This paper concerns the problem of stability analysis for delayed stochastic genetic regulatory networks. By introducing an appropriate Lyapunov-Krasovskii functional and employing delay-range partition approach, a new stability criterion is given to ensure the mean square stability of genetic regulatory networks with time-varying delays and stochastic disturbances. The stability criterion is given in the form of linear matrix inequalities, which can be easily tested by the LMI Toolbox of MATLAB. Moreover, it is theoretically shown that the obtained stability criterion is less conservative than the one in W. Zhang et al., 2012. Finally, a numerical example is presented to illustrate our theory.
1. Introduction
With the further progress of gene expression, researchers find that a gene expression is affected by other genes; conversely, it also influences others. Based on this reciprocal impact relation, gene expression forms a complex network—genetic regulatory network (GRN). GRNs are dynamical systems, which consist of an interaction of genes, proteins, and small molecules. In the past two decades, scholars have established mathematical models to represent GRNs. Basically, there are four types of GRN models, that is, Petri net model [1], Bayesian network model [2], Boolean model [3, 4], and (functional) differential equation model [5, 6]. The concentrations of mRNA and protein are described as the state variables in the functional differential equation model.
As dynamical systems, stability analysis is the first priority to explore GRNs. On the one hand, time delay inevitably occurs in GRNs due to the slow process of transcription, translation, and translocation [7]. On the other hand, internal noises of cells caused by random birth and death of the individual molecules and external noises from environmental fluctuations make the gene expression be best viewed as a stochastic process [8, 9]. So, it is very necessary to analyze the stability of GRNs with time-varying delays and stochastic disturbances [6, 10–18].
Recently, for a class of GRNs with interval time-varying delays and stochastic disturbances (see (17a), (17b) below), Wu et al. [18] established several delay-range-dependent and/or rate-dependent global stochastic asymptotical stability criteria in terms of linear matrix inequalities (LMIs) by using the stochastic analysis approach, employing some free-weighting matrices and introducing a type of Lyapunov-Krasovskii functional which includes the items like ∫-τ2-τ1∫t+θth1T(s)Ph1(s)dsdθ and ∫-τ2-τ1∫t+θth3T(s)Qh3(s)dsdθ, where P and Q are real symmetric positive definite matrices. Furthermore, in [18], the item -∫t-τ(t)t-τ1h1T(s)Ph1(s)ds in the stochastic differential of ∫-τ2-τ1∫t+θth1T(s)Ph1(s)dsdθ was first estimated by employing Leibniz-Newton formula and the inequality
(1)-2ξT(t)M∫t-τ(t)t-τ1h1(s)ds≤(τ(t)-τ1)ξT(t)MP-1MTξ(t)+∫t-τ(t)t-τ1h1T(s)Ph1(s)ds,
where M is a free-weighting matrix, and then τ(t)-τ1 was enlarged to τ2-τ1. Clearly, the conservatism would be produced because τ(t)-τ1 is enlarged to τ2-τ1. In order to reduce the conservatism, Zhang et al. [17] first estimated the item -∫t-τ(t)t-τ1h1T(s)Ph1(s)ds by using Leibniz-Newton formula and the inequality
(2)-2ξT(t)S∫t-τ(t)t-τ1h1(s)ds≤(1-α)(τ(t)-τ1)ξT(t)SP-1STξ(t)+α(τ(t)-τ1)ξT(t)NP-1NTξ(t)+∫t-τ(t)t-τ1h1T(s)Ph1(s)ds,
where S and N are free-weighting matrices and α is an adjusting parameter with 0<α<1, and then a so-called convex combination technique was employed to obtain less conservative delay-range-dependent and/or rate-dependent global stochastic asymptotical stability criteria. It should be emphasized that the Lyapunov-Krasovskii functional used in [17] includes not only the items like ∫-τ2-τ1∫t+θth1T(s)Ph1(s)dsdθ and ∫-τ2-τ1∫t+θth3T(s)Qh3(s)dsdθ, but also the items like ∫t-ατ1txT(s)Q4x(s)ds and ∫t-τ1t-ατ1xT(s)Q4x(s)ds involved in the adjustable parameter α, which can be viewed as a generalized delay-partition approach due to the adjustable parameter α. Generally, an appropriate delay-partition approach can bring less conservative stability criteria (see [19] and the references therein).
