since gene regulation is an intrinsically noisyprocess, we consider GRNs with both time delays andnoise perturbations in the following form:

de1(t) = [−Ae1(t)+B f (e2(t−d2(t)))]dt+�(e1(t),

e1(t−d1(t)),e2(t),e2(t−d2(t)))d�(t)

de2(t) = [−Ce2(t)+We1(t−d1(t))]dt.(5)

As in many studies on stochastic systems, weassume that �(·) could be estimated by

trace(��T ) ≤2∑

i=1[eTi (t)Hiei (t)+eTi (t−di (t))Hi+2

ei (t−di (t))] (6)


Y. He et al.: Stability of Genetic Regulatory Networks 627

In order to obtain the main results, the followinglemmas are needed.

Lemma 1 ([26]). For any x, y∈ Rn , matrix P>0,

−2xT y≤ xT P−1x+ yT Py

Lemma 2 ([27]). There exists a symmetric matrix Xsuch that[

P1+X Q1

∗ R1

]>0,

[P2−X Q2

∗ R2

]>0

if and only if⎡⎢⎣P1+P2 Q1 Q2

∗ R1 0

∗ ∗ R2

⎤⎥⎦>0.

III. MAIN RESULTS

First of all, denote

h1(t) = −Ae1(t)+B f (e2(t−d2(t)))

h2(t) = −Ce2(t)+We1(t−d1(t))

h3(t) = �(e1(t),e1(t−d1(t)),e2(t),e2(t−d2(t)))

h4(t) = 0

then system (5) is rewritten as

de1(t) = h1(t)dt+h3(t)d�(t)

de2(t) = h2(t)dt+h4(t)d�(t)(7)

and the following theorem is presented by employingthe improved-free-weighting matrix (IFWM) approachfor the stability of GRNs (5).

Theorem 1. For given scalars 0≤ds1≤db1, 0≤ds2≤db2, 0<�<1, �1 and �2, the system (5) is stochasticallyasymptotically stable in mean square if there exist

matrices P=[P11∗

P12P22

]>0, Q1i>0, Q2i>0, Q3i>0,

R1i>0, R2i>0, Zi>0, L1i , L2i , M1i , M2i , N1i , N2i , i=1,2, �=diag{�1,�2, . . . ,�n}≥0, and three positivescalars j>0, j =1,2,3, such that the following LMIshold ⎡

⎢⎢⎣−�1−�2−�T

2 �12 �13

∗ �22 0

∗ ∗ �33

⎤⎥⎥⎦>0 (8)

P11≤1 I, Z1≤2 I, Z2≤3 I (9)

where

�1 =⎡⎢⎣

1 0 2

∗ 3 4

∗ ∗ 99

⎤⎥⎦

1 =

⎡⎢⎢⎣

11 −AT P12−P12C P12W

∗ 22 P22W−CT R2W

∗ ∗ 33

⎤⎥⎥⎦

2 = [(P11B−AT R1B)T BTP12 0]T

3 = diag{44,55,66,−Q21,−Q22}4 = [K� 0 0 0 0]T

99 = BT R1B−2�

11 = −P11A−AT P11+Q11+AT R1A+H1

22 = −P22C−CT P22+Q12+CT R2C+H2

33 = −(1−�1)Q31+WT R2W +H3

44 = −(1−�2)Q32+H4

55 = Q21+Q31−Q11

66 = Q22+Q32−Q12

�2 = [L1 L2 −M1+N1 −M2+N2

−L1+M1 −L2+M2 −N1 −N2 0]L1 = [L11

T 0 L21T 0 0 0 0 0 0]T

M1 = [M11T 0 M21

T 0 0 0 0 0 0]T

N1 = [N11T 0 N21

T 0 0 0 0 0 0]T

L2 = [0 L12T 0 L22

T 0 0 0 0 0]T

M2 = [0 M12T 0 M22

T 0 0 0 0 0]T

N2 = [0 N12T 0 N22

T 0 0 0 0 0]T

�12 = [ds1L1 ds2L2 �d1M1 �d2M2

(1−�)d1N1 (1−�)d2N2]�22 = diag{ds1R11,ds2R12,�d1R21,�d2R22

(1−�)d1R21, (1−�)d2R22}�13 = [L1 M1 N1]�33 = diag{Z1, Z2, Z2}

with K =diag{k1,k2, . . . ,kn}, di =dbi−dsi, =1+ds12+d13, Ri =dsiR1i +di R2i , i=1,2.



