Event-based cluster synchronization of coupled genetic regulatory networks

Accepted Manuscript Event-based cluster synchronization of coupled genetic regulatory networks Dandan Yue, Zhi-Hong Guan, Tao Li, Rui-Quan Liao, Feng Liu, Qiang Lai PII: DOI: Reference:

S0378-4371(17)30325-4 http://dx.doi.org/10.1016/j.physa.2017.04.024 PHYSA 18126

To appear in:

Physica A

Received date: 10 November 2016 Revised date: 13 February 2017 Please cite this article as: D. Yue, Z.-H. Guan, T. Li, R.-Q. Liao, F. Liu, Q. Lai, Event-based cluster synchronization of coupled genetic regulatory networks, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.04.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Event-based cluster synchronization of coupled genetic regulatory networks ∗

Dandan Yuea , Zhi-Hong Guana†, Tao Lib , Rui-Quan Liaoc , Feng Liud , Qiang Laie a

College of Automation, Huazhong University of Science and Technology, Wuhan, 430074, China b School of Electronics and Information, Yangtze University, Jingzhou, 434023, China c Petroleum Engineering College, Yangtze University, Jingzhou, 434023, China d School of Automation, China University of Geosciences, Wuhan, 430074, China e School of Electrical and Electronic Engineering, East China Jiaotong University, Nanchang, 330013, China

Abstract In this paper, the cluster synchronization of coupled genetic regulatory networks with a directed topology is studied by using the event-based strategy and pinning control. An event-triggered condition with a threshold consisting of the neighbors’ discrete states at their own event time instants and a stateindependent exponential decay function is proposed. The intra-cluster states information and extracluster states information are involved in the threshold in different ways. By using the Lyapunov function approach and the theories of matrices and inequalities, we establish the cluster synchronization criterion. It is shown that both the avoidance of continuous transmission of information and the exclusion of the Zeno behavior are ensured under the presented triggering condition. Explicit conditions on the parameters in the threshold are obtained for synchronization. The stability criterion of a single GRN is also given under the reduced triggering condition. Numerical examples are provided to validate the theoretical results. Keywords pling.

1

Cluster synchronization, event-based strategy, genetic regulatory networks, directed cou-

Introduction

Genetic regulatory networks (GRNs) have aroused great interest in the fields of science and engineering. As complex dynamical systems, GRNs exhibit rich intrinsic dynamics. The interplay of the dynamic properties among individuals can result in collective behaviors, which help us exploring the biological functions at the cellular level [1]-[3]. The synchronization problem is a particular focus of GRNs. The important role of synchronization problem in theoretical researches and biological science has been witnessed in recent years [4]-[7]. The studies of synchronization of GRNs have been primarily concerning the complete synchronization. Many results on synchronization of GRNs with different features have been aquired, such as exponential synchronization of switched GRNs with delays [8], passive synchronization of Markov jump GRNs [9] and finite-time synchronization of GRNs with stochastic noise [10]. These works have carried out investigations for synchronization of GRNs, but not concerned the clustering phenomena. It is known that cells ∗ This work was partially supported by the National Natural Science Foundation of China under Grants 61633011, 61672112, 61472374, 61572084, and 61603137. † Corresponding author. Email: [email protected](Z.-H. Guan)