Note that free-weighting matrices (e.g., matrices M, S, and N above) have been employed in both [17, 18] to obtain less conservative stability criteria. However, it is emphasized in [18, Remark 2] that free-weighting matrices may produce a super-high amount of computation for the feasible solutions of LMIs. In order to overcome the disadvantage, in this paper we will propose a delay-range partition (DRP) approach to estimate accurately the item -∫t-τ(t)t-τ1h1T(s)Ph1(s)ds (see (27) and (30)), where no free-weighting matrix is involved. By employing an appropriate Lyapunov-Krasovskii functional and introducing a DRP approach, a mean square stability criterion for GRNs with time-varying delays and stochastic disturbances is first established. Then it is theoretically shown that the proposed stability criterion is less conservative than [17, Theorem 1]. Finally, a numerical example is given to illustrate the theoretical results proposed here. The main contribution of this paper can be listed as follows: (i) the Lyapunov-Krasovskii functional employed in this paper does not include the items like ∫-τ2-τ1∫t+θth3T(s)Qh3(s)dsdθ which are required in [17, 18]; (ii) the items like -∫t-τ(t)t-τ1h1T(s)Ph1(s)ds in the stochastic differential of Lyapunov-Krasovskii functional are estimated accurately by proposing a DRP approach; (iii) theoretical comparison of the stability criterion [17, Theorem 1] and the one proposed in this paper is given; and (iv) there is no free-weighting matrix involved, which reduces the computational complexity.
The rest of the paper is organized as follows. In Section 2, the model of GRNs to be studied is described. A DRP-based mean square stability criterion (Theorem 3 below) for GRNs with time-varying delays and stochastic disturbances is established in Section 3. The theoretical comparison of Theorem 3 and [17, Theorem 1] is presented in Section 4. In Section 5, an example is given to show the validity of the obtained results. Finally, in Section 6, the conclusions are drawn.
Notation. For a positive integer n, set 〈n〉={1,2,…,n}. Rn denotes the n-dimensional Euclidean space. We denote by Rm×n the set of m×n matrices over R. AT and A-1 represent the transpose and inverse of a matrix A, respectively. For real symmetric matrices X and Y, the notation X≥Y(X>Y) means that the matrix X-Y is positive semidefinite (positive definite). In is the n×n identity matrix. In a symmetric matrix, ⋆ denotes the entries implied by symmetry.
2. Model Description
The following differential equations have been used recently to describe GRNs [7]:(3a)m˙i(t)=-aimi(t)+bi(p1(t-σ(t)),p2(t-σ(t)),…,pn(t-σ(t))),(3b)p˙i(t)=-cipi(t)+dimi(t-τ(t)),i∈〈n〉,where mi(t) and pi(t) are the concentrations of the ith mRNA and protein at time t, respectively; ai>0, ci>0, and di>0 are constants, representing the degradation rate of the ith mRNA, the degradation rate of the ith protein, and the translation rate of the ith mRNA to ith protein, respectively; both σ(t) and τ(t) are transcriptional and translational delays, respectively; bi is the regulatory function of the ith gene, which is generally a nonlinear function of the variables p1(t),p2(t),…,pn(t), but it is monotonic with each variable.
For convenience, we give the following assumptions throughout the paper.
Assumption 1.
The delays σ(t) and τ(t) are differentiable functions satisfying(4a)0≤σ1≤σ(t)≤σ2,0≤τ1≤τ(t)≤τ2,(4b)σ˙(t)≤σd<+∞,τ˙(t)≤τd<+∞,where σ1, σ2, σd, τ1, τ2, and τd are constants.
Assumption 2.
The function bi is taken as
(5)bi(p1(t),p2(t),…,pn(t))=∑j=1nbij(pj(t)),
which is called SUM logic. Here, bij is a monotonic function of the Hill form; that is,
(6)bij(x)={αij(x/βj)Hj1+(x/βj)Hjiftranscriptionfactorjisanactivatorofgenei,αij11+(x/βj)Hjiftranscriptionfactorjisarepresssorofgenei,
where Hj is the Hill coefficient, βj is a scalar, and αij is a bounded constant, which denotes the dimensionless transcriptional rate of transcription factor j to gene i.
Clearly, GRN ((3a), (3b)) can be rewritten as(7a)m˙i(t)=-aimi(t)+∑j=1nwijhj(pj(t-σ(t)))+vi,(7b)p˙i(t)=-cipi(t)+dimi(t-τ(t)),i∈〈n〉, where
(8)wij={αijiftranscriptionfactorjisanactivatorofgenei,0ifthereisnoconnectionbetweenjandi,-αijiftranscriptionfactorjisarepresssorofgenei,hj(x)=(x/βj)Hj1+(x/βj)Hj,vi=∑j∈𝒱iαij,
and 𝒱i is the set of all the transcription factors j which is a repressor of gene i.
Rewriting GRN ((7a), (7b)) into compact matrix form, we obtain(9a)m˙(t)=-Am(t)+Wh(p(t-σ(t)))+v,(9b)p˙(t)=-Cp(t)+Dm(t-τ(t)),where
(10)A=diag(a1,a2,…,an),W=[wij]n×n,C=diag(c1,c2,…,cn),D=diag(d1,d2,…,dn),v=col(v1,v2,…,vn),m(t)=col(m1(t),m2(t),…,mn(t)),p(t)=col(p1(t),p2(t),…,pn(t)),h(p(t))=col(h1(p1(t)),h2(p2(t)),…,hn(pn(t))).
