Dissipativity and Error Feedback Controller Design of Time-Delay Genetic Regulatory Networks

Genetic regulatory networks (GRNs) play an important role in the development and evolution of the biological system. With the rapid development of DNA technology, further research on GRNs becomes possible. In this paper, we discuss a class of time-delay genetic regulatory networks with external inputs. Firstly, under some reasonable assumptions, using matrix measures, matrix norm inequalities, and Halanay inequalities, we give the global dissipative properties of the solution of the time-delay genetic regulation networks and estimate the parameter-dependent global attraction set. Secondly, an error feedback control system is designed for the time-delay genetic control networks. Furthermore, we prove that the estimation error of the model is as-ymptotically stable. Finally, two examples are used to illustrate the validity of the theoretical results.


Introduction
In the past few decades, research on gene regulatory networks' modeling has attracted many biologists and mathematicians. ere have been many research studies in this area, such as models based on Boolean networks [1], models based on Bayesian methods [2], and models based on differential equations [3,4]. In particular, the gene regulation networks' model based on ordinary differential equation technology is welcomed by researchers because of its simple mathematical technology and clear biological meaning. e concept of dissipative system originates from the research of control theory. In a sense, it is a generalization of Lyapunov's stability theory. ere are fruitful research results on stability issues in various research fields, such as neural network systems [5][6][7][8][9][10], biological systems [11,12], and nonlinear systems [13,14]. However, the study of stability requires the existence of a balance point. In some special cases, the balance point of the system may not exist. At this time, we consider the dissipative characteristics of the system. In the dissipative problem, we need to find a positive invariant set, and we only need to consider the dynamic characteristics of the system in the invariant set. So far, the dissipative characteristics of gene regulatory network models based on ordinary differential equation technology have not been seen in related publications.
In addition, the control system has been widely used in all aspects of human society. In the ecosystem, a large number of phenomena can be observed through the setting of the controller, and different control schemes will produce different kinds of effects. In [15], the author gives some characteristics of the linear time-varying system under the control of timevarying feedback conjugate. In [16], the authors establish the stability criterion and saddle node bifurcation of gene regulatory networks that have nothing to do with coupling delay. In [17], the authors discuss a delayed fractional-order two-gene regulatory network model that can more accurately reflect the memory and genetic characteristics of the gene network.
Based on the enlightenment obtained from the abovementioned literature research, we discuss the characteristics of time-delay gene regulation networks with external inputs. Compared with most previous related results, our novelty lies in the use of matrix measurement theory to carry out discussions. e matrix measurement strategy has the following two advantages. (1) Lyapunov function does not need to be constructed. (2) e result obtained by the matrix measurement is more accurate and less conservative than the result obtained by the matrix norm. e rest of the paper is arranged as follows. In Section 2, we give the mathematical form of the gene regulatory network model and some basic assumptions. In Section 3, the global dissipativity of the model is discussed. In Section 4, we study the asymptotic stability of the error feedback control system. In Section 5, two examples are given to verify the validity of our conclusions.
Notations: in this paper, R n represents the n-dimensional Euclidean space. A T represents the transpose of matrix A.
x ∈ R n ∖ Ω means x ∈ R n but x ∉ Ω. For real symmetric matrices A and B, A ≽ 0 indicates that A is positive semidefinite. A ≽ B represents A − B ≽ 0. λ max (A) represents the maximum eigenvalue of matrix A.

Preliminaries
e gene regulatory networks' model has the following classic form: where m i (t), p i (t) ∈ R are the concentrations of the ith mRNA and protein, respectively, a i and c i are positive constants, representing the degradation rate of the ith mRNA and protein, respectively, d i > 0 is the translation rate from the ith mRNA to the ith protein, h j (x) � ((x/β j ) H j /1 + (x/β j ) H j ) is the regulatory functions of mRNA, where H j is the Hill coefficient, β j is a scalar, σ i (t) and τ i (t) are transcriptional and translational delay with 0 < σ i (t) ≤ σ, 0 < τ i (t) ≤ τ, and q i � j∈] i α ij with ] i is the set of all the transcription factor j which is a repressor of gene i, and if factor j activates gene i, 0, if there is no connection between j and i, − α ij , if factor j represses gene i.
Assumption 1. e activation function h i (·) meets the Lipschitz condition.
e condition can be reduced to where m − i and m + i are constants. Furthermore, the condition is reduced to Assumption 1, i.e., there exists a constant l i > 0 such that, for any x, y ∈ R, the following inequalities hold: Clearly, Assumption 1 is more general and less conservative.
Let (m * , p * ) be the equilibrium point of (3), and it satisfies 0 � − Am * + Bh p * + q, where In the field of gene therapy, the effect of external forces or stimuli on the biological system is very important. For example, mRNA vaccine is to transfer RNA to human cells after relevant modification in vitro for expression and produce protein antigen, which can lead the body to produce immune response to antigens, and then expand the immune ability of the body. erefore, we consider the following time-delay GRNs with external inputs described by where u � (u 1 , u 2 , . . . , u n ) and v � (v 1 , v 2 , . . . , v n ) are control inputs for mRNA and protein, respectively. e initial conditions of system (6) are By the definition of f i (·) and Assumption 1, it satisfies Definition 1. If there exists a compact set Ω ⊂ R n , for any ε > 0 and any initial value where Ω ε is the ε-neighborhood of Ω; then, the time-delay GRNs (6) are said to be a global dissipative system. In this case, Ω is called a globally attractive set. If, for any initial value then the set is said to be positive invariant. Obviously, the globally attractive set is positive invariant.

