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Global exponential stability analysis of discrete-time genetic regulatory networks with time-varying discrete delays and unbounded distributed delays
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Global exponential stability analysis of discrete-time genetic regulatory networks with time-varying discrete delays and unbounded distributed delays Chunyan Liu a, Xin Wang b,c,∗, Yu Xue b,c,∗ a b c
School of Information Management, Heilongjiang University, Harbin 150080, P R China School of Mathematical Science, Heilongjiang University, Harbin 150080, P R China Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, Heilongjiang University, Harbin 150080, P R China
a r t i c l e
i n f o
Article history: Received 23 April 2019 Revised 22 August 2019 Accepted 16 September 2019 Available online xxx Communicated by Jun Yu Keywords: Genetic regulatory networks Global exponential stability Time-varying discrete delays Unbounded distributed delays
a b s t r a c t This paper addresses the problem of exponential stability analysis of discrete-time genetic regulatory networks (GRNs) with time-varying discrete delays and unbounded distributed delays. A novel approach is proposed to establish delay-dependent and -independent global exponential stability criteria for the class of GRNs under consideration. The main novelties lie in: (i) without any auxiliary function or Lyapunov– Krasovskii functional is required in the process of deriving the global exponential stability criteria; (ii) the obtained stability criteria improves or generalizes the existing results; (iii) the obtained stability criteria are directly represented by the systems parameters, which can be conveniently verified; and (iv) the exponential estimation of GRNs can be obtained from the derivation process of the exponential stability criteria. The effectiveness of theoretical results is illustrated by presenting several numerical examples.
1. Introduction Genetic regulatory networks (GRNs) describe the process of gene expression, including transcription and translation, which is helpful to understand the genome sequencing and the gene recognition [1–3]. Thus the study of GRNs has attracted interest of many experts and scholars in theoretical physics, control science, biology, etc (see [4–6] and the references therein). Due to the slow transcription and translation processes, functional differential/difference equations have been employed to model GRNs [6,7]. Moreover, in individual molecules, movement of mRNA from a transcription site to translation sites requires a long time, which is necessary to introduce distributed delays into functional differential/difference equation models [8–10]. As a result, the functional differential/difference equation models with discrete and/or distributed delays have been established to solve the analysis and design problems of GRNs, including the stability analysis [7,9,11–24], passivity analysis [25], Hopf bifurcation analysis [26,27], controller synthesis [28–30], observer design [31–33], state estimation [5,10,34–39], filter design [40–42], and so on. ∗ Corresponding authors at: School of Mathematical Science, Heilongjiang University, Harbin 150080, P R China. E-mail addresses: [email protected] (C. Liu), [email protected] (X. Wang), [email protected] (Y. Xue).
© 2019 Elsevier B.V. All rights reserved.
The most important property of GRNs, stability, has been studied in a great number of literature. However, to the best of authors’ knowledge, these achieved results are mainly based on the functional differential equation models (i.e., delayed continuous-time GRNs), and rarely focus on the functional difference equation models (i.e., delayed discrete-time GRNs). The stability criteria for delayed discrete-time GRNs are only established in [10,13,20–24,37,38]. The exponential stability criteria are involved in [20–22] by constructing appropriate auxiliary functions, and the asymptotic/exponential stability criteria are investigated in [10,13,23,24,37,38] by constructing appropriate Lyapunov– Krasovskii functionals. Since the construct of auxiliary function or Lyapunov–Krasovskii functional is quite difficult and requires a high degree of skill, it is necessary to find new approach to analyze stability of delayed discrete-time GRNs, which motivates the research. This paper is concerned with the problem of exponential stability analysis of discrete-time GRNs with time-varying discrete delays and unbounded distributed delays. A new approach is proposed to establish delay-dependent and -independent global exponential stability criteria for the class of GRNs under consideration. For some special cases of the discrete-time GRNs under consideration, it is theoretically proven that the global exponential stability criteria in Corollary 1 below are less conservative than ones in [20, Corollay], [21, Theorem 2] and [22, Theorem 2], and are equivalent to ones in
https://doi.org/10.1016/j.neucom.2019.09.047 0925-2312/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: C. Liu, X. Wang and Y. Xue, Global exponential stability analysis of discrete-time genetic regulatory networks with time-varying discrete delays and unbounded distributed delays, Neurocomputing, https://doi.org/10.1016/j.neucom.2019.09.047
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[20, Theorem] and [23, Theorem 2]. So the paper can be regarded as a generalized version of [20–23]. Compared with the linear matrix inequality-based stability criteria proposed in [10,13,24,37,38], the obtained stability criteria possess more simpler form, which reduces the computational complexity. In addition, several numerical examples are also provided to illustrate the effectiveness of theoretical results. The novelties of this paper are as follows: (i) the proposed approach is directly based on the definition of global exponential stability, and it does not require to construct any auxiliary function or Lyapunov–Krasovskii functional; (ii) the obtained stability criteria are directly represented by the systems parameters, which is more simple and easily be checked by using the software MATLAB; (iii) the obtained stability criteria extend or improve the existing results; and (iv) the obtained delay-dependent global exponential stability criteria can used to give the exponential estimation of GRNs. Notation. Suppose Z, R and C are sets of all integers, real numbers and complex numbers, respectively. Let Z[a, b] be the subset of Z consisting of all integers between a and b, and let Z[a, ∞ ) = ∪b>a Z[a, b]. For given positive integers p and q, let R p×q denote the set of all p × q matrices over R. Set R p = R p×1 . For A = [ai j ] ∈ R p×q and B = [bi j ] ∈ R p×q , the matrix [aij bij ], denoted by A◦B, is called the Hadamard product of A and B. Furthermore, we denote by |A| the matrix [|aij |]. 2. Problem formulation and preliminaries Consider a class of discrete-time GRNs with time-varying discrete delays, which can be described as [4]:
mi ( k + 1 ) = ai mi ( k ) +
n
bi j g j ( p j (k − σi j (k ))) + Ji ,
(1a)
where li > 0 is a known constant. ∞ ∞ Assumption 2. s=1 αs = s=1 βs = 1. Assumption 3. σi j (k ) ≤ σ¯ i j and τi (k ) ≤ τ¯i for any i, j ∈ Z[1, n] and k ∈ Z[0, ∞ ), where σ¯ i j and τ¯i are known positive constants. For A = [ai j ] ∈ R p×q and B = [bi j ] ∈ R p×q , the symbol A B (or B A) means that aij ≥ bij for all i ∈ Z[1, p] and j ∈ Z[1, q]; in particular, if aij > bij for all i ∈ Z[1, p] and j ∈ Z[1, q], then we use A B (or B ≺ A) instead of A B (or B A). Clearly, |XY| |X||Y| for p×q p×q all X ∈ R p×q and Q ∈ Rq×r . Denote by R and R the sets of all p × q nonnegative and positive matrices, respectively. Similar notations are adopted for vectors. A pair (m∗ , p∗ ) ∈ Rn × Rn is called a nonnegative equilibrium of GRN (2), if
m∗i = ai m∗i +
n
(bi j + ei j )g j ( p∗j ) + Ji ,
(3a)
j=1
p∗i = ci p∗i + di m∗i +
n
fi j m∗j , i ∈ Z[1, n], k ∈ Z[0, ∞ ),
(3b)
j=1
where m∗i and p∗i are the i-th components of m∗ and p∗ , respectively. Obviously, (3) can be written as the following compact matrix form:
(A − In )m∗ + (B + E )g( p∗ ) + J = 0, (C − In ) p∗ + (D + F )m∗ = 0, where
A = diag(a1 , a2 , . . . , an ), C = diag(c1 , c2 , . . . , cn ),
j=1
pi (k + 1 ) = ci pi (k ) + di mi (k − τi (k )), i ∈ Z[1, n], k ∈ Z[0, ∞ ),
D = diag(d1 , d2 , . . . , dn ), J = col(J1 , J2 , . . . , Jn ),
(1b) where mi (k) and pi (k) are the mRNA and protein concentrations of gene i, respectively, gi is the regulation function of gene i, τ i (k) and σ ij (k) are the time-varying translational delay and feedback regulation delay, respectively, ai > 0, ci > 0, di > 0, Ji > 0, and bi j ∈ R satisfies that bij < 0 if transcription factor j represses gene i, bij > 0 if transcription factor j activates gene i, and bi j = 0 if transcription factor j does not regulate gene i. See [4, Chapter 11] for more detailed explanations about model (1). When the unbounded distributed delays are introduced into GRN (1), the following model of discrete-time GRNs with timevarying discrete delays and unbounded distributed delays are obtained [10]:
mi ( k + 1 ) = ai mi ( k ) +
n
+
j=1
ei j
∞
αs g j ( p j (k − s )) + Ji ,
(2a)
s=1
+
j=1
fi j
∞
and diag( · ) and col( · ) refer to the diagonal matrix and column vector, respectively. The existence issue of nonnegative equilibrium of GRN (2) has been addressed in [17] when D + F is a positive diagonal matrix. In the rest of this section we make the following assumption: Assumption 4. GRN (2) has at least one nonnegative equilibrium (m∗ , p∗ ).
