# Stability Analysis of Delayed Genetic Regulatory Networks via a Relaxed Double Integral Inequality

Authors: Fu-Dong Li; Qi Zhu; Hao-Tian Xu; Lin Jiang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Time delay arising in a genetic regulatory network may cause the instability. This paper is concerned with the stability analysis of genetic regulatory networks with interval time-varying delays. Firstly, a relaxed double integral inequality, named as Wirtinger-type double integral inequality (WTDII), is established to estimate the double integral term appearing in the derivative of Lyapunov-Krasovskii functional with a triple integral term. And it is proved theoretically that the proposed WTDII is tighter than the widely used Jensen-based double inequality and the recently developed Wiringter-based double inequality. Then, by applying the WTDII to the stability analysis of a delayed genetic regulatory network, together with the usage of useful information of regulatory functions, several delay-range- and delay-rate-dependent (or delay-rate-independent) criteria are derived in terms of linear matrix inequalities. Finally, an example is carried out to verify the effectiveness of the proposed method and also to show the advantages of the established stability criteria through the comparison with some literature.


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## Body

## 1. Introduction

In the past few years, genetic regulatory networks (GRNs), which describe the interactions of many molecules (DNA, RNA, proteins, etc.), have been becoming a new research area of biological and biomedical sciences [1–4]. Mathematical modelling based on the extracted functional information from the time-series data provides a useful tool for studying gene regulation processes in living organisms [5, 6], and a large variety of formalisms have been proposed to model and simulate GRNs, such as directed graphs, Boolean networks, and nonlinear differential equations [7]. Among them, the nonlinear differential equation model can provide more detailed understanding and insights into the nonlinear dynamical behavior exhibited by GRNs [8].Since mRNAs and proteins in the GRNs may be synthesized at different locations, an important issue in modelling GRNs is that the slow processes of transcription, translation, and translocation result in sizable delays [9–11]. Time delays arising in the GRNs may lead to wrong prediction of dynamic behaviors [12, 13], which may lead to very serious consequences. The stability is essential for designing or controlling genetic regulatory networks [14]; it is of a great significance to study the influence of delays on the stability of the GRNs.Up to now, a huge number of results on the stability of the delayed GRNs have been reported in the literature (see, e.g., [15–58]). The sufficient and necessary local stability criteria were firstly given for the GRNs with constant delay in [15, 16]. However, local stability is not enough for understanding nonlinear GRNs; the globally asymptotical stability of GRNs with SUM regulatory functions has been widely investigated [17–22]. Meanwhile, by taking into account the unavoidable uncertainties caused by modelling errors and parameter fluctuations, many scholars paid attentions to the robust stability analysis of the delayed GRNs [23–36]. Moreover, both the intrinsic noise derived from the random births and deaths of individual molecules and the extrinsic noise due to environment fluctuations make the gene regulation process an intrinsically noisy process [59]. Thus, many researches aimed at the robust stability analysis of the GRNs in consideration of those noises [37–46]. Also, some results have considered both the uncertainties and the noises [47–52]. In addition, based on the definition of convergence rate index, the exponential stability problem was also studied in [53–57].On the other hand, no matter what type of stability problems is concerned, the analysis methods for finding stability criteria have always been an important topic. To the best of the authors’ knowledge, there are mainly two methods that have been used for the delayed GRNs. The first type of method is theM-matrix-based method. For example, the delay- and rate-independent stability criteria were proposed in [20], the delay-independent but rate-dependent criteria were established in [23, 44], and the delay- and rate-dependent criteria were developed in [21, 22]. The stability of the GRNs through those M-matrix-based criteria is judged by verifying whether or not a matrix is a nonsingular M-matrix. Although the computational complexity is low, those criteria are just available for slow-varying delay case [20–23, 44]. However, the time delays encountered in GRNs may be fast-varying or random changing. The M-matrix-based method is inapplicable for those cases. The second type of method is based on the framework of Lyapunov-Krasovskii functional (LKF) and linear matrix inequality (LMI). The LKF-based method can be used to handle all time delays mentioned before and it is available for not only stability analysis but also many other problems, like controller synthesis, state estimation, filter design, passivity analysis, and so on [13, 59–70]. Meanwhile, the LMI-based criteria can be easily checked through MATLAB/LMI toolbox for determining the system stability. Therefore, most existing researches for the GRNs are based on this type of method [17–19, 25–43, 45–56].The problem of stability analysis by using the LKF and the LMI is that the criterion obtained has more or less conservatism. It is well-known that the criterion with less conservatism means that it can derive an admissible maximum upper bound such that the understudied GRNs maintains global asymptotical stability. It is predictable that the form of the LKF candidate is tightly related to the conservatism of the obtained criteria. Thus, the key point of the stability analysis based on such framework is to find an LKF satisfying some requirements for ensuring the globally asymptotical stability of the GRNs.In most researches, the used LKFs were constructed by introducing delay-based single and/or double integral terms into the typical nonintegral quadratic form of Lyapunov function for delay-free systems [17, 18, 28–33, 35–42, 46–50, 53–55]. Based on a predictable fact that the conservatism-reducing of criteria can be achieved by constructing more general LKF, two types of more general LKFs have been developed to reduce the conservatism. The first one is the delay-partition-based LKFs, which is constructed by dividing the delay interval into several small subintervals and then replacing the original integral terms with multiple new integral terms based on delay subintervals. This type of LKF has been used to investigate the robust stability of various GRNs [25, 26, 51], the exponential stability of switch GRNs [56], and the stochastic stability of jumping GRNs [27, 43, 45]. The other is the augmented LKF constructed by using various state vectors (current and delayed and/or integrated state vectors, etc.) to augment the quadratic terms of original LKFs, and it has been used to derive the improved stability criteria of the GRNs [19, 34, 52].Beside the above-mentioned two types of improved LKFs, a new LKF including triple integral terms firstly developed in [71] is proved to be very useful to reduce the conservatism. However, only a few researches of the GRNs have applied such type of LKF. The LKF with triple integral terms was used to discuss the asymptotical stability of the GRNs [19, 34]. The following form of double integral term will be introduced into the derivative of the LKF with a triple integral term:(1)-∫ab∫sbyTuZyududs,Z>0.As mentioned in [72], the effective estimation of the above term is strongly linked to the conservatism of the criteria. To the best of the authors’ knowledge, for the researches referring to the triple integral term in the LKFs, most literature directly applied the Jensen-based double integral inequality (JBDII) (see (17) for details) to achieve the estimation task [34]. Although an improved integral inequality was developed in [19], it is also derived based on Jensen inequality. Very recently, a Wirtinger-based double integral inequality (WBDII) was developed to general linear time-delay system and it was proved to be less conservative than the JBDII [72]. However, such inequality has not been used to discuss the GRNs. Furthermore, the gap between term (1) and its estimated value obtained by the WBDII still leads to conservatism. Therefore, it can be expected that the results may be further improved if a new estimation method that brings tighter gap is applied for term (1). This is the motivation of the paper.This paper further investigates the delay-dependent stability of the GRNs by developing a more effective inequality to estimate the double integral term (1). The contributions of the paper are summarized as follows:(1)
A relaxed double integral inequality, that is, Wiringter-type double integral inequality (WTDII), is established to estimate the double integral term. Compared with the widely used JBDII and the recently developed WTDII, the presented WTDII is theoretically proved to be the tightest.(2)
Two less conservative stability criteria of the GRNs are derived. For the GRNs with time-varying delays satisfying different conditions, two stability criteria are, respectively, established by applying the proposed WTDII to estimate the double integral terms appearing in the derivative of the LKFs.The rest of the paper is organized as follows. Problem statements and preliminaries are presented in Section2. In Section 3, the development and the comparison of the WTDII approach are discussed in detail. Two stability criteria of the GRN with time-varying delay are derived through the WTDII in Section 4. An example is given to show the validity of the obtained results in Section 5. Finally, in Section 6, the conclusions are drawn.In the Notations, the list of notations and abbreviations used throughout this paper is shown.