Proof. Choose a Lyapunov-Krasovskii functionalcandidate to be

V (t)=4∑j=1

Vj (t) (10)

where

V1(t) =[e1(t)

e2(t)

]T [P11 P12

∗ P22

][e1(t)

e2(t)

]

V2(t) =2∑

i=1

[∫ t

t−dsiei

T (s)Q1i ei (s)ds

+∫ t−dsi

t−dbiei

T (s)Q2i ei (s)ds

+∫ t−dsi

t−di (t)ei

T (s)Q3i ei (s)ds

]

V3(t) =2∑

i=1

[∫ 0

−dsi

∫ t

t+�hTi (s)R1i hi (s)dsd�

+∫ −dsi

−dbi

∫ t

t+�hTi (s)R2i hi (s)dsd�

]

V4(t) =∫ 0

−ds1

∫ t

t+�trace{hT3 (s)Z1h3(s)}dsd�

+∫ −ds1

−db1

∫ t

t+�trace{hT3 (s)Z2h3(s)}dsd�

with P=[P11∗

P12P22

]>0, Q1i>0, Q2i>0, Q3i>0, R1i>0,

R2i>0, Zi>0, i=1,2 to be determined.By using Ito’s formula, we have

dV (t) =LV (t)dt+2[eT1 (t)P11

+eT2 (t)PT12]h3(t)d�(t) (11)

and the weak infinitesimal operator L of the stochasticprocess along the evolution of V (t) is given as

LV (t) = 2

[e1(t)

e2(t)

]T [P11 P12

∗ P22

][h1(t)

h2(t)

]

+trace{hT3 (t)P11h3(t)}+2∑

i=1[ei T (t)Q1i ei (t)

−eiT (t−dsi)(Q1i −Q2i −Q3i )ei (t−dsi)

−eiT (t−dbi)Q2i ei (t−dbi)

−(1− di (t))eiT (t−di (t))Q3i ei (t−di (t))]

+2∑

i=1

{hTi (t)[dsiR1i +di R2i ]hi (t)

−∫ t

t−dsihTi (s)R1i hi (s)ds

−∫ t−dsi

t−di (t)hTi (s)R2i hi (s)ds

−∫ t−di (t)

t−dbihTi (s)R2i hi (s)ds

}

+trace{hT3 (t)[ds1Z1+d1Z2]h3(t)}

−∫ t

t−ds1trace{hT3 (s)Z1h3(s)}ds

−∫ t−ds1

t−d1(t)trace{hT3 (s)Z2h3(s)}ds

−∫ t−d1(t)

t−db1trace{hT3 (s)Z2h3(s)}ds. (12)

For any appropriate dimension matrices Li , Mi ,Ni , i=1,2, using the Leibniz-Newton formula tosystem (7) yields

0= 2�T (t)Li

[ei (t)−ei (t−dsi)

−∫ t

t−dsihi (s)ds−

∫ t

t−dsihi+2(s)d�(s)

](13)

0= 2�T (t)Mi

[ei (t−dsi)−ei (t−di (t))

−∫ t−dsi

t−di (t)hi (s)ds−

∫ t−dsi

t−di (t)hi+2(s)d�(s)

]

(14)

0= 2�T (t)Ni

[ei (t−di (t))−ei (t−dbi)

−∫ t−di (t)

t−dbihi (s)ds−

∫ t−di (t)

t−dbihi+2(s)d�(s)

]

(15)

where

�T (t) = [e1T (t) e2T (t) e1

T (t−d1(t)) e2T (t−d2(t))

e1T (t−ds1) e2

T (t−ds2) e1T (t−db1)

e2T (t−db2) f T (e2(t−d2(t)))].



By Lemma 1, for Z1>0, Z2>0 with appropriatedimensions, the following inequalities hold

−2�T (t)L1 1

≤�T (t)L1Z1−1LT

1 �(t)+ T1 Z1 1 (16)

−2�T (t)M1 2

≤�T (t)M1Z2−1MT

1 �(t)+ T2 Z2 2 (17)

−2�T (t)N1 3

≤�T (t)N1Z2−1NT

1 �(t)+ T3 Z2 3 (18)

where

1 =∫ t

t−ds1h3(s)d�(s), 2=

∫ t−ds1

t−d1(t)h3(s)d�(s)

3 =∫ t−d1(t)

t−db1h3(s)d�(s).

