Circuits Syst Signal Process DOI 10.1007/s00034-015-0006-8

Asymptotic Stability Criteria for Genetic Regulatory Networks with Time-Varying Delays and Reaction–Diffusion Terms Yuanyuan Han · Xian Zhang · Yantao Wang

Received: 2 December 2014 / Revised: 12 February 2015 / Accepted: 13 February 2015 © Springer Science+Business Media New York 2015

Abstract This paper investigates the asymptotic stability problem for delayed genetic regulatory networks with reaction–diffusion terms under both Dirichlet boundary conditions and Neumann boundary conditions. First, by constructing a new Lyapunov– Krasovskii functional and using Jensen’s inequality, Wirtinger’s inequality, Green’s second identity and the reciprocally convex approach, we establish delay-dependent asymptotic stability criteria that do not require a restriction of the upper bounds of the delays’ derivatives being less than 1. Thus, the stability criteria that we establish are less conservative than the existing criteria and extend the range of applications of the theoretical results. In addition, it is shown that the obtained criterion under Dirichlet boundary conditions retains the information about the reaction–diffusion terms, while these do not exist in the criterion under Neumann boundary conditions. It is then theoretically presented that the stability criteria established in this paper are less conservative than the existing ones. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results. Keywords Genetic regulatory networks · Reaction–diffusion terms · Asymptotic stability · Time-varying delays

Y. Han · X. Zhang (B) · Y. Wang School of Mathematical Science, Heilongjiang University, Harbin 150080, China e-mail: [email protected] Y. Han e-mail: [email protected] Y. Wang e-mail: [email protected]

Circuits Syst Signal Process

1 Introduction In recent years, genetic regulatory network (GRN) models, such as the Bayesian model [3], the Boolean model [1,30] and the differential equations model [19,38,43], have attracted the attention of an increasing number of mathematicians and biologists. These models focus both on the dynamical behaviors of systems and on the mechanism of interactions between different genes at the molecular level. Among these models, the dynamical behaviors of GRNs are inherently captured by differential equation models; thus, it has become the hot spot issue in the field of biological systems. Models governed by systems of differential equations incorporating several biological features, such as reaction–diffusion and time delay (a frequent phenomenon in nature), have been considered. Recently, the stability issue for GRNs has received considerable attention owing to the fact that stability is the most significant and essential dynamical behavior in terms of problems’ performance. Moreover, many important results on the stability issue for GRNs have been published in the literature, for which we refer the reader to [6,12,22,29,37] and the references cited therein. It is well known that time delays are unavoidably encountered in various practical systems, such as chemical engineering systems, biological systems and economic systems [13,17,18,20,32,33,36,39,40,42]. It is worth pointing out that time delays may lead to instability or oscillation in GRNs. To be more realistic, GRNs often incorporate time delays as well, where the delays represent several biological features. Understanding the interplay of time delays is therefore a key point for many models [11, 14,15,17,41,44,45]. These models are given by systems of delay differential equations and are known as functional differential equation models. Genetic regulatory systems are assumed, implicitly, to be spatially homogeneous. However, there are some situations in which these assumptions are not appropriate. For example, it might be necessary to consider the diffusion of regulatory proteins or metabolites from one compartment to another [4,5,7,16]. In this situation, the general functional differential equation model cannot precisely describe the genetic regulatory process. We must consider variables which vary both with space and with time. Therefore, it is indispensable to introduce reaction–diffusion terms in modeling. Furthermore, the stability analysis of GRNs with reaction–diffusion terms becomes more mathematical challenging and should obtain much attention. So far, only three papers [10,23,46] analyze the dynamic properties of GRNs with time delays and reaction–diffusion terms. From the point of view of applications, it is particularly relevant to study three different types of boundary conditions, namely Dirichlet boundary conditions, Neumann boundary conditions and Robin boundary conditions. We discuss some recent results on Dirichlet boundary conditions. Based on the Lyapunov functional method, Zhou et al. [46] investigated finite-time robust stochastic stability criteria for uncertain GRNs with time-varying delays and reaction–diffusion terms. By constructing a Lyapunov–Krasovskii functional and using Friedrichs’ inequality (“2” instead of “π 2 ” in Lemma 2 below) and Greens formula to deal with the reaction–diffusion terms in the derivative, Ma et al. [23] established a delay-dependent asymptotic stability criteria for a class of delayed GRNs with reaction–diffusion terms. Han and Zhang [10] improved Ma et al’s results

Circuits Syst Signal Process

in two aspects: (1) Friedrichs’ inequality is replaced with Wirtinger’s inequality (see Lemma 2 below); and (2) the Lyapunov–Krasovskii functional is simplified on the premise that the conservativeness does not increase (see Theorem 3 below). In addition, Han and Zhang theoretically and numerically demonstrated that the stability criteria established in [10] are less conservative than the corresponding criteria in [23]. However, it should be highlighted that the stability criteria obtained in [10,23,46] are only available when the upper bounds on the delay derivatives are restricted to being less than 1. Inspired by previous research, in this paper we propose a new approach to establishing delay-dependent asymptotic stability criteria for GRNs with time-varying delays and reaction–diffusion terms under Dirichlet boundary conditions and Neumann boundary conditions. By constructing a new Lyapunov–Krasovskii functional and using Green’s second identity (see Lemma 3 below) and the reciprocally convex combination lemma (see Lemma 5 below), delay-dependent asymptotic stability criteria for a class of GRNs with time-varying delays and reaction–diffusion terms are derived. The obtained stability criterion is diffusion-dependent under Dirichlet boundary conditions and is diffusion-independent under Neumann boundary conditions. The main contributions of this paper can be reduced to the following three aspects: – Comparing the results obtained in this paper with those in [10,23], we remove the restriction on the upper bounds of the delay derivatives being less than 1. This is realized by introducing a term of the form

V0 (t, m) :=

l   k=1 Ω

∂m T (t, x) ∂m(t, x) N 1 Dk dx ∂ xk ∂ xk

into our Lyapunov–Krasovskii functional; – Green’s second identity and the reciprocally convex combination lemma are used to deal with the derivative of V0 (t, m) in t; – It is theoretically and numerically shown that the stability criteria established in this paper are less conservative than the corresponding criteria in [10,23] (see Sects. 4 and 5 below). Notation We now set some standard notation to be used throughout the paper. For real symmetric matrices X and Y , X > Y (X ≥ Y ) means that X − Y is positive definite (positive semi-definite). I is the identity matrix with appropriate dimension, ∗ denotes the symmetric terms of a symmetric matrix, and A T represents the transpose of the matrix A. Ω is a compact set in the vector space R n with smooth boundary ∂Ω, and mesΩ is the measure of Ω. Let C 1 (X × Y, R n ) be the Banach space of functions that map X × Y into R n and have continuous first derivatives. We define two norms  ·  and  · d as follows:  y(t, x) =

Ω

1/2 y(t, x)T y(t, x)dx

, ∀y(t, x) ∈ C 1 ((0, +∞) × Ω, R n ).

Circuits Syst Signal Process

 z(t, x)d =

1/2 T

sup z(t, x) z(t, x)dx

Ω −d≤t≤0

, ∀z(t, x) ∈ C 1 ([−d, 0] × Ω, R n ).