Definition 2.
If there exists a compact set Ω ⊂ R n , for any ε > 0 and any initial value en, the time-delay GRNs (6) are said to be a globally exponentially dissipative system. Definition 3. For constant matrix A � (a ij ) n×n ∈ R n×n , the matrix norms are given as For x ∈ R n , the vector norms are defined as follows: Definition 4. Suppose that A � (a ij ) n×n is a real matrix; then, the matrix measure of A is defined as where E n is a n × n identity matrix, p � 1, 2, ∞, and ‖ · ‖ p is the corresponding induced matrix norm. It can be calculated directly by Definition 4: Lemma 1 (see [18]). Let r > 0, ξ > 0, η > 0, and τ > 0, if the continuous function V(t) ≥ 0, t ∈ R, and ere exists a constant σ > 0 such that − ξ + η ≤ − σ; then, (1) When V(t 0 ) > (r/σ), the following formula is established: (2) When V(t 0 ) ≤ (r/σ), the following formula is established: where Lemma 2 (see [19]). For any A, B ∈ R n×n and p � 1, 2, ∞, the matrix measure μ p (·) has the following properties:

Global Dissipativity
In this section, we give the global dissipative properties of the studied mathematical model. (8), there exist positive constant δ 1 , and the matrix measures μ p (·), p � 1, 2, ∞, such that Π 2 � max l‖B‖ p , ‖D‖ p , l � max 1≤i≤l l i . en, the time-delay GRNs (6) are a globally dissipative system, and the set is a positive invariant set and a globally attractive set of (6).
We calculate the upper-right hand Dini derivative of V(t) along the solution of (6):
In eorem 1, parameter l i is considered only as the maximum value information. If the information of each component l i is utilized, the conservativeness of the result can be decreased. Denote en, (6) can be written in the form of matrix blocks:
Proof. Consider the upper-right hand Dini derivative of ‖z(t)‖ p with respect to time along the trajectory of (6), and we obtain (30)

State Estimation of Error Feedback Control System
In this section, we will discuss the global exponential stability of the following error dynamic system: and the considered state estimator is of the following form: where U V be an error feedback control term with the and e 2 � y(t) − y(t). K 1 and K 2 are feedback control gains, K 1 � diag k 1 , k 2 , . . . , k n and K 2 � diag k 1 ′ , k 2 ′ , . . . , k n ′ . us, the error dynamic system is given as In order to facilitate the discussion later, we write (40) for the matrix block form where Theorem 3. If f i (·) meets (8), there exist matrix K 1 and K 2 and matrix measure μ p (·), p � 1, 2, ∞, such that − μ p (K) > ‖BL‖ p + ‖DG‖ p > 0. en, the error dynamic system (40) is globally exponential stable under controller Proof. By inequality (30), we have where r � m − ne rρ � − μ p (K) − [‖BL‖ p + ‖DG‖ p ]e rρ . erefore, e(t) converges exponentially to zero with convergence rate of r, i.e., the error system is globally exponential stable.

Numerical Simulations
In this part, we give a pair of examples to illustrate the validity of our theoretical results.
Example 2. In this example, we consider the error dynamic system (40), in which the parameters are listed as follows: (4,4,4), (1, 1, 1), (1, 1, 1), and f j (x) is the same as Example 1. It is easy to obtain that when p takes any one of 1, 2, ∞, and there are μ p (K) � 2. It can be directly obtained by simple matlab programming: e above calculation results are easy to see that the conditions in eorem 3 are satisfied. at is, − μ p (K) > ‖BL‖ p + ‖DG‖ p > 0 holds. is shows that system (40) is globally exponentially stable. Figure 2 shows the trajectory of error system (40) with time when the initial values are e 1 (0) � (0.8, 1, 1.8) T and e 2 (0) � (1.5, 0.6, 2.4) T .

Conclusion
is paper proposes a sufficient condition for the global dissipative properties of GRN with external inputs. For the proof of dissipative properties, a large number of documents adopt the method of constructing Lyapunov function. However, this paper adopts the method of matrix x 4 (t) measurement, which simplifies the calculation difficulty, and the result is satisfactory. At the same time, error feedback control is given to ensure the stability of the system. And, the limitation of this kind of control is very small, which can bring great convenience in practical application. In addition, the study of the influence of Brown process and Levy process on gene regulatory networks is a challenging problem, which requires more mathematical knowledge about stochastic processes, but it is a very meaningful topic.

Data Availability
All data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.