An advanced delay-dependent approach of impulsive genetic regulatory networks besides the distributed delays , parameter uncertainties and time-varying delays ∗

Selvakumar Pandiselvi, Raja Ramachandran, Jinde Caoc,1, Grienggrai Rajchakit, Aly R. Seadawy, Ahmed Alsaedi Department of Mathematics, Alagappa University, Karaikudi-630 004, India Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi-630 004, India School of Mathematics, Southeast University, Nanjing 211189, China jdcao@seu.edu.cn Department of Mathematics, Maejo University, Chiang Mai, Thailand Department of Mathematics and Statistics, Taibah University, Medina, 41 477, Saudi Arabia Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

by the distributed delays and time-varying delays.By taking the time-varying delays into account, the stability criteria are granted by using the delay-dependent approach, convex combination and free-weighting matrix method combined with Jensen's inequality.Finally, numerical simulations are worn to show the less conservativeness of the attained results.The significant of the manuscript is given as follows: (i) An advanced delay-dependent genetic regulatory networks with parameter uncertainties, which includes distributed delays and impulsive effects are investigated using delay-dependent approach.(ii) Based on the contemporary Lyapunov-Krasovskii functional and integral inequality techniques, some sufficient conditions for asymptotical stability of delaydependent genetic regulatory networks are derived in the form of LMIs.In addition, compared to the existing results, the derived outcomes are different and advanced.(iii) In this chapter, the feasibility of the obtained LMIs for asymptotic stability can be easily solved by the aid of MATLAB LMI control toolbox.(iv) By handled the time-varying delay and distributed time-varying delay terms in our concerned genetic regulatory networks, the allowable upper bounds of timedelays are maximum in comparison with some existing literatures, see Table 1 in Example 2. This can be expressed that the approach developed in this chapter is more effective and less conserved.
The remaining things of this work is classified well as follows: In Section 2, GRNs with distributed delays and impulses are described, and we introduced some assumptions and lemmas for proving our required criteria.In Section 3, we define an advanced Lyapunov-Krasovskii functional, which is in triple integral form, and derived sufficient conditions, which can be expressed in the form of LMIs.Additionally, two mathematical examples are shown in Section 4 to demonstrate the advantages of our stability conditions.Lastly, conclusions are shown in Section 5.
Notations.The superscript "T" act as the transpose of matrix.R n indicates the Euclidean space with n dimensions, and R n×m denotes the set of all n×m real matrices.I means the identity matrix of appropriate dimensions.diag{•} is the diagonal matrix.The symbol " * " denotes the symmetric term.In this paper, the matrices are assumed to be with appropriate dimensions.
(1) Nonlinear Anal.Model.Control, 23 (6):  Here m i (T ) and p i (T ) are the concentrations of mRNAs and proteins, respectively.g 1i and g 2i are the degradation rates of mRNAs and proteins, respectively.h 2i defines the translation rate, ξ(T ) and η(T ) are the transcriptional and translational delay, respectively.The regulatory function is defined as h 1i , which is nonlinear, and the sum logic is h 1i (p 1 (T ), p 2 (T ), . . ., p n (T )) = n j=1 h ij (p j (T )), which is in [12,40].In [6], a monotone function of the Hill form h ij (p j (T )) is defined as where j is the transcription factor, β ij is a bounded constant, γ j is a positive scalar, H f j is the Hill coefficient.Therefore, Eq. ( 1) can be changed accordingly as where and U i is the basel rate, which is defined as Equation (2) changed into the compact matrix form, we have where ) is bounded with H f j 1 and have the continuous derivatives for x 0. Completely the direct algebraic directions, we have Let (m * , p * ) is an equilibrium point of the GRN (3).Then we have (3) will be rewritten as where , the initial functions ψ(•) and π(•) are continuously differentiable on [ , 0].Now, we discuss the following impulsive genetic regulatory networks with distributed delays and time-varying delays: Here r and l are constants.Ĵ1 (y(T )) = (J 11 (y 1 (T )), . . ., J 1n (y n (T ))) T denotes the activation function, T k denotes the sequence of time, which satisfies 0 The impulses are denoted by x( k ) and y( k ).D 1 , D 2 ∈ R n are the sudden change effects of the state of the above system.Assumption 1.A monotonically increasing function fi (•), i ∈ {1, 2, . . ., n}, with saturation satisfies , where q i are known constants.