2 Model Description and Preliminaries This paper considers the following delayed GRNs with reaction–diffusion terms: ⎧   ∂m i (t, x) l ∂ ∂m i (t, x) ⎪ ⎪ Dik − ai m = k=1 i (t, x) ⎪ ⎪ ⎪ ∂t ∂ xk ⎪

n ∂ x k ⎨ + j=1 wi p j (t − σ (t), j g j (  x)) + qi , (1)

∂ p (t, x) ∂ ∂ p (t, x) ⎪ i i l ∗ ⎪ D − c = p (t, x) ⎪ i i ik k=1 ⎪ ⎪ ∂t ∂ xk ∂ xk ⎪ ⎩ i (t − τ (t), x), i = 1, 2, . . . , n, + bi m  where x = [ x1 x2 . . . xl ]T ∈ Ω ⊂ R l , Ω = {x |xk | ≤ L k , k = 1, 2, . . . , l}, L k is ∗ > 0 denote the diffusion rate matrices; m i (t, x) and a constant; Dik > 0 and Dik pi (t, x) are the concentrations of mRNA and protein at the ith node, respectively; ai and ci are degradation rates of the mRNA and protein, respectively; bi is a constant; W := [wi j ] ∈ R n×n represents the coupling matrix, which is defined as follows: ⎧ if j is an activator of gene i, ⎨ γi j , wi j = −γi j , if j is a repressor of gene i, ⎩ 0, if there is no link from gene j to i, where γi j is the dimensionless transcriptional rate of transcription factor j to gene i; sH g j is the activation function of the form g j (s) = 1+s H , where H is the Hill coefficient; qi =  j∈Ii γi j , Ii is the set of all the nodes which are repressors of gene i; and σ (t) and τ (t) are time-varying delays satisfying 0 ≤ τ (t) ≤ τ , τ˙ (t) ≤ μ1 , 0 ≤ σ (t) ≤ σ , σ˙ (t) ≤ μ2 ,

(2)

where τ , σ , μ1 and μ2 are nonnegative real numbers. From the form of gi , it is easy to see that gi is a monotonically nondecreasing functions with saturation and that there exists ξi > 0 such that gi (yi ) − gi (z i ) 0≤ ≤ ξi yi − z i for all yi , z i ∈ R with yi = z i . The initial conditions associated with GRN (1) are given as follows: m i (s, x) = φi (s, x), x ∈ Ω, s ∈ [−d, 0], i = 1, 2, . . . , n, pi (s, x) = φi∗ (s, x), x ∈ Ω, s ∈ [−d, 0], i = 1, 2, . . . , n, where d = max{σ , τ }, and φi (s, x), φi∗ (s, x) ∈ C 1 ([−d, 0] × Ω, R).

Circuits Syst Signal Process

In this paper, the following two types of boundary conditions are considered: (i) Dirichlet boundary conditions m i (t, x) = 0, x ∈ ∂Ω, t ∈ [−d, +∞), pi (t, x) = 0, x ∈ ∂Ω, t ∈ [−d, +∞); (ii) Neumann boundary conditions ∂m i (t, x) = 0, x ∈ ∂Ω, t ∈ [−d, +∞), ∂n ∂ pi (t, x) = 0, x ∈ ∂Ω, t ∈ [−d, +∞), ∂n where n is the outer normal vector of ∂Ω. Now, we assume that m ∗ (x) = [ m ∗1 (x) m ∗2 (x) · · · m ∗n (x) ]T and p ∗ (x) = [ p1∗ (x) p2∗ (x) · · · p ∗ (x)n ]T are the unique solution of GRN (1), that is, ⎧   n l 

∂ ∂m ∗ (x) ⎪ ∗ ⎪ 0 = D − a m + wi j g j ( p ∗j ) + qi , ⎪ k i i ⎨ ∂ xk k=1 ∂ x k j=1  ∗ (x)  ⎪ l

∂ ∂ p ⎪ ⎪ ⎩0 = Dk∗ − ci pi∗ + bi m i∗ , i = 1, 2, . . . , n. ∂ xk k=1 ∂ x k

(3)

Obviously, the transformations m i = m i −m i∗ (x) and pi = pi − pi∗ (x), i = 1, 2, . . . , n transform GRN (1) into the following matrix form:   ⎧ l

∂m(t, x) ∂ ∂m(t, x) ⎪ ⎪ = D − Am(t, x) k ⎪ ⎪ ∂t ∂ xk ⎪ k=1 ∂ x k ⎪ ⎨ +W f ( p(t − σ (t), x)),   l

∂ p(t, x) ∂ ⎪ ∗ ∂ p(t, x) ⎪ ⎪ Dk − C p(t, x) = ⎪ ⎪ ∂t ∂ xk ⎪ k=1 ∂ x k ⎩ +Bm(t − τ (t), x), where A = diag(a1 , a2 , . . . , an ), C = diag(c1 , c2 , . . . , cn ), B = diag(b1 , b2 , . . . , bn ), Dk = diag(D1k , D2k , . . . , Dnk ),

(4)

Circuits Syst Signal Process ∗ ∗ ∗ Dk∗ = diag(D1k , D2k , . . . , Dnk ),

m(t, x) = [ m 1 (t, x) m 2 (t, x) · · · m n (t, x) ]T , p(t, x) = [ p1 (t, x) p2 (t, x) · · · pn (t, x) ]T , f ( p(t −σ (t), x)) = [ f 1 ( p1 (t − σ (t), x)) f 2 ( p2 (t − σ (t), x)) · · · f n ( pn (t − σ (t), x))]T , f i ( pi (t − σ (t), x)) = gi ( pi (t − σ (t), x) + pi∗ ) − gi ( pi∗ ),

i = 1, 2, . . . , n.

From the relationship between f i and gi , we obtain f i (yi ) ≤ ξi , ∀yi ∈ R, yi = 0, i = 1, 2, . . . , n, yi

f i (0) = 0, 0 ≤ namely that

f (0) = 0, f T (y)( f (y) − K y) ≤ 0, ∀y ∈ R n ,

(5)

where K = diag(ξ1 , ξ2 , . . . , ξn ) > 0. To obtain our main results, we introduce the following definition and lemmas. Definition 1 [8] The trivial solution of GRN (4) is said to be asymptotically stable, if for any ε > 0, there exists a δ(ε) > 0, such that: (i) the solution m(t, x) and p(t, x) satisfies: m(t, x)2 ≤ ε,  p(t, x)2 ≤ ε when the initial conditions φ(t, x) and φ ∗ (t, x) satisfy φ(t, x)2d ≤ δ(ε), φ ∗ (t, x)2d ≤ δ(ε); and (ii) m(t, x) → 0 and p(t, x) → 0, when t → ∞. Lemma 1 (Jensen’s Inequality) [9] For any constant matrix M T = M > 0 of appropriate dimension, any scalars a and b with a < b, and a vector function x : [a, b] → R n such that the integrals below are well defined, the following inequality holds: 

b

 x (s)ds M T

a

b

 x(s)ds ≤ (b − a)

a

b

x T (s)M x(s)ds.

a

Lemma 2 (Wirtinger’s Inequality) [31] Let f (v) be a real-valued function defined on [a, b] ⊂ R with f (a) = f (b) = 0. If f (v) ∈ C 1 [a, b], then  a

b

f 2 (v)dv ≤

(b − a)2 π2

 a

b

[ f (v)]2 dv.

Circuits Syst Signal Process

Lemma 3 (Green’s Second Identity) [34] If Ω is a bounded C 1 open set in R n and μ, υ ∈ C 2 (Ω), then   ∂υ ∂μ μ dS, μΔυdx = υΔμdx + −υ ∂n ∂n Ω Ω ∂Ω







∂μ where ∂υ ∂n and ∂n are the directional derivatives of υ and μ in the direction of the outward pointing normal n to the surface element dS, respectively.  

Note that lk=1 ∂∂xk Dk ∂∂xk can be regarded as a Laplacian operator which is formally self-adjoint and differential in the L 2 inner product for functions with a Dirichlet boundary or a Neumann boundary. As a result, the following lemma follows directly from Lemma 3.