According to Ito isometry [28], it is clear that

E{ T1 Z1 1} =∫ t

t−ds1trace{hT3 (s)Z1h3(s)}ds (19)

E{ T2 Z2 2} =∫ t−ds1

t−d1(t)trace{hT3 (s)Z2h3(s)}ds (20)

E{ T3 Z2 3} =∫ t−d1(t)

t−db1trace{hT3 (s)Z2h3(s)}ds (21)

The following equations hold for any appropriatedimension matrices Xi ,Yi , i=1,2:

dsi�T (t)Xi�(t)−

∫ t

t−dsi�T (t)Xi�(t)ds=0 (22)

di�T (t)Yi�(t)−

∫ t−dsi

t−di (t)�T (t)Yi�(t)ds

−∫ t−di (t)

t−dbi�T (t)Yi�(t)ds=0. (23)

In addition, from the sector condition (4), thefollowing inequality holds for any �=diag{�1,�2, . . .,�n}≥0[

e2(t−d2(t))

f (e2(t−d2(t)))

]T [0 �K

∗ −2�

]

[e2(t−d2(t))

f (e2(t−d2(t)))

]≥0 (24)

Then, combining conditions (13)–(24) to equation(12) and using conditions (6) and (9) yield

E{LV (t)}≤�T (t)(�1+�2+�T

2 +�3+�4)�(t)

−2∑

i=1

⎧⎨⎩

∫ t

t−dsi

[�(t)

hi (s)

]T [Xi Li

∗ R1i

][�(t)

hi (s)

]ds

+∫ t−dsi

t−di (t)

[�(t)

hi (s)

]T [Yi Mi

∗ R2i

][�(t)

hi (s)

]ds

+∫ t−di (t)

t−dbi(t)

[�(t)

hi (s)

]T [Yi Ni

∗ R2i

][�(t)

hi (s)

]ds

⎫⎬⎭(25)

where �3=∑2i=1[dsiXi +diYi ], �4= L1Z

−11 LT

1 +M1Z

−12 MT

1 +N1Z2−1NT

1 with �1 and �2 beingdefined in Theorem 1.

Therefore, if the following inequalities for i=1,2hold

�1+�2+�T2 +�3+�4<0 (26)[

Xi Li

∗ R1i

]≥0,

[Yi Mi

∗ R2i

]≥0,

[Yi Ni

∗ R2i

]≥0 (27)

we have E{dV (t)}=E{LV (t)}<0, which indicates thesystem (5) is stochastically asymptotically stable inmean square.

For i=1,2, from the conditions (26) and (27), wehave

−�1−�2−�T2 −�3−�4>0 (28)

dsi

[Xi Li

∗ R1i

]>0, �di

[Yi Mi

∗ R2i

]>0

(1−�)di

[Yi Ni

∗ R2i

]>0. (29)

Applying Lemma 2 six times and the Schur Comple-ment, the conditions (28) and (29) are equivalent to LMI(8). This completes the proof. �



Remark 1. Compared with the existing Lyapunov-Krasovskii functional [11, 12],

2∑i=1

[∫ t−dsi

t−dbiei

T (s)Q2i ei (s)ds

+∫ t−dsi

t−di (t)ei

T (s)Q3i ei (s)ds

]

is introduced instead of

2∑i=1

[∫ t

t−dbiei

T (s)Q2i ei (s)ds

+∫ t

t−di (t)ei

T (s)Q3i ei (s)ds

],

which uses the information on the lower bound ofdelays effectively. As mentioned in [29], the newLyapunov-Krasovskii functional is favorable for thereduction of conservativeness. Additionally, in [12],extra nonnegative terms eT (t)(�(t)−�m)N R−1

2 NT e(t)

and eT (t)(�(t)−�m)T R−14 T T e(t) were added to the

weak infinitesimal operator of the Lyapunov-Krasovskiifunctional to bound it. Besides, the lower bounds oftwo time delays in the integral terms of the weakinfinitesimal operator of Lyapunov-Krasovskii func-tional were enlarged as time delays themselves. Thus,the conservativeness of results was raised.

Remark 2. By the Schur complement, the conditions(26) and (27) can be written as the form of LMIsdirectly, which we can take as the conclusion. However,the dimensions of decision variables Xi ,Yi , i=1,2 areso large that it will influence the computing speed to agreat extent.

Remark 3. In fact, the inequalities in (27) are treatedequivalently as strict inequalities when we seek thefeasible solution using the LMI toolbox in MATLAB.For this reason, it makes little difference to the resultsthat strict inequalities (29) take the place of inequali-ties (27).

Remark 4. In addition, there are often some unavoid-able uncertainties in modeling GRNs because ofmodeling errors and parameter fluctuations. Similar to[30, 32], a robust stability condition can be obtainedbased on Theorem 1 for GRNs with parameter uncer-tainties which are norm-bound.