Let (m*,p*) be an equilibrium point of ((9a), (9b)); that is, it is a solution of the following equation:
(11)-Am*+Wh(p*)+v=0,-Cp*+Dm*=0.
For convenience, we shift the equilibrium point (m*,p*) to the origin by using the transformations x(t)=m(t)-m* and y(t)=p(t)-p*; then we have(12a)x˙(t)=-Ax(t)+Wf(y(t-σ(t))),(12b)y˙(t)=-Cy(t)+Dx(t-τ(t)),where f(y(t))=h(y(t)+p*)-h(p*).
From the relationship between h and f, one can easily find that f satisfies the following sector condition:
(13)fi(0)=0,li-≤fi(s)s≤li+,i∈〈n〉,∀0≠s∈R,
where li- and li+ are a pair of nonnegative scalars and fi(s) is the ith entry of f(s). Since hi is a monotonically increasing and differentiable function with saturation, we have to choose li- as zero or a small positive number. Let L-=diag(l1-,l2-,…,ln-) and L+=diag(l1+,l2+,…,ln+).
As shown in [15–17] the gene regulation is an intrinsically noisy process. For this reason, in this paper, we consider a class of GRNs with both time delays and noise disturbances by the following model:(14a)dx(t)=[-Ax(t)+Wf(y(t-σ(t)))]dt+H(x(t),x(t-τ(t)),y(t),y(t-σ(t)))dω(t),(14b)dy(t)=[-Cy(t)+Dx(t-τ(t))]dt,where ω(t) is an m-dimensional Brown motion, m≥1, and H(x(t),x(t-τ(t)),y(t),y(t-σ(t))) is the noise intensity matrix at time t such that
(15)trace(HHT)≤xT(t)H1x(t)+yT(t)H2y(t)+xT(t-τ(t))H3x(t-τ(t))+yT(t-σ(t))H4y(t-σ(t)),
where Hi(i=1,2,3,4) are real symmetric positive semidefinite matrices.
For simplicity, set
(16)h1(t)=-Ax(t)+Wf(y(t-σ(t))),h2(t)=-Cy(t)+Dx(t-τ(t)),h3(t)=H(x(t),x(t-τ(t)),y(t),y(t-σ(t))).
Then, GRN ((14a), (14b)) can be represented as(17a)dx(t)=h1(t)dt+h3(t)dω(t),(17b)dy(t)=h2(t)dt.
3. Stability Criterion
In the following theorem, we will propose a DRP approach to present an asymptotical stability criterion in the mean square sense for GRNs with time-varying delays and stochastic disturbance.
Theorem 3.
For given scalars α∈(0,1), τ2>τ1>0, σ2>σ1>0, τd and σd, and positive integers s1 and s2, under the conditions (15) and ((4a), (4b)), we can conclude that GRN ((14a), (14b)) is asymptotically stable in the sense of mean square, if there exist a scalar ρ>0 and matrices UiT=Ui>0(i=1,2), VjT=Vj>0, WjT=Wj>0 (j=1,2,…,7), and Λ:=diag(λ1,λ2,…,λn)>0 such that the following LMIs hold:
(18)U1≤ρI,(19)Ψk,l:=Ψ1+Ψ2+Ψ3+Ψ4+Ψ5+Ψ6+Ψk+Ψl<0,∀k∈〈s1〉,l∈〈s2〉,
where
(20)Ψ1=Ψ~1+Ψ~1T+ρ(e1TH1e1+e3TH2e3+e7TH3e7+e9TH4e9),Ψ~1=e1TU1(-Ae1+We13)+e7TU2(-Ce7+De3),Ψ2=e1T(V1+V5)e1+e4T(V2-V1)e4+e5T(V3-V2)e5+e2T[(1+ατd)V4-(1-ατd)V3]e2-(1-τd)e3TV4e3-e6TV5e6,Ψ3=e7T(W1+W5)e7+e10T(W2-W1)e10+e11T(W3-W2)e11+e8T[(1+ασd)W4-(1-ασd)W3]e8-(1-σd)e9TW4e9-e12TW5e12,Ψ4=(-Ae1+We13)T(τ1V6+τ12V7)(-Ae1+We13)+(-Ce7+De3)T(σ1W6+σ12W7)(-Ce7+De3),Ψ5=Ψ~5+Ψ~5T,Ψ~5=e13T(L+)-ΛL-e9-e13T(L+)-Λe13-e9TΛL-e9+e9TΛe13,Ψ6=-1ατ1(e4-e1)TV6(e4-e1)-1(1-α)τ1(e5-e4)TV6(e5-e4)-1ασ1(e10-e7)W6(e10-e7)-1(1-α)σ1(e11-e10)TW6(e11-e10),Ψk=-s1kατ12(e2-e5)TV7(e2-e5)-s1(1-α)kτ12(e3-e2)TV7(e3-e2)-s1(s1-k+1)τ12(e6-e3)TV7(e6-e3),Ψl=-s2lασ12(e8-e11)TW7(e8-e11)-s2(1-α)lσ12(e9-e8)TW7(e9-e8)-s2(s2-l+1)σ12(e12-e9)TW7(e12-e9),ei=[0⋯0︸numberi-1In0⋯0︸numbern-i],i=1,2,…,n,σ12=σ2-σ1,τ12=τ2-τ1.