## 2. Problem Formulation and Preliminary

This section describes the problem to be investigated and gives some necessary preliminaries.


### 2.1. Problem Formulation

The following nonlinear differential equations have been used recently to describe the GRNs with time-varying feedback regulation delays and translational delays [28]:(2)m˙it=-aimit+bip1t-σt,p2t-σt,…,pnt-σt,p˙it=-cipit+dimit-τtas shown in Figure 1, where mit and pit are the concentrations of the ith mRNA and protein, respectively. ai>0 and ci>0 are the positive real numbers that represent the degradation rate of the ith mRNA and protein, respectively. di>0 is the positive real number that represents the translating rate from mRNA i to protein i. bi is the regulatory function of the ith gene. σt and τt are the transcriptional and translational delays, respectively.Figure 1
GRNs with time-varying feedback regulation delays and translational delays.Since each transcription factor acts additively to regulate the gene, it is usual to assume that the regulatory functionbi satisfies the following SUM logic [37]:(3)bip1t,p2t,…,pnt=∑j=1nbijpjtand bij is a monotonic function of the Hill form; that is, (4)bij=αij1+x/βjHj,if transcription factorjrepresses geneiαijx/βjHj1+x/βjHj,if transcription factorjactivates genei,where αij is bounded constant that denotes the dimensionless transcriptional rate of transcription factor j to gene i,  βj is a positive scalar, and Hj is the Hill coefficient that represents the degree of cooperativity.The transcriptional and translational delays,σt and τt, are assumed to satisfy the following two different conditions.Case 1.
τ
t and σt satisfy(5)0≤τ1≤τt≤τ2,0≤σ1≤σt≤σ2,τ˙t≤τd,σ˙t≤σd.Case 2.
τ
t and σt satisfy(6)0≤τ1≤τt≤τ2,0≤σ1≤σt≤σ2.Clearly, based on (3), GRN (2) can be rewritten as [19](7)m˙it=-aimit+∑j=1nwijgjpjt-σt+li,p˙it=-cipit+dimit-τt,where li=∑j∈Viαij with Vi being the set of all the transcription factors j which are repressors of gene i; wij=αij if transcription factor j activates gene i,  wij=0 if there is no connection between j and i, and wij=-αij if transcription factor j represses gene i; and gjx=x/βjHj/1+x/βjHj,  x≥0 is a monotonically increasing function satisfying(8)ρ≤gjs1-gjs2s1-s2≤ρiwith ρ=mins≥0⁡g˙js=0 and(9)ρi=maxs≥0⁡g˙js=Hj-1Hj-1/HjHj+1Hj+1/Hj4βjHj.GRN (7) can be expressed as the following vector-matrix form:(10)m˙t=-Amt+Wgpt-σt+l,p˙t=-Cpt+Dmt-τt,where mt=m1t,m2t,…,mntT,  pt=p1t,p2t,…,pntT,  A=diaga1,a2,…,an>0,  C=diagc1,c2,…,cn>0,  D=diagd1,d2,…,dn>0,  gpt=g1p1t,g2p2t,…,gnpntT,  W=wijn×n, and l=l1,l2,…,ln.Letm∗,p∗ be the equilibrium point (steady state) of (10); that is, -Am∗+Wgp∗+l=0 and -Cp∗+Dm∗=0. Using the transformations xt=mt-m∗ and yt=pt-p∗, one can shift the equilibrium point m∗,p∗ to the origin and rewrite (10) as the following GRN:(11)x˙t=-Axt+Wfyt-σt,y˙t=-Cyt+Dxt-τt,where fs=f1s,f2s,…,fnsT and fiyt=giyt+p∗-gip∗ with fi0=0. Then, (12)fis1-fis2s1-s2=gis1+p∗-gis2+p∗s1+p∗-s2+p∗. Thus, it follows from (8) and fi0=0 that(13)0≤fis1-fis2s1-s2≤ρi,s1≠s2,(14)0≤fiss≤ρi,s≠0.This paper aims to analyze the asymptotical stability of GRN (2) and to determine the delay bounds, named as maximal admissible delay bounds (MADBs), under which the GRN is asymptotically stable. In order to achieve this aim, this paper will develop a new double integral inequality (i.e., WTDII) for estimating the double integral term (1) so as to derive some less conservative stability criteria.


### 2.2. Preliminaries

Several lemmas used to obtain the main results are given as follows.For the estimation of single integral term, the most popular technique is Wirtinger-based inequality, shown as Lemma1.Lemma 1 (Wirtinger-based inequality [73]).
For symmetric positive-definite matrixR∈Rn×n, scalars a<b, and vector ω:a,b↦Rn such that the integration concerned is well defined, the following inequality holds:(15)∫abωTsRωsds≥1b-aχaχbTR003Rχaχb,where χa=∫abωsds and χb=χa-2/b-a∫ab∫asωududs=-χa+2/b-a∫ab∫sbωududs.The auxiliary function-based integral inequality, which encompasses the Wirtinger-based inequality, has been developed in recent years.Lemma 2 (auxiliary function-based integral inequality [74]).
For symmetric positive-definite matrixR∈Rn×n, scalars a<b, and vector ω:a,b↦Rn such that the integration concerned is well defined, the following inequality holds(16)b-a∫abω˙TsRω˙sds≥χ1TRχ1+3χ2TRχ2+5χ3TRχ3,where χ1=ωb-ωa,  χ2=ωb+ωa-2/b-a∫abωsds, and χ3=ωb-ωa+6/b-a∫abωsds-12/b-a2∫ab∫sbωududs.For the estimation of double integral term, the JBDII is widely applied in [71], and, with its improvement, the WBDII was developed in [72] very recently, respectively shown as Lemmas 3 and 4.Lemma 3 (Jensen-based double integral inequality (JBDII) [71]).
For symmetric positive-definite matrixZ∈Rn×n, scalars a<b, and vector ν:a,b↦Rn such that the integration concerned is well defined, the following inequality holds:(17)b-a22∫ab∫sbνTuZνududs≥χ4TZχ4,where χ4=∫ab∫sbνududs.Lemma 4 (Wirtinger-based double integral inequality (WBDII) [72]).
For symmetric positive-definite matrixZ∈Rn×n, scalars a<b, and vector ν:a,b↦Rn such that the integration concerned is well defined, the following inequality holds:(18)b-a22∫ab∫sbνTuZνududs≥χ4TZχ4+2χ5TZχ5,where χ5=-χ4+3/b-a∫ab∫sb∫θbνududθds with χ4 given in Lemma 3.For time-varying delay, when using the integral inequality, the reciprocally convex lemma is needed, and its simple form can be reformulated as Lemma5.Lemma 5 (reciprocally convex combination lemma [75]).
For any vectorsβ1 and β2, symmetric matrix R, any matrix S, and real scalar 0≤α≤1 satisfying RS∗R≥0, the following inequality holds:(19)1αβ1TRβ1+11-αβ2TRβ2≥β1β2TRS∗Rβ1β2.


