Improved Delay-Dependent Stability Criterion for Genetic Regulatory Networks with Interval Time-Varying Delays via New Lyapunov Functionals

In this work, the stability analysis problem of the genetic regulatory networks (GRNs) with interval time-varying delays is presented. In the previous works, the constructions of Lyapunov functional have usually been in simple Lyapunov functional, augmented Lyapunov functional, and multiple integral Lyapunov functional. erefore, we introduce new Lyapunov functionals expressed in terms of delay product functions. New delay-dependent sufficient conditions for the genetic regulatory networks (GRNs) are established in the terms of linear matrix inequalities (LMIS). In addition, a numerical example is provided to illustrate the effectiveness of the theoretical results.


Introduction
In recent years, genetic regulatory networks (GRNs) have achieved popularity in the biological and biomedical applications since it can effectively reflect the living organisms of molecular and cellular levels [1,2]. e researchers in various fields have wildly emphasized on GRNs. e study of living organism processes is one of the main challenges in our postgenomic era. Mathematical modelling of GRNs is a tool in the study of mechanism of control gene expression in an organism. e platforms of the studied model have various types such as Boolean models [3], differential equation model [4], stochastic equations [5,6], and fractional-order dynamical systems [7]. In Boolean models, the expression of each gene is described to be either ON or OFF, no intermediate activity level is ever taken into consideration, and the state of gene is determined by the Boolean function of the states of other related genes [3]. In contrast, differential equation models described the continuous dynamical behaviors of GRNS between the concentration of gene product such as messenger ribonucleic acids (mRNAs) and proteins as the unknown functions. Consequently, differential equation models have drawn the attention of research studies to describe the gene regulatory process of organisms on the molecular level.
In experimental study in biochemistry, the gene expression was studied which consists of transcription and translation processes. As we known that the transcription and translation processes are slow reaction, time delay is inevitable in genetic regulatory network. In a biochemistry experiment on mice, it has been proved that there exists the time lag of about 15 minutes in the peaks between the mRNA molecules and the proteins of the gene Hes1 [8].
us, time delay has been attended in the study of mathematical modelling of GRNs.
Time delays in mathematical systems have received attention from researchers in the stability analysis which provide a poor performance or even instability of the relevant system [9][10][11][12][13][14][15]. In this reason, the stability analysis problems of genetic regulatory network with time delay have been reported in the literature [6,8,[16][17][18][19][20][21][22][23][24][25][26][27][28]. ey proposed new stability condition and reduced the possible conservativeness caused by the time delay. First, the local stability criteria for GRNs with constant delay were established in [8]. However, it is not sufficient to describe the behavior dynamic of nonlinear GRNs. e global asymptotic stability of GRN with SUM regulatory functions has been studied [16,20,21,25,26]. An increasing number of research studies about more complex modelled GRNs has been received attention in the following years, e.g., GRNs with distributed delayed [27], GRNs with nonlinear disturbance [17], leakage delay involved GRNS [22], and impulsive perturbation [24]. e stability problems of GRNs, such as stochastic stability [19], robust stability [18], exponential convergence analysis [28], and discrete-time stochastic stability [6,23] have been also reported in the literature. e stability analysis of systems with time-varying delays is the major problem of stability in systems. e conservatism stability condition and the maximum allowable delay bound have been recognized as the most important index. As we know that there are many methods to reduce the conservatism in stability criterions such as the development of the integral inequalities, i.e., the Jensens inequality, the Wirtinger-based inequality [16,20], delay partitioning approach, and free-weighting matrix were used in [29], generalized zero equalities [30], and Lyapunov function approach [31]. As we know, in order to get the less conservative results, the main efforts were concentrated on two directions. One of the most popular approach is Lyapunov function. e construction of Lyapunov functions generally has two types, i.e., augmented Lyapunov functional approach (ALFA) and multiple integral Lyapunov functional approach (MILFA), which have been mainly utilized to propose new Lyapunov functionals [30][31][32][33][34][35][36][37][38]. e ALFA is more state information into the vector of the positive quadratic term, for example, ξ T (t)Pξ(t) with ξ(t) including x(t), y(t), f(x(t)), t t− τ 1 x(s)ds, and t t− σ 1 x(s)ds, while the MILFA is used in multiple integral terms of a positive quadratic term as a Lyapunov function. In [32,33], they established a new Lyapunov functionals which are delay product-type functionals and lead to less conservative result in time-varying delay systems. Furthermore, the construction of MILFA also led to least conservatism and decision variables for uncertain systems with interval timevarying delay [30][31][32][33][34][35][36][37][38].
Motivated by the above discussion, the most important concerns are the improved stability conditions for genetic regulatory networks with time-varying delays. In this paper, we will present new Lyapunov functions which are extended from [32,33], by adding two quadratic terms. One is double integral terms of a positive quadratic term and the other one is a quadratic term which does not need to meet positive definite and would relax the stability conditions. Moreover, the development of double integral inequality is utilized to derive sufficient conditions in terms of linear matrix inequalities (LMIS). Finally, a numerical example is provided to illustrate the effectiveness of the theoretical results.
Notations: throughout this paper, R n and R m×n denote the n-dimensional Euclidean space and the set of all m × n real matrices, respectively. S n + denotes a set of positive definite matrices with n x n dimensions. X > 0 means that the matrix X is a real symmetric positive definite matrix. I denotes the identity matrix with appropriate dimensions, diag . . .
{ } denotes column matrix. e * in the matrix represents the elements below the main diagonal of a symmetric matrix. Sym X { } indicates indicates X + X T For X ∈ R m×n .