1

can communicate by quorum sensing. In a interactive population, clustering is a commonly observed phenomenon. The coupled repressilators can present multistability [11]. An ensemble of cellular oscillators generate clustering patterns due to the influence of coupling or noise [12]. In theoretical researches, there is a synchronization issue called cluster synchronization. It is interpreted that the individuals are divided into several clusters and each cluster synchronizes to one state which is different from another [13]. In practice, cluster synchronization is an important research aspect in biological control. The cluster synchronization has been extensively investigated in complex networks. Many techniques have been employed to study the cluster synchronization, such as adaptive control, pinning control, impulsive control [14]-[17], and so on. Wu et al. investigated the cluster synchronization by pinning some nodes, and they provided a method to construct the coupling matrix and appropriate control gains to satisfy the synchronization condition [18]. To the best of our knowledge, there are few analytical results on the cluster synchronization of GRNs with an exception of [19]. The authors investigated the exponential cluster synchronization of genetic oscillators with impulse, coupling and delay in [19]. The relationships among the coupling strength, the bound of delay and the exponential decay rate were revealed. But in these studies, the controller and the coupling states were required to update in real time and not related to the dynamical evolution of systems. Hence, the cluster synchronization issue regarding coupled GRNs needs to be further explored. Most of the above studies utilized the time-triggered techniques. In practical network environment, the artificial nodes have limited capabilities of computation and communication. It is particularly necessary to efficiently utilize the resources to achieve desired behaviors. In view of the advantages in reducing energy consumption and lowering cost, the event-based strategy has been applied to the cooperative control of complex dynamical systems [20, 21]. The discrete-instant states were adopted to design the eventtriggered condition in [22, 23], and continuous communication among nodes was avoided. The considered topologies were general directed graphs, yet they both required controlling all nodes in the systems. In [24], the pinning strategy and event-triggered mechanism were first combined to drive the network to achieve complete synchronization. In [25], by pinning some followers and using the triggering schedule, the nodes in several clusters were synchronized to different equilibria of the virtual leader. These methods further reduced energy-consumption, but they defaulted that the nodes could obtain the neighbors’ states without request in the directed graph. In [26], the cluster synchronization of neural networks was further investigated by way of event-triggered and self-triggered mechanisms. The self-triggered technique avoided continuously verifying the triggering condition, which saved more cost for the system, but the threshold in the triggering schedule was state-independent. The event-based idea has been used to infer the regulatory relationships among genes from the gene expression data in [27]. It provided a practical validity for the event-triggered technique that was applied to GRNs. Lately, an event-based estimator has been designed to estimate the concentrations of gene products [28], and the authors derived some sufficient conditions which ensured the stability and H∞ performance of the error system. The Zeno phenomenon was naturally circumvented since the considered model was discretized. The event-based strategy has been gradually concerned in the researches of GRNs. However, there are still few (if any) works that utilizes the discrete event-instants states to design the threshold in triggering scheme for the cluster synchronization. Motivated by the above discussions, the cluster synchronization of coupled GRNs by using the eventbased strategy and pinning control is studied in this paper. Different from the general sampled-data control, the update of the presented event-based approach depends on the dynamic evolution of systems. This helps avoiding unnecessary energy consumption. Specifically, the main contributions include: Firstly, compared with existing literatures on cluster synchronization via event-based strategies [25, 26], not only the connection topology that we consider is directed (i.e., the coupling matrix is asymmetric), but also only the neighbors’ states at their own event-instants are needed in the triggering condition, which abandons the assumption that the nodes can get the neighbors’ states without request. Moreover, different from the event-triggered rules for cluster synchronization with sampling-based or state-independent thresholds [25, 26], we develop a triggering condition in which the threshold consists of neighbors’ discrete states at their event-instants and a state-independent function. This triggering condition not only avoids continuously transmitting the information but also excludes the Zeno behavior. The triggering scheme is improved that the intra-cluster 2

information and extra-cluster information are involved in the threshold in individual ways, which not only ensures cluster synchronization but also in a sense decreases the computational load due to reducing using the connection matrix. Finally, the cluster synchronization criterion is given, and the appropriate coupling strengths, control gains and triggering gains are presented by easy-verified algebraic condition. The remaining part of this paper is organized as follows. In Section 2, some mathematical preliminaries are given, and an event-triggered condition for the coupled and pinning controlled GRNs that uses the event-based schedule is proposed. In Section 3, the main results, that is, the detailed analysis for the cluster synchronization are presented. Numerical examples are provided to illustrate our theoretical results in Section 4. Conclusions are drawn in Section 5. Notations: N, N+ and R represent the sets of non-negative integers, positive integers and real numbers, respectively. k·k denotes 2-norm of a vector or matrix. A matrix B > 0 (B < 0) represents that B is a positive (negative) definite matrix. 1n refers to a vector whose elements are all 1. ⊗ stands for the Kronecker product.

2

Preliminaries and problem formulation

In this section, we will give some useful definitions and lemmas and then present our problem on the GRNs which are coupled and pinning controlled using the event-based strategies. The event-triggered condition will be given.

2.1

Mathematical preliminaries

Definition 2.1 [15] Consider G = (Gij ) ∈ RN ×N . If P i = 1, 2, . . . , N ; (i) Gij ≥ 0, i 6= j, Gii = − N j=1,j6=i Gij , (ii) G is irreducible;

then we say that G ∈ M1 . Definition 2.2 [15] For an N × N matrix    G= 

G11 G21 .. .

G12 G22 .. .

··· ··· .. .

G1m G2m .. .

Gm1 Gm2 · · ·

Gmm

    

with Gl1 l2 ∈ RLl1 ×Ll2 (l1 , l2 = 1, 2, . . . , m), if each non-diagonal block Gl1 l2 (l1 6= l2 ) is a zero-row-sum matrix, and each diagonal block Gl1 l1 ∈ M1 , then we say that G ∈ M2 . Lemma 2.1 [15] Suppose that G ∈ M1 and σ > 0. Then, a positive definite diagonal matrix P = diag(α1 , α2 , . . . , αN ) exists such that P (G − Υ) + (G − Υ)T P < 0, where Υ = diag(σ, 0, . . . , 0). Lemma 2.2 [18] For a matrix A = (aij ) ∈ Rd1 ×d2 , xT Ay ≤ ℘(A)(xT x + y T y) holds for any x ∈ Rd1 and y ∈ Rd2 , where ℘(A) = 12 max{d1 , d2 } maxi,j |aij |. Remark 2.1 Since the practical transmission of information is usually directed, in this paper, we consider that the topology graph is directed, that is, the coupling matrix is asymmetric.

