A Razumikhin approach to stability and synchronization criteria for fractional order time delayed gene regulatory networks

1 Department of Mathematics, Alagappa University, Karaikudi-630 004, India 2 Department of Mathematics, Near East University TRNC, Mersin 10, Turkey 3 Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi-630 004, India 4 Department of Mathematics, Cankaya University, Ankara 06530, Turkey, and Institute of Space Sciences, Magurele-Bucharest, R 76900, Romania 5 School of Mathematics, Southeast University, Nanjing 211189, China 6 Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea 7 Faculty of Automatic Control, Electronics and Computer Science, and Department of Automatic Control, and Robotics, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland


Introduction
Fractional calculus, which is the general development of old calculus, has a remarkable ability to examine the world around us through the areas of both applied and pure mathematics, respectively. It has also gained considerable attention from the genetics, physics, chemistry, and computer research communities. The best way to make discoveries in mathematics is by adding some new theories to evaluate the current results. Frequently the results may fail. But, often it may pave a way to a new field of research works. Similarly, the fractional-order calculus is a perfect answer to the nonsensical question "what if the derivative order is non-integer?" by Leibnitz during the late sixteenth century. One of the most important properties of the differential fractional equation is its ability to track a motion of an object continuously and instantly of non-local nature. Besides, it contains more memory of the systems. The fractional-order models are easier to understand the complexity of the dynamic system with greater precision contrast to the integer-order differential models. Due to the memory properties, several researchers are integrating the memory properties into nonlinear dynamical systems, and lot of important results about fractional order nonlinear dynamical systems have been reported in recent literature, see Ref [3,11,12].
In an organism, gene expression is regulated by RNA, DNA, protein, and tiny molecules. Gene Regulatory Network (GRNs) defined the interconnections between these two. GRNs are viewed as complex networks. Each gene is regarded as a node, and the regulatory link between these genes is known as a relation between the nodes. To unleash the cure for deadly diseases such as cancer and AIDS, a greater understanding of the complex networks of GRNs is essential. Time delays in both biological and artificial neural networks are unavoidable because of the limited speed of information processing. Generally, there are typically two types of gene regulatory networks, such as the boolean model and the continuous model. The continuous model is commonly used for the study of GRNs, and several important results about GRNs with time delays had been well documented, see Ref [1,16,19].
On the other hand, stability theory is the flexible branch of science and engineering that deals with the behavioral effect for linear and nonlinear systems of dynamic structures. The investigation on various stability problems of time-delayed GRNs (TDGRNs) are accounted [20,24,26,36]. Paper by Luo et al., has analyzed the existence and Lagrange stability of TDGRNs in Lyapunov's sense based on novel algebraic method and stability theory [20]. In [24], the authors demonstrated the delay-dependent finite time stability issues of TDGRNs with impulses based on the LMI approach and Lyapunov stability theory. In [26], the problem of the stability criterion of TDGRNs with impulsive effects was analyzed. By employing the LMI techniques, convex combination approach, and Lyapunov-Krasovskii functional, the sufficient conditions to assure the global asymptotic stability analysis of the proposed TDGRNs model. In [36], the authors researched the global exponential delay-dependent stability criterion of TDGRNs under distributed delays based on LMI techniques and Lyapunov-Krasovskii functional approach.
Nowadays, the synchronization of dynamic systems is advancing as a dominant research field and has drawn a great deal of interest from researchers of diverse field. Its application found in many fields like secure communication, image, and signaling process. Many types of synchronization results are available in recent works including Mittag-Leffler, asymptotic, quasi and pinning synchronization, and so forth [7-10, 17, 18, 30, 33]. GRN synchronization is essential for knowing the synergic behavior between the more than one gene networks through the connections of gene signals and their products. The advantages of researching GRN synchronization are acquiring knowledge about a gene's internal processes even at cellular levels.