Proof.
Let β(t)=τ1+α(τ(t)-τ1) and γ(t)=σ1+α(σ(t)-σ1). Choose a Lyapunov-Krasovskii functional candidate as
(21)V(t)=∑i=14Vi(t),
where
(22)V1(t)=xT(t)U1x(t)+yT(t)U2y(t),V2(t)=∫t-ατ1txT(s)V1x(s)ds+∫t-τ1t-ατ1xT(s)V2x(s)ds+∫t-β(t)t-τ1xT(s)V3x(s)ds+∫t-τ(t)t-β(t)xT(s)V4x(s)ds+∫t-τ2txT(s)V5x(s)ds,V3(t)=∫t-ασ1tyT(s)W1y(s)ds+∫t-σ1t-ασ1yT(s)W2y(s)ds+∫t-γ(t)t-σ1yT(s)W3y(s)ds+∫t-σ(t)t-γ(t)yT(s)W4y(s)ds+∫t-σ2tyT(s)W5y(s)ds,V4(t)=∫-τ10∫t+θth1T(s)V6h1(s)dsdθ+∫-τ2-τ1∫t+θth1T(s)V7h1(s)dsdθ+∫-σ10∫t+θth2T(s)W6h2(s)dsdθ+∫-σ2-σ1∫t+θth2T(s)W7h2(s)dsdθ,
and the matrices UiT=Ui>0 (i=1,2), VjT=Vj>0, and WjT=Wj>0 (j=1,2,…,7) are taken from a feasible solution to (18) and (19). By Itô’s formula, we can obtain the following stochastic differential:
(23)dV(t)=∑i=14ℒVi(t)dt+2xT(t)U1h3(t)dω(t),
where ℒ is the weak infinitesimal operator and
(24)ℒV1(t)=2xT(t)U1h1(t)+2yT(t)U2h2(t)+h3T(t)U1h3(t)≤ηT(t)Ψ1η(t),(25)ℒV2(t)=xT(t)(V1+V5)x(t)+xT(t-ατ1)(V2-V1)x(t-ατ1)+xT(t-τ1)(V3-V2)x(t-τ1)+(1-β˙(t))xT(t-β(t))(V4-V3)x(t-β(t))-(1-τ˙(t))xT(t-τ(t))V4x(t-τ(t))-xT(t-τ2)V5x(t-τ2)≤ηT(t)Ψ2η(t),(26)ℒV3(t)=yT(t)(W1+W5)y(t)+yT(t-ασ1)(W2-W1)y(t-ασ1)+yT(t-σ1)(W3-W2)y(t-σ1)+(1-γ˙(t))yT(t-γ(t))(W4-W3)y(t-γ(t))-(1-σ˙(t))yT(t-σ(t))W4y(t-σ(t))-yT(t-σ2)W5y(t-σ2)≤ηT(t)Ψ3η(t),(27)ℒV4(t)=h1T(t)(τ1V6+τ12V7)h1(t)-∫t-τ1th1T(s)V6h1(s)ds-∫t-τ2t-τ1h1T(s)V7h1(s)ds+h2T(t)(σ1W6+σ12W7)h2(t)-∫t-σ1th2T(s)W6h2(s)ds-∫t-σ2t-σ1h2T(s)W7h2(s)ds≤ηT(t)Ψ4η(t)-1ατ1∫t-ατ1th1T(s)dsV6∫t-ατ1th1(s)ds-1(1-α)τ1∫t-τ1t-ατ1h1T(s)dsV6∫t-τ1t-ατ1h1(s)ds-1β(t)-τ1∫t-β(t)t-τ1h1T(s)dsV7∫t-β(t)t-τ1h1(s)ds-1τ(t)-β(t)∫t-τ(t)t-β(t)h1T(s)dsV7∫t-τ(t)t-β(t)h1(s)ds-1τ2-τ(t)∫t-τ2t-τ(t)h1T(s)dsV7∫t-τ2t-τ(t)h1(s)ds-1ασ1∫t-ασ1th2T(s)dsW6∫t-ασ1th2(s)ds-1(1-α)σ1∫t-σ1t-ασ1h2T(s)dsW6∫t-σ1t-ασ1h2(s)ds-1γ(t)-σ1∫t-γ(t)t-σ1h2T(s)dsW7∫t-γ(t)t-σ1h2(s)ds-1σ(t)-γ(t)∫t-σ(t)t-γ(t)h2T(s)dsW7∫t-σ(t)t-γ(t)h2(s)ds-1σ2-σ(t)∫t-σ2t-σ(t)h2T(s)dsW7∫t-σ2t-σ(t)h2(s)ds.