Problem Formulations
Genetic regulatory networks (GRNs) are composed of a number of genes that interact and regulate the expression of other genes by proteins (the gene product). e dynamic behavior of genetic regulatory networks with variable delays can be described by the following state equations [4]: where m i (t) and p i (t) denote the concentration of mRNA and protein of the ith node at time t. a i and c i are positive real numbers that present the degradation rates of mRNA and protein, respectively. d i is the translation rate. σ(t) and τ(t) are transcriptional and translational delays, respectively, and function f i denotes the regulatory function or transcription function, which is generally nonlinear function but has a form of monotonicity of each variable. It is usual to assume that the regulatory function satisfies the following SUM logic [26]: And the function b ij (p j (t)) is generally expressed by a monotonic function of the Hill form: if transcription factor j is an activator of gene i, where h j is the Hill coefficient, β j is a positive scalar, and α ij is a bounded constant denoting the dimensionless transcriptional rate of transcription factor j to gene i. We note that us, GRNs (1) can be rewritten as where g j (x) � (x/β i ) h j /1 + (x/β i ) h j is a monotonically increasing function, J i � j∈V i α ij , and V i is the set of all the transcription factor j which is a repressor of gene i. e matrix W � (w ij ) ∈ R n×n is the coupling matrix of GRNs, which is defined as follows: In compact matrix form, (5) can be written as where Let (m * , p * ) be the equilibrium point of GRNs (7), Shifting equilibrium point (m * , p * ) to the origin, using the following transformation (7) can be transformed into the following form: where f(y(t)) � g(p(t) + p * ) − g(p * ). Since the function g is monotonically increasing function with saturation, it satisfies that, for all a, b ∈ R with a ≠ b, When g(·) is differentiable, the above inequality is equivalent to 0 ≤ g d(a)/d(a) ≤ k. From the relationship of f(·) and g(·), we know that f(·) satisfies the sector condition [39], or equivalently Assumption 1. e assumptions of time-delay conditions are where 0 < τ 1 ≤ τ 2 , 0 < σ 1 ≤ σ 2 , τ d > 0, and σ d > 0.
Assumption 2. Let g i : R ⟶ R, i � 1, . . . , n, be monotonically increasing functions with saturation and moreover satisfy Remark 1. e modelling of GRNs is largely dependent on powerful tools of mathematics theory. e differential equation model has drawn a lot of research attention since variables in gene dynamics are usually the concentrations of gene product (messenger ribonucleic acids (mRNAs) and proteins). It has been shown that the time delays may play an important role in the predictions of the dynamics of the mRNA and protein concentrations; GRNs models without consideration of time delays may provide wrong predictions. erefore, it is significant to study the stability of delayed GRNs and sufficient stability conditions. However, the conservatism stability condition and the maximum allowable delay bound have been recognized as the most important index. erefore, in this research, we have improved delay dependent stability criterion for genetic regulatory Mathematical Problems in Engineering network (GRNs) with interval-time varying delays via new Lyapunov function by constructing a new Lyapunov functionals in Section 4. In addition, we have established one integral and double integral inequalities to estimate. erefore, the results had been established that the stability criterion lead to less conservativeness.
To proceed, the following lemmas are introduced which will be useful for further derivations.
Lemma 1 (Jensen integral inequalities, see [40]). Let w: [a, b] ⟶ R n be a differentiable function. For any given matrix R > 0 and a ≤ s ≤ b, the following inequalities hold: Lemma 2 (Wirtinger-based integral inequality, see [41]). Let w: [a, b] ⟶ R n be a differentiable function. For any given matrix R > 0 and a ≤ s ≤ b, the following inequality holds: where Lemma 3 (relaxed double integral inequality, see [35]). Let w: [a, b] ⟶ R n be a differentiable function. For any given matrix R > 0 and a ≤ s ≤ b, the following inequality holds: where Lemma 4 (Schur complement, see [42]).