Set x(k ) = m(k ) − m∗ and y(k ) = p(k ) − p∗ , k ∈ Z[0, ∞ ). Due to (2) and (3), we have
xi ( k + 1 ) = ai xi ( k ) +
pi (k + 1 ) = ci pi (k ) + di mi (k − τi (k )) n
g( p∗ ) = col(g1 ( p∗1 ), g2 ( p∗2 ), . . . , gn ( p∗n )),
Remark 1. By using an approach similar to one in [17], it can be concluded that if B + E 0 and D + F 0, then GRN (2) has at least one nonnegative equilibrium.
bi j g j ( p j (k − σi j (k )))
j=1 n
B = [bi j ], E = [ei j ], F = [ f i j ],
n
bi j g˜ j (y j (k − σi j (k )))
j=1
βs m j (k − s ), i ∈ Z[1, n], k ∈ Z[0, ∞ ),
(2b)
s=1
+
where eij , fij , α s ≥ 0 and β s ≥ 0 are known constants. For convenience, let’s make the following assumptions: Assumption 1. The regulation functions gi (i = 1, 2, . . . , n ) satisfy
gi ( s1 ) − gi ( s2 ) gi ( 0 ) = 0 , 0 ≤ ≤ li , ∀s1 , s2 ≥ 0, s1 = s2 , s1 − s2
n
ei j
j=1
∞
αs g˜ j (y j (k − s )),
(5a)
s=1
yi (k + 1 ) = ci yi (k ) + di xi (k − τi (k )) +
n j=1
fi j
∞
βs x j (k − s ), i ∈ Z[1, n], k ∈ Z[0, ∞ ),
(5b)
s=1
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where g˜ j (· ) = g j (· + p∗ ) − g j ( p∗ ). Clearly, (0,0) is an equilibrium of system (5), and g˜ satisfies the following inequality:
|g˜ j (s )| ≤ l j |s|, ∀s ≥ 0, j ∈ Z[1, n].
(ψ , ϕ )∞ = max
sup
s∈Z(−∞,0]
where
Aγ = A − e−γ In , eγ σ¯ = [eγ σ¯ i j ], eγ τ¯ = diag(eγ τ¯1 , eγ τ¯2 , . . . ,eγ τ¯n ),
(6)
Let C (Z(−∞, 0], Rn ) refer to the linear space over R consisting of all functions φ : Z(−∞, 0] → Rn satisfying sups∈Z(−∞,0] φ (s ) < ∞. Define the norm · on Rn × Rn by (a, b) = (a22 + b|22 )1/2 for any a, b ∈ Rn , and the norm · ∞ on C (Z(−∞, 0], Rn ) × C (Z(−∞, 0], Rn ) by
Cγ = eγ τ¯ ◦ D + β|F |, Dγ = C − e−γ In ,
α=
s∈Z(−∞,0]
Definition 1. The zero equilibrium of system (5) is said to be globally exponentially stable, if there exist scalars K > 0 and γ > 0 such that every solution (x(k), y(k)), starting from ϕ , ψ ∈ C (Z(−∞, 0], Rn ), satisfies
(x(k ), y(t )) ≤ Ke−γ k (ψ , ϕ )∞ , ∀k ∈ Z[0, ∞ ). This paper aims at proposing a novel approach to establish delay-dependent and -independent global exponential stability criteria for the zero equilibrium of system (5) (i.e., the unique nonnegative equilibrium of GRN (2)). It is worth emphasizing that the proposed approach does not involve the construct of any Lyapunov–Krasovskii functional or auxiliary function, which is different ones in [10,13,20–24,37,38]. A matrix M ∈ Rn×n is called a Metzler matrix if all off-diagonal elements of M are nonnegative. Let In be the identity matrix in Rn×n , and λ(M ) = {z ∈ C : det(zIn − M ) = 0}. The spectral abscissa of M is defined by s(M) := max{Reλ: λ ∈ λ(M)}, and the spectral radius of M is defined by ρ (M) := max {|λ|: λ ∈ λ(M)}. We end this section by introducing the following lemma which is helpful to present the main contributions of this paper. Lemma 1. [43] Let A0 ∈ Rn×n be a Metzler matrix and B0 , C0 , D0 ∈ Rn ×n . Then the following statements (i)–(iii) are equivalent: (i) ρ (D0 ) < 1 and s(A0 + B0 (In − D0 )−1C0 ) < 0. (ii) A0 x + B0 y ≺ 0 and C0 x + D0 y ≺ y for some x, y ∈ Rn . (iii) s(A0 ) < 0 and ρ (C0 (−A0 )−1 B0 + D0 ) < 1. 3. Global exponential stability criteria
αs eγ s , β =
∞
βs eγ s ,
s=1
The proof of Theorem 1 will be given in Appendix A below. Combining Theorem 1 and Lemma 1, one can easily derive the following conclusion. Theorem 2. For given positive scalars γ ≤ γ 0 , σ¯ i j , τ¯i and lj (i, j ∈ Z[1, n] ), let Assumptions 1–5 be satisfied. Then GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable, if one of the following statements (a)–(c) holds: (a) Aγ v˜ + Bγ u˜ ≺ 0 and Cγ v˜ + (Dγ + In )u˜ ≺ u˜ for some u˜, v˜ ∈ Rn . (b) ρ (Dγ + In ) < 1 and s(Aγ − Bγ D−1 γ Cγ ) < 0. (c) s(Aγ ) < 0 and ρ (Dγ + In − Cγ A−1 γ Bγ ) < 1. Theorem 2 provides delay-dependent sufficient conditions under which GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable. To present delay-independent sufficient conditions, we define
A˜ = A − In , B˜ = (|B| + |E | )L, C˜ = D + |F |. Theorem 3. For given positive scalars lj ( j ∈ Z[1, n] ), let Assumptions 1, 2, 4 and 5 be satisfied. Then GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable, if one of the following statements (a)–(c) holds: (a) A˜ v˜ + B˜u˜ ≺ 0 and C˜v˜ + C u˜ ≺ u˜ for some u˜, v˜ ∈ Rn . (b) ρ (C) < 1 and s(A˜ + B˜(In − C )−1C˜ ) < 0. (c) s(A˜ ) < 0 and ρ (C − C˜A˜ −1 B˜ ) < 1. ∞ γ s and Proof. Assumption 5 ensures that the series s=1 αs e ∞ γ s are uniformly convergent for γ ≤ γ . This, together β e 0 s=1 s with Assumption 2, implies that
lim+
γ →0
In this section we will investigate sufficient conditions under which the zero equilibrium of system (5) is globally exponentially stable, that is, GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable. To this end, we require the following assumption: ∞ γ0 s < +∞ Assumption 5. There exists γ 0 > 0 such that s=1 αs e ∞ and s=1 βs eγ0 s < +∞. Assumption 5 guarantees the uniform convergence of the series ∞ γ s and ∞ β eγ s for γ ≤ γ , which is indispensable to 0 s=1 αs e s=1 s our approach. Now it is the place to present the following conclusion which is one of the main contributions of this paper. Theorem 1. For given positive scalars γ ≤ γ 0 , σ¯ i j , τ¯i and lj (i, j ∈ Z[1, n] ), let Assumptions 1–5 be satisfied. Then GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable, if there exist u˜, v˜ ∈ Rn such that
Cγ v˜ + Dγ u˜ 0,
∞
and the operator ◦ means the Hadamard product of matrices.
for any ψ , ϕ ∈ C (Z(−∞, 0], Rn ), where · 2 is the Euclidean norm on Rn .
Aγ v˜ + Bγ u˜ 0,
Bγ = eγ σ¯ ◦ |B| + α|E | L, L = diag(l1 , l2 , . . . , ln ),
s=1
ψ (s )2 , sup ϕ (s )2
3
(7)
(8)
∞ s=1
αs eγ s = lim+ γ →0
∞
βs eγ s = 1.
s=1
Furthermore, limγ →0+ Aγ = A˜ , limγ →0+ Bγ = B˜, limγ →0+ Cγ = C˜ and limγ →0+ Dγ = C − In . This, together with Theorem 2, completes the proof. Remark 2. Compared with [20–22], the system model under consideration involves the unbounded distributed delays, which brings extreme difficult in theoretical derivation and numerical computation, so that we have to add Assumptions 2 and 5. Moreover, the time-varying discrete delays complicate the addressed problem, which is not concerned in [20–22]. Remark 3. The conditions in Theorem 1 and (a) in Theorems 2 and 3 can be easily realized by employing the toolbox YALMIP of MATLAB. The testify of the condition (b) or (c) in Theorems 2 and 3 is concerned with the computation of a constant matrix, which can be proceeded by using the function eig of MATLAB for the loworder case, and the iterative methods, including the power method and the inverse power method, for the high-order case. We end the section by the following corollary which gives the global exponential stability criteria for the nonnegative equilibrium of GRN (1).
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Corollary 1. For given positive scalars lj ( j ∈ Z[1, n] ), let Assumption 1 be satisfied. Then GRN (1) has a unique nonnegative equilibrium which is globally exponentially stable, if one of the following statements (a)–(c) holds: (a) (In − A )v˜ |B|Lu˜ and (In − C )u˜ Dv˜ for some u˜, v˜ ∈ (b) ρ (C) < 1 and s(A + |B|L(In − C )−1 D ) < 1. (c) s(A) < 1 and ρ (C + D(In − A )−1 |B|L ) < 1.
Rn .