Some significant results about synchronization for time-delayed GRNs (TDGRNs) had been studied in recent years. For example, by means of observer-based non-fragile and linear feedback control, LMI techniques, and Lyapunov-Krasovskii method, Ali et al. demonstrate some sufficient criteria for the global asymptotic synchronization issues of TDGRNs under uncertainty [2]. By exploiting finite-time control techniques, robust analysis, and theory of finite-time stability, Jiang et al. investigate stochastic synchronization in finite time analysis for TDGRNs under parameter uncertainties [14]. Depending on the pinning control strategy, some famous inequality approaches, matrix theory, and event-triggered condition, Yue et al.analyze the cluster synchronization analysis for GRNs with coupling terms [35].
The research works in fractional order gene regulatory with delay arguments has been undergone exciting development in recent years, and some meaningful scientific results had been obtained. By using a hybrid control approach, the authors experimented with the bifurcation analysis for FODGRNs [15]. By utilizing the principle of Banach contraction mapping and absolute Lyapunov functional with 1-norm, the authors exhibited the several stability criteria of fractional order GRNs [25]. Depending on the principle of Banach contraction, Lyapunov functional with 1-norm, linear feedback, and adaptive feedback techniques, the sufficient criteria to ensure the finite time delay-independent synchronization problem of considered FODGRNs via Razumikhin approach [23]. By employing the diffusion and stability theory, the authors have demonstrated with the issues of local stability and instability criteria for bifurcation diffusion FODGRNs [29]. Unfortunately, there is no work done on the existence, stability, and synchronization for FODGRNs via Razumikhin approach and quadratic Lyapunov approach, this situation motivates further discussion for global Mittag-Leffler stability and adaptive synchronization of FODGRNs. The essential theme of this manuscript lies in the following aspects: 1) By means of homeomorphism theory and Cauchy Schwartz inequality, a sufficient condition is presented to ascertain the existence and uniqueness of the equilibrium point for FODGRNs. 2) Based on fractional Lyapunov method, fractional-order Razumikhin theorem, and some traditional inequality techniques, a sufficient condition is established for global Mittag-Leffler stability of the proposed networks. 3) According to feedback control technique, two kinds feedback controllers are designed to guarantee the synchronization of a class of master-slave fractional order time delayed gene regulatory networks. One is linear feedback control, which is better and simpler to execute over the other controls. Another one is adaptive feedback control, which is designed to prevent the high feedback gains and it is regarded as the more versatile one. Since, it can adjust the coupling weights by itself. 4) The proposed results in this paper are still true for global exponential stability and synchronization of integer-order GRNs with time delay effects, and these results do not discuss in the previous works of literature.
The scheme of this paper is as planned out as follows. We present the key concepts about the calculus of fractional order, essential lemmas, and the system description in Section 2. In Section 3 and Section 4, we present the main results of this manuscript. In Section 5, we include the numerical results and its simulations. Lastly, we draw some conclusions in Section 6.
Notations: The required notations are displayed as follows: R m refers to the space of m-dimensional space. A set of all m × m real matrix is described by R m×m . sign(·) indicate the signum function. E λ,µ and E λ,1 refers to the two parameter and one parameter Mittag-Leffler functions, respectively. For any matrix B = b pq m×m , |B| = |b pq | m×m . The greatest and smallest eigenvalues of matrix B is represented by Φ M and Φ m , respectively. The symmetric term in a matrix is displayed by . The operator norm of a matrix B is denoted by B = Φ M B T B . Γ(·) is the gamma function. C [−τ, 0], R m indicate the group of continuous functions from [−τ, 0] to R m , where time lag τ > 0 and the signum function is referred by sign(·).