Improved Integral Inequalities
In this section, we propose the following results which will be used to the development of the inequality.
Lemma 5 (extended relaxed one integral inequality, see [42]). Let w: and any matrices S i ∈ R 2n×2n , i � 1, 2, the following integral inequality holds: where Proof. By setting a function f(s, where a and b are constants, then the following equations are derived: Furthermore, based on Schur complement, for symmetric matrices R i > 0, i � 1, 2, and any matrices X j , j � 1, 2, 3, 4, with appropriate dimensions, the following inequalities hold: which lead to Define matrices X i , i � 1, 2, 3, 4, as follows: where . en, the following equivalent relations are taken into account similar to [42]: Mathematical Problems in Engineering 5 To sum up with multiplication, we have Mathematical Problems in Engineering us, is completes the proof inequality (20).
□ Remark 2. Consider the inequality in Lemma 4 [42] can be written in the following form: e advantage of Lemma 5 can be concluded as follows: (1) Equation (20) can be reduced to Lemma 4 [42] when S 1 � S 2 and the lower bound of time delay is zeros, i.e., α(t) ∈ [0, h]. However, Lemma 4 [42] cannot be used to estimate in the time-delay interval [42] can be written as βS 1 + αS 2 which leads to exactitude in the estimation of equation (20). (3) e matrices R 1 � R 2 can deal with Lemma 4 [42].
Lemma 5 can be applied in case of the different matrices, in which Lemma 4 [42] cannot be estimated in this case.

Remark 3.
From the estimation of Corollary 5 [41] and reciprocally convex combination lemma [43], the estimation of integral (20) is as follows: When (20), two aspects can be shown: which is relaxed to βS 1 + αS 2 is substituted in (20). (2) e estimation gap between (20) and (30), which is calculated from the right-hand side of (20) and (30), is, respectively, defined by Γ 1 and Γ 2 ; then, is can be shown that an upper bound of inequality (20) is more than (30). So, we can see that (20) is less conservative than (30).
and any matrices S, and the following integral inequality holds: where Mathematical Problems in Engineering 7 Proof. By , where a and b are constants, then the following equations are derived: Furthermore, based on Schur complement, for symmetric matrices R i > 0, i � 1, 2, and any matrices X j , j � 1, 2, 3, 4, with appropriate dimensions, the following inequalities hold: which lead to . Define matrices X i , i � 1, 2, 3, 4, as follows: where L i , i � 1, 2, 3, 4, are appropriate dimensional matrices, Carrying out simple algebraic calculation, then Mathematical Problems in Engineering us, is completes the proof of inequality.
□ Remark 4. From Lemma 3, the following double integral inequality can be written as Compared with inequality (32), two aspects can be shown: (1) If S 1 � 0 and S 2 � 0, then (32) is reduced to (40) which it is a more general form. Based on Schur complement, (32) can be written as (i) It provides the freedom matrices S 1 and S 2 , which lead to reduced conservativeness. (2) e right-hand side of (32) has information of nonlinear delayed terms α 2 1 (t) and α 2 2 (t). Time-delay terms appear in this term. It is the advantage to deal with a larger time delay system which can help to reduce conservatism.

