Finite-Time Stability of Fractional-Order Time-Varying Delays Gene Regulatory Networks with Structured Uncertainties and Controllers

In this paper, we investigate a class of fractional-order time-varying delays gene regulatory networks with structured uncertainties and controllers (DFGRNs). Our contributions lie in three aspects: first, a necessary and sufficient condition on the existence of the solution for the DFGRNs is given by using the properties of the Riemann–Liouville fractional derivative and Caputo’s fractional derivative; second, the unique solution of the DFGRNs is proved under given initial function and certain condition; third, some novel sufficient conditions on finite-time stability of the DFGRNs are established by using a generalized Gronwall inequality and norm technique, and some conclusions on the finite-time stability of the DFGRNs with memory state-feedback controllers are reached, and those conditions and conclusions depend on the fractional order of the DFGRNs. One of the most interesting findings is that the “estimated time” of the finite-time stability is indeed related to the structured uncertainties, state-feedback controllers, time delays, and the fractional order of the system.


Introduction
Genetic regulatory networks (GRNs), which describe the interaction functions in gene expressions between DNAs, RNAs, proteins, and small molecules in an organism, are fundamental and important biological networks. e analysis and control of GRNs involve two aspects: first, understanding the widespread phenomena in living organisms and providing potential routes to prolong life span, cure cancer and diabetes, and so on; second, potential application of GRNs in the development of related disciplines, such as synthetic biology, network medicine, and personalized medicine [1][2][3][4][5].
With the development of sequencing technology, more and more genes and their regulatory sites are discovered. e structures and functions of a large number of genes are confirmed through experimental techniques, and even the regulatory mechanisms of the gene expressions controlled by some proteins are also identified [2]. Some qualitative models were proposed for distinguishing these regulatory mechanisms, such as directed graphs, Boolean networks, generalized logical networks, and rule-based formalisms [2]. However, it is difficult to know the biochemical reaction mechanisms' underlying regulatory interactions when qualitatively handling great quantities of experimental data. erefore, studying GRNs needs more accurately quantitative model. Based on experimental data, some simple genetic networks have been constructed, for example, genetic repressilator network [5], negative feedback GRNs [6], and genetic switch network [7]. e results in these experiments show that quantitative mathematical modeling approaches on dynamical systems have been great tools in providing insights on the mechanisms underlying the structure and the behaviors of GRNs [8][9][10][11][12].
Using integer-order differential equation to model GRNs is a classical method. However, the fractional-order differential equations are more suitable for modeling the gene regulatory mechanism. Ji et al. [13] applied the particle swarm optimization technique in modeling the fractionalorder GRNs with eight real target genes. e experimental results confirmed that the fractional-order model has achieved much lower fitting error on test data than integerorder model. Other studies also revealed that the fractionalorder systems have excellent performance in describing the memory and hereditary properties of various processes in GRNs, which could be far better than the integer-order ones [8][9][10][11].
Due to slow biochemical processes such as gene transcription, translation, and transportation (the synthesis of mRNAs and proteins at nucleus and cytoplasm, respectively, in eukaryotic cells), time delays are omnipresent in GRNs [14]. Many nonlinear differential equations with time delays have been proposed to model general GRNs, and the important role of time delays in dynamics of GRNs is now widely accepted [15][16][17][18]. Actually, time delays often degrade the system performance or destabilize the system [1,19,20]; even GRN models without time delay may generate wrong predictions [21]. As time delays often change with time and their precise measurement is difficult in real GRNs, the dynamics of fractional-order linear and nonlinear systems with time-varying delays has attracted increasing interest, and the results show that it is naturally of better practical significance than those with constant delays [21][22][23][24][25].
In addition, in order to avoid undesirable states associated with disease, the control of GRNs is often regarded as developing therapeutic intervention strategies for some diseases [26,27]. And many literatures focus on the research of control in the dynamic system [28][29][30][31][32][33]. In [32], the authors obtained some stabilization results for neural networks with leakage delay by designing state-feedback controller. Ebihara et al. [33] discovered that exact robust control is indeed attained for discrete-time linear systems by designing periodically time-varying memory state-feedback controller. erefore, it is necessary to consider the controller for the DFGRNs.