Asymptotic stability criterion
In this portion, we discuss the asymptotic stability criterion for impulsive GRNs with distributed delays and time-varying delays by using matrix analysis techniques and Lyapunov stability theory.
On the other hand, from ( 6) and Theorem 1 conditions, we note that System (5) with impulsive effect is globally asymptotically stable.Hence, the proof is completed.
Proof.The proof follows from Theorem 1.Then, the integral terms are different compared with the existing works [33,34].
Remark 2. In this work, some convex combination technique and free-weighting matrix method is approached.Because, convex combination method helps us to reduce the decision variables in LMIs, which is the relevance lemma of Jensen's inequality and freeweighting matrix assist to decrease the conservatism of stability criterion than the existing literature.
Remark 3. As much as know, all the existing results concerning the dynamical behaviors of genetic regulatory networks [20,33,41] have not considered the global asymptotic stability performance in the mean square and time-varying delayed situation, which are investigated via LMI approach in this paper.Therefore, our conclusions are new when compared to the previous results.
Remark 4. In this paper, we also consider the relationship between time-varying delays and their upper bounds.In order to obtain the maximum upper bounds of distributed delays and time-varying delays, we used some inequality techniques, see Example 1.Hence, the techniques and methods used in this paper may lead to less conservative criterions.To this evident, Table 1 shows the maximum upper bound of ξ, which guarantees the global asymptotic stability of the addressed genetic networks (5).These tables demonstrate the effectiveness of our proposed method.

Numerical simulations
In this portion, twin examples with simulations are provided to demonstrate the usefulness of the obtained results.
By Theorem 1 we can obtain the following feasible parameters.From Table 1 our work is more effective and less conservative than the existing works.Due to space consideration, we only provide a part of the feasible solutions here.proteins with impulsive effects are illustrated in Fig. 1  The regulatory function is taken as g(y) = y 2 /(1 + y 2 ).It can be easily checked that the derivative of g(y) is less than 0.65.Assume that the feedback regulation delay η(T ) = 2 and the translation delay ξ(T ) = 2. Then η 1 = 0.3, η 2 = 0.5, ξ 1 = 0.45, ξ 2 = 2.5, λ = 0.2 and δ = 0.4 can be obtained.
By Theorem 2 we can obtain the following feasible parameters.Due to space consideration, we only provide a part of the feasible solutions here.

Conclusions
In this work, we have investigated the global asymptotic stability problem for a class of uncertain genetic regulatory networks with distributed delays, time-varying delays and impulses.By constructing new Lyapunov-Krasovskii functional with triple integral terms, sufficient stability analysis has been rooted in terms of LMIs.By applying convex combination technique and free-weighting matrix method, conservatism of the stability criteria have been diminished greatly.Lastly, the feasibility and advantages of the developed results have been demonstrated by the numerical simulation examples.
In the near future, we plan to work with stabilization of stochastic genetic regulatory networks with leakage and impulsive effects in finite-time stable sense.Also, we will try to present a real life model to justify our theoretical concepts for the considered GRN.

Remark 1 .
In the Lyapunov-Krasovskii functional, the triple integral terms )U 4 ẏ(s) ds dµ dθ are introduced with hope to reduce the less conservativeness of the advanced results.In addition, the improved vector υ(T ) consists of the terms y(s) ds T .

Figure 1 .
Figure 1. mRNA and Protein concentrations with impulsive effects.

Figure 2 .
Figure 2. mRNA and Protein concentrations without impulsive effects.

From
Theorem 2 one can conclude that the continuous-time GRNs(23) with impulsive effects are globally asymptotically stable.The concentrations of mRNAs and proteins with impulsive effects are illustrated in Fig.3with the initial conditions x(0) = [0.01− 0.01] T , y(0) = [0.3− 0.2] T , and the concentrations of mRNAs and proteins without impulsive effects are illustrated in Fig. 4 with the initial conditions x(0) = [0.01−0.02] T and y(0) = [0.30.1] T .

Figure 3 .
Figure 3. mRNA and Protein concentrations with impulsive effects.

Figure 4 .
Figure 4. mRNA and Protein concentrations without impulsive effects.

Table 1 .
Comparisons of upper bounds of time-delay ξ(T ) for various ξ 1 .