Basic tools and research problem
This section comprises of the rudimentary fractional-order definitions, lemmas which are further employed in the subsequent section.

Basic tools of Caputo-fractional operator
Definition 2.1 [22] The λ − th fractional order for integral function (t) is denoted as: Definition 2.2 [22] The λ − th Caputo type fractional order for a function (t) is denoted as: where t ≥ t 0 and m − 1 < λ < m ∈ Z + . Lemma 2.3 [13] For 0 < λ < 1, (t) ∈ R m be a continuously vector valued differentiable function, then for any t ≥ t 0 where X ∈ R m×m is a positive definite symmetric matrix.

Research problem
We consider a class of Caputo-sense FODGRNs in this manuscript as follows: where p = 1, 2, .., m, 0 < λ < 1 signifies the fractional order, g p (t) ∈ R m and h p (t) ∈ R m indicate the concentrations of mRNA and protein of pth node at time t, respectively. a p and c p are degradation velocities of mRNA and protein molecule, respectively. Moreover, d p represents the translation rate.
The time lags are denoted as σ 1 > 0 and σ 2 > 0. b pq is coupling matrix. Besides, the functions f q (·) represents the nonlinear protein feedback regulation, which are commonly indicated in the Hill form as where H q is the Hill coefficients and α y signifies non-negative constants. The coupling matrix of the network B = (b pq ) m×m ∈ R m×m are represents as follows: q is a initiator of gene p.
Presently, we define G p as G p = y∈Ĝ b pq , whereĜ indicate the set of all repressor of gene p. It's significant to mention that Caputo's definition was the most celebrated definition due to its properties such as derivative of constant is zero. Further, the Cauchy problem defined in the sense of Caputo's definition has an interpretation of integer-order initial values. Therefore, the initial values combined with FODGRNs (2.1) in the sense of Caputo type can be described as: where ω p (t), p (t) ∈ C [−τ, 0], R m and its norm is defined by The vector form of FODGRNs (2.1) is given as In the development of main results, the following Assumption and Lemma's are important.
Then, Υ(g) is homeomorphism of R m .

Existence and global Mittag-Leffler stability
In this part, we will derive the existence and stability of FODGRNs (2.1) by using the following Definitions.
Proof. According to Definition 3.1, it easy to obtain which prove that if the existence of equilibrium point g p , h p of FODGRNs (2.1) is unique, so we only to establish the existence of unique equilibrium point h p .
In the following, we will demonstrate that Υ(h) is homeomorphism of R m onto itself based on Lemma 2.10. That is (i). If h p k p , then According to Assumption 1, it follows that Based on inequality (3.2) and (3.5), it follows that |h p − k p | = 0, which leads to a contradiction with our assumption.
where ξ min = min 1≤p≤m {ξ p }. Then, we have By means of famous Cauchy-Schwartz inequality and the above inequality (3.6), we obtain Based on above discussions, we see that Υ(h) → +∞ as h → +∞. In view of Lemma 2.10, Υ(h) is homeomorphism on R m , which indicates, the existence of equilibrium point g p , h p of FODGRNs (2.1) is unique, and the proof of Theorem 3.3 is ended.
Remark 3.4 There exist other methods to obtain the existence of equilibrium, such as Schauder's fixed point theorem, Banach fixed point theorem, Browner's fixed point theorem and Krasnoselskii fixed point theorem, and Homotopy invariance theorem. In [23,25], based on the theory of fractional calculus, the contraction mapping principle and the norm-1 properties, the existence and uniqueness of the equilibrium point of the fractional order genetic regulatory networks is discussed. Different from above mentioned references [23,25], we have discussed the existence and uniqueness by homeomorphism theory.
Transform g p , h p of FODGRNs (2.1) to origin via the transformation v p (t) = g p (t) − g p and w p (t) = h p (t) − h p for p = 1, 2, .., m. Then, the FODGRNs error system is: is given as Theorem 3.5 Under Assumption 1, the existence of equilibrium point g , h of FODGRNs (2.1) is globally Mittag-Leffler stable if there exist two positive diagonal matrices X ∈ R m×m and Y ∈ R m×m such that Proof. According to conditions (3.9) and (3.10) from Theorem 3.5 that there exist two positive scalars γ 1 > 0 and γ 2 > 0 such that Consider the following subsequent Lyapunov-Razumikhin functional: Noting that (3.14) Then, based on Lemma 2.3, and the Caputo-derivative of H v(t), w(t) with respect to FODGRNs error system (3.7), we have From Assumption 1 and Lemma 2.7, we sustain By using famous inequality 2|µ 1 For any function v(t) and w(t) that hold the following Razumikhin criteria, see Ref [27,32] H we get where γ m = min{γ 1 , γ 2 }. According to inequality (3.14), we get According to inequality (3.19) and based on Lemma 2.5, we have Then, by using inequality (3.14), we get where N ≥ 0 and N = 0 satisfy only if ω(θ) = g for −σ 2 ≤ θ ≤ 0 and (θ) = h for −σ 1 ≤ θ ≤ 0, respectively. Therefore, based on Definition 3.2, the existence of equilibrium point g , h of FODGRNs (2.1) is globally Mittag-Leffler stable, and the proof of Theorem 3.5 is ended.
If σ 1 = σ 2 = 0, then system (2.1) becomes the following form: (3.20) Corollary 3.6 Under Assumption 1, the existence of equilibrium point g , h of (3.20) is globally Mittag-Leffler stable if there exist two positive diagonal matrices X ∈ R m×m and Y ∈ R m×m such that Remark 3.7 This study is a first attempt on the global Mittag-Leffler stability criterion FODGRNs. This analysis takes into account for feedback regulation time delay σ 1 and translation time delay σ 2 .
The main difficulties of this study is how to deal with time-delay terms. To overcome this difficulty, we adopt Razumikhin condition. In [25,28], many sufficient criteria guaranteeing the global Mittag-Leffler stability of FODGRNs are obtained in terms of algebraic inequalities. Compared with other researches by employing algebraic inequalities method to obtain the global Mittag-Leffler stability criteria in [25,28], our results, in terms of norm matrices, are very easy to verified with help of MATLAB toolbox in practice.