3

2.2

Problem formulation

Consider a single genetic regulatory network, the dynamics of ith node can be described as  ˜ u(t) ˙ = −K1 u(t) + W1 f˜(v(t)) + L, v(t) ˙ = −K2 v(t) + W2 u(t),

(2.1)

where u(t) = (u1 (t), u2 (t), . . . , un1 (t))T ∈ Rn1 and v(t) = (v1 (t), v2 (t), . . . , vn1 (t))T ∈ Rn1 denote the concentrations of mRNAs and proteins, respectively, Ks = diag(ks1 , ks2 , . . . , ksn1 ) > 0, s = 1, 2, represent the degradation rates of them, f˜(v(t)) = (f˜1 (v1 (t)), f˜2 (v2 (t)), . . . , f˜n1 (vn1 (t)))T , with f˜i1 (vi1 (t)) (i1 = 1, 2, . . . , n1 ) being the nonlinear monotonic increasing functions which represents the transcriptional regulation and often takes the Hill form, W2 = diag(w21 , w22 , . . . , w2n1 ) > 0 denotes the translation rate, W1 = (w1,i1 i2 )n1 ×n1 is defined as: if the transcription factor of gene i2 activates the transcription of gene i1 , then w1,i1 i2 > 0; if the transcription factor of gene i2 represses the transcription of gene i1 , then w1,i1 i2 < 0; otherwise, ˜ = (L ˜1, L ˜2, . . . , L ˜ n )T denotes the basal transcription rate, and w1,i1 i2 =P0, where i2 = 1, 2, . . . , n1 . L 1 ˜ Li1 = − i2 ∈W3 w1,i1 i2 with W3 being the sets of repressors of gene i1 . In this paper, we consider that f˜i1 (·) satisfies the following sector condition: there exists ξi1 > 0 such that f˜i (a) − f˜i1 (b) ≤ ξi1 , ∀a 6= b ∈ R. (2.2) 0≤ 1 a−b Recall the commonly used increasing regulatory function of Hill form f˜i1 (vi1 ) =



1+

vi1 ηi1



hi

1

vi1 ηi1

hi , 1

where ηi1 is the transcription coefficient and hi1 is the Hill coefficient. It is clear that this function satisfies the above sector condition. Let U (t) = (uT (t), v T (t))T . System (2.1) becomes U˙ (t) = AU (t) + Bf (U (t)) + L, where A=



−K1 On1 W2 −K2



,

B=



W1 O n1



,

L=



˜ L 0 n1

(2.3) 

,

f (U (t)) = f˜(v(t)),

with On1 denoting an n1 × n1 zero matrix and 0n1 denoting an n1 -dimension zero vector. Denote n = 2n1 . Based on (2.3), we present the model of coupled GRNs as follows: x˙i (t) = Axi (t) + Bf (xi (t)) + L +

N X

Gij Dxj (t), i = 1, 2, . . . , N

(2.4)

j=1

where xi (t) = (xi1 (t), xi2 (t), . . . , xin (t))T ∈ Rn is the state vector, A, B, L and f (·) are the same as those in (2.3), D = (dk¯m ¯ )n×n > 0 is the inner linking matrix of two networks, G = (Gij )N ×N is the coupling configuration matrix that describes the topological structure of the coupled networks, and Gij is defined as: if node i obtains the directed information from node j, Gij 6= 0; otherwise, Gij = 0. Suppose that the networks (2.4) are driven to realize desired states with m clusters which is denoted by H1 , H2 , . . . , Hm , where Hl = {pl−1 + 1, pl−1 + 2, . . . , pl }, with l = 1, 2, . . . , m, p0 = 0 and pm = N . A connection between two nodes in the same (different) cluster is called intra-cluster (extra-cluster) coupling, and the state information that a node receives from neighbors in the same (different) cluster is called intra-cluster (extra-cluster) information. We say that a network realizes the cluster synchronization, if lim kxi (t) − xj (t)k = 0 for ¯i = ¯j, and lim kxi (t) − xj (t)k = 6 0 for ¯i 6= ¯j, where ¯i denotes the index of the t→∞

cluster to which node i belongs.