## 2.1. Problem Formulation

The following nonlinear differential equations have been used recently to describe the GRNs with time-varying feedback regulation delays and translational delays [28]:(2)m˙it=-aimit+bip1t-σt,p2t-σt,…,pnt-σt,p˙it=-cipit+dimit-τtas shown in Figure 1, where mit and pit are the concentrations of the ith mRNA and protein, respectively. ai>0 and ci>0 are the positive real numbers that represent the degradation rate of the ith mRNA and protein, respectively. di>0 is the positive real number that represents the translating rate from mRNA i to protein i. bi is the regulatory function of the ith gene. σt and τt are the transcriptional and translational delays, respectively.Figure 1
GRNs with time-varying feedback regulation delays and translational delays.Since each transcription factor acts additively to regulate the gene, it is usual to assume that the regulatory functionbi satisfies the following SUM logic [37]:(3)bip1t,p2t,…,pnt=∑j=1nbijpjtand bij is a monotonic function of the Hill form; that is, (4)bij=αij1+x/βjHj,if transcription factorjrepresses geneiαijx/βjHj1+x/βjHj,if transcription factorjactivates genei,where αij is bounded constant that denotes the dimensionless transcriptional rate of transcription factor j to gene i,  βj is a positive scalar, and Hj is the Hill coefficient that represents the degree of cooperativity.The transcriptional and translational delays,σt and τt, are assumed to satisfy the following two different conditions.Case 1.
τ
t and σt satisfy(5)0≤τ1≤τt≤τ2,0≤σ1≤σt≤σ2,τ˙t≤τd,σ˙t≤σd.Case 2.
τ
t and σt satisfy(6)0≤τ1≤τt≤τ2,0≤σ1≤σt≤σ2.Clearly, based on (3), GRN (2) can be rewritten as [19](7)m˙it=-aimit+∑j=1nwijgjpjt-σt+li,p˙it=-cipit+dimit-τt,where li=∑j∈Viαij with Vi being the set of all the transcription factors j which are repressors of gene i; wij=αij if transcription factor j activates gene i,  wij=0 if there is no connection between j and i, and wij=-αij if transcription factor j represses gene i; and gjx=x/βjHj/1+x/βjHj,  x≥0 is a monotonically increasing function satisfying(8)ρ≤gjs1-gjs2s1-s2≤ρiwith ρ=mins≥0⁡g˙js=0 and(9)ρi=maxs≥0⁡g˙js=Hj-1Hj-1/HjHj+1Hj+1/Hj4βjHj.GRN (7) can be expressed as the following vector-matrix form:(10)m˙t=-Amt+Wgpt-σt+l,p˙t=-Cpt+Dmt-τt,where mt=m1t,m2t,…,mntT,  pt=p1t,p2t,…,pntT,  A=diaga1,a2,…,an>0,  C=diagc1,c2,…,cn>0,  D=diagd1,d2,…,dn>0,  gpt=g1p1t,g2p2t,…,gnpntT,  W=wijn×n, and l=l1,l2,…,ln.Letm∗,p∗ be the equilibrium point (steady state) of (10); that is, -Am∗+Wgp∗+l=0 and -Cp∗+Dm∗=0. Using the transformations xt=mt-m∗ and yt=pt-p∗, one can shift the equilibrium point m∗,p∗ to the origin and rewrite (10) as the following GRN:(11)x˙t=-Axt+Wfyt-σt,y˙t=-Cyt+Dxt-τt,where fs=f1s,f2s,…,fnsT and fiyt=giyt+p∗-gip∗ with fi0=0. Then, (12)fis1-fis2s1-s2=gis1+p∗-gis2+p∗s1+p∗-s2+p∗. Thus, it follows from (8) and fi0=0 that(13)0≤fis1-fis2s1-s2≤ρi,s1≠s2,(14)0≤fiss≤ρi,s≠0.This paper aims to analyze the asymptotical stability of GRN (2) and to determine the delay bounds, named as maximal admissible delay bounds (MADBs), under which the GRN is asymptotically stable. In order to achieve this aim, this paper will develop a new double integral inequality (i.e., WTDII) for estimating the double integral term (1) so as to derive some less conservative stability criteria.


## 2.2. Preliminaries

Several lemmas used to obtain the main results are given as follows.For the estimation of single integral term, the most popular technique is Wirtinger-based inequality, shown as Lemma1.Lemma 1 (Wirtinger-based inequality [73]).
For symmetric positive-definite matrixR∈Rn×n, scalars a<b, and vector ω:a,b↦Rn such that the integration concerned is well defined, the following inequality holds:(15)∫abωTsRωsds≥1b-aχaχbTR003Rχaχb,where χa=∫abωsds and χb=χa-2/b-a∫ab∫asωududs=-χa+2/b-a∫ab∫sbωududs.The auxiliary function-based integral inequality, which encompasses the Wirtinger-based inequality, has been developed in recent years.Lemma 2 (auxiliary function-based integral inequality [74]).
For symmetric positive-definite matrixR∈Rn×n, scalars a<b, and vector ω:a,b↦Rn such that the integration concerned is well defined, the following inequality holds(16)b-a∫abω˙TsRω˙sds≥χ1TRχ1+3χ2TRχ2+5χ3TRχ3,where χ1=ωb-ωa,  χ2=ωb+ωa-2/b-a∫abωsds, and χ3=ωb-ωa+6/b-a∫abωsds-12/b-a2∫ab∫sbωududs.For the estimation of double integral term, the JBDII is widely applied in [71], and, with its improvement, the WBDII was developed in [72] very recently, respectively shown as Lemmas 3 and 4.Lemma 3 (Jensen-based double integral inequality (JBDII) [71]).
For symmetric positive-definite matrixZ∈Rn×n, scalars a<b, and vector ν:a,b↦Rn such that the integration concerned is well defined, the following inequality holds:(17)b-a22∫ab∫sbνTuZνududs≥χ4TZχ4,where χ4=∫ab∫sbνududs.Lemma 4 (Wirtinger-based double integral inequality (WBDII) [72]).
For symmetric positive-definite matrixZ∈Rn×n, scalars a<b, and vector ν:a,b↦Rn such that the integration concerned is well defined, the following inequality holds:(18)b-a22∫ab∫sbνTuZνududs≥χ4TZχ4+2χ5TZχ5,where χ5=-χ4+3/b-a∫ab∫sb∫θbνududθds with χ4 given in Lemma 3.For time-varying delay, when using the integral inequality, the reciprocally convex lemma is needed, and its simple form can be reformulated as Lemma5.Lemma 5 (reciprocally convex combination lemma [75]).
For any vectorsβ1 and β2, symmetric matrix R, any matrix S, and real scalar 0≤α≤1 satisfying RS∗R≥0, the following inequality holds:(19)1αβ1TRβ1+11-αβ2TRβ2≥β1β2TRS∗Rβ1β2.


## 3. A Relaxed Double Integral Inequality and Its Advantages

This section develops a new integral inequality, that is, the WTDII, to estimate the double integral terms existing. The comparison of the WTDII and the existing double integral inequalities is also given.Based on the technique of integral in parts, the following WTDII is given.Lemma 6.
For symmetric positive-definite matrixZ∈Rn×n, scalars a<b, and vector ν:a,b↦Rn such that the integration concerned is well defined, the following inequality holds:(20)b-a22∫ab∫sbνTuZνududs≥χ4TZχ4+8χ5TZχ5,where χ4 and χ5 are defined in Lemmas 3 and 4.Proof.
For a functionλu=k1+k2u, the calculation through integration by parts leads to (21)∫ab∫sbλuνududs=λa∫ab∫sbνududs+2k2∫ab∫sb∫θbxududθds. By setting λa=-1,  2k2=3/b-a, that is, λu=-a-2b/2b-a+3/2b-au, the above equality is rewritten as(22)∫ab∫sbλuνududs=χ5.Then the following equality is obtained for any vector χ0 and any matrix M:(23)∫ab∫sbλuχ0TMνududs=χ0TMχ5.
Similarly, the following equalities are derived:(24)∫ab∫sbχ0TLνududs=χ0TLχ4,∫ab∫sbχ0TLR-1LTχ0duds=b-a22χ0TLR-1LTχ0,∫ab∫sbχ0TLR-1MTλuχ0duds=0,∫ab∫sbλ2uχ0TMR-1MTχ0duds=b-a216χ0TMR-1MTχ0. Therefore, using the above five equalities and the Schur complement derives the following equality:(25)∫ab∫sbχ0λuχ0νuTLZ-1LTLZ-1MTL∗MZ-1MTM∗∗Zχ0λuχ0νududs=∫ab∫sbνTuRνududs+Symχ0TLχ4+χ0TMχ5+b-a22χ0T8LZ-1LT+MZ-1MT8χ0≥0.
By lettingχ0T=χ4T,χ5T,  L=-2/b-a2Z,0T, and M=-16/b-a20,Z, that is, χ0TL=-2/b-a2χ4TZ and χ0TM=-16/b-a2χ5TZ, then (25) leads to(26)∫ab∫sbνTuZνududs≥2b-a2χ4TZχ4+8χ5TZχ5. Thus (20) holds. This completes the proof.Remark 7.
Based on the comparison of the proposed WTDII (20) with the widely used JBDII (17) and the recently developed WBDII (18), it can be found that WTDII (20) provides the tightest estimation value of the double integral term (1). More specifically, compared with the widely used JBDII (17), the extra positive term 8χ5TZχ5 reduces the gap between the original double integral term (1) and its estimated value; and, compared with the recently developed WBDII (18), the extra positive term 6χ5TZχ5 reduces the estimation gap. As mentioned in [72–74], it is helpful to reduce the conservatism by reducing such estimation gap. Therefore, the proposed WTDII (20) will lead to less conservative criteria than the ones derived by JBDII (17) [19] or WBDII (18).By settingνu=ω˙u, the following lemma can be directly obtained from Lemma 6.Lemma 8.
For symmetric positive-definite matrixZ∈Rn×n, scalars a<b, and vector ω˙:a,b↦Rn such that the integration concerned is well defined, the following inequality holds:(27)∫ab∫sbω˙TuZω˙ududs≥2θ1TZθ1+16θ2TZθ2,where θ1=1/b-aχ4|ν(u)=ω˙(u)=ω(b)-∫abω(s)/b-ads and θ2=1/b-aχ5|ν(u)=ω˙(u)=-1/2ω(b)-∫abω(s)/b-ads+3∫ab∫sbω(u)/(b-a)2duds.