4. Theoretical comparisons In this section, we will give theoretical comparisons of the global exponential stability criteria for the nonnegative equilibrium of GRN (1) presented in this paper and literature. When GRN (1) is viewed as a semi-discretized version of continuous-time GRN, its parameters possess the following special form [20–22]:
σi j (k ) ≡ σˆ i j /h, τi (k ) ≡ τˆi /h,
(9a)
ai = e−aˆi h , ci = e−cˆi h ,
(9b)
bi j =
1 − e−aˆi h ˆ 1 − e−cˆi h ˆ 1 − e−aˆi h bi j , di = di , Ji = Jˆi , i, j ∈ Z[1, n], aˆi cˆi aˆi
(9c)
where h is the discretization step-size, · represents the integral function, and aˆi , cˆi , dˆi , Jˆi ∈ R , bˆ i j ∈ R and σˆ i j , τˆi ∈ Z[1, ∞ ) are the system parameters of original continuous-time GRN. The following proposition is a simple overview of several results in literature.
which is equivalent to
1 > C + D(In − A )−1 L|B|1 , where · 1 is the maximum row sum norm of matrices. Hence 1 > ρ (C + D(In − A )−1 L|B| ). Noting that the matrix C + D(In − A )−1 L|B| is similar to C + D(In − A )−1 |B|L and 0 < ai < 1 for all i ∈ Z[1, n], we can derive that (c) in Corollary 1 is true. Case 4: (iv) in Proposition 1 is true. Then so is (iii) in Proposition 1. By Case 3, it follows that (c) in Corollary 1 is true. Remark 4. Theorem 4 and its proof show that the global exponential stability criteria in Corollary 1 are less conservative than ones in [20, Corollay], [21, Theorem 2] and [22, Theorem 2], and are equivalent to one in [20, Theorem]. Remark 5. Similar to the proof of Theorem 4, one can derive that Corollary 1 is equivalent to [23, Theorem 2]. Finally, we further emphasize the merits of the proposed approach by the following two remarks. Remark 6. Different from the approaches proposed in [20–23], our approach has the following advantages: • The approach is available for GRNs containing time-varying discrete delays and/or distributed delays; • The approach can provide delay-dependent global exponential stability criteria; • The obtained delay-dependent global exponential stability criteria can used to give the exponential estimation of GRNs.
Proposition 1. For given positive scalars lj ( j ∈ Z[1, n] ), let Assumption 1 be satisfied. Then GRN (1) subject to (9) has a unique nonnegative equilibrium which is globally exponentially stable, if one of the following statements (i)–(iv) is satisfied:
So, our future work is to extend the approach proposed in this paper to the more general system models, for examples, complex networks [44,45], time-varying systems [46], neutral systems [47– 50], multi-agent systems [51,52], and switched systems [53,54].
(i) [20, Theorem] There exist u˜, v˜ ∈ Rn such that cˆi u˜i > dˆi v˜ i and aˆi v˜ i > nj=1 |bˆ i j |l j u˜ j for all i ∈ Z[1, n]. (ii) [20, Corollay] cˆi > dˆi and aˆi > nj=1 |bˆ i j |l j for all i ∈ Z[1, n]. (iii) [21, Theorem 2] aˆi cˆi > dˆi li nj=1 |bˆ i j | for all i ∈ Z[1, n]. (iv) [22, Theorem 2] aˆi > dˆi and cˆi > li n |bˆ i j | for all i ∈ Z[1, n].
Remark 7. For a class of stochastic GRNs with leakage, discrete, and distributed delays, Pandiselvi, Raja and Cao et al. [10] established an exponential stability criterion in the form of linear matrix inequalities by utilizing the Lyapunov–Krasovskii functional approach. The approach proposed in [10] is also available to investigate exponential stability of GRN (2), however, it will result in an exponential stability criterion described by a set of linear matrix inequalities, which requires more compute time to verify than one obtained in this paper.
j=1
Theorem 4. For given positive scalars lj ( j ∈ Z[1, n] ), let Assumption 1 be satisfied. Assume that the parameters in GRN (1) possess the special form (9). If one of (i)–(iv) in Proposition 1 is true, then so is one of (a)–(c) in Corollary 1. Proof. Case 1: (i) in Proposition 1 is true. Then
5. Illustrative examples In this section we will present the effectiveness of the established global exponential stability criteria by three illustrative examples.
dˆi di di u˜i > v˜ i = v˜ i = v˜ i 1 − ci cˆi 1 − e−cˆi h and
Example 1. Consider GRN (2) with
n n n |bˆ i j |l j |bi j |l j |bi j |l j v˜ i > u˜ j = u˜ j = u˜ j . ˆ − a h i 1 − ai aˆi 1 − e j=1 j=1 j=1
A = diag(0.6, 0.6 ), C = D = diag(0.3, 0.2 ), F = diag(0.09, 0.27 ),
This, together with 0 < ai < 1 and 0 < ci < 1, implies that (a) in Corollary 1 is true. Case 2: (ii) in Proposition 1 is true. Then (i) in Proposition 1 holds by taking v˜ i = u˜i = 1 for all i ∈ Z[1, n]. Due to Case 1, it is concluded that (a) in Corollary 1 is true. Case 3: (iii) in Proposition 1 is true. Then
1>
dˆi li cˆi
n j=1
|bˆ i j | aˆi
=
di li 1 − e−cˆi h
n j=1
|bi j | 1−
e−aˆi h
for all i ∈ Z[1, n], and hence
1 > ci + di (1 − ai )−1 li
n j=1
|bi j |, i ∈ Z[1, n]
=
di li 1 − ci
n j=1
|bi j | 1 − ai
B=
−0.2 0.1
0.5 0.2 , E= 0.2 0.1
J = col(0.2, 0 ), gi (s ) =
−0.2 , 0.2
s2 , s ∈ [0, ∞ ), i ∈ Z[1, 2], 1 + s2
αs = βs = (1 − e−2 )e−2(s−1) , s ∈ Z[1, ∞ ). Clearly, Assumption 2 is satisfied. If we choose l1 = l2 = 0.65, then Assumption 1 is satisfied. Following the discussion in [9, Section 4], one can easily obtain a nonnegative equilibrium, (m∗ , p∗ ), of the GRN under consideration as follows:
m∗ = col(0.5003, 0.0361 ), p∗ = col(0.2788, 0.0212 ), i.e., Assumption 4 is satisfied.
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5
Fig. 1. Trajectories of mRNA and protein concentrations (Example 1).
Fig. 2. Trajectories of mRNA and protein concentrations (Example 1).
∞ γs = Take γ = 0.2. Then it is easy to derive that s=1 αs e ∞ γ s = 1.2652, i.e., Assumption 5 is satisfied. By employing s=1 βs e the function eig of MATLAB, it is easy to verify that the condition (b) of Theorem 3 holds, and hence the GRN under consideration has a unique nonnegative equilibrium which is globally exponentially stable. Since the condition (b) in Theorem 3 is delayindependent, the resulted global exponential stability is independent of the choice of delays σ ij (k) and τ i (k), i, j ∈ Z[1, 2]. Furthermore, when σ ij (k) ≡ 10, τ i (k) ≡ 12 (i, j ∈ Z[1, 2]), m(s) ≡ φ and p(s) ≡ ψ for all s ∈ Z(−∞, 0], we take 100 values of (φ , ψ ) by using the function rand in MATLAB. For all of these values, the simulation results present that the trajectories of mRNAs and proteins converge to the unique nonnegative equilibrium point (m∗ , p∗ ), which concludes that the GRN under consideration here is globally stable. Partial simulation results are shown in Figs. 1–3. This explains the theoretical results presented in Theorem 3. Example 2. Consider GRN (2) with
A = diag(0.1, 0.1 ), C = D = diag(0.1, 0.2 ),
B=
−0.2 0.1
0.5 0.02 , E= 0.2 0.01
J = col(0.2, 0 ), gi (s ) =
−0.02 −0.01 , F= −0.02 0.01
s2 , s ∈ [0, ∞ ), i ∈ Z[1, 2], 1 + s2
αs = βs = (1 − e−2 )e−2(s−1) , s ∈ Z[1, ∞ ),
0.02 , 0.03
σi j (k ) = ri j + si j sin(kπ /2 ), τi (k ) = pi + qi cos(kπ /2 ), where r11 = p2 = 3, s11 = r12 = r21 = s12 = s21 = s22 = q1 = q2 = 1 and r22 = p1 = 2. Clearly, Assumption 2 is satisfied. Furthermore, Assumptions 1 and 3 are satisfied by taking l1 = l2 = 0.65, σ¯ 11 = τ¯2 = 4, σ¯ 12 = σ¯ 21 = 2 and σ¯ 22 = τ¯1 = 3. Based on the discussion in [9, Section 4], a nonnegative equilibrium, (m∗ , p∗ ), of the GRN under consideration can be obtained as follows:
m∗ = col(0.2221, 0.0 0 01 ), p∗ = col(0.0222, 0.0 028 ), i.e., Assumption 4 is satisfied. For γ = 0.3, by direct computation it is derived that ∞ γ s = ∞ β eγ s = 1.4281, that is, Assumption 5 is satiss=1 αs e s=1 s fied. By using the function eig of MATLAB, it is easy to verify that the condition (b) of Theorem 2 holds, so the GRN under consideration has a unique nonnegative equilibrium which is globally exponentially stable. For σ ij (k) and τ i (k) (i, j ∈ Z[1, 2]) given above, when m(s) ≡ φ and p(s) ≡ ψ for all s ∈ Z(−∞, 0], we take 100 values of (φ , ψ ) by using the function rand in MATLAB. For all of these values, the simulation results present that the trajectories of mRNAs and proteins converge to the unique nonnegative equilibrium point (m∗ , p∗ ), which concludes that the GRN under consideration here is globally stable. Partial simulation results are shown in Figs. 4–6.