t→∞

4

The synchronized dynamics is expressed as y˙ l (t) = Ayl (t) + Bf (yl (t)) + L,

(2.5)

where yl (t) = (yl1 (t), yl2 (t), . . . , yln (t))T ∈ Rn , l = 1, 2, . . . , m. Since the GRN is often capable of presenting multiple steady states, we consider the equilibria as the synchronized states which satisfy (2.5). For convenience, we denote the equilibria as yl , l = 1, 2, . . . , m. To realize the cluster synchronization, we exert the pinning control on some nodes xi (t), i ∈ V = {p0 + 1, p1 + 1, . . . , pm−1 + 1}. Both the control and coupling update by utilizing the event-based strategies. Considering that the transmission of information is directed, the states that node i uses to update the events are its own states and the neighbors’ states at their own latest event time instants. This is consistent with the assumption that the coupling matrix is asymmetric. Let {tki }∞ ki =1 be the sequence of event-triggered instants, at which ith node updates its own controller and transmits the state xi (tki ) to others. For t ∈ [tki , tki +1 ), the coupled and controlled genetic networks can be written as  N P    x˙ i (t) = Axi (t) + Bf (xi (t)) + L + Gij Dxj (tkj ) − σl D(xi (tki ) − yl ),    j=1 i = pl−1 + 1, l = 1, 2, . . . , m, (2.6)   N P    Gij Dxj (tkj ), i 6= pl−1 + 1,  x˙ i (t) = Axi (t) + Bf (xi (t)) + L + j=1

where σl > 0 is the feedback gain, tkj is the latest triggering instant of node j, i.e. kj = arg maxkj ∈N tkj , where tkj ≤ t, with t0 = 0. The controlled GRNs (2.6) are said to achieve cluster synchronization, if lim kxi (t) − yl k = 0 for each node i ∈ Hl , l = 1, 2, . . . , m. t→∞

It is worth to point out that the goal of cluster synchronization is to drive each cluster to synchronize to one equilibrium that is different from another (m > 1), which can also be seen from the definition of cluster synchronization of GRNs (2.6). It implies that there is a one-one mapping between the clusters and desired synchronized equilibria. In accordance with the goal (or definition) of cluster synchronization, we group the nodes into clusters with the same number as that of the desired synchronized equilibria, i.e., the number l of clusters is also the number of different synchronized equilibria, which can be seen from the subscript of (2.5). In this way, when cluster synchronization is achieved, synchronization is realized only within each cluster not in extra clusters. Define the measurement error as ei (t) = xi (tki ) − xi (t). For each node i, we design the event-triggered condition q kei (t)k < µi φi (tki , tkj1 ) + θi ϕi (tki , tkj2 ) + γexp(−̟t), (2.7)

2

P

2

P



G x (t ) where φi (tki , tkj1 ) = j1 ∈H¯i ,j1 6=i |sgn(Gij1 )| xj1 (tkj1 ) − xi (tki ) and ϕi (tki , tkj2 ) = j2 ∈H

, ij j k 2 2 j / ¯i 2 jι = 1, 2, . . . , N , ι = 1, 2, µi > 0, θi > 0, γ > 0, ̟ > 0, with sgn(·) denoting the sign function. Once the triggering condition (2.7) is broken, ei (t) is reset to zero at the event time instant. The next triggering time instant of node i is expressed as o n q tki +1 = inf t > tki : kei (t)k ≥ µi φi (tki , tkj1 ) + θi ϕi (tki , tkj2 ) + γexp(−̟t) .

For node i, taking advantage of the feature that the desired states of nodes in the same clusters are consistent, we use the state information of neighbors in the same cluster with node i to design the first item of the threshold in the triggering condition (2.7). Moreover, we avoid using the intra-cluster coupling matrix G¯i¯i and the inner linking matrix D to calculate the threshold. Besides, D is also not used in the second item, i.e., the item that involves extra-cluster information. Both these points make the practical implementation be easier to some extent. The gains µi , θi , and γ, ̟ determine the threshold and node i’s event time instants. 5

Algorithm 1 Determination of the event-triggered times for all nodes Set ki = 0, t = t0 = 0, i = 1, 2,¼, N . while t £ tˆ if the triggering condition (2.7) is satisfied at t ki = ki , tki = tki ; else (i.e., (2.7) is violated at t (> tki )) ki = ki + 1, tki = t , and ei (tki ) = 0; end t = t + h; end