## 4. Delay-Dependent Stability Analysis of GRN

This section derives delay-dependent stability criteria of GRN (2) by constructing the LKF with triple integral terms and applying the proposed WTDII (20) to estimate the double integral terms appearing in its derivative.The following notations are introduced at first for simplifying the representation of subsequent parts:(28)τ1τt=τt-τ1,τ2τt=τ2-τt,σ1σt=σt-σ1,σ2σt=σ2-σt,xτ1t=xt-τ1,yσ1t=yt-σ1,xτt=xt-τt,yσt=yt-σt,xτ2t=xt-τ2,yσ2t=yt-σ2,v1t=∫t-τ1txsτ1ds,v4t=∫t-τ1t∫stxuτ12duds,v2t=∫t-τtt-τ1xsτ1τtds,v5t=∫t-τtt-τ1∫st-τ1xuτ1τ2tduds,v3t=∫t-τ2t-τtxsτ2τtds,v6t=∫t-τ2t-τt∫st-τtxuτ2τ2tduds,v7t=∫t-σ1txsσ1ds,v10t=∫t-σ1t∫stxuσ12duds,v8t=∫t-σtt-σ1xsσ1σtds,v11t=∫t-σtt-σ1∫st-σ1xuσ1σ2tduds,v9t=∫t-σ2t-σtxsσ2σtds,v12t=∫t-σ2t-σt∫st-σtxuσ2σ2tduds,ζt=xTt,xTt-τ1,xTt-τt,xTt-τ2,v1Tt,v2Tt,…,v6Tt,yTt,yTt-σ1,yTt-σt,yTt-σ2,v7Tt,v8Tt,…,v12Tt,fTyt,fTyt-σ1,fTyt-σt,fTyt-σ2T,(29)ex=-A,0n×21n,W,0n×n,ey=0n×2n,D,0n×7n,-C,0n×13n,e0=0n×24n,ei=0n×i-1n,In×n,0n×24-in,i=1,2,…,24,Σ=diagρ1,ρ2,…,ρn.