For any scalars a, b with a<b, it follows from (16) that ∫abh1(t)dt=x(b)-x(a)-∫abh3(t)dω(t) and ∫abh2(t)dt=y(b)-y(a), and hence(28a)ℰ(∫abh1T(t)dtVj∫abh1(t)dt)≥[x(b)-x(a)]TVj[x(b)-x(a)],j=6,7,(28b)∫abh2T(t)dtWj∫abh2(t)dt=[y(b)-y(a)]TWj[y(b)-y(a)],j=6,7,where ℰ represents the mathematical expectation operator. Next, from the sector condition (13), we can obtain that
(29)0≤-2[(L+)-f(y(t-σ(t)))-y(t-σ(t))]T×Λ[f(y(t-σ(t)))-L-y(t-σ(t))]=ηT(t)Ψ5η(t).
When τ(t)∈[τ1+((k-1)/s1)τ12,τ1+(k/s1)τ12] for some positive integer k∈〈s1〉 and σ(t)∈[σ1+((l-1)/s2)σ12,σ1+(l/s2)σ12] for some positive integer l∈〈s2〉, it is easy to see that
(30)1β(t)-τ1≥s1kατ12,1τ(t)-β(t)≥s1(1-α)kτ12,1τ2-τ(t)≥s1(s1-k+1)τ12,(31)1γ(t)-σ1≥s2lασ12,1σ(t)-γ(t)≥s2(1-α)lσ12,1σ2-σ(t)≥s2(s2-l+1)σ12.
Then, the combination of (21)–(31) gives
(32)ℰℒV(t)≤ηT(t)Ψk,lη(t),
where
(33)η(t)=col(,y(t-σ1),y(t-σ2),f(y(t-σ(t)))y(t-σ1),y(t-σ2),f(y(t-σ(t)))x(t),x(t-β(t)),x(t-τ(t)),x(t-ατ1),x(t-τ1),x(t-τ2),y(t),y(t-γ(t)),y(t-σ(t)),y(t-ασ1),y(t-σ1),y(t-σ2),f(y(t-σ(t)))).
Due to (19), we have ℰℒV(t)<0, and hence GRN ((14a), (14b)) is asymptotically stable in the mean square sense.
Remark 4.
In the above theorem a DRP approach has been proposed to establish an asymptotic mean square stability criterion for GRN ((14a), (14b)). Both DRP approach and the so-called piecewise analysis method (see, e.g., [20]) divide the delay-varying intervals into some parts with equal length. Then DRP approach enlarges the expectation of weak infinitesimal operator of the same Lyapunov-Krasovskii functional in every subinterval, while the piecewise analysis method constructs different Lyapunov-Krasovskii functional in every subinterval.
Remark 5.
Comparing with the Lyapunov-Krasovskii functionals employed in Theorem 3 and [17, Theorem 1], we remove the items ∫-τ2-τ1∫t+θttrace[f1T(s)Z2f1T(s)]dsdθ and ∫-σ2-σ1∫t+θttrace[f2T(s)Z4f2T(s)]dsdθ, which is required in [17]. This will reduce the number of LMI variables to be solved, and hence Theorem 3 requires less computer time than [17, Theorem 1]. Furthermore, it will be shown in the next section that Theorem 3 is certainly less conservative than [17, Theorem 1].
4. Theoretical Comparisons
In this part we will offer a theoretical comparison on conservativeness of Theorem 3 and [17, Theorem 1]. For this reason, we introduce [17, Theorem 1] as follows.
Lemma 6 (see [17, Theorem 1]).