Lemma 4 Let N1 > 0 and N2 > 0 be a pair of diagonal matrices. Then 

∂m T (s, x)  ∂ N1 ∂t ∂ xk l

Ω



k=1

 ∂m(t, x) Dk dx ∂ xk

   l  ∂m(t, x) ∂ ∂ = Dk dx, m (t, x)N1 ∂ xk ∂ xk ∂t Ω 

T

k=1



∂ p T (s, x) ∂t

Ω

N2

k=1

 =

l 

p T (t, x)N2

Ω

∂ ∂ xk



Dk∗

 ∂ p(t, x) dx ∂ xk

   l  ∂ p(t, x) ∂ ∂ Dk∗ dx. ∂ xk ∂ xk ∂t k=1

Lemma 5 (Reciprocally Convex Combination Lemma) [25] Let f 1 , f 2 , . . . , f N : R m → R have positive finite values in an open subset D of R m . Then the reciprocally convex combination of f i over D satisfies  1   f i (t) = f i (t) + max gi, j (t) gi, j (t) αi i αi =1}

min

{αi :αi >0,

i

(6)

i = j

i

subject to  gi j : R m → R, g j,i (t) = gi, j (t),

 f i (t) gi, j (t) ≥ 0. gi, j (t) f j (t)

(7)

3 Stability Criteria In this section, we will investigate a stability criterion for GRN (4) under Dirichlet boundary conditions.

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Theorem 1 For given scalars τ , σ , μ1 and μ2 satisfying (2), the trivial solution of GRN (4) under Dirichlet boundary conditions is asymptotically stable if there exist matrices Q iT = Q i > 0 (i = 1, . . . , 5) and R Tj = R j > 0 ( j = 1, 2), diagonal matrices P j > 0, Λ j > 0, N j > 0 ( j = 1, 2), and matrices G 1 and G 2 of appropriate sizes, such that the following linear matrix inequalities (LMIs) hold: 

 R1 G 1 ≥ 0, G 1T R1 ⎡ Ξ11 Ξ12 Ξ := ⎣ ∗ Ξ22 ∗ ∗



R2 G 2 G 2T R2 ⎤

 ≥ 0,

Ξ13 Ξ23 ⎦ < 0, Ξ33

where ⎡

⎤ Φ1 G 1T R1 − G 1T Ξ11 = ⎣ ∗ −Q 2 − R1 R1 − G 1 ⎦ , ∗ ∗ Φ2 ⎡ ⎤ 0 0 0 0 0 0 0⎦, Ξ12 = ⎣ 0 T B P2 0 0 0 ⎡ ⎤ P1 W −A T N1 0 0 0 ⎦, Ξ13 = ⎣ 0 T 0 0 B N2 ⎤ ⎡ T G2 R2 − G 2T K Λ1 Φ3 ⎥ ⎢ ∗ −Q 4 − R2 R2 − G 2 0 ⎥, Ξ22 = ⎢ ⎦ ⎣ ∗ 0 ∗ Φ4 ∗ ∗ ∗ Q 5 − 2Λ1 ⎤ ⎡ T 0 0 −C N2 ⎥ ⎢ 0 0 0 ⎥, Ξ23 = ⎢ ⎦ ⎣ K Λ2 0 0 0 0 0 ⎡ ⎤ W T N1 0 Φ5 ⎦, Ξ33 = ⎣ ∗ τ 2 R1 − 2N1 0 ∗ ∗ σ 2 R2 − 2N2 1 Φ1 = − π 2 P1 D L − 2P1 A + Q 1 + Q 2 − R1 , 2 Φ2 = (μ1 − 1)Q 1 − 2R1 + G 1 + G 1T , 1 Φ3 = − π 2 P2 D ∗L − 2P2 C + Q 3 + Q 4 − R2 , 2 Φ4 = (μ2 − 1)Q 3 − 2R2 + G 2 + G 2T , Φ5 = (μ2 − 1)Q 5 − 2Λ2 ,

(8)

(9)

Circuits Syst Signal Process

 l l  D1k  D L = diag , L 2k k=1 k=1  l l  D∗  1k ∗ , D L = diag L 2k k=1 k=1

l  D2k Dnk , . . . , 2 Lk L 2k k=1 ∗ D2k

L 2k

l ∗  Dnk

,...,

k=1

 , 

L 2k

,

∗ , A, B, C, W and K are the same with previous ones. and L k , Dik , Dik

Proof Define a Lyapunov–Krasovskii functional candidate for GRN (4) as V (t, m, p) =

4 

Vi (t, m, p),

i=1

where

 V1 (t, m, p) =



Ω

+ +

m T (t, x)P1 m(t, x)dx +

l   k=1 Ω

l  

∂ p T (t, x) ∂ p(t, x) N2 Dk∗ dx, ∂ xk ∂ xk

k=1 Ω t



+

m T (s, x)Q 1 m(s, x)dsdx

t−τ (t)  t

Ω

Ω

p T (t, x)P2 p(t, x)dx

∂m T (t, x) ∂m(t, x) N 1 Dk dx ∂ xk ∂ xk

  V2 (t, m, p) =

Ω

m T (s, x)Q 2 m(s, x)dsdx

t−τ t

  + 

Ω

t−σ (t)  t

+ Ω   t

V3 (t, m, p) =

Ω

p T (s, x)Q 4 p(s, x)dsdx,

t−σ

f T ( p(s, x))Q 5 f ( p(s, x))dsdx,

t−σ (t)  0 t



V4 (t, m, p) =τ

Ω

−τ

 

+σ

p T (s, x)Q 3 p(s, x)dsdx

Ω

∂m T (s, x) ∂m(s, x) R1 dsdθ dx ∂s ∂s

t+θ 0  t −σ

t+θ

∂ p T (s, x) ∂ p(s, x) R2 dsdθ dx. ∂s ∂s

Then, computing the derivatives of Vi (t, m, p) (i = 1, 2, 3, 4) along the solution of GRN (4), we can get ∂ V1 (t, m, p) =2 ∂t



 m (t, x)P1 T

Ω

  l  ∂ ∂m(t, x) Dk ∂ xk ∂ xk k=1

Circuits Syst Signal Process

−Am(t, x) + W f ( p(t − σ (t), x))] dx  l   ∂  ∂ p(t, x)  T Dk∗ p (t, x)P2 +2 ∂ xk ∂ xk Ω k=1

−C p(t, x) + Bm(t − τ (t), x)] dx   l   ∂m(t, x) ∂m T (t, x) ∂ dx N 1 Dk +2 ∂ xk ∂ xk ∂t Ω k=1

l  

+2 ∂ V2 (t, m, p) = ∂t

k=1 Ω



∂ p T (t, x) ∂ N2 Dk∗ ∂ xk ∂ xk



 ∂ p(t, x) dx. ∂t

(10)

m T (t, x)(Q 1 + Q 2 )m(t, x)dx Ω  − m T (t − τ , x)Q 2 m(t − τ , x)dx Ω  − (1 − τ˙ (t)) m T (t − τ (t), x)Q 1 m(t − τ (t), x)dx Ω  + p T (t, x)(Q 3 + Q 4 ) p(t, x)dx Ω − p T (t − σ , x)Q 4 p(t − σ , x)dx Ω  − (1 − σ˙ (t)) p T (t − σ (t), x)Q 3 p(t − σ (t), x)dx, (11) Ω  ∂ V3 (t, m, p) = − (1 − σ˙ (t)) f T ( p(t − σ (t), x))Q 5 f ( p(t − σ (t), x))dx ∂t Ω  T + f ( p(t, x))Q 5 f ( p(t, x))dx, (12) Ω

∂ V4 (t, m, p) = − τ ∂t

  Ω



t

t−τ

∂m T (s, x) ∂m(s, x) R1 dsdx ∂s ∂s

∂m T (t, x) ∂m(t, x) R1 dx ∂t ∂t Ω   t ∂ p T (s, x) ∂ p(s, x) −σ R2 dsdx ∂s ∂s Ω t−σ  ∂ p T (t, x) ∂ p(t, x) R2 dx. + σ2 ∂t ∂t Ω + τ2