Theorem 1 gives a delay- and rate-dependentrobust stability criterion for time-varying delayssatisfying (2). Note that a delay-dependent and rate-independent criterion for time-varying delays satisfying

(2) without any restriction on the derivative of delayscan be derived as follows by choosing Q3i =0, i=1,2in Theorem 1.

Corollary 1. For given scalars 0≤ds1≤db1, 0≤ds2≤db2 and 0<�<1, the system (5) is stochasticallyasymptotically stable in mean square if there exist

matrices P=[P11 P12

∗ P22

]>0, Q1i>0, Q2i>0, R1i>0,

R2i>0, Zi>0, L1i , L2i , M1i , M2i , N1i , N2i , i=1,2,�=diag{�1,�2, . . . ,�n}≥0, and three positive scalars j>0, j =1,2,3, such that the following LMIs hold

⎡⎢⎢⎣

−�1−�2−�T2 �12 �13

∗ �22 0

∗ ∗ �33

⎤⎥⎥⎦>0 (30)

P11≤1 I, Z1≤2 I, Z2≤3 I (31)

where

�1 =⎡⎢⎣

1 0 2

∗ 3 4

∗ ∗ 99

⎤⎥⎦

1 =

⎡⎢⎢⎣

11 12 13

∗ 22 23

∗ ∗ WT R2W +H3

⎤⎥⎥⎦

3 = diag{H4,Q21−Q11,Q22−Q12,77,88}

and other parameters are defined in Theorem 1.

IV. NUMERICAL EXAMPLES

In this section, two examples are presented to illus-trate the effectiveness and the merits of the proposedresults.

Example 1. Consider a small sized genetic network(shown as Fig. 1) in the form of state equation (5) withparameters described as

A =C=diag{0.9,0.9,0.9,0.9,0.9}W = diag{1,1,1,1,1}



1 2

5

3 4

Fig. 1. A genetic network model(↑: activation; ᵀ: repression).

�(e1(t),e1(t−d1(t)),e2(t),e2(t−d2(t)))

= 0.05

⎡⎢⎢⎢⎢⎢⎢⎢⎣

e11(t)+e11(t−d1(t))+e21(t)+e21(t−d2(t))

e12(t)+e12(t−d1(t))+e22(t)+e22(t−d2(t))

e13(t)+e13(t−d1(t))+e23(t)+e23(t−d2(t))

e14(t)+e14(t−d1(t))+e24(t)+e24(t−d2(t))

e15(t)+e15(t−d1(t))+e25(t)+e25(t−d2(t))

⎤⎥⎥⎥⎥⎥⎥⎥⎦

and according to the definition of links in Section II, wecan obtain the coupling matrix B and l of this networkas follows

B = �×

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 −1 1 0 0

−1 0 0 1 1

0 1 0 0 0

1 −1 0 0 0

0 0 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

,

l = �×

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1

1

0

1

0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

.

Here, we set �=0.25. The time delays d1(t) and d2(t)are assumed to be

d1(t)=0.2+0.4|sin(t)|, d2(t)=0.4+0.6|sin(t)|.Therefore, ds1=0.2,db1=0.6,ds2=0.4,db2=1.

Let H1=H2=H3=H4=0.1I , �=1/3, g j (x)=x2/(1+x2), j =1,2, . . . ,n, i.e., the Hill coefficientis 2. By computing, we can find k j =max{ f j (x)}=max{g j (x)}=0.65, j=1,2, . . . ,n and K =diag{0.65,0.65,0.65,0.65,0.65}.

Under the condition m(t)=0 and p(t)=0, theunique equilibrium point of this network is

m∗ =[0.4055,0.5952,0.1690,0.4803,0.1231]T

p∗ =[0.4505,0.6613,0.1878,0.5336,0.1368]T

0 5 10 15 20–0.5

0

0.5

1

1.5

2

2.5

t

e 1 (t

)

Fig. 2. Trajectories of the concentrations of mRNA e1(t).

0 5 10 15 20–0.5

0

0.5

1

1.5

2

t

e 2 (t

)

Fig. 3. Trajectories of the concentrations of protein e2(t).

For simplicity, we shift the equilibrium point to theorigin.

By solving the LMIs in Corollary 1 using the LMItoolbox in MATLAB, we can easily get feasible solu-tions of the LMIs (30) and (31), which illustrates thatthis network is mean square asymptotically stable.

Figures 2 and 3 show the trajectories of thevariables e1(t) and e2(t) with the initial condi-tions [2,1.4,1.4,1,0.7]T and [1.6,1.4,1.2,0.6,0.4]T ,respectively.