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Fig. 3. Trajectories of mRNA and protein concentrations (Example 1).
Fig. 4. Trajectories of mRNA and protein concentrations (Example 2).
Fig. 5. Trajectories of mRNA and protein concentrations (Example 2).
Moreover, if we use p2 = 4 instead of p2 = 3 in this example, then the condition (b) of Theorem 2 is not true, which implies that Theorem 2 is not available in this case. So, the global exponential stability criteria provided in Theorem 2 is delay-dependent.
bˆ 11 = −0.2, bˆ 12 = 0.5, bˆ 21 = 0.1, bˆ 22 = 0.2, l1 = l2 = 0.65. From (9), we obtain
A = diag(0.9704, 0.9704 ), C = diag(0.9900, 0.9802 ),
Example 3. Consider GRN (1) subject to (9), where
h = 0.1, aˆ1 = aˆ2 = 0.3, cˆ1 = dˆ1 = 0.1, cˆ2 = dˆ2 = 0.2,
D = diag(0.0100, 0.0198 ), B =
−0.0197 0.0099
0.0493 . 0.0197
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7
Fig. 6. Trajectories of mRNA and protein concentrations (Example 2).
It is easy to verify that [20, Corollay], [21, Theorem 2] and [22, Theorem 2] are not applicable to judge the global exponential stability of the considered GRN. However, by applying [20, Theorem], Corollary 1 or [23, Theorem 2], we can insert that the considered GRN is globally exponentially stable. This illustrates the theoretical results presented in Remark 4.
Expenses Program of Colleges and Universities in Heilongjiang Province (Grant nos. RCCXYJ201813 and RCCX201717). The authors would like to thank the associate editor and the anonymous reviewers for their very helpful comments and suggestions, which greatly improves the original version of the paper. Appendix A. The proof of Theorem 1
6. Conclusions For a class of discrete-time GRNs with time-varying discrete delays and unbounded distributed delays, the problem of exponential stability analysis colorredhas been investigated by proposing a novel approach which does not require to construct any auxiliary function or Lyapunov–Krasovskii functional. Several delaydependent and -independent global exponential stability criteria are established for the class of GRNs under consideration. The obtained stability criteria are only related to the systems parameters, which is simple and can easily be verified by using the software MATLAB. For some special cases of the discrete-time GRNs under consideration, it has been theoretically shown that the global exponential stability criteria in Corollary 1 are less conservative than those in [20, Corollay], [21, Theorem 2] and [22, Theorem 2], and are equivalent to those in [20, Theorem] and [23, Theorem 2]. So the paper can be viewed as a generalization of [20–23]. Furthermore, compared with the linear matrix inequality-based stability criteria in [10,13,24,37,38], the obtained stability criteria possess more simpler forms which reduce the computational complexity. Moreover, several numerical examples are also presented to illustrate the effectiveness of theoretical results. 7. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Choose K1 > 0 such that
K1 u˜ col(1, 1, . . . , 1 ), K1 v˜ col(1, 1, . . . , 1 ). For any but fixed initial functions ϕ , ψ ∈ C (Z(−∞, 0], Rn ) of system (5), define
u(k ) = K1 (ψ , ϕ )∞ e−γ k u˜, k ∈ Z,
(A.1a)
v(k ) = K1 (ψ , ϕ )∞ e−γ k v˜ , k ∈ Z.
(A.1b)
Now we indsert that
|x ( k )| v ( k ), |y ( k )| u ( k ), ∀k ∈ Z,
(A.2)
where
x(k ) = col(x1 (k ), x2 (k ), . . . , xn (k )), y(k ) = col(y1 (k ), y2 (k ), . . . , yn (k )). Indeed, from the definition of · ∞ and the choice of K1 , it is clear that
|x(q )| v(q ), |y(q )| u(q ), ∀q ∈ Z(−∞, 0].
(A.3)
If there exists a nonnegative integer q such that (A.2) holds for k ≤ q, but (A.2) does not hold for k = q + 1, then without loss of generality, we can assume that
|x(k )| v(k ), |y(k )| u(k ), ∀k ∈ Z(−∞, q],
(A.4a)
|xi0 (q + 1 )| > vi0 (q + 1 ) for some i0 ∈ Z[1, n].
(A.4b)
It follows from (A.4b), (5a) and (6) that Acknowledgment This work was supported in part by the National Social Science Foundation of China (Grant no. 17BTQ052), the National Natural Science Foundation of China (Grant nos. 61703148, 61673098 and 61873306), and the Natural Science Foundation of Heilongjiang Province (Grant nos.QC2018083). the Basic Research Operating
vi0 ( q + 1 ) < |xi0 ( q + 1 )| ≤ ai0 |xi0 ( q )| +
n
|bi0 j ||g˜(y j (q − σi0 j (q )))|
j=1
+
n ∞
αs |ei0 j ||g˜(y j (q − s ))|
j=1 s=1
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≤ ai0 |xi0 ( q )| +
n
|bi0 j |l j |y j (q − σi0 j (q ))|
j=1
+
n ∞
αs |ei0 j |l j |y j (q − s )|.
j=1 s=1
By using (A.4a), one can easily derive that
vi0 ( q + 1 ) < ai0 vi0 ( q ) +
n
|bi0 j |l j u j (q − σi0 j (q ))
j=1
+
n ∞
αs |ei0 j |l j u j (q − s ),
(A.5)
j=1 s=1
where ui (q) and vi (q) are the i-th components of u(q) and v(q), respectively. The combination of (A.1) and (A.5) gives
vi0 (q + 1 ) < ai0 K1 (ψ , ϕ )∞ e−γ q v˜ i0 +
n
|bi0 j |l j K1 (ψ , ϕ )∞ e−γ (q−σi0 j (q)) u˜ j
j=1
+
n ∞
αs |ei0 j |l j K1 (ψ , ϕ )∞ e−γ (q−s) u˜ j
j=1 s=1
≤ K1 (ψ , ϕ )∞ e−γ q
×
ai0 v˜ i0 +
n
|bi0 j |e
γ σ¯ i0 j
+ α|ei0 j | l j u˜ j ,
(A.6)
j=1
where u˜i and v˜ i are the i-th components of u˜ and v˜ , respectively. Using (7) and (A.1b), we obtain that
vi0 (q + 1 ) < K1 (ψ , ϕ )∞ e−γ (q+1) v˜ i0 = vi0 (q + 1 ),
(A.7)
a contradiction. Therefore, (A.2) holds for k ∈ Z. It follows from (A.1) and (A.2) that
(x(k ), y(k )) = (x(k )22 + y(k )22 ) 2 1
≤ (v(k )22 + u(k )22 ) 2 1
= K1 e−γ k (ψ , ϕ )∞ (v˜ 22 + u˜22 ) 2 , ∀k ∈ Z[0, ∞ ). 1
1
Let K = K1 (v˜ 22 + u˜22 ) 2 , then
(x(k ), y(t )) ≤ Ke−γ k (ψ , ϕ )∞ , ∀k ∈ Z[0, ∞ ). This, together with the arbitrariness of ϕ , ψ ∈ C(Z(−∞, 0], Rn ), implies that the zero equilibrium of system (5) is globally exponentially stable, and hence GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable. This completes the proof. References [1] H. De Jong, Modeling and simulation of genetic regulatory systems: a literature review, J. Comput. Biol. 9 (1) (2002) 67–103. [2] M.B. Elowitz, S. Leibler, A synthetic oscillatory network of transcriptional regulators, Nature 403 (20) (20 0 0) 335–338. [3] N. Friedman, Inferring cellular networks using probabilistic graphical models, Science 303 (5659) (2004) 799–805. [4] X. Zhang, Y. Wang, L. Wu, Analysis and design of delayed genetic regulatory networks, Vol. 207 of Studies in Systems, Decision and Control, Springer, Cham, Switzerland, 2019. [5] X.B. Wan, Z.D. Wang, M. Wu, X.H. Liu, State estimation for discrete time-delayed genetic regulatory networks with stochastic noises under the round-robin protocols, IEEE Trans. Nanobiosci. 17 (2) (2018) 145–154. [6] L. Chen, K. Aihara, Stability of genetic regulatory networks with time delay, IEEE Trans. Circuits Syst. I Regul. Pap. 49 (5) (2002) 602–608. [7] X. Zhang, L. Wu, S.C. Cui, An improved integral inequality to stability analysis of genetic regulatory networks with interval time-varying delays, IEEE/ACM Trans. Comput. Biol. Bioinf. 12 (2) (2015) 398–409. [8] W. He, J. Cao, Robust stability of genetic regulatory networks with distributed delay, Cognit. Neurodyn. 2 (4) (2008) 355–361.
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[54] R. Yang, L. Li, P. Shi, Dissipativity-based two-dimensional control and filtering for a class of switched systems, IEEE Trans. Syst. Man Cybern. Syst. (2019), doi:10.1109/TSMC.2019.2916417. Published online. Chunyan Liu received Ph.D. degree in Informatics from Jilin University in China in 2016. She is currently a Professor in information management school of Heilongjiang University, Harbin, China. Her research interests include information analysis and forecast, social networks and dynamic systems. She has authored more than 40 research papers.