From Green formula, Dirichlet boundary conditions and Lemma 2, we have

2

l   k=1 Ω

m T (t, x)P1

∂ ∂ xk

 Dk

 ∂m(t, x) dx ∂ xk

(13)

Circuits Syst Signal Process

=2

l   k=1 Ω

l  

−2 =2

∂ ∂ xk

k=1 Ω

  ∂m(t, x) m T (t, x)P1 Dk dx ∂ xk

∂m T (t, x) ∂m(t, x) P1 Dk dx ∂ xk ∂ xk

  ∂m(t, x) l m T (t, x)P1 Dk · n ds ∂ xk ∂Ω k=1

l   k=1

l  

−2

k=1 Ω

= −2 ≤−

l  

k=1 Ω  π2

2

Ω

∂m T (t, x) ∂m(t, x) P1 Dk dx ∂ xk ∂ xk ∂m T (t, x) ∂m(t, x) P1 Dk dx ∂ xk ∂ xk

m T (t, x)P1 D L m(t, x)dx,

(14)

where

    ∂m(t, x) l ∂m T (t, x) ∂m(t, x) T T m (t, x)P1 Dk = m(t, x)P1 D1 , . . . , m (t, x)P1 Dl . ∂ xk ∂ x1 ∂ xl k=1

Similarly,  2

Ω

p T (t, x)P2

   l  π2 ∂ ∂ p(t, x) Dk∗ dx ≤ − p T (t, x)P2 D ∗L p(t, x)dx. ∂ xk ∂ xk 2 Ω k=1

(15) The combination of (10), (14), (15) gives   π2 ∂ T V1 (t, m, p) = 2 m (t, x)P1 − D L m(t, x) ∂t 4 Ω −Am(t, x) + W f ( p(t − σ (t), x))] dx   π2 p T (t, x)P2 − D ∗L p(t, x) +2 4 Ω − C p(t, x) + Bm(t − τ (t), x)] dx   l   ∂m(t, x) ∂m T (t, x) ∂ dx N 1 Dk +2 ∂ xk ∂ xk ∂t Ω k=1

+2

l   k=1 Ω

∂ p T (t, x) ∂ N2 Dk∗ ∂ xk ∂ xk



 ∂ p(t, x) dx. ∂t

(16)

Circuits Syst Signal Process

On the other side, from Lemmas 1 and 5, it follows that   t ∂m T (s, x) ∂m(s, x) R1 dsdx −τ ∂s ∂s Ω t−τ   t−τ (t) ∂m T (s, x) ∂m(s, x) = −τ R1 dsdx ∂s ∂s Ω t−τ   t ∂m T (s, x) ∂m(s, x) R1 dsdx −τ ∂s ∂s Ω t−τ (t)    R1 G 1 ≤ ζ1 dx, −ζ1T T G Ω 1 R1

(17)

where ζ1 = [m T (t − τ (t), x) − m T (t − τ , x) m T (t, x) − m T (t − τ (t), x)]T . Similarly, −σ

 

t

Ω

t−σ

= −σ

  Ω

 

∂ p T (s, x) ∂ p(s, x) R2 dsdx ∂s ∂s t−σ (t)

t−σ t

∂ p T (s, x) ∂ p(s, x) R2 dsdx ∂s ∂s

∂ p T (s, x) ∂ p(s, x) R2 dsdx ∂s ∂s Ω t−σ (t)    R2 G 2 ≤ ζ dx, −ζ2T G 2T R2 2 Ω −σ

(18)

where ζ2 = [ p T (t − σ (t), x) − p T (t − σ , x) p T (t, x) − p T (t − σ (t), x)]T . Finally, for the diagonal matrices Λ1 > 0 and Λ2 > 0, the following inequalities (19) and (20) can be obtained from (5). (19) 2 f T ( p(t, x))Λ1 f ( p(t, x)) − 2 p T (t, x)K Λ1 f ( p(t, x)) ≤ 0, 2 f T ( p(t − σ (t), x))Λ2 f ( p(t − σ (t), x)) −2 p T (t − σ (t), x)K Λ2 f ( p(t − σ (t), x)) ≤ 0. For diagonal matrices N1 > 0 and N2 > 0, it is easy to see that  l   ∂  ∂m(t, x)  ∂m T (s, x) Dk 2 N1 ∂t ∂ xk ∂ xk Ω k=1  ∂m(t, x) dx = 0 − Am(t, x) + W f ( p(t − σ (t), x) − ∂t

(20)

(21)

and  2

Ω

 l  ∂  ∂ p(t, x)  ∂ p T (s, x) Dk∗ N2 ∂t ∂ xk ∂ xk k=1

− C p(t, x) + Bm(t − τ (t), x) −

 ∂ p(t, x) dx = 0. ∂t

(22)

Circuits Syst Signal Process

According to Lemma 4, Green formula and Dirichlet boundary conditions, we have  2

∂m T (s, x)  ∂ N1 ∂t ∂ xk l

Ω

 Dk

k=1

 ∂m(t, x) dx ∂ xk

   l  ∂m(t, x) ∂ ∂ =2 Dk dx m (t, x)N1 ∂ xk ∂ xk ∂t Ω 

T

k=1

l  

= −2

∂m T (t, x)

k=1 Ω

∂ xk

∂ N 1 Dk ∂ xk



 ∂m(t, x) dx. ∂t

(23)

Similarly,  2

Ω

∂ p T (s, x)  ∂ N2 ∂t ∂ xk l

k=1

 =2

Ω

= −2

p T (t, x)N2



Dk∗

 ∂ p(t, x) dx ∂ xk

   l  ∂ p(t, x) ∂ ∂ Dk∗ dx ∂ xk ∂ xk ∂t k=1

l   k=1 Ω

∂ p T (t, x) ∂ N2 Dk∗ ∂ xk ∂ xk



 ∂ p(t, x) dx. ∂t

(24)

Combining (11)∼(13) and (16)∼ (24) results in  ∂ ∂ V (t, m, p) = Vi (t, m, p) ∂t ∂t i=1  ς T (t, x)Ξ ς (t, x)dx ≤ 4

Ω

≤ −λmin (−Ξ )(m(t, x)2 +  p(t, x)2 ) ≤ 0,

(25)

where

 ς (t, x) = m T (t, x) m T (t − τ , x) m T (t − τ (t), x) p T (t, x) p T (t − σ , x) p T (t − σ (t), x) f T ( p(t, x)) f T ( p(t − σ (t), x))

∂m T (t,x) ∂ p T (t,x) ∂t ∂t

T

.

From (25), it follows that ∂ V (t, m, p) ≤ −λmin (−Ξ )m(t, x)2 , ∂t ∂ V (t, m, p) ≤ −λmin (−Ξ ) p(t, x)2 . ∂t Integrating two sides of the inequalities in (26) from 0 to t, we have

(26)

Circuits Syst Signal Process



t

0



t

0

 t ∂ −λmin (−Ξ )m(s, x)2 ds, V (s, m, p)ds ≤ ∂s 0  t ∂ −λmin (−Ξ ) p(s, x)2 ds. V (s, m, p)ds ≤ ∂s 0

(27)

Namely, 

t

V (t, m, p) ≤ 

−λmin (−Ξ )m(s, x)2 ds + V (0, m(0, x), p(0, x)),

0 t

V (t, m, p) ≤

−λmin (−Ξ ) p(s, x)2 ds + V (0, m(0, x), p(0, x)).

(28)

0

Therefore, m(t, x)2 → 0,  p(t, x)2 → 0(t → ∞). Namely, m(t, x) → 0, p(t, x) → 0(t → ∞). From (28), we get V (t, m, p) ≤ V (0, m(0, x), p(0, x)).