According to the event-triggered condition, we can determine the event-triggered times (denoted as ki ) for node i (i = 1, 2, . . . , N ) in any time interval [t0 , tˆ], where tˆ > t0 = 0. It is clear that at t0 = 0, the triggering condition (2.7) is satisfied. When an event happens depends on (2.7). Every time the condition (2.7) is violated, the event occurs and the states are triggered to update to those at this instant, and this instant is the event-triggered time-instant. The event-triggered times of node i, i.e., ki is updated to ki + 1. Until tˆ, the value of ki is the total event-triggered times of node i. The event-triggered times of all the other nodes are similarly counted. The Algorithm 1 is presented to clearly show how the event-triggered times of all nodes are determined. For accuracy, we set h > 0 to be small enough during practical implementation. We point out that after an event happens at tki , node i (i = 1, 2, . . . , N ) immediately transmits the new state at the event-instant to others. For t ∈ [tki , tki +1 ], the triggering condition is verified based on the states of neighbors and node i at their latest event-instants and the state of node i at t, but not using those of neighbors at t. Thus, node i does not request the neighbors’ real-time states, and the neighbors are also not needed to respond in real time. This avoids continuous transmission of information. We define the synchronization error zi (t) = xi (t) − y¯i , i = 1, 2, . . . , N . Then, one has  N P   ¯  Gij Dxj (tkj ) − σl Dzi (tki ),  z˙i (t) = Azi (t) + B f (zi (t)) +   j=1 i = pl−1 + 1, l = 1, 2, . . . , m, (2.8)   N P   z˙ (t) = Az (t) + B f¯(z (t)) +  Gij Dxj (tkj ), i 6= pl−1 + 1, i i  i j=1

 T ¯ ¯ ¯ where zi (t) = (zi1 (t), zi2 (t), . . . , zin (t))T , f¯(zi (t)) := f˜1 (zi(1+n1 ) (t)), f˜2 (zi(2+n1 ) (t)), . . . , f˜n1 (zin (t)) , with ¯ f˜i3 (zi(i3 +n1 ) (t)) = f˜i3 (xi(i3 +n1 ) (t)) − f˜i3 (y¯i(i3 +n1 ) ), i3 = 1, 2, . . . , n1 . It follows from (2.2) that 0≤

3

f¯˜i3 (zi(i3 +n1 ) (t)) ≤ ξ i3 , zi(i3 +n1 ) (t)

∀zi(i3 +n1 ) (t) 6= 0 ∈ R, i = 1, 2, . . . , N.

(2.9)

Event-based cluster synchronization

In this section, we shall derive the cluster synchronization criterion for the coupled and pinning controlled GRNs (2.6) with directed connections by using the event-triggered strategy. For convenience, before carrying out specific analysis, we give some notations which will be used, and the undefined parameters are given later. 6

Λ1 = ~ + χ + λ ¯, where

Λ2 = P ⊗ B + IN ⊗

~ = P ⊗ (A + AT ) + (PG + G T P − 2PΥ) ⊗ D,

On1 ΓK


T

(3.1)

,

χ = diag(χ1 , χ2 , . . . , χm ),

λ ¯ = ∆InN ,

P is given in the following paragraph, On1 is an n1 × n1 zero matrix, and Γ is an n1 × n1 positive definite diagonal matrix and given in Theorem 3.1, with Υ = diag( σ1 , 0, . . . , 0, σ2 , 0, . . . , 0, σm , 0, . . . , 0) ∈ RN ×N , |{z} |{z} |{z} p0 +1

p1 +1

m X

δ ∆= 1−δ

l=1

pm−1 +1

1 2℘((Pl G˜l ) ⊗ D) + max1≤l≤m ζl β

Xm Xpl δ = 4µ(N + p∗ − 1) + 2θ( l=1

κ=−

λmax (Λ) , λmax (P)

!

i=pl−1 +1

,

χl = [2℘((Pl G˜l ) ⊗ D) + β]In(pl −pl−1 ) ,

K = diag(ξ1 , ξ2 , . . . , ξn1 ),

2

˜i

Gl ) satisfying δ < 1.

ℓ = kAk + max {ξi3 } kBk > 0. 1≤i3 ≤n1

(3.2) (3.3)

For Gll ∈ M1 , we denote G¯ll = Gll − Υl , where Υl = diag(σl , 0, . . . , 0). From Lemma 2.1, we know that positive definite diagonal matrix Pl = diag(αpl−1 +1 , αpl−1 +2 , . . . , αpl ) (l = 1, 2, . . . , m) exists. This ensures that Pl G¯ll + G¯llT Pl < 0 holds. Denote P = diag(P1 , P2 , . . . , Pm ). Based on these preparations, we give the following result. Theorem 3.1 For network (2.6), suppose that G ∈ M2 . If there exists a positive definite diagonal matrix Γ = diag(ϑ1 , ϑ2 , . . . , ϑn1 ) such that   Λ1 Λ2 < 0, Λ= ΛT2 −2IN ⊗ Γ where Λ1 , Λ2 are given in (3.1), then network (2.6) achieves cluster synchronization exponentially under the event-triggered condition (2.7) with ̟ < min{κ, 2ℓ}, where κ, ℓ are given in (3.3). Furthermore, the network does not exhibit Zeno behavior. Proof: Let x ¯l (t) = (xTpl−1 +1 (t), xTpl−1 +2 (t), . . . , xTpl (t))T , y¯l = 1pl −pl−1 ⊗ yl , z¯l (t) = (zpTl−1 +1 (t), zpTl−1 +2 (t), zl (t)) = (f¯T (zpl−1 +1 (t)), f¯T (zpl−1 +2 (t)), . . . , f¯T (zpl (t)))T , x ¯l3 (t¯kl3 ) = (xTpl −1 +1 (tkpl −1 +1 ), . . . , zpTl (t))T , f¯(¯ 3 3 T T T T xpl −1 +2 (tkpl −1 +2 ), . . . , xpl (tkpl )), z¯l (t˜kl ) = (zpl−1 +1 (tkpl−1 +1 ), zpl−1 +2 (tkpl−1 +2 ), . . . , zpTl (tkpl ))T , e¯l (t) = 3