### 4.1. Stability of GRN (2) with Delay Satisfying (5)

For GRN (2) with a delay satisfying (5), the following stability criterion is derived by using the proposed WTDII (27), together with Lemmas 1, 2, and 5, to estimate the derivative of the LKF.Theorem 9.
For given scalarsτi,  σi,i=1,2,  τd, and σd, GRN (2) with the time delay satisfying (5) and regulatory function satisfying (3) is asymptotically stable, if there exist symmetric matrices P>0,  Qi>0,  Rj>0,  Zk>0,  i=1,2,…,6,  j=1,2,…,5, and k=1,2,…,4; diagonal matrices Λ1>0,Λ2>0,  Hj>0,j=1,2,…,4,  Ulk>0,l=1,2,…,4, and k=l+1,…,4; and any matrices Si,i=1,2, such that the following LMIs hold:(30)R~2i+1Si∗R~2i+1>0,i=1,2,(31)Ψ1=Ξτtτt=τ1+∑i=18Ξi≤0,(32)Ψ2=Ξτtτt=τ2+∑i=18Ξi≤0,where τ12=τ2-τ1,  σ12=σ2-σ1, and(33)Ξτt=-τ1τte6TR2e6+32e9-e6TR22e9-e6-τ2τte7TR2e7+32e10-e7TR22e10-e7,(34)Ξ1=Ξ11+Ξ11T,(35)Ξ11=e1e11TPexey+Σe11-e21TΛ1+e21TΛ2ey,(36)Ξ2=e1TQ1e1-e2TQ1-Q2-Q3e2-e4TQ2e4-1-τde3TQ3e3,(37)Ξ3=Ξ31+Ξ32+Ξ33,(38)Ξ31=exTτ12R1+τ122R3ex+τ12e1TR2e1,(39)Ξ32=E1TR~1E1,R~1=diagR1,3R1,5R1,(40)Ξ33=E2E3TR~3S1∗R~3E2E3,R~3=diagR3,3R3,5R3,(41)Ξ4=Ξ41+Ξ42+Ξ43,(42)Ξ41=exTτ122Z1+τ22-τ122Z2-τ1τ12Z2ex,(43)Ξ42=-2e1-e5TZ1e1-e5-163e8-e12-e5TZ13e8-e12-e5,(44)Ξ43=-2e2-e6TZ2e2-e6-163e9-e22-e6TZ23e9-e22-e6-2e3-e7TZ2e3-e7-163e10-e32-e7TZ23e10-e32-e7,(45)Ξ5=e11e21TQ4e11e21+e12e22TQ5+Q6-Q4e12e22-e14e24TQ5e14e24-1-σde13e23TQ6e13e23,(46)Ξ6=Ξ61+Ξ62+Ξ63,(47)Ξ61=eyTσ12R4+σ122R5ey,(48)Ξ62=E4TR~4E4,R~4=diagR4,3R4,5R4,(49)Ξ63=E5E6TR~5S2∗R~5E5E6,R~5=diagR5,3R5,5R5,(50)Ξ7=Ξ71+Ξ72+Ξ73,(51)Ξ71=eyTσ122Z3+σ22-σ122Z4-σ1σ12Z4ey,(52)Ξ72=-2e11-e15TZ3e11-e15-163e18-e112-e15TZ33e18-e112-e15,(53)Ξ73=-2e12-e16TZ4e12-e16-163e19-e122-e16TZ43e19-e122-e16-2e13-e17TZ4e13-e17-163e20-e132-e17TZ43e20-e132-e17,(54)Ξ8=Ξ81+Ξ81T,(55)Ξ81=∑i=14Σe1i-e2iTHie2i+∑i=14∑j=i+14Σe1i-e1j-e2i-e2jTUije2i-e2j,(56)E1=e1-e2e1+e2-2e5e1-e2+6e5-12e8,(57)E2=e2-e3e2+e3-2e6e2-e3+6e6-12e9,(58)E3=e3-e4e3+e4-2e7e3-e4+6e7-12e10,(59)E4=e11-e12e11+e12-2e15e11-e12+6e15-12e18,(60)E5=e12-e13e12+e13-2e16e12-e13+6e16-12e19,(61)E6=e13-e14e13+e14-2e17e13-e14+6e17-12e20.Proof.
Construct the following LKF candidate:(62)Vt=∑i=17Vit,where(63)V1t=xtytTPxtyt+∑i=1n∫0yiλ1iρis-fis+λ2ifisds,V2t=∫t-τ1txTsQ1xsds+∫t-τ2t-τ1xTsQ2xsds+∫t-τtt-τ1xTsQ3xsds,V3t=τ1∫-τ10∫t+θtx˙TsR1x˙sdsdθ+∫-τ2-τ1∫t+θtxTsR2xs+τ12x˙TsR3x˙sdsdθ,V4t=∫-τ10∫θ0∫t+stx˙TuZ1x˙ududsdθ+∫-τ2-τ1∫θ-τ1∫t+stx˙TuZ2x˙ududsdθ,V5t=∫t-σ1tysfysTQ4ysfysds+∫t-σ2t-σ1ysfysTQ5ysfysds+∫t-σtt-σ1ysfysTQ6ysfysds,V6t=σ1∫-σ10∫t+θty˙TsR4y˙sdsdθ+σ12∫-σ2-σ1∫t+θty˙TsR5y˙sdsdθ,V7t=∫-σ10∫θ0∫t+sty˙TuZ3y˙ududsdθ+∫-σ2-σ1∫θ-σ1∫t+sty˙TuZ4y˙ududsdθand P>0,  Qi>0,  Rj>0,  Zk>0,  i=1,2,…,6,  j=1,2,…,5, and  k=1,2,…,4 are the symmetric positive-definite matrices and Λi=diagλi1,λi2,…,λin>0,  i=1,2, are the symmetric positive-definite diagonal matrices.
Calculating the derivative of the LKF along the solutions of GRN (11) yields(64)V˙t=∑i=17V˙it,where(65)V˙1t=2xtytTPx˙ty˙t+2Σyt-fytTΛ1+fTytΛ2y˙t=ζTtΞ11+Ξ11Tζt,V˙2t=xTtQ1xt+xτ1TtQ2+Q3-Q1xτ1t-xτ2TtQ2xτ2t-1-τ˙txτTtQ3xτt≤xTtQ1xt+xτ1TtQ2+Q3-Q1xτ1t-xτ2TtQ2xτ2t-1-τdxτTtQ3xτt=ζTtΞ2ζt,V˙3t=x˙Ttτ12R1+τ122R3x˙t+τ12xTtR2xt-τ1∫t-τ1tx˙TsR1x˙sds-∫t-τ2t-τ1xTsR2xs+τ12x˙TsR3x˙sds,V˙4t=x˙Ttτ122Z1+τ22-τ122Z2-τ1τ12Z2x˙t-∫t-τ1t∫stx˙TuZ1x˙ududs-∫t-τ2t-τ1∫st-τ1x˙TuZ2x˙ududs,V˙5t=ytfytTQ4ytfyt+yσ1tfyt-σ1TQ5+Q6-Q4yσ1tfyt-σ1-yσ2tfyt-σ2TQ5yσ2tfyt-σ2-1-σ˙tyσtfyt-σtTQ6yσtfyt-σt≤ζTtΞ5ζt,V˙6t=y˙Ttσ12R4+σ122R5y˙t-σ1∫t-σ1ty˙TsR4y˙sds-σ12∫t-σ2t-σ1y˙TsR5y˙sds,V˙7t=y˙Ttσ122Z3+σ22-σ122Z4-σ1σ12Z4y˙t-∫t-σ1t∫sty˙TuZ3y˙ududs-∫t-σ2t-σ1∫st-σ1y˙TuZ4y˙ududs,where Ξ11,  Ξ2, and Ξ5 are defined in (35), (36), and (45), respectively.
Using Lemma2 to estimate the R1-dependent single integral terms in V˙3t yields(66)-τ1∫t-τ1tx˙TsR1x˙sds≤-η1TtR~1η1t=ζTtΞ32ζt, where R~1 and Ξ32 are defined in (39) and (67)η1t=xt-xτ1txt+xτ1t-2v1txt-xτ1t+6v1t-12v4t.
Using Lemma1 to estimate the R2-dependent single integral terms in V˙3t yields(68)-∫t-τ2t-τ1xTsRxsds=-∫t-τtt-τ1xTsRxsds-∫t-τ2t-τtxTsRxsds≤-τ1τtv2TtR2v2t+32v5t-v2tTR22v5t-v2t-τ2τtv3TtR2v3t+32v6t-v3tTR22v6t-v3t=ζTtΞτtζt, where Ξτt is defined in (33).
Using Lemmas2 and 5, together with (30), to estimate the R3-dependent single integral terms in V˙3t yields(69)-τ12∫t-τ2t-τ1x˙TsR3x˙sds=-τ12∫t-τtt-τ1x˙TsR3x˙sds-τ12∫t-τ2t-τtx˙TsR3x˙sds≤-τ12τt-τ1η2TtR~3η2t-τ12τ2-τtη3TtR~3η3t≤-η2tη3tTR~3S1∗R~3η2tη3t=ζTtΞ33ζt, where R~3 and Ξ33 are defined in (40) and (70)η2t=xτ1t-xτtxτ1t+xτt-2v2txτ1t-xτt+6v2t-12v5t,η3t=xτt-xτ2txτt+xτ2t-2v3txτt-xτ2t+6v3t-12v6t.
Using Lemma8 to estimate the Z1-dependent double integral terms in V˙4t yields(71)-∫t-τ1t∫stx˙TuZ1x˙ududs≤-2xt-v1tTZ1xt-v1t+163v4t-xt2-v1tTZ13v4t-xt2-v1t=ζTtΞ42ζt,where Ξ42 is defined in (43).
Using Lemma8 to estimate the Z2-dependent double integral terms in V˙4t yields(72)-∫t-τ2t-τ1∫st-τ1x˙TuZ2x˙ududs=-∫t-τtt-τ1∫st-τ1x˙TuZ2x˙ududs-∫t-τ2t-τt∫st-τ1x˙TuZ2x˙ududs≤-∫t-τtt-τ1∫st-τ1x˙TuZ2x˙ududs-∫t-τ2t-τt∫st-τtx˙TuZ2x˙ududs≤-2xτ1t-v2tTZ2xτ1t-v2t-163v5t-xτ1t2-v2tTZ23v5t-xτ1t2-v2t-2xτt-v3tTZ2xτt-v3t-163v6t-xτt2-v3tTZ23v6t-xτt2-v3t=ζTtΞ43ζt,where Ξ43 is defined in (44).
Similarly, using Lemmas2, 5, and 8 to estimate the single and double integral terms in V˙6t and V˙7t yields(73)-σ1∫t-σ1tx˙TsR4x˙sds≤ζTtΞ62ζt,-σ12∫t-σ2t-σ1x˙TsR5x˙sds≤ζTtΞ63ζt,-∫t-σ1t∫stx˙TuZ3x˙ududs≤ζTtΞ72ζt,-∫t-σ2t-σ1∫st-σ1x˙TuZ4x˙ududs≤ζTtΞ73ζt,where Ξ62,  Ξ63,  Ξ72,   and Ξ73 are defined in (48)–(53).
Taking into account the assumption of the activation function, (13) and (14), the following inequalities hold [76, 77]: (74)his=2Σys-fysTHifys≥0,uijs1,s2=2Σys1-ys2-fys1-fys2TUij×fys1-fys2≥0, where Hi,i=1,2,…,4, and Uij,i=1,2,…,4,j=i+1,…,4, are the symmetric diagonal matrices. Thus, the following inequality holds:(75)Ht+Ut=h1t+h2t-σ1+h3t-σt+h4t-σ2+u12t,t-σ1+u13t,t-σt+u14t,t-σ2+u23t-σ1,t-σt+u24t-σ1,t-σ2+u34t-σt,t-σ2=ζTtΞ8ζt≥0, where Ξ8 is defined in (54).
Finally, combining (64), (65), (66), (68), (69), (71), (72), (73), and (75) yields(76)V˙t≤ζTtΞτt+∑i=18Ξiζt, where the related notations are defined in (31).
Therefore, if LMIs (31) and (32) hold, then the following holds for a sufficiently small scalar ϵ>0 based on convex combination method [78, 79]:(77)V˙t≤-ϵxt2+yt2 which shows the asymptotical stability of GRN (2) with time delay satisfying (5). This completes the proof.