Since the modeling of GRNs is underlined with the realworld gene expression time-series data, some limitations of the current experimental techniques in GRNs make the modeling errors and parameter fluctuations unavoidable. Moreover, some point out that the system parameters identified with the experimental data may construct an unknown but bounded time-varying function, and this unknown nature is referred to as the structural uncertainty or the parametric uncertainty, also known as variation or fluctuation [34]. As is known, the structural uncertainties in GRNs may lead to the poor performance or even instability in real genetic networks [28,[34][35][36][37]. In [28], the authors studied the robust stabilization and state-feedback controller design for a class of integer-order GRNs with time-varying delays (DGRNs) and structured uncertainties and established some delay-dependent stability results by using some matrix techniques. erefore, taking into account the structural uncertainties while investigating the dynamical behaviors of DFGRNs is essential.
Since the expression of gene and mRNA-translated protein is accomplished in a much relatively short period, in recent decades, some scholars have paid more attention to the finite-time stability of GRNs [4,38]. For example, Wu et al. [4] investigated the finite-time stability associated with a class of integer-order GRNs by designing adaptive controllers. Wang et al. [38] established some new sufficient conditions of the finite-time stability for a class of integerorder uncertain GRNs with time-varying delays. Lazarević [22] investigated the finite-time stability for fractionalorder nonlinear differential equation with time-varying delays by using generalized Gronwall inequality and the classical Bellman-Gronwall inequality, respectively. Phat and anh [23] established some new sufficient conditions of robust finite-time stability for a class of nonlinear fractional-order differential systems with time-varying delays. Wang et al. [39] considered a class of nonlinear fractional-order systems with constant delays and studied the existence and uniqueness of the solution for this kind of systems by using relevant properties of the fractional derivative.
However, the discussions on the existence and uniqueness of the solutions and the finite-time stability results for the fractional-order uncertain GRNs with timevarying delays and controllers seem rare.
From above discussions, we focus on the existence and uniqueness of the solution and the finite-time stability for a class of DFGRNs with structured uncertainties and controllers. e remainder of this paper is organized as follows. In Section 2, we give the model description, some definitions, and related properties on fractional calculus. In Section 3, we discuss the existence and uniqueness of the solution and give some sufficient criteria on the finite-time stability for the DFGRNs. In Section 4, we perform some numerical simulations, which support our findings. In Section 5, we briefly review and summarize the main results.
We will focus on a class of DFGRNs with structured uncertainties and controllers, which is established as follows: where m(t) � m 1 (t), m 2 (t), . . . , m n (t) T , A � diag a 1 , a 2 , . . . , a n , H � diag e 1 , e 2 , . . . , e n , in which C D q t represents Caputo's fractional derivative and q ∈ (0, 1), m i (t), p i (t) ∈ R are the concentrations of mRNA and protein of the ith node, respectively. e parameters a i > 0 and c i > 0 are the decay rates of mRNA and protein, respectively; d i > 0 are the translation rates; e i ≥ 0 are the translation rates. Both f j (p j (t)) and g j (p j (t − τ 1 (t))) represent the feedback regulation of the protein on the transcription. Generally, each one of the two functions is a nonlinear function but has a form of monotonicity with its variable. As a monotonic increasing or decreasing regulatory function, f j and g j are usually of the Michaelis-Menten or Hill forms [21]. B i � j∈I i b ij + j∈I i b ij , where b ij and b ij are bounded constants which are, respectively, the dimensionless transcriptional rates of transcription factor j to i at time t and t − τ 1 (t), and I i , I i , respectively, are the set of all the j where the transcription factor j is a repressor of gene i at time t and t − τ 1 (t). W � (w ij ) ∈ R n×n , K � (k ij ) ∈ R n×n are the coupling matrices of the gene network, which are defined as follows: if transcription factor j is an activator of gene i, if there is no link from node j to i.
(ii) Assumption (II): the functions F, G satisfy the following inequalities: where L 1 , L 2 are positive constants. Next, we give some definitions and lemmas.
Definition 1 (see [40]). e fractional integral of order q for a function f(t) is defined as Definition 2 (see [40]). Caputo's fractional derivative of order q for a function f is defined by where t ≥ a and n is a positive integer such that n − 1 < q < n.
Definition 3 (see [40]). e Riemann-Liouville fractional derivative of order q for a function f is defined as (11) where t ≥ a and n is a positive integer such that n − 1 < q < n.
For convenience, we choose the notation (6) is a vector (m(t), p(t)) T composed of continuous functions