(29)

Noting that φi (s, x), φi∗ (s, x) ∈ C 1 ([−d, 0] × Ω, R), namely, there exist nonnegative real numbers α, α ∗ , β and β ∗ such that       ∂φi (s, x)     ≤ α,  ∂φi (s, x)  ≤ α ∗ ,    ∂t ∂ xk   ∗  ∗     ∂φi (s, x)     ≤ β,  ∂φi (s, x)  ≤ β ∗ .    ∂t ∂ xk  Therefore, we can get n   i=1

Ω

N1i

l 

 Dik

k=1

∂m i (0, x) ∂ xk

2 dx ≤ mes(Ω)

n  l 

N1i Dik (α ∗ )2

(30)

i=1 k=1

and τ

 

0

Ω

−τ



1 ∂m T (s, x) ∂m(s, x) R1 dsdθ dx ≤ λmax (R1 )α 2 τ 3 mes(Ω). ∂s ∂s 2

0 θ

(31)

Similarly, n   i=1

Ω

N2i

l  k=1

∗ Dik



∂ pi (0, x) ∂ xk

2 dx ≤ mes(Ω)

n  l 

∗ N2i Dik (β ∗ )2

(32)

i=1 k=1

and   σ

Ω

0

−σ

 θ

0

1 ∂ p T (s, x) ∂ p(s, x) R2 dsdθ dx ≤ λmax (R2 )β 2 σ 3 mes(Ω). ∂s ∂s 2

(33)

Circuits Syst Signal Process

Obviously, there exist nonnegative real numbers M1 and M2 , such that the following inequalities hold  mes(Ω)  mes(Ω)

n  l  i=1 k=1 n  l 

1 N1i Dik (α ∗ )2 + λmax (R1 )α 2 τ 3 2 ∗ N2i Dik (β ∗ )2

i=1 k=1



1 + λmax (R2 )β 2 σ 3 2

= M1 φ(t, x)2d ,

(34)

 = M2 φ ∗ (t, x)2d . (35)

Therefore, we have  V (0, m(0, x), p(0, x)) =

m T (0, x)P1 m(0, x)dx Ω  + p T (0, x)P2 p(0, x)dx +

Ω n 

+

N1i

Ω

i=1

  +

l 

N2i

Ω

k=1

0

−τ (0) 0

Ω

 Dik

k=1

n   i=1

l 

∗ Dik



∂m i (0, x) ∂ xk ∂ pi (0, x) ∂ xk

2 dx

2 dx

m T (s, x)Q 1 m(s, x)dsdx

  +

−τ  0

Ω

−σ (0)  0 T

Ω

−σ  0

 +

 +

 +

m T (s, x)Q 2 m(s, x)dsdx

Ω

p (s, x)Q 4 p(s, x)dsdx

 Ω

−τ 0

  +σ

f T ( p(s, x))Q 5 f ( p(s, x))dsdx

−σ (0)  0 0

Ω

+τ

p T (s, x)Q 3 p(s, x)dsdx

Ω

−σ

θ



θ

0

∂m T (s, x) ∂m(s, x) R1 dsdθ dx ∂s ∂s ∂ p T (s, x) ∂ p(s, x) R2 dsdθ dx ∂s ∂s

≤λ11 φ(t, x)2d + λ12 φ ∗ (t, x)2d ,

(36)

where λ11 = λmax (P1 ) + τ¯ λmax (Q 1 ) + τ¯ λmax (Q 2 ) + M1 , λ12 = λmax (P2 ) + σ¯ λmax (Q 3 ) + σ¯ λmax (Q 4 ) + σ¯ λmax (Q 5 )λmax (K T K ) + M2 .

Circuits Syst Signal Process

On the other hand, V (t, m, p) ≥ λmin (P1,2 )m(t, x)2 ,

(37)

V (t, m, p) ≥ λmin (P1,2 ) p(t, x) ,

(38)

2

where λmin (P1,2 ) denotes the minimum eigenvalue of diag(P1 , P2 ). It follows from (29), (36), (37) and (38) that we have m(t, x)2 ≤

λ11 φ(t, x)2d + λ12 φ ∗ (t, x)2d , λmin (P1,2 )

 p(t, x)2 ≤

λ11 φ(t, x)2d + λ12 φ ∗ (t, x)2d . λmin (P1,2 )

For any ε > 0, there exists δ := min{

ελmin (P1,2 ) ελmin (P1,2 ) , 2λ12 }, 2λ11

(39)

such that

m(t, x)2 ≤ ε,  p(t, x)2 ≤ ε when φ(t, x)2d ≤ δ(ε), φ ∗ (t, x)2d ≤ δ(ε). Hence, by Definition 1, it is easy to see that the trivial solution of GRN (4) under Dirichlet boundary conditions is asymptotically stable. The proof is completed.   Remark 1 It should be emphasized that the last two terms in V1 (t, m, p) are introduced to remove the restrictions μ1 < 1 and μ2 < 1 required in [10,23]. Remark 2 To establish a less conservative stability criterion for GRN (4), Lemmas 1 and 2 are used to deal with the integral terms  Ω

m T (t, x)P1

  l  ∂ ∂m(t, x) Dk dx ∂ xk ∂ xk k=1

and   Ω

t

t−τ

∂m T (s, x) ∂m(s, x) R1 dsdx, ∂s ∂s

respectively. Remark 3 The so-called reciprocally convex approach, proposed in [25], is used to deal with the term   t ∂m T (s, x) ∂m(s, x) R1 dsdx, ∂s ∂s Ω t−τ which reduces the conservativeness of the resulting stability criterion.

Circuits Syst Signal Process

The stability criterion for the trivial solution of GRN (4) under Neumann boundary conditions is given by the following theorem. Theorem 2 For given scalars τ , σ , μ1 and μ2 satisfying (2), the trivial solution of GRN (4) under Neumann boundary conditions is asymptotically stable if there exist matrices Q iT = Q i > 0 (i = 1, . . . , 5) and R Tj = R j > 0 ( j = 1, 2), diagonal matrices P j > 0, Λ j > 0, N j > 0 ( j = 1, 2), and matrices G 1 and G 2 of appropriate sizes, such that the following LMIs holds     R1 G 1 R2 G 2 ≥ 0, ≥ 0, (40) G 1T R1 G 2T R2 ⎡ ⎤ 11 Ξ12 Ξ13 Ξ := ⎣ ∗ Ξ 22 Ξ23 ⎦ < 0, Ξ (41) ∗ ∗ Ξ33 where ⎡

⎤ 1 Φ G 1T R1 − G 1T 11 = ⎣ ∗ −Q 2 − R1 R1 − G 1 ⎦ , Ξ ∗ ∗ Φ2 ⎤ ⎡ T 3 Φ G2 R2 − G 2T K Λ1 ⎥ ⎢ 0 ⎥, 22 = ⎢ ∗ −Q 4 − R2 R2 − G 2 Ξ ⎦ ⎣ ∗ 0 ∗ Φ4 ∗ ∗ ∗ Q 5 − 2Λ1 1 = −2P1 A + Q 1 + Q 2 − R1 , Φ 3 = −2P2 C + Q 3 + Q 4 − R2 , Φ Ξ12 , Ξ13 , Ξ23 , Ξ33 , A, C and K are the same with ones used in Theorem 1. Proof We use the Lyapunov–Krasovskii functional V (t, m, p) from Theorem 1. From Green formula and Neumann boundary conditions, we have 2

l   k=1 Ω

=2

l   k=1 Ω

−2 =2

m T (t, x)P1 ∂ ∂ xk

l   k=1 Ω

∂ ∂ xk

 Dk

 ∂m(t, x) dx ∂ xk

  ∂m(t, x) m T (t, x)P1 Dk dx ∂ xk

∂m T (t, x) ∂m(t, x) P1 Dk dx ∂ xk ∂ xk

  ∂m(t, x) l m T (t, x)P1 Dk · n ds ∂ xk ∂Ω k=1

l   k=1

−2

l   k=1 Ω

∂m T (t, x) ∂m(t, x) P1 Dk dx ∂ xk ∂ xk

Circuits Syst Signal Process

= −2

l   k=1 Ω

∂m T (t, x) ∂m(t, x) P1 Dk dx ∂ xk ∂ xk

≤ 0. Similarly, 2

 Ω

(42)

p T (t, x)P2

l

k=1

∂ ∂ xk



 Dk∗ ∂ p(t,x) dx ≤ 0. ∂ xk

(43)