Xin Wang received the B.S. and M.S. degrees in School of Mathematical Science from Heilongjiang University, Harbin, China, in 2008 and 2011, respectively, and the Ph.D. degree in navigation guidance and control from Northeastern University, Shenyang, China, in 2016. He is currently a Lecturer with the School of Mathematical Science, Heilongjiang University, Harbin, China. His research interests include fault diagnosis, fault-tolerant control, and multi-agent coordination.
Yu Xue received the B.S. degree in Automation from Harbin University of Science and Technology, China in 2002; the PhD degree in Control Theory and Control Engineering from Harbin Institute of Technology, China in 20 07. From April 20 07 to August 2009, she is an Engineer in Sichuan Academy of Aerospace Technology. From September 2009 to May 2013, she is a Senior Engineer in Sichuan Academy of Aerospace Technology. In June 2013, she joined Heilongjiang University. Her research interests include stability analysis of delayed dynamic systems, robust control and genetic regulatory networks.
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Global exponential stability analysis of discrete-time genetic regulatory networks with time-varying discrete delays and unbounded distributed delays Chunyan Liu a, Xin Wang b,c,∗, Yu Xue b,c,∗ a b c
School of Information Management, Heilongjiang University, Harbin 150080, P R China School of Mathematical Science, Heilongjiang University, Harbin 150080, P R China Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, Heilongjiang University, Harbin 150080, P R China
a r t i c l e
i n f o
Article history: Received 23 April 2019 Revised 22 August 2019 Accepted 16 September 2019 Available online xxx Communicated by Jun Yu Keywords: Genetic regulatory networks Global exponential stability Time-varying discrete delays Unbounded distributed delays
a b s t r a c t This paper addresses the problem of exponential stability analysis of discrete-time genetic regulatory networks (GRNs) with time-varying discrete delays and unbounded distributed delays. A novel approach is proposed to establish delay-dependent and -independent global exponential stability criteria for the class of GRNs under consideration. The main novelties lie in: (i) without any auxiliary function or Lyapunov– Krasovskii functional is required in the process of deriving the global exponential stability criteria; (ii) the obtained stability criteria improves or generalizes the existing results; (iii) the obtained stability criteria are directly represented by the systems parameters, which can be conveniently verified; and (iv) the exponential estimation of GRNs can be obtained from the derivation process of the exponential stability criteria. The effectiveness of theoretical results is illustrated by presenting several numerical examples.
1. Introduction Genetic regulatory networks (GRNs) describe the process of gene expression, including transcription and translation, which is helpful to understand the genome sequencing and the gene recognition [1–3]. Thus the study of GRNs has attracted interest of many experts and scholars in theoretical physics, control science, biology, etc (see [4–6] and the references therein). Due to the slow transcription and translation processes, functional differential/difference equations have been employed to model GRNs [6,7]. Moreover, in individual molecules, movement of mRNA from a transcription site to translation sites requires a long time, which is necessary to introduce distributed delays into functional differential/difference equation models [8–10]. As a result, the functional differential/difference equation models with discrete and/or distributed delays have been established to solve the analysis and design problems of GRNs, including the stability analysis [7,9,11–24], passivity analysis [25], Hopf bifurcation analysis [26,27], controller synthesis [28–30], observer design [31–33], state estimation [5,10,34–39], filter design [40–42], and so on. ∗ Corresponding authors at: School of Mathematical Science, Heilongjiang University, Harbin 150080, P R China. E-mail addresses: [email protected] (C. Liu), [email protected] (X. Wang), [email protected] (Y. Xue).
© 2019 Elsevier B.V. All rights reserved.
The most important property of GRNs, stability, has been studied in a great number of literature. However, to the best of authors’ knowledge, these achieved results are mainly based on the functional differential equation models (i.e., delayed continuous-time GRNs), and rarely focus on the functional difference equation models (i.e., delayed discrete-time GRNs). The stability criteria for delayed discrete-time GRNs are only established in [10,13,20–24,37,38]. The exponential stability criteria are involved in [20–22] by constructing appropriate auxiliary functions, and the asymptotic/exponential stability criteria are investigated in [10,13,23,24,37,38] by constructing appropriate Lyapunov– Krasovskii functionals. Since the construct of auxiliary function or Lyapunov–Krasovskii functional is quite difficult and requires a high degree of skill, it is necessary to find new approach to analyze stability of delayed discrete-time GRNs, which motivates the research. This paper is concerned with the problem of exponential stability analysis of discrete-time GRNs with time-varying discrete delays and unbounded distributed delays. A new approach is proposed to establish delay-dependent and -independent global exponential stability criteria for the class of GRNs under consideration. For some special cases of the discrete-time GRNs under consideration, it is theoretically proven that the global exponential stability criteria in Corollary 1 below are less conservative than ones in [20, Corollay], [21, Theorem 2] and [22, Theorem 2], and are equivalent to ones in
https://doi.org/10.1016/j.neucom.2019.09.047 0925-2312/© 2019 Elsevier B.V. All rights reserved.
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[20, Theorem] and [23, Theorem 2]. So the paper can be regarded as a generalized version of [20–23]. Compared with the linear matrix inequality-based stability criteria proposed in [10,13,24,37,38], the obtained stability criteria possess more simpler form, which reduces the computational complexity. In addition, several numerical examples are also provided to illustrate the effectiveness of theoretical results. The novelties of this paper are as follows: (i) the proposed approach is directly based on the definition of global exponential stability, and it does not require to construct any auxiliary function or Lyapunov–Krasovskii functional; (ii) the obtained stability criteria are directly represented by the systems parameters, which is more simple and easily be checked by using the software MATLAB; (iii) the obtained stability criteria extend or improve the existing results; and (iv) the obtained delay-dependent global exponential stability criteria can used to give the exponential estimation of GRNs. Notation. Suppose Z, R and C are sets of all integers, real numbers and complex numbers, respectively. Let Z[a, b] be the subset of Z consisting of all integers between a and b, and let Z[a, ∞ ) = ∪b>a Z[a, b]. For given positive integers p and q, let R p×q denote the set of all p × q matrices over R. Set R p = R p×1 . For A = [ai j ] ∈ R p×q and B = [bi j ] ∈ R p×q , the matrix [aij bij ], denoted by A◦B, is called the Hadamard product of A and B. Furthermore, we denote by |A| the matrix [|aij |]. 2. Problem formulation and preliminaries Consider a class of discrete-time GRNs with time-varying discrete delays, which can be described as [4]:
mi ( k + 1 ) = ai mi ( k ) +
n
bi j g j ( p j (k − σi j (k ))) + Ji ,
(1a)
where li > 0 is a known constant. ∞ ∞ Assumption 2. s=1 αs = s=1 βs = 1. Assumption 3. σi j (k ) ≤ σ¯ i j and τi (k ) ≤ τ¯i for any i, j ∈ Z[1, n] and k ∈ Z[0, ∞ ), where σ¯ i j and τ¯i are known positive constants. For A = [ai j ] ∈ R p×q and B = [bi j ] ∈ R p×q , the symbol A B (or B A) means that aij ≥ bij for all i ∈ Z[1, p] and j ∈ Z[1, q]; in particular, if aij > bij for all i ∈ Z[1, p] and j ∈ Z[1, q], then we use A B (or B ≺ A) instead of A B (or B A). Clearly, |XY| |X||Y| for p×q p×q all X ∈ R p×q and Q ∈ Rq×r . Denote by R and R the sets of all p × q nonnegative and positive matrices, respectively. Similar notations are adopted for vectors. A pair (m∗ , p∗ ) ∈ Rn × Rn is called a nonnegative equilibrium of GRN (2), if
m∗i = ai m∗i +
n
(bi j + ei j )g j ( p∗j ) + Ji ,
(3a)
j=1
p∗i = ci p∗i + di m∗i +
n
fi j m∗j , i ∈ Z[1, n], k ∈ Z[0, ∞ ),
(3b)
j=1
where m∗i and p∗i are the i-th components of m∗ and p∗ , respectively. Obviously, (3) can be written as the following compact matrix form:
(A − In )m∗ + (B + E )g( p∗ ) + J = 0, (C − In ) p∗ + (D + F )m∗ = 0, where
A = diag(a1 , a2 , . . . , an ), C = diag(c1 , c2 , . . . , cn ),
j=1
pi (k + 1 ) = ci pi (k ) + di mi (k − τi (k )), i ∈ Z[1, n], k ∈ Z[0, ∞ ),
D = diag(d1 , d2 , . . . , dn ), J = col(J1 , J2 , . . . , Jn ),
(1b) where mi (k) and pi (k) are the mRNA and protein concentrations of gene i, respectively, gi is the regulation function of gene i, τ i (k) and σ ij (k) are the time-varying translational delay and feedback regulation delay, respectively, ai > 0, ci > 0, di > 0, Ji > 0, and bi j ∈ R satisfies that bij < 0 if transcription factor j represses gene i, bij > 0 if transcription factor j activates gene i, and bi j = 0 if transcription factor j does not regulate gene i. See [4, Chapter 11] for more detailed explanations about model (1). When the unbounded distributed delays are introduced into GRN (1), the following model of discrete-time GRNs with timevarying discrete delays and unbounded distributed delays are obtained [10]:
mi ( k + 1 ) = ai mi ( k ) +
n
+
j=1
ei j
∞
αs g j ( p j (k − s )) + Ji ,
(2a)
s=1
+
j=1
fi j
∞
and diag( · ) and col( · ) refer to the diagonal matrix and column vector, respectively. The existence issue of nonnegative equilibrium of GRN (2) has been addressed in [17] when D + F is a positive diagonal matrix. In the rest of this section we make the following assumption: Assumption 4. GRN (2) has at least one nonnegative equilibrium (m∗ , p∗ ).