Novel Lyapunov Functionals
In this section, we propose two novel Lyapunov functionals, which are the main contributions of this paper. Proposition 1. For system (10) with given scalars σ 1 , σ 2 , τ 1 , and τ 2 and positive definite matrices M i and N i , i � 1, 2, 3, 4, the following function can be Lyapunov functional candidate: where Proof. By Lemma 1, V J (t) is a positive definite function which completes the proof. □ Remark 5. e construction of Lyapunov functionals is usually in the form of ALFA and MILFA. e special formed function (42) in Proposition 1 is based on Jensen integral inequality and relax double integral inequality, respectively. It is obtained in the following points: . ese terms would play an important role in relaxing the stability condition.
Remark 6. e new Lyapunov functional has different forms compared with [32,33], i.e., there are two double integral ese terms are increasing the information in stability conditions which can help in delay conditions for GRNs with interval time-varying delays.

Main Results
In this section, we analyse the asymptotic stability of genetic regulatory networks (10) with time-varying delays. e main theorem given below shows that the stability criteria can be expressed in the terms of the feasibility of the linear matrix inequalities (LMIs).
For simplification, the following vectors and matrices are defined for later use: dsdθ,

Mathematical Problems in Engineering
Proof. We first consider the following Lyapunov functional candidate: where V 0 (t) � 4 i�1 V i (t) and V J (t) is defined in Proposition 1: Moreover, we consider the following simple calculation results: where 14 Mathematical Problems in Engineering en, taking time derivatives of V(t) along the trajectory of system (10) yields: where Mathematical Problems in Engineering And where

16
Mathematical Problems in Engineering So, the time derivative of V J (t) can be rewritten as follows: where Mathematical Problems in Engineering Utilizing Lemmas 2 and 3, the single integral terms and the double integral terms of V .
, respectively, can be as follows: where According to (11), it follows that there exist diagonal matrices H 1 and H 2 ; then, Mathematical Problems in Engineering where Utilizing Lemmas 5 and 6, then the single and double integrals in (80) are estimated as follows: Mathematical Problems in Engineering Combining (82)-(84), we have
Remark 8. Utilizing Lemma 5 in double integral terms (84) and (85) affects LMI condition (91) which is in the form of nonlinear time-delayed terms τ 2 (t) and σ 2 (t). e stability criterion is obtained unfeasible by MATLAB LMI tools. However, it can be easy to find negative conditions for a quadratic function with time-varying delays by using the convex function [44].

Remark 9.
e estimation of the derivative of the new Lyapunov functions V 3 (t) and V 4 (t) are considered together which appear in (82)-(85). By employing extended relaxed one integral inequality (Lemma 5) and the extended relaxed double integral inequality (Lemma 6), it ensures that the conservativeness in stability condition is reduced compared with [32,33]. Moreover, the estimation of the double integral terms in the derivative forms (84) and (85) guarantees our approach to be less conservative than one in [35]. e effectiveness of this method will be demonstrated in numerical simulation.

Remark 10.
e stability criterion of eorem 1 in the form LMIs (45)-(51) can be easy to examine by using LMI toolbox in MATLAB [45].

Remark 11.
e improved stability conditions by constructing new Lyapunov functionals are based on LMIs and the dimension of the LMIs depends on the number of the genes in GRNs. us, the computational burden problem goes up. is problem is the issue in studying needs of LMI optimization in applied mathematics and the optimization research. Hence, further new techniques are developed to reduce the conservativeness caused by the time delays such as the delay-fractioning approach.

Numerical Example
In this section, we provide numerical example with a simulation to demonstrate the effectiveness of our results.   respect to various τ 1 which is obtained in Table 1. Moreover, the time-varying delays τ(t) and σ(t) are assumed to τ(t) � 5.4sin 2 (5/18t) + 0.1 and σ(t) � 0.2sin 2 (3.5t) + 0.1. So, the trajectories of mRNA and proteins are given in Figure 1

Conclusions
In this paper, the stability analysis problem for genetic regulatory networks (GRNs) with time-varying delays is studied. e new Lyapunov functionals have been established for deriving the stability criterion for genetic regulatory networks with time-varying delays to reduce the conservativeness of the stability condition. is paper focuses on the construction of new Lyapunov functionals based on Jensen's inequality and relaxed double integral inequality. By employing Lemmas 5 and 6, new delay dependent sufficient conditions are expressed in the terms of linear matrix inequalities (LMIs) to ensure that it is asymptotically stable for GRNs with time-varying delays. Finally, a numerical example was given to illustrate the effectiveness of the theoretical result and to show less conservativeness than some existing results in the literature.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.