3

3

3

T (t))T , e(t) = (¯ eT1 (t), e¯T2 (t), z1T (t), z¯2T (t), . . . , z¯m (eTpl−1 +1 (t), eTpl−1 +2 (t), . . . , eTpl (t))T , l, l3 = 1, 2, . . . , m. z(t) = (¯ z1 (t)), f¯T (¯ z2 (t)), . . . , f¯T (¯ zm (t)))T . Then, (2.8) can be rewritten as . . . , e¯Tm (t))T , f¯(z(t)) = (f¯T (¯

z¯˙ l (t) = (Ipl −pl−1 ⊗ A)¯ zl (t) + (Ipl −pl−1 ⊗ B)f¯(¯ zl (t)) +

m X

l3 =1

(Gll3 ⊗ D)¯ xl3 (t¯kl3 ) − (Υl ⊗ D)¯ zl (t˜kl ),

We construct a Lyapunov function as follows: V (t) =

m X l=1

zl (t). z¯lT (t)(Pl ⊗ In )¯

7

l = 1, 2, . . . , m.

(3.4)

Taking the derivative of V (t) along the trajectory of system (3.4), we obtain m X

V˙ (t) = 2

z¯lT (t)(Pl ⊗ In )[(Ipl −pl−1 ⊗ A)¯ zl (t) + (Ipl −pl−1 ⊗ B)f¯(¯ zl (t))

l=1 m X

+

l3 =1

zl (t˜kl )]. (Gll3 ⊗ D)¯ xl3 (t¯kl3 ) − (Υl ⊗ D)¯

(3.5)

P p l3 Since G ∈ M2 , we have j=p Gij Dy¯j = 0n for i ∈ Hl , where 0n is an n-dimension vector. It leads l3 −1 +1 to (Gll3 ⊗ D)¯ yl3 = 0n(pl −pl−1 ) , where 0n(pl −pl−1 ) is an n(pl − pl−1 )-dimension vector. Then, we can derive m m m m P P P P (Gll3 ⊗ D)¯ xl3 (t¯kl ) = (Gll3 ⊗ D) (¯ el3 (t) + z¯l3 (t)) + (Gll3 ⊗ D)¯ yl3 = (Gll3 ⊗ D)(¯ el3 (t) + z¯l3 (t)). 3

l3 =1

l3 =1

l3 =1

l3 =1

Besides, it is obvious that zi (tki ) = ei (t) + zi (t). This yields z¯l (t˜kl ) = e¯l (t) + z¯l (t). Then, (3.5) becomes V˙ (t) = 2

m X l=1

z¯lT (t)(Pl ⊗ In )[(Ipl −pl−1 ⊗ A)¯ zl (t) + (Ipl −pl−1 ⊗ B)f¯(¯ zl (t)) +

−(Υl ⊗ D)¯ zl (t) +

m X

l3 =1

m X

l3 =1

(Gll3 ⊗ D)¯ zl3 (t)

(Gll3 ⊗ D)¯ el3 (t) − (Υl ⊗ D)¯ el (t)].

(3.6)

By calculations, we derive 2

m X

l=1 T

zl (t) + z¯lT (t)(Pl ⊗ In )[(Ipl −pl−1 ⊗ A)¯

m X

l3 =1

(Gll3 ⊗ D)¯ zl3 (t) − (Υl ⊗ D)¯ zl (t)]

= z (t)[P ⊗ (A + AT ) + (PG + G T P − 2PΥ) ⊗ D]z(t).