Set x(k ) = m(k ) − m∗ and y(k ) = p(k ) − p∗ , k ∈ Z[0, ∞ ). Due to (2) and (3), we have
xi ( k + 1 ) = ai xi ( k ) +
pi (k + 1 ) = ci pi (k ) + di mi (k − τi (k )) n
g( p∗ ) = col(g1 ( p∗1 ), g2 ( p∗2 ), . . . , gn ( p∗n )),
Remark 1. By using an approach similar to one in [17], it can be concluded that if B + E 0 and D + F 0, then GRN (2) has at least one nonnegative equilibrium.
bi j g j ( p j (k − σi j (k )))
j=1 n
B = [bi j ], E = [ei j ], F = [ f i j ],
n
bi j g˜ j (y j (k − σi j (k )))
j=1
βs m j (k − s ), i ∈ Z[1, n], k ∈ Z[0, ∞ ),
(2b)
s=1
+
where eij , fij , α s ≥ 0 and β s ≥ 0 are known constants. For convenience, let’s make the following assumptions: Assumption 1. The regulation functions gi (i = 1, 2, . . . , n ) satisfy
gi ( s1 ) − gi ( s2 ) gi ( 0 ) = 0 , 0 ≤ ≤ li , ∀s1 , s2 ≥ 0, s1 = s2 , s1 − s2
n
ei j
j=1
∞
αs g˜ j (y j (k − s )),
(5a)
s=1
yi (k + 1 ) = ci yi (k ) + di xi (k − τi (k )) +
n j=1
fi j
∞
βs x j (k − s ), i ∈ Z[1, n], k ∈ Z[0, ∞ ),
(5b)
s=1
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where g˜ j (· ) = g j (· + p∗ ) − g j ( p∗ ). Clearly, (0,0) is an equilibrium of system (5), and g˜ satisfies the following inequality:
|g˜ j (s )| ≤ l j |s|, ∀s ≥ 0, j ∈ Z[1, n].
(ψ , ϕ )∞ = max
sup
s∈Z(−∞,0]
where
Aγ = A − e−γ In , eγ σ¯ = [eγ σ¯ i j ], eγ τ¯ = diag(eγ τ¯1 , eγ τ¯2 , . . . ,eγ τ¯n ),
(6)
Let C (Z(−∞, 0], Rn ) refer to the linear space over R consisting of all functions φ : Z(−∞, 0] → Rn satisfying sups∈Z(−∞,0] φ (s ) < ∞. Define the norm · on Rn × Rn by (a, b) = (a22 + b|22 )1/2 for any a, b ∈ Rn , and the norm · ∞ on C (Z(−∞, 0], Rn ) × C (Z(−∞, 0], Rn ) by
Cγ = eγ τ¯ ◦ D + β|F |, Dγ = C − e−γ In ,
α=
s∈Z(−∞,0]
Definition 1. The zero equilibrium of system (5) is said to be globally exponentially stable, if there exist scalars K > 0 and γ > 0 such that every solution (x(k), y(k)), starting from ϕ , ψ ∈ C (Z(−∞, 0], Rn ), satisfies
(x(k ), y(t )) ≤ Ke−γ k (ψ , ϕ )∞ , ∀k ∈ Z[0, ∞ ). This paper aims at proposing a novel approach to establish delay-dependent and -independent global exponential stability criteria for the zero equilibrium of system (5) (i.e., the unique nonnegative equilibrium of GRN (2)). It is worth emphasizing that the proposed approach does not involve the construct of any Lyapunov–Krasovskii functional or auxiliary function, which is different ones in [10,13,20–24,37,38]. A matrix M ∈ Rn×n is called a Metzler matrix if all off-diagonal elements of M are nonnegative. Let In be the identity matrix in Rn×n , and λ(M ) = {z ∈ C : det(zIn − M ) = 0}. The spectral abscissa of M is defined by s(M) := max{Reλ: λ ∈ λ(M)}, and the spectral radius of M is defined by ρ (M) := max {|λ|: λ ∈ λ(M)}. We end this section by introducing the following lemma which is helpful to present the main contributions of this paper. Lemma 1. [43] Let A0 ∈ Rn×n be a Metzler matrix and B0 , C0 , D0 ∈ Rn ×n . Then the following statements (i)–(iii) are equivalent: (i) ρ (D0 ) < 1 and s(A0 + B0 (In − D0 )−1C0 ) < 0. (ii) A0 x + B0 y ≺ 0 and C0 x + D0 y ≺ y for some x, y ∈ Rn . (iii) s(A0 ) < 0 and ρ (C0 (−A0 )−1 B0 + D0 ) < 1. 3. Global exponential stability criteria
αs eγ s , β =
∞
βs eγ s ,
s=1
The proof of Theorem 1 will be given in Appendix A below. Combining Theorem 1 and Lemma 1, one can easily derive the following conclusion. Theorem 2. For given positive scalars γ ≤ γ 0 , σ¯ i j , τ¯i and lj (i, j ∈ Z[1, n] ), let Assumptions 1–5 be satisfied. Then GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable, if one of the following statements (a)–(c) holds: (a) Aγ v˜ + Bγ u˜ ≺ 0 and Cγ v˜ + (Dγ + In )u˜ ≺ u˜ for some u˜, v˜ ∈ Rn . (b) ρ (Dγ + In ) < 1 and s(Aγ − Bγ D−1 γ Cγ ) < 0. (c) s(Aγ ) < 0 and ρ (Dγ + In − Cγ A−1 γ Bγ ) < 1. Theorem 2 provides delay-dependent sufficient conditions under which GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable. To present delay-independent sufficient conditions, we define
A˜ = A − In , B˜ = (|B| + |E | )L, C˜ = D + |F |. Theorem 3. For given positive scalars lj ( j ∈ Z[1, n] ), let Assumptions 1, 2, 4 and 5 be satisfied. Then GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable, if one of the following statements (a)–(c) holds: (a) A˜ v˜ + B˜u˜ ≺ 0 and C˜v˜ + C u˜ ≺ u˜ for some u˜, v˜ ∈ Rn . (b) ρ (C) < 1 and s(A˜ + B˜(In − C )−1C˜ ) < 0. (c) s(A˜ ) < 0 and ρ (C − C˜A˜ −1 B˜ ) < 1. ∞ γ s and Proof. Assumption 5 ensures that the series s=1 αs e ∞ γ s are uniformly convergent for γ ≤ γ . This, together β e 0 s=1 s with Assumption 2, implies that
lim+
γ →0
In this section we will investigate sufficient conditions under which the zero equilibrium of system (5) is globally exponentially stable, that is, GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable. To this end, we require the following assumption: ∞ γ0 s < +∞ Assumption 5. There exists γ 0 > 0 such that s=1 αs e ∞ and s=1 βs eγ0 s < +∞. Assumption 5 guarantees the uniform convergence of the series ∞ γ s and ∞ β eγ s for γ ≤ γ , which is indispensable to 0 s=1 αs e s=1 s our approach. Now it is the place to present the following conclusion which is one of the main contributions of this paper. Theorem 1. For given positive scalars γ ≤ γ 0 , σ¯ i j , τ¯i and lj (i, j ∈ Z[1, n] ), let Assumptions 1–5 be satisfied. Then GRN (2) has a unique nonnegative equilibrium which is globally exponentially stable, if there exist u˜, v˜ ∈ Rn such that
Cγ v˜ + Dγ u˜ 0,
∞
and the operator ◦ means the Hadamard product of matrices.
for any ψ , ϕ ∈ C (Z(−∞, 0], Rn ), where · 2 is the Euclidean norm on Rn .
Aγ v˜ + Bγ u˜ 0,
Bγ = eγ σ¯ ◦ |B| + α|E | L, L = diag(l1 , l2 , . . . , ln ),
s=1
ψ (s )2 , sup ϕ (s )2
3
(7)
(8)
∞ s=1
αs eγ s = lim+ γ →0
∞
βs eγ s = 1.
s=1
Furthermore, limγ →0+ Aγ = A˜ , limγ →0+ Bγ = B˜, limγ →0+ Cγ = C˜ and limγ →0+ Dγ = C − In . This, together with Theorem 2, completes the proof. Remark 2. Compared with [20–22], the system model under consideration involves the unbounded distributed delays, which brings extreme difficult in theoretical derivation and numerical computation, so that we have to add Assumptions 2 and 5. Moreover, the time-varying discrete delays complicate the addressed problem, which is not concerned in [20–22]. Remark 3. The conditions in Theorem 1 and (a) in Theorems 2 and 3 can be easily realized by employing the toolbox YALMIP of MATLAB. The testify of the condition (b) or (c) in Theorems 2 and 3 is concerned with the computation of a constant matrix, which can be proceeded by using the function eig of MATLAB for the loworder case, and the iterative methods, including the power method and the inverse power method, for the high-order case. We end the section by the following corollary which gives the global exponential stability criteria for the nonnegative equilibrium of GRN (1).