### 4.2. Stability of GRN (11) with Delay Satisfying (6)

For some cases, the change rates of the time-varying delays are unmeasurable, that is, time delay satisfying (6). For this case, the following stability criterion can be derived by using the proposed WTDII (27), together with Lemmas 1, 2, 5, and 8, to estimate the derivative of the LKF.Theorem 10.
For given scalarsτi and σi,  i=1,2, GRN (2) with the time delay satisfying (6) and regulatory function satisfying (3) is asymptotically stable, if there exist symmetric matrices P>0,  Qi>0,  Rj>0,  Zk>0,  i=1,2,4,5,  j=1,2,…,5, and  k=1,2,…,4; diagonal matrices Λ1>0,  Λ2>0,  Hj>0,  j=1,2,3,4, Ulk>0,  l=1,2,…,4, and  k=l+1,…,4; and any matrices Si,  i=1,2, such that the following LMIs hold:(78)R~2i+1Si∗R~2i+1>0,i=1,2,Ψ3=Ξτtτt=τ1+∑i=1,3,4,6,7Ξi+Ξ¯2+Ξ¯5≤0,Ψ4=Ξτtτt=τ2∑i=1,3,4,6,7Ξi+Ξ¯2+Ξ¯5≤0,where Ξi,  i=1,3,4,6,7, are defined in Theorem 9 and (79)Ξ¯2=e1TQ1e1-e2TQ1-Q2e2-e4TQ2e4,Ξ¯5=e11e21TQ4e11e21+e12e22TQ5-Q4e12e22-e14e24TQ5e14e24.Proof.
The above stability criterion can be obtained by settingQ3=0 and Q6=0 in Theorem 9.


### 4.3. Some Remarks

This part gives some remarks for the above criteria.Remark 11.
During the proof of the above two stability criteria, the double integral terms arising in the derivative of the LKFs are estimated by using the proposed WTDII, that is, Lemma8. As discussed in Section 3, the WTDII is tighter than the widely used JBDII (17), which was used for the GRN [19, 34], and the recently developed WBDII (18), which has not been used for the GRN. Thus, the proposed criteria are less conservative than the ones reported in [19, 34].Remark 12.
Compared with the literature, more information of regulatory function has been used during the proof of criteria. Specifically, in the literature, only (14) is used during the estimation of the derivative of the LKF, while, in this paper, extra information of regulatory function (13) is also used for estimating task. It has been proved in [77] that such additional information is helpful to reduce the conservatism.Remark 13.
The conditions given in Theorems9 and 10 are in the form of LMI. Such LMI conditions can be easily checked by using MATLAB/LMI toolbox [80]. One can refer to [81–83] for more details.Remark 14.
Although this paper has just investigated the asymptotical stability, the proposed method can be extended to the robust stability analysis by taking into account the parameter uncertainties and/or noises of the GRNs. Moreover, the proposed method can also be extended to other problems discussed in Section2, like controller synthesis, state estimation, filter design, passivity analysis, and so on [13, 59–70].