Synchronization criteria
FODGRNs (2.1) acts as the master system and the slave system is where p = 1, 2, .., m, u p (t) ∈ R m and v p (t) ∈ R m indicate the concentrations of mRNA and protein of p-node at time t, respectively. x p (t) and y p (t) are suitable controller and all others are same as one of system (2.1). The initial values of FODGRNs (4.1) can be described as: , R m and its norm is defined by Two types control like linear feedback control and adaptive feedback control, respectively, are designed as follows: and for p = 1, 2, .., m, k p > 0, l p > 0,k p > 0,l p > 0 are suitable constants, ξ p (t) and η p (t) are adaptive coupling weights.
In the development of synchronization criteria, the following Definition's are significant.
Proof. The proof of Theorem 4.4 is similar to the proof Theorem 3.5. Hence the proof of the above Theorem 4.4 is skipped.
Proof. Consider the following subsequent Lyapunov-Razumikhin functional: wherek p andl p holds the LMI of condition (i) in Theorem 4.6. Based on Lemma 2.3, Lemma 2.6, Lemma 2.9 and Assumption 1, we obtain Combined with condition (i) and (ii) from Theorem 4.6 of (4.6) and Razumikhin theorem for fractional systems, see Ref [4] that where γ = Ω − ζζ 3 − ρζ 3 > 0. Taking integer order integration of (4.7) on both sides, we have which leads to Noting that where δ m = min Φ m X , Φ m Y and δ M = max Φ M X , Φ M Y . From (4.8) and (4.9), we sustain where Lemma 2.8 has been used.
If lim t→+∞ t t 0 v(θ) 2 + w(θ) 2 dθ = +∞, then by using well known L'Hospital rule, we get (4.10) Based on Lemma 2.4, we take the fractional integral of both sides of (4.7) from t 0 to t, one can get Combined with (4.5) and (4.11), we establish which proves that v(t) 2 + w(t) 2 must be bounded, then it follows from (4.10), there exist a T > 0 such that  (i). There exist two constants ζ 1 > 0 and ζ 2 > 0 such that where X, Y,K andL are already defined in Theorem 4.6.
Remark 4.8 For time delayed GRNs, there are many findings regarding stability and synchronization criteria, see Ref [10,20,26,30,33]. Yet these results are discussed mainly in the case of integer-order. Consequently, the stability and synchronization of FODGRNs via 2-norm method properties are not studied by anyone. Therefore our research work, in the sense of innovation, is completely distinct from previous ones.

Numerical examples
This section, providing three numerical simulations to verify the superiority and benefits of the presented main results.
Remark 5.4 As we can see that the linear feedback controller (4.2) is simpler than the adaptive feedback controller (4.3), but the control strengths of the adaptive feedback controller (4.3) is smaller than those of linear feedback controller (4.2). Adaptive synchronization is superior to the synchronization in general.

Conclusions
In this manuscript, the stability and synchronization for fractional-order gene regulatory networks wit time-delay effects has been investigated in brief. Under some inequality techniques, Razumikhin approach and fractional order Lyapunov method, the globally Mittag-Leffler stability of proposed FODGRNs is proved. Moreover, the suitable controllers were designed to ensure the several synchronization for addressing FODGRNs in terms of LMIs. Further, three numerical simulations are provided. Our future research work will be generalized to state estimator design for fuzzy non-integer order gene regulatory networks with time delays and impulsive effects.