(3.7)

In view of (2.9), one has 0 ≤ 2

n1 h N X X i=1 i3 =1

  i ¯ ¯ ϑi3 ξi3 zi(i3 +n1 ) (t) − f˜i3 (zi(i3 +n1 ) (t)) f˜i3 (zi(i3 +n1 ) (t))

 = z (t) IN ⊗ T

On1 ΓK


T 

−2f¯T (z(t))(IN ⊗ Γ)f¯(z(t)),

f¯(z(t)) + f¯T (z(t)) IN ⊗

On1 ΓK





z(t) (3.8)

where ϑi3 > 0, i3 = 1, 2, . . . , n1 . Based on the event-triggered condition (2.7), we deduce the following inequalities:

X

2

|sgn(Gij1 )| xj1 (tkj1 ) − xi (tki ) + θi

2

Gij2 xj2 (tkj2 )

+ γexp(−̟t) j2 ∈H / ¯i j1 ∈H¯i ,j1 6=i

2

X X

G (z (t) + e (t)) |sgn(Gij1 )| kzj1 (t) − zi (t) + ej1 (t) − ei (t)k2 + θi = µi ij2 j2 j2

j2 ∈H / ¯i j1 ∈H¯i ,j1 6=i

kei (t)k2 ≤ µi

X

+γexp(−̟t) X |sgn(Gij1 )| (kzi (t)k2 + kzj1 (t)k2 + kei (t)k2 + kej1 (t)k2 ) ≤ 4µi j1 ∈H¯i ,j1 6=i

2

+2θi G˜¯ii (kz(t)k2 + ke(t)k2 ) + γexp(−̟t),

where G˜¯ii is the (i − p¯i−1 )th row of G˜¯i , and G˜¯i =

G¯i1 · · ·

8

G¯i(¯i−1)

Op¯i −p¯i−1

G¯i(¯i+1) · · ·

(3.9) G¯im




with

Op¯i −p¯i−1 being a (p¯i − p¯i−1 ) × (p¯i − p¯i−1 ) zero matrix, i = 1, 2, . . . , N . It follows from (3.9) that ke(t)k2 =

Xm Xpl

i=pl−1 +1

l=1 Xm

kei (t)k2

(maxpl−1 +1≤i≤pl µi )(Nl − 1 + pl − pl−1 )(k¯ zl (t)k2 + k¯ el (t)k2 )

2 Xpl

+2 θi G˜li (kz(t)k2 + ke(t)k2 ) + N γexp(−̟t) i=pl−1 +1 l=1

2 Xm Xpl

˜i 2 2 ≤ 4µ(N + p∗ − 1)(kz(t)k2 + ke(t)k2 ) + 2θ(

Gl )(kz(t)k + ke(t)k )

≤ 4

l=1 Xm

l=1

i=pl−1 +1

+N γexp(−̟t), P where Nl = maxpl−1 +1≤i≤pl ( j1 ∈H¯i ,j1 6=i |sgn(Gij1 )|), N = max1≤l≤m Nl , µ = max1≤i≤N µi , p∗ = max {pl − 1≤l≤m

pl−1 }, θ = max1≤i≤N θi . It is easy to get

δkz(t)k2 + N γexp(−̟t) , 1−δ

2 P Pp l

˜i where δ = 4µ(N + p∗ − 1) + 2θ( m l=1 i=pl−1 +1 Gl ) < 1. Then, by Lemma 2.2, one derives ke(t)k2 ≤

2

= 2

m X

l=1 m X l=1

≤ 2

m X l=1

z¯lT (t)(Pl ⊗ In )[

m X

l3 =1

(3.10)

(Gll3 ⊗ D)¯ el3 (t) − (Υl ⊗ D)¯ el (t)]

el (t)] [¯ zlT (t)((Pl G˜l ) ⊗ D)e(t) + z¯lT (t)((Pl (Gll − Υl )) ⊗ D)¯ 1 zl (t) zl (t) + eT (t)e(t)) + [β z¯lT (t)¯ {℘((Pl G˜l ) ⊗ D)(¯ zlT (t)¯ 2

1 el (t)]} + e¯Tl (t)((Pl (Gll − Υl )) ⊗ D)T ((Pl (Gll − Υl )) ⊗ D)¯ β m m X X 1 2℘((Pl G˜l ) ⊗ D) + max ζl ]eT (t)e(t), zl (t) + [ [2℘((Pl G˜l ) ⊗ D) + β]¯ zlT (t)¯ ≤ β 1≤l≤m

(3.11)

l=1

l=1

  where β > 0, ζl = λmax ((Pl (Gll − Υl )) ⊗ D)T ((Pl (Gll − Υl )) ⊗ D) > 0. Substituting (3.10) into (3.11), we get 2

m X l=1

where ℵ =

z¯lT (t)(Pl ⊗ In )[

m X

l3 =1

Nγ 1−δ

m P

l=1

(Gll3 ⊗ D)¯ el3 (t) − (Υl ⊗ D)¯ el (t)] ≤ z T (t)(χ + λ ¯)z(t) + ℵexp(−̟t),

2℘((Pl G˜l ) ⊗ D) + β1 max1≤l≤m ζl



(3.12)

> 0.