## 4.1. Stability of GRN (2) with Delay Satisfying (5)

For GRN (2) with a delay satisfying (5), the following stability criterion is derived by using the proposed WTDII (27), together with Lemmas 1, 2, and 5, to estimate the derivative of the LKF.Theorem 9.
For given scalarsτi,  σi,i=1,2,  τd, and σd, GRN (2) with the time delay satisfying (5) and regulatory function satisfying (3) is asymptotically stable, if there exist symmetric matrices P>0,  Qi>0,  Rj>0,  Zk>0,  i=1,2,…,6,  j=1,2,…,5, and k=1,2,…,4; diagonal matrices Λ1>0,Λ2>0,  Hj>0,j=1,2,…,4,  Ulk>0,l=1,2,…,4, and k=l+1,…,4; and any matrices Si,i=1,2, such that the following LMIs hold:(30)R~2i+1Si∗R~2i+1>0,i=1,2,(31)Ψ1=Ξτtτt=τ1+∑i=18Ξi≤0,(32)Ψ2=Ξτtτt=τ2+∑i=18Ξi≤0,where τ12=τ2-τ1,  σ12=σ2-σ1, and(33)Ξτt=-τ1τte6TR2e6+32e9-e6TR22e9-e6-τ2τte7TR2e7+32e10-e7TR22e10-e7,(34)Ξ1=Ξ11+Ξ11T,(35)Ξ11=e1e11TPexey+Σe11-e21TΛ1+e21TΛ2ey,(36)Ξ2=e1TQ1e1-e2TQ1-Q2-Q3e2-e4TQ2e4-1-τde3TQ3e3,(37)Ξ3=Ξ31+Ξ32+Ξ33,(38)Ξ31=exTτ12R1+τ122R3ex+τ12e1TR2e1,(39)Ξ32=E1TR~1E1,R~1=diagR1,3R1,5R1,(40)Ξ33=E2E3TR~3S1∗R~3E2E3,R~3=diagR3,3R3,5R3,(41)Ξ4=Ξ41+Ξ42+Ξ43,(42)Ξ41=exTτ122Z1+τ22-τ122Z2-τ1τ12Z2ex,(43)Ξ42=-2e1-e5TZ1e1-e5-163e8-e12-e5TZ13e8-e12-e5,(44)Ξ43=-2e2-e6TZ2e2-e6-163e9-e22-e6TZ23e9-e22-e6-2e3-e7TZ2e3-e7-163e10-e32-e7TZ23e10-e32-e7,(45)Ξ5=e11e21TQ4e11e21+e12e22TQ5+Q6-Q4e12e22-e14e24TQ5e14e24-1-σde13e23TQ6e13e23,(46)Ξ6=Ξ61+Ξ62+Ξ63,(47)Ξ61=eyTσ12R4+σ122R5ey,(48)Ξ62=E4TR~4E4,R~4=diagR4,3R4,5R4,(49)Ξ63=E5E6TR~5S2∗R~5E5E6,R~5=diagR5,3R5,5R5,(50)Ξ7=Ξ71+Ξ72+Ξ73,(51)Ξ71=eyTσ122Z3+σ22-σ122Z4-σ1σ12Z4ey,(52)Ξ72=-2e11-e15TZ3e11-e15-163e18-e112-e15TZ33e18-e112-e15,(53)Ξ73=-2e12-e16TZ4e12-e16-163e19-e122-e16TZ43e19-e122-e16-2e13-e17TZ4e13-e17-163e20-e132-e17TZ43e20-e132-e17,(54)Ξ8=Ξ81+Ξ81T,(55)Ξ81=∑i=14Σe1i-e2iTHie2i+∑i=14∑j=i+14Σe1i-e1j-e2i-e2jTUije2i-e2j,(56)E1=e1-e2e1+e2-2e5e1-e2+6e5-12e8,(57)E2=e2-e3e2+e3-2e6e2-e3+6e6-12e9,(58)E3=e3-e4e3+e4-2e7e3-e4+6e7-12e10,(59)E4=e11-e12e11+e12-2e15e11-e12+6e15-12e18,(60)E5=e12-e13e12+e13-2e16e12-e13+6e16-12e19,(61)E6=e13-e14e13+e14-2e17e13-e14+6e17-12e20.Proof.
Construct the following LKF candidate:(62)Vt=∑i=17Vit,where(63)V1t=xtytTPxtyt+∑i=1n∫0yiλ1iρis-fis+λ2ifisds,V2t=∫t-τ1txTsQ1xsds+∫t-τ2t-τ1xTsQ2xsds+∫t-τtt-τ1xTsQ3xsds,V3t=τ1∫-τ10∫t+θtx˙TsR1x˙sdsdθ+∫-τ2-τ1∫t+θtxTsR2xs+τ12x˙TsR3x˙sdsdθ,V4t=∫-τ10∫θ0∫t+stx˙TuZ1x˙ududsdθ+∫-τ2-τ1∫θ-τ1∫t+stx˙TuZ2x˙ududsdθ,V5t=∫t-σ1tysfysTQ4ysfysds+∫t-σ2t-σ1ysfysTQ5ysfysds+∫t-σtt-σ1ysfysTQ6ysfysds,V6t=σ1∫-σ10∫t+θty˙TsR4y˙sdsdθ+σ12∫-σ2-σ1∫t+θty˙TsR5y˙sdsdθ,V7t=∫-σ10∫θ0∫t+sty˙TuZ3y˙ududsdθ+∫-σ2-σ1∫θ-σ1∫t+sty˙TuZ4y˙ududsdθand P>0,  Qi>0,  Rj>0,  Zk>0,  i=1,2,…,6,  j=1,2,…,5, and  k=1,2,…,4 are the symmetric positive-definite matrices and Λi=diagλi1,λi2,…,λin>0,  i=1,2, are the symmetric positive-definite diagonal matrices.
Calculating the derivative of the LKF along the solutions of GRN (11) yields(64)V˙t=∑i=17V˙it,where(65)V˙1t=2xtytTPx˙ty˙t+2Σyt-fytTΛ1+fTytΛ2y˙t=ζTtΞ11+Ξ11Tζt,V˙2t=xTtQ1xt+xτ1TtQ2+Q3-Q1xτ1t-xτ2TtQ2xτ2t-1-τ˙txτTtQ3xτt≤xTtQ1xt+xτ1TtQ2+Q3-Q1xτ1t-xτ2TtQ2xτ2t-1-τdxτTtQ3xτt=ζTtΞ2ζt,V˙3t=x˙Ttτ12R1+τ122R3x˙t+τ12xTtR2xt-τ1∫t-τ1tx˙TsR1x˙sds-∫t-τ2t-τ1xTsR2xs+τ12x˙TsR3x˙sds,V˙4t=x˙Ttτ122Z1+τ22-τ122Z2-τ1τ12Z2x˙t-∫t-τ1t∫stx˙TuZ1x˙ududs-∫t-τ2t-τ1∫st-τ1x˙TuZ2x˙ududs,V˙5t=ytfytTQ4ytfyt+yσ1tfyt-σ1TQ5+Q6-Q4yσ1tfyt-σ1-yσ2tfyt-σ2TQ5yσ2tfyt-σ2-1-σ˙tyσtfyt-σtTQ6yσtfyt-σt≤ζTtΞ5ζt,V˙6t=y˙Ttσ12R4+σ122R5y˙t-σ1∫t-σ1ty˙TsR4y˙sds-σ12∫t-σ2t-σ1y˙TsR5y˙sds,V˙7t=y˙Ttσ122Z3+σ22-σ122Z4-σ1σ12Z4y˙t-∫t-σ1t∫sty˙TuZ3y˙ududs-∫t-σ2t-σ1∫st-σ1y˙TuZ4y˙ududs,where Ξ11,  Ξ2, and Ξ5 are defined in (35), (36), and (45), respectively.
Using Lemma2 to estimate the R1-dependent single integral terms in V˙3t yields(66)-τ1∫t-τ1tx˙TsR1x˙sds≤-η1TtR~1η1t=ζTtΞ32ζt, where R~1 and Ξ32 are defined in (39) and (67)η1t=xt-xτ1txt+xτ1t-2v1txt-xτ1t+6v1t-12v4t.
Using Lemma1 to estimate the R2-dependent single integral terms in V˙3t yields(68)-∫t-τ2t-τ1xTsRxsds=-∫t-τtt-τ1xTsRxsds-∫t-τ2t-τtxTsRxsds≤-τ1τtv2TtR2v2t+32v5t-v2tTR22v5t-v2t-τ2τtv3TtR2v3t+32v6t-v3tTR22v6t-v3t=ζTtΞτtζt, where Ξτt is defined in (33).
Using Lemmas2 and 5, together with (30), to estimate the R3-dependent single integral terms in V˙3t yields(69)-τ12∫t-τ2t-τ1x˙TsR3x˙sds=-τ12∫t-τtt-τ1x˙TsR3x˙sds-τ12∫t-τ2t-τtx˙TsR3x˙sds≤-τ12τt-τ1η2TtR~3η2t-τ12τ2-τtη3TtR~3η3t≤-η2tη3tTR~3S1∗R~3η2tη3t=ζTtΞ33ζt, where R~3 and Ξ33 are defined in (40) and (70)η2t=xτ1t-xτtxτ1t+xτt-2v2txτ1t-xτt+6v2t-12v5t,η3t=xτt-xτ2txτt+xτ2t-2v3txτt-xτ2t+6v3t-12v6t.
Using Lemma8 to estimate the Z1-dependent double integral terms in V˙4t yields(71)-∫t-τ1t∫stx˙TuZ1x˙ududs≤-2xt-v1tTZ1xt-v1t+163v4t-xt2-v1tTZ13v4t-xt2-v1t=ζTtΞ42ζt,where Ξ42 is defined in (43).
Using Lemma8 to estimate the Z2-dependent double integral terms in V˙4t yields(72)-∫t-τ2t-τ1∫st-τ1x˙TuZ2x˙ududs=-∫t-τtt-τ1∫st-τ1x˙TuZ2x˙ududs-∫t-τ2t-τt∫st-τ1x˙TuZ2x˙ududs≤-∫t-τtt-τ1∫st-τ1x˙TuZ2x˙ududs-∫t-τ2t-τt∫st-τtx˙TuZ2x˙ududs≤-2xτ1t-v2tTZ2xτ1t-v2t-163v5t-xτ1t2-v2tTZ23v5t-xτ1t2-v2t-2xτt-v3tTZ2xτt-v3t-163v6t-xτt2-v3tTZ23v6t-xτt2-v3t=ζTtΞ43ζt,where Ξ43 is defined in (44).
Similarly, using Lemmas2, 5, and 8 to estimate the single and double integral terms in V˙6t and V˙7t yields(73)-σ1∫t-σ1tx˙TsR4x˙sds≤ζTtΞ62ζt,-σ12∫t-σ2t-σ1x˙TsR5x˙sds≤ζTtΞ63ζt,-∫t-σ1t∫stx˙TuZ3x˙ududs≤ζTtΞ72ζt,-∫t-σ2t-σ1∫st-σ1x˙TuZ4x˙ududs≤ζTtΞ73ζt,where Ξ62,  Ξ63,  Ξ72,   and Ξ73 are defined in (48)–(53).
Taking into account the assumption of the activation function, (13) and (14), the following inequalities hold [76, 77]: (74)his=2Σys-fysTHifys≥0,uijs1,s2=2Σys1-ys2-fys1-fys2TUij×fys1-fys2≥0, where Hi,i=1,2,…,4, and Uij,i=1,2,…,4,j=i+1,…,4, are the symmetric diagonal matrices. Thus, the following inequality holds:(75)Ht+Ut=h1t+h2t-σ1+h3t-σt+h4t-σ2+u12t,t-σ1+u13t,t-σt+u14t,t-σ2+u23t-σ1,t-σt+u24t-σ1,t-σ2+u34t-σt,t-σ2=ζTtΞ8ζt≥0, where Ξ8 is defined in (54).
Finally, combining (64), (65), (66), (68), (69), (71), (72), (73), and (75) yields(76)V˙t≤ζTtΞτt+∑i=18Ξiζt, where the related notations are defined in (31).
Therefore, if LMIs (31) and (32) hold, then the following holds for a sufficiently small scalar ϵ>0 based on convex combination method [78, 79]:(77)V˙t≤-ϵxt2+yt2 which shows the asymptotical stability of GRN (2) with time delay satisfying (5). This completes the proof.