Based on (42) and (43), the proof follows in a manner similar to that for Theorem 1.   Remark 4 The numbers of decision variables in the LMI conditions in [23, Theorem 2] and Theorem 2 are 4.5n 2 + 10.5n and 5.5n 2 + 9.5n, respectively, and hence the computational complexity of Theorem 2 is slightly greater than that of [23, Theorem 2]. However, the conservativeness of Theorem 2 is certainly less than [23, Theorem 2] (see Remark 8 below). Remark 5 Owing to Lemma 2, the information about the reaction–diffusion terms is remained in the stability criterion presented in Theorem 1. However, because Lemma 2 is invalid under Neumann boundary conditions, the information about reaction– diffusion terms cannot be included into the stability criterion given in Theorem 2. Remark 6 The stability conditions in Theorems 1 and 2 are given in the form of LMIs. Thus, they can be easily verified using MATLAB, YALMIP [21] or SeDuMi 1.3 [35]. 4 Theoretical Comparisons In this section, we further illustrate that the stability criterion given in Theorem 1 is less conservative than those in [23, Theorem 1] and [10, Theorem 1]. To this end, we first state [23, Theorem 1] and [10, Theorem 1]: Theorem 3 [10, Theorem 1] For given scalars τ , σ , μ1 and μ2 satisfying 0 ≤ τ (t) ≤ τ , τ˙ (t) ≤ μ1 < 1, 0 ≤ σ (t) ≤ σ , σ˙ (t) ≤ μ2 < 1,

(44)

GRN (4) under Dirichlet boundary conditions is asymptotically stable if there exist T = Q i > 0 (i = 1, 2, 3) and diagonal matrices P j > 0, Λ j > 0 matrices Q i ( j = 1, 2), such that the following LMI (45) is feasible.  Ξ :=

Ξ1 0

0 Ξ2

 < 0,

where ⎡

Φ1 Ξ1 = ⎣ 0 (P1 W )T

⎤ 0 P1 W Φ2 K Λ2 ⎦ , T (K Λ2 ) Φ3

(45)

Circuits Syst Signal Process

⎡

⎤ Φ 4 (P2 B)T 0 Ξ2 = ⎣ P2 B Φ 5 K Λ1 ⎦ , T 0 (K Λ1 ) Q 3 − 2Λ1 1 2 1 , Φ 2 = (μ2 − 1) Q 2 , Φ 1 = − π P1 D L − 2P1 A + Q 2 3 − 2Λ2 , Φ 4 = (μ1 − 1) Q 1 , Φ 3 = (μ2 − 1) Q 1 2 2 , Φ 5 = − π P2 D ∗L − 2P2 C + Q 2  l  l l  D1k   D2k Dnk , ,..., D L = diag , L 2k k=1 L 2k L 2k k=1 k=1  l  l l ∗ ∗  D∗   D D 1k 2k nk , ,..., D ∗L = diag . 2 2 2 L L L k k k k=1 k=1 k=1 Theorem 4 [23, Theorem 1] For given scalars τ , σ , μ1 and μ2 satisfying (44), GRN (4) is asymptotically stable under Dirichlet boundary conditions if there exist matrices Q iT = Q i > 0 (i = 1, . . . , 5), RiT = Ri > 0, TiT = Ti > 0 (i = 1, 2), and diagonal matrices P j > 0, Λ j > 0 ( j = 1, 2), such that the following LMIs (46) and (47) are feasible. ⎡

χ1 ⎢∗ ⎢ Ξ1 = ⎢ ⎢∗ ⎣∗ ∗ ⎡ χ4 ⎢∗ ⎢ Ξ2 = ⎢ ⎢∗ ⎣∗ ∗

0 −Q 3 ∗ ∗ ∗

0 P1 W 0 0 (μ2 − 1)Q 2 K Λ2 ∗ χ3 ∗ ∗

(P2 B)T χ2 ∗ ∗ ∗

0 0 −Q 4 ∗ ∗

0 K Λ1 0 χ5 ∗

⎤ T1 −T1 ⎥ ⎥ ⎥ < 0, 0 ⎥ 0 ⎦ − τ1 R1 ⎤

0 T2 ⎥ ⎥ −T2 ⎥ ⎥ < 0, 0 ⎦ − σ1 R2

where χ1 = −2P1 D L − 2P1 A + Q 1 + Q 3 + τ R1 , χ2 = −2P2 D ∗L − 2P2 C + Q 2 + Q 4 + σ R2 , χ3 = (μ2 − 1)Q 5 − 2Λ2 ,  l l  D1k  , D L = diag L 2k k=1 k=1  l l  D∗  1k ∗ , D L = diag L 2k k=1 k=1

χ4 = (μ1 − 1)Q 1 , χ5 = Q 5 − 2Λ1 ,  l  D2k Dnk ,..., , L 2k L 2k k=1  l ∗ ∗  D2k Dnk ,..., . L 2k L 2k k=1

(46)

(47)

Circuits Syst Signal Process

To show that the stability criterion given in Theorem 3 (that is, [10, Theorem 1]) is less conservative than that in Theorem 4 (that is, [23, Theorem 1]), Han and Zhang [10] investigated the following theorem. Theorem 5 [10, Theorem 3] If the LMIs (46) and (47) are feasible, then the LMI (45) is feasible. Now we are in a position to state the following theorem, which claims that the stability criterion obtained in Theorem 1 is less conservative than that in [10, Theorem 1], and hence also less conservative than that in [23, Theorem 1] by Theorem 5. Theorem 6 If the LMI (45) is feasible, then the LMIs (8) and (9) are feasible. Proof The LMI (45) can be rewritten as  Ξ :=

Ξ11 Ξ12 T Ξ Ξ12 22

 < 0,

where ⎡

⎡ ⎤ Φ1 0 0 0 Ξ11 = ⎣ 0 Φ 4 (P2 B)T ⎦ , Ξ12 = ⎣ 0 0 0 P2 B Φ5 ⎤ ⎡ Φ2 0 K Λ2 3 − 2Λ1 Ξ22 = ⎣ 0 Q 0 ⎦. T 0 Φ3 (K Λ2 )

0 0 K Λ1

⎤ P1 W 0 ⎦, 0

Then there exist sufficiently small positive numbers εi (i = 1, 2, 5, 6) such that ⎡