However, it should be pointed out that, the condi-tions in [9, 10] are not applicable due to the fact thatd1(t) and d2(t) are not continuously differentiable.

Example 2. Consider the genetic network describedin Fig. 1. Let d2(t)=0.4+1.2sin2(t), d1(t) is unknownhere, H1=H3=0, H2=H4=0.1I , and the otherparameters are the same as in Example 1.



Table I. Admissible upper bound db1 with various ds1.

ds1

�1 Method 0.2 0.5 0.8 1

0.7 Theorem 1 in [12] 0.6305 0.9151 1.2069 1.4058Theorem 1 0.9333 1.2244 1.5234 1.7234

unknown Theorem 1 in [15] 0.2668 0.5668 0.8668 1.0668Theorem 2 in [12] 0.6179 0.8500 1.2003 1.3996

Corollary 1 0.9285 1.2209 1.5202 1.7202

For �1=0.7 and �1 unknown, the corre-sponding admissible upper bounds of d1(t) obtainedby Theorem 1 and Corollary 1 are shown with givends1 in Table I and �=1/3. Meanwhile, results obtainedby Theorem 1 in [15] and theorems in [12] are alsopresented in Table I.

Note that Theorem 1 in [15] is rate-independentand the conditions in [9, 10] are not applicable to thisexample because d2(t)>1.

V. CONCLUSIONS

In this paper, the stability problem has beenanalyzed for a class of GRNs with both interval time-varying delays and stochastic noise. Based on theLyapunov-Krasovskii method and the IFWM approach,some delay-dependent criteria are derived to guaranteethe GRNs to be asymptotically stable in mean square.The proposed results are less conservative than theexisting ones and two examples are given finally toverify the effectiveness and the merits of our results.

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Yong He received his BS andMS degrees in Applied Mathe-matics from Central South Univer-sity, Changsha, China in 1991and 1994, respectively. In July1994, He joined the staff of theuniversity, where he is currently aprofessor in the School of Informa-tion Science and Engineering. Hereceived his PhD degree in Control

Theory and Control Engineering from Central SouthUniversity in 2004. From January 2005 to March 2006,he was a research fellow in the Department of Elec-trical and Computer Engineering, National Universityof Singapore. From March 2006 to January 2007, hewas a research fellow in the Faculty of Advanced Tech-nology, University of Glamorgan, UK. He received theGuan Zhao-Zhi Award of the 26th Chinese ControlConference in 2007 (jointly with M. Wu, G.P. Liu andJ.H. She). Professor He is a senior member of the IEEE.

His current research interests include robustcontrol and its applications, networked control, andprocess control.

Ling-Yun Fu received her Bach-elor’s degree in Informationand Computing Science fromCentral South University(CSU),Changsha, China in 2008. She iscurrently pursuing her Master’sdegree in Control Science andEngineering at the School of Infor-mation Science and Engineering,CSU.



Her current research interests include robustcontrol and its applications, genetic regulatory networks.

Jin Zeng received her BS degreeof Engineering from Central SouthUniversity, Changsha, China in2009. She is now pursuing herMS degree in Control Scienceand Engineering at Central SouthUniversity, Changsha, China. Herresearch interests include robustcontrol, filter design, and geneticregulatory networks.

Min Wu (SM’08) received hisBS and MS degrees in Engi-neering from Central SouthUniversity, Changsha, China, in1983 and 1986, respectively, andhis PhD degree in Engineering

from Tokyo Institute of Technology, Tokyo, Japan, in1999. Since July 1986, he has been a Faculty Memberwith Central South University, where he is currentlya Professor of Automatic Control Engineering in theSchool of Information Science and Engineering. Hewas a Visiting Scholar with the Department of Elec-trical Engineering, Tohoku University, Sendai, Japan,from 1989 to 1990, a Visiting Research Scholar withthe Department of Control and Systems Engineering,Tokyo Institute of Technology, from 1996 to 1999, anda Visiting Professor with the School of Mechanical,Materials, Manufacturing Engineering and Manage-ment, University of Nottingham, Nottingham, U.K.,from 2001 to 2002. Professor Wu is a senior memberof the IEEE, and a member of the Nonferrous MetalsSociety of China and the Chinese Association ofAutomation. He received the IFAC Control EngineeringPractice Prize Paper Award in 1999 (together with M.Nakano and J.-H. She).

His current research interests include robustcontrol and its applications, process control, andintelligent systems.


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Stability of genetic regulatory networks with interval time-varying delays and stochastic perturbation