Definition 4. A mild solution of DFGRN (3) with initial condition
satisfying

is a nonnegative and nondecreasing continuous function, v(t) ≤ M (constant), and u(t) is a nonnegative and locally integrable function with
In addition, if a(t) is a nondecreasing function, then

e Existence and Uniqueness of the Mild Solution of DFGRNs
Proof. We firstly give the sufficient condition of the existence of the mild solution to DFGRN (3). (15), applying RL D q t and property (ii) of Lemma 1, we obtain

Complexity
According to the property (iii) of Lemma 1 and 0 < q < 1, we get .
From (16) and (17), we have We secondly give the necessary condition of the existence of the mild solution to DFGRN (3).
When t ∈ [− τ * , 0], the solution of DFGRN (3) is In the case of 0 < q < 1, from property (i) of Lemma 1, we can obtain e proof is completed.

Remark 2.
In the proof of eorem 3, if we use the "classical" Bellman-Gronwall inequality instead of the generalized Gronwall inequality, we can get the following result. e uncertain DFGRN with controllers given by (3) satisfying the initial condition (6) is finite-time stable with respect to δ, ε, α 1 , J 0 , δ < ε, if assumptions (I) and (II) hold and the following condition is satisfied: Remark 3. If we take u 1 (t) ≡ 0, u 2 (t) ≡ 0, ∀t ∈ J 0 in system (3), the above results turn into the following conclusion. e uncertain DFGRN (3) satisfying the initial condition (6) is finite-time stable with respect to δ, ε, J 0 , δ < ε. If assumptions (I) and (II) hold, the following condition is satisfied:

Finite-Time Stability of DFGRNs with Memory State-Feedback Controllers.
We consider the following memory state-feedback controllers on DFGRN (3): where c i , i � 1, 2, 3, 4 are the gain matrices of

hold, then the uncertain DFGRN (3) with memory statefeedback controllers given by (43) satisfying the initial condition (6) is finite-time stable with respect to δ, ε, J
Proof. Similar to eorem 1 and eorem 2, it is easy to prove that DFGRN (43) has a mild solution satisfying the following integral equation: Using the norm ‖(·)‖, we have 10 Complexity From (43), by using assumptions (I) and (II), we have (46) and (47), we obtain Hence, .
□ Remark 4. Similar to Remark 2, we can get the following result.
Complexity e uncertain DFGRN (3) with memory state-feedback controller given by (43) satisfying the initial condition (6) is finite-time stable with respect to δ, ε, J 0 , δ < ε, if assumptions (I) and (II) hold and the following condition is satisfied: Remark 5. We can obtain the same conclusion as eorem 3 and eorem 4 if the inequalities in assumption (II) are Remark 6. All the results in Remarks 1-4 are still new.