11 Ξ ⎢Ξ T ⎢ 12  := ⎢ 0 Ξ ⎢ ⎣Ξ T 14 T Ξ 15

12 Ξ 22 Ξ T Ξ 23 0 T Ξ 25

14 0 Ξ  Ξ23 0 33 Ξ 34 Ξ T Φ 3 Ξ 34 45 0 Ξ

⎤ 15 Ξ 25 ⎥ Ξ ⎥ 0 ⎥ ⎥ < 0, 45 ⎦ Ξ 55 Ξ

(48)

where ⎡

⎡ ⎡ ⎤ ⎤ ⎤ 0 P1 W Φ 1 + ε1 I 0 0 11 = ⎣ 12 = ⎣ 0 ⎦ , Ξ 14 = ⎣ 0 ⎦ , ∗ −ε1 I 0 ⎦ , Ξ Ξ TP 0 B ∗ ∗ Φ4 2 ⎤ ⎡ −ε5 A T 0 ⎣ 22 = Φ 5 + ε2 I, Ξ  23 = [ 0 0 K Λ1 ], 0 0 ⎦, Ξ Ξ15 = T 0 ε6 B ⎡ ⎡ ⎤ ⎤ −ε2 I 0 0 0 ⎦, Ξ 25 = [ 0 −ε6 C T ], Ξ 33 = ⎣ ∗ Φ 2 34 = ⎣ K Λ2 ⎦ , Ξ 0 0 ∗ ∗ Q 3 − 2Λ1

Circuits Syst Signal Process

  0 45 = [ ε5 W T 0 ], Ξ 55 = −2ε5 I Ξ . 0 −2ε6 I Furthermore, there exist sufficiently small positive numbers εi (i = 3, 4) such that ⎡ Ξ11 T ⎢Ξ ⎢ 12 Ξˇ := ⎢ ⎢ 0T ⎣Ξ  14 T Ξ 15

12 Ξ 22 Ξ T Ξ 23 0 T Ξ 25

0 23 Ξ 33 Ξ 34 Ξ 0

14 Ξ 0 34 Ξ Φ3 45 Ξ

15 ⎤ Ξ 25 ⎥ Ξ ⎥ 0 ⎥ ⎥ < 0, 45 ⎦ Ξ Ξˇ 55

(49)

where   0 −2ε5 I + τ 2 ε3 I Ξˇ 55 = . 0 −2ε6 I + σ 2 ε4 I Due to ⎡

⎤ −ε3 I 0 ε3 I ⎣ 0 −ε3 I ε3 I ⎦ ≤ 0 ε3 I ε3 I −2ε3 I from (49), we have

⎡ and

⎤ −ε4 I 0 ε4 I ⎣ 0 −ε4 I ε4 I ⎦ ≤ 0, ε4 I ε4 I −2ε4 I

⎤ Ξ 11 Ξ 12 Ξ 13 Ξ := ⎣ ∗ Ξ 22 Ξ 23 ⎦ < 0, ∗ ∗ Ξ 33 ⎡

(50)

where ⎡

⎤ −ε3 I 0 ε3 I 11 + ⎣ 0 −ε3 I ε3 I ⎦ , Ξ 11 = Ξ ε3 I ε3 I −2ε3 I     15 ,  14 Ξ Ξ 12 = Ξ12 0 , Ξ 13 = Ξ ⎡ 0 ε4 I −ε4 I   ⎢ 0 23 22 Ξ Ξ I ε4 I −ε 4 +⎢ Ξ 22 = T  ⎣ ε4 I ε4 I −2ε4 I Ξ23 Ξ33 0 0 0      25 Ξ45 Φ 0 Ξ . , Ξ 23 =  Ξ 33 = T3 Ξ34 0 Ξ45 Ξˇ 55

⎤ 0 0⎥ ⎥, 0⎦ 0

Set G 1 = 0, G 2 = 0, Q 2 = ε1 I , Q 4 = ε2 I , N1 = ε5 I , N2 = ε6 I , R1 = ε3 I , 1 , Q 3 = Q 2 , Q 5 = Q 3 . Then the (50) becomes (9), and hence the R2 = ε4 I , Q 1 = Q LMIs (8) and (9) are feasible. This completes the proof.   Remark 7 It follows from Theorems 5 and 6 that the stability criterion obtained in this paper is less conservative than those in [10,23]. This will be illustrated by a numerical example in the next section.

Circuits Syst Signal Process

Remark 8 As with the previous discussion, one can easily show that the stability criterion presented in Theorem 2 is less conservative than that given in [23, Theorem 2]. 5 Illustrative Examples To demonstrate the effectiveness of the results obtained in this paper, we give two numerical examples in this section. Example 1 Consider the reaction–diffusion-delayed GRN (4) with the following parameters: A = diag(0.2, 1.1, 1.2), B = diag(1.0, 0.4, 0.7), C = diag(0.3, 0.7, 1.3), L 1 = L 2 = L 3 = 1, ⎡ ⎤ 0 0 −0.5 0 ⎦, W = ⎣ −0.5 0 0 −0.5 0 D1 = D2 = D3 = diag(0.1, 0.1, 0.1), D1∗ = D2∗ = D3∗ = diag(0.2, 0.2, 0.2). 2

x Let f i (x) = 1+x 2 , i = 1, 2, . . . , n. Then, it is easy to verify that (5) is satisfied when K = 0.65I . Case 1: Dirichlet boundary conditions. In Table 1, when μ1 = μ2 ∈ {0.83, 0.93, 0.94, 0.99, 1}, we list the maximum delay upper bounds obtained by applying [23, Theorem 1], [10, Theorem 1] and Theorem 1 to the above system. From Table 1, it is clear that (i) when μ1 = μ2 ≥ 0.94, the LMI conditions presented in [23, Theorem 1] and [10, Theorem 1] are infeasible, while the LMI conditions in Theorem 1 of this paper are feasible; and (ii) when μ1 = μ2 = 0.83, all LMI conditions discussed here are feasible, and all delay upper bounds from [23, Theorem 1], [10, Theorem 1] and Theorem 1 are ∞. This is because the obtained stability conditions in Theorems [10, Theorem 1] and [23, Theorem 1] are in essence delayindependent, although τ¯ and σ¯ appear in the stability conditions in [23, Theorem 1]. In addition, we conclude from Theorems 5 and 6 that the upper bound obtained from Theorem 1 is certainly not less than those obtained from Theorems [10, Theorem 1] and [23, Theorem 1]. Therefore, the proposed Theorem 1 reduces the conservatism of the stability criteria in [23, Theorem 1] and [10, Theorem 1]. This illustrates the theoretical results presented previously.

Table 1 Upper bounds on τ¯ = σ¯ with different μ1 = μ2 = μ (Case 1) Cases

μ = 0.83

μ = 0.93

μ = 0.94

μ = 0.99

μ=1

[23, Theorem 1]

∞

–

[10, Theorem 1]

∞

∞

–

–

–

–

–

Theorem 1

∞

∞

6.8216

–

4.0050

3.9616

Circuits Syst Signal Process

When τ = σ = 3 and μ1 = μ2 = 3, by using the Toolbox YALMIP in MATLAB to solve the LMIs (8) and (9), we obtain the following feasible solution: ⎡

⎤ 0.0394 −0.0002 0.0011 Q 1 = ⎣ −0.0002 0.0597 0.0006 ⎦ , 0.0011 0.0006 0.0603 ⎡ ⎤ 1.5946 −0.0423 −0.0335 0.0121 ⎦ , Q 2 = ⎣ −0.0423 2.5766 −0.0335 0.0121 3.2462 ⎡ ⎤ 0.0152 −0.0002 0.0021 Q 3 = ⎣ −0.0002 0.0849 0.0012 ⎦ , 0.0021 0.0012 0.0302 ⎡ ⎤ 0.8267 0.0045 0.1032 Q 4 = ⎣ 0.0045 2.6262 0.0303 ⎦ , 0.1032 0.0303 3.3310 ⎡ ⎤ 0.0364 −0.0017 0.0060 Q 5 = ⎣ −0.0017 0.1799 0.0007 ⎦ , 0.0060 0.0007 0.0662 ⎡ ⎤ 0.5962 0.0006 0.0034 R1 = ⎣ 0.0006 0.2501 0.0017 ⎦ , 0.0034 0.0017 0.4758 ⎡ ⎤ 0.1116 −0.0002 −0.0038 0.0024 ⎦ , R2 = ⎣ −0.0002 0.3501 −0.0038 0.0024 0.2572 ⎡ ⎤ −0.4736 0.0026 −0.0089 G 1 = ⎣ −0.0020 −0.1805 −0.0025 ⎦ , 0.0121 −0.0002 −0.3863 ⎡ ⎤ −0.0859 0.0001 0.0132 G 2 = ⎣ −0.0006 −0.2406 −0.0036 ⎦ , −0.0054 0.0009 −0.2215 P1 = diag(2.2603, 2.3562, 3.8594), P2 = diag(0.6655, 1.8248, 2.3676), Λ1 = diag(0.9738, 1.2714, 1.2538), Λ2 = diag(0.6806, 1.6023, 1.7822), N1 = diag(4.5254, 2.2323, 3.8136), N2 = diag(1.0589, 2.8764, 2.2495). Case 2: Neumann boundary conditions. When τ = σ = 2 and μ1 = μ2 = 2, by using the Toolbox YALMIP in MATLAB to solve the LMIs (40) and (41), we obtain the following feasible solution:

Circuits Syst Signal Process

⎡

⎤ 1.0389 0.3407 0.9695 Q 1 = ⎣ 0.3407 0.9294 0.5177 ⎦ , 0.9695 0.5177 7.7951 ⎡ ⎤ 41.7901 2.8719 1.3432 4.9196 ⎦ , Q 2 = ⎣ 2.8719 33.2893 1.3432 4.9196 259.0009 ⎡ ⎤ 0.1642 0.1524 0.8938 Q 3 = ⎣ 0.1524 1.7086 0.8103 ⎦ , 0.8938 0.8103 10.6278 ⎡ ⎤ 3.2325 0.3606 −0.4024 1.2725 ⎦ , Q 4 = ⎣ 0.3606 34.6128 −0.4024 1.2725 306.4690 ⎡ ⎤ 0.4919 0.4291 2.1119 Q 5 = ⎣ 0.4291 4.9452 1.7387 ⎦ , 2.1119 1.7387 20.8385 ⎡ ⎤ 160.2664 −1.7993 −2.7894 R1 = ⎣ −1.7993 10.9976 −0.7374 ⎦ , −2.7894 −0.7374 81.7270 ⎡ ⎤ 13.4466 −0.7363 −2.6117 32.9799 −0.2836 ⎦ , R2 = ⎣ −0.7363 −2.6117 −0.283687.6663 ⎡ ⎤ −76.4554 3.1296 14.3556 −9.6075 0.9731 ⎦ , G 1 = ⎣ 5.2155 7.6107 1.1604 −70.7894 ⎡ ⎤ −5.6264 1.4487 8.7938 0.8851 ⎦ , G 2 = ⎣ 2.1965 −23.6846 3.5598 0.3868 −78.2569 P1 = diag(390.2954, 55.2897, 432.0640), P2 = diag(21.3919, 106.6573, 489.7395), Λ1 = diag(1.7727, 33.1116, 91.1365), Λ2 = diag(12.3035, 92.9105, 467.0758), N1 = diag(420.4140, 46.6980, 301.6333), N2 = diag(62.2517, 135.0285, 331.2812). However, the LMI conditions presented in [23, Theorem 2] are infeasible. Therefore, the stability criterion obtained in Theorem 2 is less conservative than that in [23, Theorem 2]. On the other hand, the numbers of decision variables in [23, Theorem 2] and Theorem 2 are 72 and 78, respectively. Hence, the computational complexity of Theorem 2 is slightly greater than that of [23, Theorem 2]. To further demonstrate the effectiveness of our results, we now give another example.

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Example 2 When l = n = 1, GRN (4) simplifies to ⎧   ⎨ ∂m(t,x) = ∂ D1 ∂m(t,x) − Am(t, x) + W f ( p(t − σ (t), x)), ∂t ∂x  ∂x  ⎩ ∂ p(t,x) = ∂ D ∗ ∂ p(t,x) − C p(t, x) + Bm(t − τ (t), x). 1 ∂x ∂t ∂x

(51)

We choose the values of parameters in (51) as follows, A = 0.2,

B = 1, C = 0.3,

W = −0.5,

D1 = 0.1,

D1∗

L 1 = 1, = 0.2.

When μ1 = μ2 = 2, K = 0.65 and τ = σ = 1, for Dirichlet boundary conditions, by using the Toolbox YALMIP in MATLAB to solve the LMIs (8) and (9), we obtain the following feasible solution: Q 1 = 0.1970, Q 2 = 1.8428, Q 3 = 0.1410, Q 4 = 0.9088, Q 5 = 0.5139, R1 = 3.0854, R2 = 1.5898, G 1 = −0.5888, G 2 = −0.2178, P1 = 6.0303, P2 = 1.7936, Λ1 = 1.5442, Λ2 = 3.2518, N1 = 3.1152, N2 = 1.6924. However, the LMI conditions presented in [23, Theorem 1] and [10, Theorem 1] are infeasible. Further, when σ (t) = τ (t) ∈ {1, 5}, the state responses of GRN (51) are given in Figures 1, 2, 3 and 4. It is seen from Figures 1, 2, 3 and 4 that GRN (51) is

0.2

mRNA concentration

0.15 0.1 0.05 0 −0.05 −0.1 −0.15 30 25

4

20 3

15

t

2

10 1

5 0

Fig. 1 The trajectory of m(t, x)

0

x

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0.5

protein concentration

0.4 0.3 0.2 0.1 0 −0.1 −0.2 30 25

4

20

t

3

15 2

10

x

1

5 0

0

Fig. 2 The trajectory of p(t, x)

1

0.5

0

−0.5

−1 30 4

20 3

protein concentration

2

10 1 0

Fig. 3 The trajectory of m(t, x)

0

t

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0.4

0.2

0

−0.2

−0.4 30 4

20 3

mRNA concentration

2

10 1 0

t

0

Fig. 4 The trajectory of p(t, x)

stable when σ (t) = τ (t) = 1 and is instable when σ (t) = τ (t) = 5. Thus, larger delays may lead to instability of GRNs.

6 Conclusions This paper investigates asymptotic stability criteria for the trivial solution of the delayed GRN (4) with reaction–diffusion terms under Dirichlet boundary conditions and Neumann boundary conditions, respectively. By constructing a new Lyapunov functional and using several existing results (Jensen’s inequality, Wirtinger’s inequality, Green’s second identity and the reciprocally convex combination lemma), we established asymptotic stability criteria that are theoretically demonstrated to be less conservative than the corresponding criteria in [10,23]. On the one hand, Theorems 1 and 2 in this paper remove the restrictions μ1 < 1 and μ2 < 1 required in [10,23] and thus extend the range of application of the theoretical results. On the other hand, comparing Theorems 1 and 2 of this paper, Theorem 1 retains information about the reaction–diffusion terms, whereas Theorem 2 does not. The results of two numerical examples illustrate the correctness and effectiveness of the theoretical results of this paper. It is worth mentioning that various related problems fall under our research interests. At present, fuzzy logic control has attracted a great deal of attention as a simple and effective approach to controlling certain complex nonlinear systems [2,24,26–28].

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Hence, extending the results in this paper to T-S fuzzy systems with distributed delays or stochastic delays will be left for future work. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (11371006), the National Natural Science Foundation of Heilongjiang Province (F201326, A201416), the fund of Heilongjiang Province Innovation Team Support Plan (2012TD007), the Fund of Key Laboratory of Electronics Engineering, College of Heilongjiang Province,(Heilongjiang University), P.R. China, and the Fund of Heilongjiang Education Committee (12541603). The authors thank the anonymous referees for their helpful comments and suggestions, which greatly improved this paper. The authors also thank Dr. Jinliang Wang at Heilongjiang University for checking for grammatical and compositional errors in the previous version.

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