When L-=0 and L+=K, GRN ((14a), (14b)) subject to ((4a), (4b)) is asymptotically stable in the mean square sense, if there exist positive definite matrices Pi, Ri(i=1,2), Qj(j=1,2,…,8), and Zk(k=1,2,…,6), a diagonal positive matrix Λ, matrices Si, Ji, Ni, Mi, Ui, Vi, Li, and Ti(i=1,2) of appropriate sizes, and positive scalars ρ1, ρ2, and ρ3 such that the following LMIs hold:
(34)P1≤ρ1I,Z3≤ρ2I,Z4≤ρ3I,[Ω11Ω12Ω13Ψi14⋆-Ω2200⋆⋆-Ω330⋆⋆⋆-Ψi44]<0,i=1,2,3,4,
where
(35)Ψ114=[τ12(1-α)Sτ12αNσ12(1-α)Lσ12αU02n×n02n×n02n×n02n×n],Ψ214=[τ12(1-α)Sτ12αNσ12T02n×n02n×n02n×n],Ψ314=[τ12Jσ12(1-α)Lσ12αU02n×n02n×n02n×n],Ψ414=[τ12Jσ12T02n×n02n×n],S=[0S1TS2T0n×10n]T,N=[N1TN2T0n×11n]T,L=[0n×7nL1TL2T0n×4n]T,U=[0n×6nU1TU2T0n×5n]T,T=[0n×6nT1T0T2T0n×4n]T,J=[J1T0J2T0n×10n]T,Ψ144=diag(τ12(1-α)Z2,τ12αZ2,σ12(1-α)Z6,σ12αZ6),Ψ244=diag(τ12(1-α)Z2,τ12αZ2,σ12Z6),Ψ344=diag(τ12Z2,σ12(1-α)Z6,σ12αZ6),Ψ444=diag(τ12Z2,σ12Z6),Ω11=[Σ11Γ1Tϕ1Γ2Tϕ2⋆-ϕ10⋆⋆-ϕ2],Σ11=[Θ1Θ3⋆Θ2],Ω22=diag(τ1Z1,σ1Z5),Ω12=[τ1Mσ1V02n×n02n×n],M=[M1T0M2T0n×10n]T,V=[0n×6nV1T0V2T0n×4n]T,Ω13=[SNMJ02n×n02n×n02n×n02n×n],Ω33=diag(Z4,Z4,Z3,Z4),Θ1=[Π11-N1M2T+J10N1-M1-J1⋆Π22-S1+S2T0N20⋆⋆Π330-M2-J2⋆⋆⋆Π4400⋆⋆⋆⋆Π550⋆⋆⋆⋆⋆-R1],Θ3=[000000P1W0000000DTP2000000000000000000000000000],Π11=-P1A-ATP1+Q4+(ρ1+τ1ρ2+τ12ρ3)H1+M1+M1T+R1,Π22=(1+ατd)Q1-(1-ατd)Q2+S1+S1T-N2-N2T,Π33=-(1-τd)Q1+(ρ1+τ1ρ2+τ12ρ3)H2+J2+J2T-S2-S2T,Π44=Q3-Q4,Π55=Q2-Q3,Γ1=[00D000-C000000],Γ2=[-A00000000000W],ϕ1=σ1Z5+σ12Z6,ϕ2=τ1Z1+τ12Z2,Θ2=[Ξ11-U1V2T+T10U1-V1-T10⋆Ξ22-L1+L2T0U200⋆⋆Ξ330-V2-T2Λ⋆⋆⋆Ξ44000⋆⋆⋆⋆Ξ5500⋆⋆⋆⋆⋆-R20⋆⋆⋆⋆⋆⋆-2ΛK-1],Ξ11=-P2C-CTP2+Q8+(ρ1+τ1ρ2+τ12ρ3)H3+V1+V1T+R2,Ξ22=(1+ασd)Q5-(1-ασd)Q6+L1+L1T-U2-U2T,Ξ33=-(1-σd)Q5+(ρ1+τ1ρ2+τ12ρ3)H4-L2-L2T+T2+T2T,Ξ44=Q7-Q8,Ξ55=Q6-Q7.
In order to show that Theorem 3 is less conservative than [17, Theorem 1], the following propositions are required.
Proposition 7.
Let ΣT=Σ, Jj, Xj (j=1,2,3), and WT=W>0 be given real matrices of appropriate sizes. For given scalars α∈(0,1) and c>0, set
(36)S1=[(1-α)J1αJ2],T1=diag((1-α)W,αW),S2=J3,T2=W,Σ0=J1X1+J2X2+J3X3.
If
(37)[Σ+Σ0+Σ0TcSi⋆-Ti]<0,i=1,2,
then there exists a (sufficiently large) positive integer s such that
(38)Σ-sk(1-α)c-1X1TWX1-skαc-1X2TWX2-ss-k+1c-1X3TWX3<0,∀k∈〈s〉.
Proof.
It follows from (37) and the Schur complementary lemma [21] that
(39)Σ+Σ0+Σ0T+cSiTi-1SiT<0,i=1,2,
and hence there exists a (sufficiently large) positive integer s such that
(40)Σ+Σ0+Σ0T+s+1scSiTi-1SiT<0,i=1,2.
For an arbitrary but fixed k∈〈s〉, one can derive from (36) and (40) that
(41)Σ+J1X1+X1TJ1T+ksc(1-α)J1W-1J1T+J2X2+X2TJ2T+kscαJ2W-1J2T+J3X3+X3TJ3T+s-k+1scJ3W-1J3T<0.
Since
(42)J1X1+X1TJ1T+k(1-α)scJ1W-1J1T+sk(1-α)c-1X1TWX1≥0,J2X2+X2TJ2T+kαscJ2W-1J2T+skαc-1X2TWX2≥0,J3X3+X3TJ3T+s-k+1scJ3W-1J3T+ss-k+1c-1X3TWX3≥0,
we obtain from (41) that (38) holds. The proof is completed.
Proposition 8.