## 4.2. Stability of GRN (11) with Delay Satisfying (6)

For some cases, the change rates of the time-varying delays are unmeasurable, that is, time delay satisfying (6). For this case, the following stability criterion can be derived by using the proposed WTDII (27), together with Lemmas 1, 2, 5, and 8, to estimate the derivative of the LKF.Theorem 10.
For given scalarsτi and σi,  i=1,2, GRN (2) with the time delay satisfying (6) and regulatory function satisfying (3) is asymptotically stable, if there exist symmetric matrices P>0,  Qi>0,  Rj>0,  Zk>0,  i=1,2,4,5,  j=1,2,…,5, and  k=1,2,…,4; diagonal matrices Λ1>0,  Λ2>0,  Hj>0,  j=1,2,3,4, Ulk>0,  l=1,2,…,4, and  k=l+1,…,4; and any matrices Si,  i=1,2, such that the following LMIs hold:(78)R~2i+1Si∗R~2i+1>0,i=1,2,Ψ3=Ξτtτt=τ1+∑i=1,3,4,6,7Ξi+Ξ¯2+Ξ¯5≤0,Ψ4=Ξτtτt=τ2∑i=1,3,4,6,7Ξi+Ξ¯2+Ξ¯5≤0,where Ξi,  i=1,3,4,6,7, are defined in Theorem 9 and (79)Ξ¯2=e1TQ1e1-e2TQ1-Q2e2-e4TQ2e4,Ξ¯5=e11e21TQ4e11e21+e12e22TQ5-Q4e12e22-e14e24TQ5e14e24.Proof.
The above stability criterion can be obtained by settingQ3=0 and Q6=0 in Theorem 9.


## 4.3. Some Remarks

This part gives some remarks for the above criteria.Remark 11.
During the proof of the above two stability criteria, the double integral terms arising in the derivative of the LKFs are estimated by using the proposed WTDII, that is, Lemma8. As discussed in Section 3, the WTDII is tighter than the widely used JBDII (17), which was used for the GRN [19, 34], and the recently developed WBDII (18), which has not been used for the GRN. Thus, the proposed criteria are less conservative than the ones reported in [19, 34].Remark 12.
Compared with the literature, more information of regulatory function has been used during the proof of criteria. Specifically, in the literature, only (14) is used during the estimation of the derivative of the LKF, while, in this paper, extra information of regulatory function (13) is also used for estimating task. It has been proved in [77] that such additional information is helpful to reduce the conservatism.Remark 13.
The conditions given in Theorems9 and 10 are in the form of LMI. Such LMI conditions can be easily checked by using MATLAB/LMI toolbox [80]. One can refer to [81–83] for more details.Remark 14.
Although this paper has just investigated the asymptotical stability, the proposed method can be extended to the robust stability analysis by taking into account the parameter uncertainties and/or noises of the GRNs. Moreover, the proposed method can also be extended to other problems discussed in Section2, like controller synthesis, state estimation, filter design, passivity analysis, and so on [13, 59–70].


## 5. Illustrative Example

In this section, an example will be presented to illustrate the effectiveness of our results. As mentioned in Section2, the important aim of the stability analysis of delayed GRNs is to determine the MADBs. And the stability criterion that provides bigger MADBs is less conservative than the one that gives smaller ones. Therefore, the advantages of the proposed criteria are demonstrated via the comparison of the MADBs calculated by various criteria. Moreover, the index of the number of variables (NoV) is applied to show the complexity of criteria.Example 1.
For the GRN model which is theoretically predicted and experimentally investigated inEscherichia coli in [4], the genetic network is composed of three repressilators(lacl,tetR, andcl) which form a cyclic negative feedback loop, each repressor protein inhibits the transcription of its downstream repressor gene, as shown in Figure 2, the protein of lacl represses the gene transcription of tetR, and the protein of tetR inhibits the gene transcription of cl simultaneously, and, finally, the transcription of lacl is inhibited by cl, which completes the cycle.
The kinetics of the genetic network are modelled as the GRN (2) with the following parameters [19]: (80)A=diag3,3,3,C=diag2.5,2.5,2.5,W=00-2.5-2.5000-2.50,D=0.80.80.8,bix=x21+x2,i=1,2,…,n.Figure 2
The repressilator network.It follows from (9) and (29) that (81)Σ=diag338,338,338.(
1) Calculation Results. The first study case is that the changing rates of the time-varying delays are measurable; that is, delays satisfy (5). Assume that σ1=0.1,  σ2=0.3,  σd=0.7, and  τd=1.5 [19], and the MADBs of τ2 with respect to various τ1 obtained by the proposed criteria are given in Table 1, where the MADBs reported in the literature are also listed for comparison.Table 1
The MADBs ofτ2 for various τ1 and the NoVs of various criteria.
Criteria
NoVs
τ
1
0.1
0.5
1
[29, 31, 34]
—
<5.5
<5.9
<6.4
[19]
40.5n2+16.5n
5.5
5.91
6.41
Theorem9
32n2+22n
9.2681
9.6682
10.1681The second study case is that the changing rates of the time-varying delays are nonmeasurable; that is, delays satisfy (6). Assume that σ1=1 and  σ2=2, and the MADBs of τ2 with respect to various τ1 obtained by the proposed criteria, together with the ones provided by the least literature [19], are given in Table 2.Table 2
The MADBs ofτ2 for various τ1 and the NoVs of various criteria.
Criteria
NoVs
τ
1
0
1
2
[19]
38
n
2
+
15
n
2.3101
3.3101
4.3102
[19]
29.5
n
2
+
20.5
n
4.1647
5.1647
6.1646Moreover, the NoVs of criteria reported in the least literature [19] and that of criteria established in this paper are also given in tables to compare the computation complexity.From the results in the tables, it can be easily found that the proposed stability criteria can provide the larger MADBs for two cases compared to those given in the existing literature. It shows that the proposed criteria are indeed less conservative than the ones reported in the literature. On the other hand, it is found that the NoV of the proposed criteria (Theorem9) is smaller than the one reported in [19], 40.5n2+16.5n-32n2+22n=8.5n2-5.5n>0 and 38n2+15n-29.5n2+20.5n=8.5n2-5.5n>0 for any n. Both of those observations show the advantages of the proposed criterion.(
2) Simulation Verification. From the given parameters, the equilibrium points of the GRN can be obtained as (82)m∗=0.7840,0.7840,0.7840,p∗=0.2509,0.2509,0.2509.Simulation studies for the following two types of time-varying delays are carried out.Case 1.
The initial conditionsmt=0.70,0.85,0.80T,  t∈-10.1681,0, and pt=0.15,0.20,0.30T,  t∈-0.3,0, and the following delays satisfy σ1=0.1,  σ2=0.3,  σd=0.7,  τ1=1,  τ2=10.1681, and  τd=1.5:(83)τt=9.1681sin2⁡0.1636t+1,σt=0.2sin2⁡3.5t+0.1.Case 2.
The initial conditionsmt=0.70,0.85,0.80T,  t∈-6.1746,0, and pt=0.15,0.20,0.30T,  t∈-2,0, and the random delays satisfy σ1=1,  σ2=2,  τ1=2,  τ2=6.1746.Based on Tables1 and 2, the GRN with the above delays, respectively, is stable. The trajectories of the concentrations of mRNA and protein are shown in Figures 3 and 4. The results show that they are stable at their equilibrium points.Figure 3
The trajectories of concentrations of mRNA and protein for Case1.Figure 4
The trajectories of concentrations of mRNA and protein for Case2.


## 6. Conclusions

This paper has investigated the stability of the GRN with time-varying delays, and its contributions have been revealed from two aspects. The novel WTDII has been developed for the estimation of the double integral terms, and it has been also proved to be tighter than the widely used JBDII and the recently developed WBDII for the same task. Then, with benefit from the WTDII, two LMI-based stability criteria with less conservatism have been derived for checking the stability of the GRN with time delays. Finally, the advantages of the proposed inequality and the established criteria have been verified through an example.



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