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Corollary 1. For given positive scalars lj ( j ∈ Z[1, n] ), let Assumption 1 be satisfied. Then GRN (1) has a unique nonnegative equilibrium which is globally exponentially stable, if one of the following statements (a)–(c) holds: (a) (In − A )v˜ |B|Lu˜ and (In − C )u˜ Dv˜ for some u˜, v˜ ∈ (b) ρ (C) < 1 and s(A + |B|L(In − C )−1 D ) < 1. (c) s(A) < 1 and ρ (C + D(In − A )−1 |B|L ) < 1.
Rn .
4. Theoretical comparisons In this section, we will give theoretical comparisons of the global exponential stability criteria for the nonnegative equilibrium of GRN (1) presented in this paper and literature. When GRN (1) is viewed as a semi-discretized version of continuous-time GRN, its parameters possess the following special form [20–22]:
σi j (k ) ≡ σˆ i j /h, τi (k ) ≡ τˆi /h,
(9a)
ai = e−aˆi h , ci = e−cˆi h ,
(9b)
bi j =
1 − e−aˆi h ˆ 1 − e−cˆi h ˆ 1 − e−aˆi h bi j , di = di , Ji = Jˆi , i, j ∈ Z[1, n], aˆi cˆi aˆi
(9c)
where h is the discretization step-size, · represents the integral function, and aˆi , cˆi , dˆi , Jˆi ∈ R , bˆ i j ∈ R and σˆ i j , τˆi ∈ Z[1, ∞ ) are the system parameters of original continuous-time GRN. The following proposition is a simple overview of several results in literature.
which is equivalent to
1 > C + D(In − A )−1 L|B|1 , where · 1 is the maximum row sum norm of matrices. Hence 1 > ρ (C + D(In − A )−1 L|B| ). Noting that the matrix C + D(In − A )−1 L|B| is similar to C + D(In − A )−1 |B|L and 0 < ai < 1 for all i ∈ Z[1, n], we can derive that (c) in Corollary 1 is true. Case 4: (iv) in Proposition 1 is true. Then so is (iii) in Proposition 1. By Case 3, it follows that (c) in Corollary 1 is true. Remark 4. Theorem 4 and its proof show that the global exponential stability criteria in Corollary 1 are less conservative than ones in [20, Corollay], [21, Theorem 2] and [22, Theorem 2], and are equivalent to one in [20, Theorem]. Remark 5. Similar to the proof of Theorem 4, one can derive that Corollary 1 is equivalent to [23, Theorem 2]. Finally, we further emphasize the merits of the proposed approach by the following two remarks. Remark 6. Different from the approaches proposed in [20–23], our approach has the following advantages: • The approach is available for GRNs containing time-varying discrete delays and/or distributed delays; • The approach can provide delay-dependent global exponential stability criteria; • The obtained delay-dependent global exponential stability criteria can used to give the exponential estimation of GRNs.
Proposition 1. For given positive scalars lj ( j ∈ Z[1, n] ), let Assumption 1 be satisfied. Then GRN (1) subject to (9) has a unique nonnegative equilibrium which is globally exponentially stable, if one of the following statements (i)–(iv) is satisfied:
So, our future work is to extend the approach proposed in this paper to the more general system models, for examples, complex networks [44,45], time-varying systems [46], neutral systems [47– 50], multi-agent systems [51,52], and switched systems [53,54].
(i) [20, Theorem] There exist u˜, v˜ ∈ Rn such that cˆi u˜i > dˆi v˜ i and aˆi v˜ i > nj=1 |bˆ i j |l j u˜ j for all i ∈ Z[1, n]. (ii) [20, Corollay] cˆi > dˆi and aˆi > nj=1 |bˆ i j |l j for all i ∈ Z[1, n]. (iii) [21, Theorem 2] aˆi cˆi > dˆi li nj=1 |bˆ i j | for all i ∈ Z[1, n]. (iv) [22, Theorem 2] aˆi > dˆi and cˆi > li n |bˆ i j | for all i ∈ Z[1, n].
Remark 7. For a class of stochastic GRNs with leakage, discrete, and distributed delays, Pandiselvi, Raja and Cao et al. [10] established an exponential stability criterion in the form of linear matrix inequalities by utilizing the Lyapunov–Krasovskii functional approach. The approach proposed in [10] is also available to investigate exponential stability of GRN (2), however, it will result in an exponential stability criterion described by a set of linear matrix inequalities, which requires more compute time to verify than one obtained in this paper.
j=1
Theorem 4. For given positive scalars lj ( j ∈ Z[1, n] ), let Assumption 1 be satisfied. Assume that the parameters in GRN (1) possess the special form (9). If one of (i)–(iv) in Proposition 1 is true, then so is one of (a)–(c) in Corollary 1. Proof. Case 1: (i) in Proposition 1 is true. Then
5. Illustrative examples In this section we will present the effectiveness of the established global exponential stability criteria by three illustrative examples.
dˆi di di u˜i > v˜ i = v˜ i = v˜ i 1 − ci cˆi 1 − e−cˆi h and
Example 1. Consider GRN (2) with
n n n |bˆ i j |l j |bi j |l j |bi j |l j v˜ i > u˜ j = u˜ j = u˜ j . ˆ − a h i 1 − ai aˆi 1 − e j=1 j=1 j=1
A = diag(0.6, 0.6 ), C = D = diag(0.3, 0.2 ), F = diag(0.09, 0.27 ),
This, together with 0 < ai < 1 and 0 < ci < 1, implies that (a) in Corollary 1 is true. Case 2: (ii) in Proposition 1 is true. Then (i) in Proposition 1 holds by taking v˜ i = u˜i = 1 for all i ∈ Z[1, n]. Due to Case 1, it is concluded that (a) in Corollary 1 is true. Case 3: (iii) in Proposition 1 is true. Then
1>
dˆi li cˆi
n j=1
|bˆ i j | aˆi
=
di li 1 − e−cˆi h
n j=1
|bi j | 1−
e−aˆi h
for all i ∈ Z[1, n], and hence
1 > ci + di (1 − ai )−1 li
n j=1
|bi j |, i ∈ Z[1, n]
=
di li 1 − ci
n j=1
|bi j | 1 − ai
B=
−0.2 0.1
0.5 0.2 , E= 0.2 0.1
J = col(0.2, 0 ), gi (s ) =
−0.2 , 0.2
s2 , s ∈ [0, ∞ ), i ∈ Z[1, 2], 1 + s2
αs = βs = (1 − e−2 )e−2(s−1) , s ∈ Z[1, ∞ ). Clearly, Assumption 2 is satisfied. If we choose l1 = l2 = 0.65, then Assumption 1 is satisfied. Following the discussion in [9, Section 4], one can easily obtain a nonnegative equilibrium, (m∗ , p∗ ), of the GRN under consideration as follows:
m∗ = col(0.5003, 0.0361 ), p∗ = col(0.2788, 0.0212 ), i.e., Assumption 4 is satisfied.
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Fig. 1. Trajectories of mRNA and protein concentrations (Example 1).
Fig. 2. Trajectories of mRNA and protein concentrations (Example 1).
∞ γs = Take γ = 0.2. Then it is easy to derive that s=1 αs e ∞ γ s = 1.2652, i.e., Assumption 5 is satisfied. By employing s=1 βs e the function eig of MATLAB, it is easy to verify that the condition (b) of Theorem 3 holds, and hence the GRN under consideration has a unique nonnegative equilibrium which is globally exponentially stable. Since the condition (b) in Theorem 3 is delayindependent, the resulted global exponential stability is independent of the choice of delays σ ij (k) and τ i (k), i, j ∈ Z[1, 2]. Furthermore, when σ ij (k) ≡ 10, τ i (k) ≡ 12 (i, j ∈ Z[1, 2]), m(s) ≡ φ and p(s) ≡ ψ for all s ∈ Z(−∞, 0], we take 100 values of (φ , ψ ) by using the function rand in MATLAB. For all of these values, the simulation results present that the trajectories of mRNAs and proteins converge to the unique nonnegative equilibrium point (m∗ , p∗ ), which concludes that the GRN under consideration here is globally stable. Partial simulation results are shown in Figs. 1–3. This explains the theoretical results presented in Theorem 3. Example 2. Consider GRN (2) with
A = diag(0.1, 0.1 ), C = D = diag(0.1, 0.2 ),
B=
−0.2 0.1
0.5 0.02 , E= 0.2 0.01
J = col(0.2, 0 ), gi (s ) =
−0.02 −0.01 , F= −0.02 0.01
s2 , s ∈ [0, ∞ ), i ∈ Z[1, 2], 1 + s2
αs = βs = (1 − e−2 )e−2(s−1) , s ∈ Z[1, ∞ ),
0.02 , 0.03
σi j (k ) = ri j + si j sin(kπ /2 ), τi (k ) = pi + qi cos(kπ /2 ), where r11 = p2 = 3, s11 = r12 = r21 = s12 = s21 = s22 = q1 = q2 = 1 and r22 = p1 = 2. Clearly, Assumption 2 is satisfied. Furthermore, Assumptions 1 and 3 are satisfied by taking l1 = l2 = 0.65, σ¯ 11 = τ¯2 = 4, σ¯ 12 = σ¯ 21 = 2 and σ¯ 22 = τ¯1 = 3. Based on the discussion in [9, Section 4], a nonnegative equilibrium, (m∗ , p∗ ), of the GRN under consideration can be obtained as follows:
m∗ = col(0.2221, 0.0 0 01 ), p∗ = col(0.0222, 0.0 028 ), i.e., Assumption 4 is satisfied. For γ = 0.3, by direct computation it is derived that ∞ γ s = ∞ β eγ s = 1.4281, that is, Assumption 5 is satiss=1 αs e s=1 s fied. By using the function eig of MATLAB, it is easy to verify that the condition (b) of Theorem 2 holds, so the GRN under consideration has a unique nonnegative equilibrium which is globally exponentially stable. For σ ij (k) and τ i (k) (i, j ∈ Z[1, 2]) given above, when m(s) ≡ φ and p(s) ≡ ψ for all s ∈ Z(−∞, 0], we take 100 values of (φ , ψ ) by using the function rand in MATLAB. For all of these values, the simulation results present that the trajectories of mRNAs and proteins converge to the unique nonnegative equilibrium point (m∗ , p∗ ), which concludes that the GRN under consideration here is globally stable. Partial simulation results are shown in Figs. 4–6.