Let ΩT=Ω, Ji, Li, Xi, Yi (i=1,2,3), WT=W>0, and ZT=Z>0 be given real matrices of appropriate sizes and ci>0 (i=1,2) and α∈(0,1) given scalars. Set
(43)S11=[c1(1-α)J1c1αJ2c2(1-α)L1c2αL2],S12=[c1(1-α)J1c1αJ2c2L3],S21=[c1J3c2(1-α)L1c2αL2],S22=[c1J3c2L3],T11=diag(c1(1-α)W,c1αW,c2(1-α)Z,c2αZ),T12=diag(c1(1-α)W,c1αW,c2Z),T21=diag(c1W,c2(1-α)Z,c2αZ),T22=diag(c1W,c2Z),Ω1=J1X1+J2X2+J3X3,Ω2=L1Y1+L2Y2+L3Y3.
If
(44)[Ω+Ω1+Ω1T+Ω2+Ω2TSij⋆-Tij]<0,i,j=1,2,
then there exists a pair of (sufficiently large) positive integers s1 and s2 such that
(45)Ω-s1k(1-α)c1-1X1TWX1-s1kαc1-1X2TWX2-s1s1-k+1c1-1X3TWX3-s2l(1-α)c2-1Y1TZY1-s2lαc2-1Y2TZY2-s2s2-l+1c2-1Y3TZY3<0,∀k∈〈s1〉,l∈〈s2〉.
Proof.
It follows from (44) that
(46)Ω+Ω1+Ω1T+Ω2+Ω2T+SijTij-1SijT<0,i,j=1,2,
and hence
(47)Ω+Ω1+Ω1T+Ω2+Ω2T+δSi1Ti1-1Si1T+(1-δ)Si2Ti2-1Si2T<0,i=1,2,∀δ∈[0,1];
that is,
(48)[Ωδ+Ω1+Ω1Tc1S^i⋆T^i]<0,i=1,2,∀δ∈[0,1],
where
(49)S^1=[(1-α)J1αJ2],T^1=diag((1-α)W,αW),S^2=J3,T^2=W,Ωδ=Ω+Ω2+Ω2T+δc2S~1T~1-1S~1T+(1-δ)c2S~2T~2-1S~2T,S~1=[(1-α)L1αL2],T~1=diag((1-α)Z,αZ),S~2=L3,T~2=Z.
Applying Proposition 7 to Σ=Ωδ, Σ0=Ω1, Si=S^i, Ti=T^i, and c=c1, we obtain that
(50)Ωδ-s1k(1-α)c1-1X1TWX1-s1kαc1-1X2TWX2-s1s1-k+1×c1-1X3TWX3<0,∀δ∈[0,1],∀k∈〈s1〉,δ∈[0,1]
for some (sufficiently large) positive integer s1.
By the Schur complementary lemma, one can easily derive from (50) that
(51)[Ωk+Ω2+Ω2Tc2S~i⋆-T~i]<0,i=1,2,∀k∈〈s1〉,
where
(52)Ωk=Ω-s1k(1-α)c1-1X1TWX1-s1kαc1-1X2TWX2-s1s1-k+1c1-1X3TWX3.
Again applying Proposition 7 to Σ=Ωk, Σ0=Ω2, Si=S~i, Ti=T~i, and c=c2, one can complete the proof.
Proposition 9.
Let p,q∈Rn and M∈Rn×n satisfying MT=M>0. Then
(53)(p+q)TM(p+q)≤1αpTMp+11-αqTMq,∀α∈(0,1).
Proof.
One has
(54)1αpTMp+11-αqTMq-(p+q)TM(p+q)=1-ααpTMp+α1-αqTMq-pTMq-qTMp=(1-ααp-α1-αq)TM(1-ααp-α1-αq)≥0.
Now it is time to show that Theorem 3 is less conservative than [17, Theorem 1] in theory.
Theorem 10.
Set L-=0 and L+=K. If the LMIs in (34) are feasible, then the LMIs in (18) and (19) are feasible.
Proof.
Set ρ=ρ1, Ui=Pi(i=1,2), Vj=Q5-j, Wj=Q9-j(j=1,2,3,4), V5=R1, W5=R2, Vk=Zk-5, and Wk=Zk-1 (k=6,7). Then it follows from (34) and the Schur complementary lemma that (18) holds and
(55)Σ11+Ψ4+τ1MV6-1MT+σ1VW6-1VT+S~ijT~ijS~ijT<0,aaaaii,j=1,2,
where
(56)S~11=[τ12(1-α)Sτ12αNσ12(1-α)Lσ12αU],S~12=[τ12(1-α)Sτ12αNσ12T],S~21=[τ12Jσ12(1-α)Lσ12αU],S~22=[τ12Jσ12T],T~11=diag(τ12(1-α)V7,τ12αV7,σ12(1-α)W7,σ12αW7),T~12=diag(τ12(1-α)V7,τ12αV7,σ12W7),T~21=diag(τ12V7,σ12(1-α)W7,σ12αW7),T~22=diag(τ12V7,σ12W7),
and S,N,L,U,T,J,M,V,Ψ4, and Σ11 are defined as noted previously.