Numerical Examples
In this section, some numerical examples are given to illustrate the effectiveness of above theoretical results. In the following examples, the functions f j and g j are taken as the Hill form. And in the Adams-Bashforth-Moulton predictor-corrector scheme [42], the step length is h � 0.1.
Example 2. Consider the following DFGRNs of three mRNA and protein nodes with structured uncertainties and without controller: Using the same parameters in Example 1, we similarly get η 4 � 4.3172, ζ 1 � 5.3845, ζ 5 � 7.0183. When t < 0.3585, we have

Remark 7.
It is worthy to note that in a special case of DFGRN (62) without structured uncertainties, it is proved that in the sense of infinite stability, (62) is globally asymptotically stable [16].
Example 3. Consider the following DFGRNs of three mRNA and protein nodes with memory state-feedback controllers and without structured uncertainties: Using the same parameters in Example 1, we similarly obtain η 5 � 4.1799, ζ 3 � 5.2009, ζ 5 � 7.0183. When t < 0.3697, we can get  If we adopt constant time-delay τ 1 (t) � τ 2 (t) � 2 and q � 0.4 in DFGRN (66), then system (66) is finite-time stable, and the "estimated time" of finite-time stability is 0.0315. e transient states of the variables m i (t) and p i (t)(i � 1, 2, 3) of DFGRN (66) with q � 0.4 are shown in Figure 6.
In order to investigate the effects of structured uncertainties, controllers and time delays on the stability of the DFGRNs, we calculate the "estimated time" T e of finite-time stability for above four examples and the corresponding FGRNs with different fractional-order q; the results are shown in Tables 1 and 2, respectively.
From Table 1 or Table 2, we have the following conclusions: (i) e effect of the controllers: comparing column 2 with 3 (or column 4 with 5), we can know that the controllers can shorten the "estimated time" of finite-time stability under the same conditions of fractional-order q and structured uncertainties. (ii) e effect of the structured uncertainties: comparing column 3 with 5, we can know that the structured uncertainties can shorten the "estimated time" of finite-time stability under the same fractional-order it q.   order q will be useful to decrease the "estimated time" of finite-time stability for DFGRNs or FGRNs. (v) e effect of time delays: comparing Table 1 with Table 2, we can know that the "estimated time" of finite-time stability is reduced under the same fractional-order q when considering time delays.

Concluding Remarks
is paper deals with the existence and uniqueness of the solution and the finite-time stability for a class of DFGRNs with structured uncertainties and controllers. In particular, we design the memory state-feedback controllers for DFGRNs with structured uncertainties and give the sufficient conditions for the system to achieve the finite-time stability.
It should be pointed out that the conditions of finitetime stability in the present paper are dependent on the fractional-order q, which is more different from the previous stability results for the case of integer order, i.e., the finite-time stability is independent of the integer order.
In addition, from the numerical results, we find that all of the controllers, uncertain terms, fractional-order q and time delays can affect the "estimated time" of finite-time stability. Particularly, (i) the size of "estimated time" of finite-time stability with controllers is shorter than the case without controller but only with structured uncertainties, which means that the controllers are more beneficial for controlling the "estimated time" than the structured uncertainties; (ii) the size of "estimated time" of finite-time stability with time delays is shorter than the case without time delays, which means that time delays degrade the GRN performance.
If we take ΔA(t) � ΔW(t) � ΔK(t) � ΔC(t) � Δ D (t) � Δ H(t) � ΔQ 1 (t) � ΔQ 2 (t) � 0 and controllers terms u 1 (t) � u 2 (t) � 0, meanwhile, in the special case constant time delay, system (3) convert to (2.2) in [16], and we find that numerically: as t ⟶ + ∞, DFGRN (62) in this paper is unstable; however, DFGRN (4.1) in [16] is globally asymptotically stable, which means that the structured uncertainty can change the stability of DFGRNs. Furthermore, from Remark 8, we know that DFGRN (66) is finite-time stable, while the corresponding system (4.1) in [16] is infinite-time unstable, which means that an infinite-time unstable system can change to a finite-time stable one under extra conditions. e analytical study on above questions is desirable in the future.

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 they have no conflicts of interest.