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Fig. 3. Trajectories of mRNA and protein concentrations (Example 1).
Fig. 4. Trajectories of mRNA and protein concentrations (Example 2).
Fig. 5. Trajectories of mRNA and protein concentrations (Example 2).
Moreover, if we use p2 = 4 instead of p2 = 3 in this example, then the condition (b) of Theorem 2 is not true, which implies that Theorem 2 is not available in this case. So, the global exponential stability criteria provided in Theorem 2 is delay-dependent.
bˆ 11 = −0.2, bˆ 12 = 0.5, bˆ 21 = 0.1, bˆ 22 = 0.2, l1 = l2 = 0.65. From (9), we obtain
A = diag(0.9704, 0.9704 ), C = diag(0.9900, 0.9802 ),
Example 3. Consider GRN (1) subject to (9), where
h = 0.1, aˆ1 = aˆ2 = 0.3, cˆ1 = dˆ1 = 0.1, cˆ2 = dˆ2 = 0.2,
D = diag(0.0100, 0.0198 ), B =
−0.0197 0.0099
0.0493 . 0.0197
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Fig. 6. Trajectories of mRNA and protein concentrations (Example 2).
It is easy to verify that [20, Corollay], [21, Theorem 2] and [22, Theorem 2] are not applicable to judge the global exponential stability of the considered GRN. However, by applying [20, Theorem], Corollary 1 or [23, Theorem 2], we can insert that the considered GRN is globally exponentially stable. This illustrates the theoretical results presented in Remark 4.
Expenses Program of Colleges and Universities in Heilongjiang Province (Grant nos. RCCXYJ201813 and RCCX201717). The authors would like to thank the associate editor and the anonymous reviewers for their very helpful comments and suggestions, which greatly improves the original version of the paper. Appendix A. The proof of Theorem 1
6. Conclusions For a class of discrete-time GRNs with time-varying discrete delays and unbounded distributed delays, the problem of exponential stability analysis colorredhas been investigated by proposing a novel approach which does not require to construct any auxiliary function or Lyapunov–Krasovskii functional. Several delaydependent and -independent global exponential stability criteria are established for the class of GRNs under consideration. The obtained stability criteria are only related to the systems parameters, which is simple and can easily be verified by using the software MATLAB. For some special cases of the discrete-time GRNs under consideration, it has been theoretically shown that the global exponential stability criteria in Corollary 1 are less conservative than those in [20, Corollay], [21, Theorem 2] and [22, Theorem 2], and are equivalent to those in [20, Theorem] and [23, Theorem 2]. So the paper can be viewed as a generalization of [20–23]. Furthermore, compared with the linear matrix inequality-based stability criteria in [10,13,24,37,38], the obtained stability criteria possess more simpler forms which reduce the computational complexity. Moreover, several numerical examples are also presented to illustrate the effectiveness of theoretical results. 7. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Choose K1 > 0 such that
K1 u˜ col(1, 1, . . . , 1 ), K1 v˜ col(1, 1, . . . , 1 ). For any but fixed initial functions ϕ , ψ ∈ C (Z(−∞, 0], Rn ) of system (5), define
u(k ) = K1 (ψ , ϕ )∞ e−γ k u˜, k ∈ Z,
(A.1a)
v(k ) = K1 (ψ , ϕ )∞ e−γ k v˜ , k ∈ Z.
(A.1b)
Now we indsert that
|x ( k )| v ( k ), |y ( k )| u ( k ), ∀k ∈ Z,
(A.2)
where
x(k ) = col(x1 (k ), x2 (k ), . . . , xn (k )), y(k ) = col(y1 (k ), y2 (k ), . . . , yn (k )). Indeed, from the definition of · ∞ and the choice of K1 , it is clear that
|x(q )| v(q ), |y(q )| u(q ), ∀q ∈ Z(−∞, 0].
(A.3)
If there exists a nonnegative integer q such that (A.2) holds for k ≤ q, but (A.2) does not hold for k = q + 1, then without loss of generality, we can assume that
|x(k )| v(k ), |y(k )| u(k ), ∀k ∈ Z(−∞, q],
(A.4a)
|xi0 (q + 1 )| > vi0 (q + 1 ) for some i0 ∈ Z[1, n].
(A.4b)
It follows from (A.4b), (5a) and (6) that Acknowledgment This work was supported in part by the National Social Science Foundation of China (Grant no. 17BTQ052), the National Natural Science Foundation of China (Grant nos. 61703148, 61673098 and 61873306), and the Natural Science Foundation of Heilongjiang Province (Grant nos.QC2018083). the Basic Research Operating
vi0 ( q + 1 ) < |xi0 ( q + 1 )| ≤ ai0 |xi0 ( q )| +
n
|bi0 j ||g˜(y j (q − σi0 j (q )))|
j=1
+
n ∞
αs |ei0 j ||g˜(y j (q − s ))|
j=1 s=1
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≤ ai0 |xi0 ( q )| +
n
|bi0 j |l j |y j (q − σi0 j (q ))|
j=1
+
n ∞
αs |ei0 j |l j |y j (q − s )|.
j=1 s=1
By using (A.4a), one can easily derive that
vi0 ( q + 1 ) < ai0 vi0 ( q ) +
n
|bi0 j |l j u j (q − σi0 j (q ))
j=1
+
n ∞
αs |ei0 j |l j u j (q − s ),
(A.5)
j=1 s=1
where ui (q) and vi (q) are the i-th components of u(q) and v(q), respectively. The combination of (A.1) and (A.5) gives
vi0 (q + 1 ) < ai0 K1 (ψ , ϕ )∞ e−γ q v˜ i0 +
n
|bi0 j |l j K1 (ψ , ϕ )∞ e−γ (q−σi0 j (q)) u˜ j
j=1
+
n ∞
αs |ei0 j |l j K1 (ψ , ϕ )∞ e−γ (q−s) u˜ j
j=1 s=1
≤ K1 (ψ , ϕ )∞ e−γ q
×
ai0 v˜ i0 +
n
|bi0 j |e
γ σ¯ i0 j
+ α|ei0 j | l j u˜ j ,
(A.6)
j=1
where u˜i and v˜ i are the i-th components of u˜ and v˜ , respectively. Using (7) and (A.1b), we obtain that
vi0 (q + 1 ) < K1 (ψ , ϕ )∞ e−γ (q+1) v˜ i0 = vi0 (q + 1 ),
(A.7)
a contradiction. Therefore, (A.2) holds for k ∈ Z. It follows from (A.1) and (A.2) that
(x(k ), y(k )) = (x(k )22 + y(k )22 ) 2 1
≤ (v(k )22 + u(k )22 ) 2 1
= K1 e−γ k (ψ , ϕ )∞ (v˜ 22 + u˜22 ) 2 , ∀k ∈ Z[0, ∞ ). 1
1
Let K = K1 (v˜ 22 + u˜22 ) 2 , then
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[54] R. Yang, L. Li, P. Shi, Dissipativity-based two-dimensional control and filtering for a class of switched systems, IEEE Trans. Syst. Man Cybern. Syst. (2019), doi:10.1109/TSMC.2019.2916417. Published online. Chunyan Liu received Ph.D. degree in Informatics from Jilin University in China in 2016. She is currently a Professor in information management school of Heilongjiang University, Harbin, China. Her research interests include information analysis and forecast, social networks and dynamic systems. She has authored more than 40 research papers.
Xin Wang received the B.S. and M.S. degrees in School of Mathematical Science from Heilongjiang University, Harbin, China, in 2008 and 2011, respectively, and the Ph.D. degree in navigation guidance and control from Northeastern University, Shenyang, China, in 2016. He is currently a Lecturer with the School of Mathematical Science, Heilongjiang University, Harbin, China. His research interests include fault diagnosis, fault-tolerant control, and multi-agent coordination.
Yu Xue received the B.S. degree in Automation from Harbin University of Science and Technology, China in 2002; the PhD degree in Control Theory and Control Engineering from Harbin Institute of Technology, China in 20 07. From April 20 07 to August 2009, she is an Engineer in Sichuan Academy of Aerospace Technology. From September 2009 to May 2013, she is a Senior Engineer in Sichuan Academy of Aerospace Technology. In June 2013, she joined Heilongjiang University. Her research interests include stability analysis of delayed dynamic systems, robust control and genetic regulatory networks.
Please cite this article as: C. Liu, X. Wang and Y. Xue, Global exponential stability analysis of discrete-time genetic regulatory networks with time-varying discrete delays and unbounded distributed delays, Neurocomputing, https://doi.org/10.1016/j.neucom.2019.09.047