By simple computation one can derive from (55) that
(57)[Ω~+Ω~1+Ω~1T+Ω~2+Ω~2TS~ij⋆-T~ij]<0,i,j=1,2,
where
(58)Ω~1=S(e2-e3)+N(e5-e2)+J(e3-e6),Ω~2=L(e8-e9)+U(e11-e8)+T(e9-e12),Ω~=Ψ~+V(e7-e11)+(e7-e11)TVT+M(e1-e5)+(e1-e5)TMT+τ1MV6-1MT+σ1VW6-1VT,Ψ~=Ψ1+Ψ2+Ψ3+Ψ4+Ψ5,
and Ψi(i=1,2,…,5) are defined as in (19).
By Proposition 8, there exists a pair of (sufficiently large) positive integers s1 and s2 such that
(59)Ω~+Ψk+Ψl<0,∀k∈〈s1〉,l∈〈s2〉,
where Ψk and Ψl are defined as in (19). This, together with Proposition 9, implies that
(60)Ω~≥Ψ~-τ1-1(e1-e5)TV6(e1-e5)-σ1-1(e7-e11)TW6(e7-e11)≥Ψ~+Ψ6,
and hence the LMIs in (19) are feasible. The proof is completed.
Remark 11.
In Theorem 10, it has been theoretically investigated that Theorem 3 is certainly less conservative than [17, Theorem 1]. On the other hand, the numbers of LMI variables to be solved in Theorem 3 and [17, Theorem 1] are 8n2+9n+1 and 25n2+10n+3, respectively, which implies that Theorem 3 will require less computer time than [17, Theorem 1].
5. An Illustrative Example
In this section, a numerical example is given to illustrate the effectiveness and less conservativeness of our theoretical results.
We consider a delayed GRN with stochastic disturbances, with the parameters described as
(61)A=diag(3,3,3),C=diag(2.5,2.5,2.5),D=diag(0.8,0.8,0.8),W=[00-2.5-2.5000-2.50].
Let G1=G2=G3=G4=0.4I and f(x)=x2/(1+x2), which means that L-=0 and L+=K=0.65I. When s1=s2=10, for τ1=σ1=1, τ2=σ2=10, τd=σd=0.5, and α=0.1, by using the MATLAB Toolbox, inequalities (18) and (19) have feasible solutions. By the theorem, the system is asymptotically stable in the mean square sense. Here, some of the solution matrices are given as follows:
(62)U1=[3.92970.00000.00000.00003.92970.00000.00000.00003.9297],U2=[7.8311-0.0000-0.0000-0.00007.8311-0.0000-0.0000-0.00007.8311],
and show the trajectories of the x(t) and y(t) in Figure 1.
State trajectories of the GRN in the considered example.
When τ1=σ1=1 and s1=s2=10, the maximal allowable upper bounds of τ2=σ2 for different values of τd=σd obtained by Theorem 3 and [17, Theorem 1] are shown in Table 1. It can be seen from Table 1 that Theorem 3 is the less conservative than [17, Theorem 1].
The maximal allowable bounds of τ2=σ2 for different τd=σd.
τd=σd
0.8
1
1.5
[17, Theorem 1]
1.5602
1.5500
1.5450
Theorem 3
2.095
2.053
2.047
6. Conclusions
In this paper, the stability problem for a class of GRNs with time-varying delays and stochastic disturbances has been investigated. By constructing an appropriate Lyapunov-Krasovskii functional and proposing a DRP approach, a mean square stability criterion is given in terms of LMIs, which can be easily tested by the LMI Toolbox of MATLAB. It is theoretically shown that the proposed result is less conservative than [17, Theorem 1]. Moreover, the number of LMI variables in this paper is more less than the one in [17, Theorem 1]. A numerical example has been provided to illustrate the theoretical results given in this paper.
Extending the idea of this paper to other system models, including singular delayed systems [19, 22–24], stochastic systems [25], Markovian jump systems [20, 26], and genetic regulatory networks [5, 10, 27], is under consideration.
Conflict of Interests
The authors declare that there is no commercial or associative interest that represents a conflict of interest in connection with the work submitted.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (11371006), the fund of Heilongjiang Province Innovation Team Support Plan (2012TD007), the fund of Heilongjiang University Innovation Team Support Plan (Hdtd2010-03), the fund of Key Laboratory of Electronics Engineering, College of Heilongjiang Province (Heilongjiang University), China, the Heilongjiang University Innovation Fund for Graduates, and the 2014 Scientific and Technological Research Programs of Education Commission of Heilongjiang Province. The authors thank the anonymous referees for their helpful comments and suggestions which improve greatly this paper.
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