Substituting (3.7), (3.8) and (3.12) into (3.6) yields V˙ (t) ≤ Z T (t)ΛZ(t) + ℵexp(−̟t) λmax (Λ) ≤ V (t) + ℵexp(−̟t), λmax (P)

max (Λ) where Z(t) = (z T (t), f¯T (z(t)))T . Denote κ = − λλmax (P) > 0. Since ̟ < κ, by simple calculations, we can

derive V (t) ≤ (V (0) +

2ℵ κ−̟ )exp(−̟t),

which leads to kz(t)k ≤

√

ωexp(− 9

̟t ), 2

(3.13)

2

2ℵ + λmin (P)(κ−̟) > 0. Thus, the controlled network (2.6) is exponentially cluster where ω = λmaxλ(P)kz(0)k min (P) synchronized. In the following, we shall prove that the Zeno behavior is excluded. For node i, we suppose that t ∈ [tki , tki +1 ). q Denote νi (tki , tkj1 , tkj2 ) = µi φi (tki , tkj1 ) + θi ϕi (tki , tkj2 ) + γexp(−̟t). From (2.8), one has

zi (t) − zi (tki ) =

Z

t

(Azi (τ ) + B f¯(zi (τ )) +

tki

N X j=1

Gij Dxj (tkj ) − σi∗ Dzi (tki ))dτ,

∗ where as: if i = pl−1 + 1, σi∗ = σl ; otherwise, σi∗ = 0, l = 1, 2, . . . , m. According to (2.9), one

σi is expressed

¯

has f (zi (τ )) ≤ max {ξi3 } kzi (τ )k. By using the Newton-Leibniz formula and the property of integral, 1≤i3 ≤n1

we derive

Z

kzi (t) − zi (tki )k ≤

Note that

N P

j=1

t

tki



Gij Dxj (tkj ) =

υi (tki , tkj ) =

(kAk + max {ξi3 } kBk) kzi (τ ) − zi (tki )k 1≤i3 ≤n1

 XN

∗

+ Azi (tki ) + Gij Dxj (tkj ) − σi Dzi (tki ) + max {ξ } kBk kz (t )k dτ. i i k i

1≤i3 ≤n1 3 j=1

max

t∈[tki ,tki +1 )

N P

j=1

Gij Dzj (tkj ). Let

  XN

∗

Azi (tk ) +

G Dz (t ) − σ Dz (t ) + max {ξ } kBk kz (t )k . ij j i i i k k k i 3 i j i i

j=1

1≤i3 ≤n1

υi (tk ,tk )

j i From the Gr¨ onwall inequality and the fact kei (t)k = kzi (tki ) − zi (t)k, one deduces that kei (t)k ≤ ℓ × (exp(ℓ(t − tki )) − 1) . When an event happens at the instant tki +1 , the threshold in (2.7) is reached. Then, one has

 υi (tki , tkj ) 

− = ν (t , t , t ) ≤ ) exp(ℓ(t − t )) − 1 . (3.14)

ei (t− i k k k k i j1 j2 i ki +1 ki +1 ℓ √ ̟t From (3.13), we have kzi (tki )k ≤ ωexp(− 2ki ), which leads to

XN

υi (tki , tkj ) ≤ kzi (tki )k (kAk + max {ξi3 } kBk + σi∗ kDk) + G Dz (t ) ij j kj ∗

j=1 1≤i3 ≤n1

≤

√

ωexp(−

N X ̟tkj∗ √ ̟tki kGij Dk ωexp(− )(ℓ + σi∗ kDk) + ), 2 2

(3.15)

j=1

P

where tkj∗ = arg maxtkj ≤t N j=1 Gij Dzj (tkj ) , j = 1, 2, . . . , N , with kj being defined in (2.6). It follows from (3.15) that √ N X υi (tki , tkj ) ω kGij Dk) (> 0). (3.16) ≤ (ℓ + σi∗ kDk + ℓ ℓ j=1

Note that at t =

t− ki +1 ,

the inequality

νi (tki , tkj1 , tkj2 ) ≥

√

γexp(−

̟ tk +1 ) 2 i

(3.17)

holds. Let Tki = tki +1 − tki . From (3.14), (3.16) and (3.17), we derive √

√ N X ω ̟ kGij Dk) (exp(ℓTki ) − 1) . (ℓ + σi∗ kDk + γexp(− tki +1 ) ≤ 2 ℓ j=1

10

(3.18)

If lim tki = T < ∞, one has 0 < t→∞

√

γ exp(− ̟ 2 T ) ≤ 0, which contradicts with the fact that γ > 0. Hence, we

say that lim tki = ∞. Since ̟ < 2ℓ, by (3.18), one deduces the following relations: t→∞

0<

√

̟ γexp(− Tki ) ≤ 2 ≤

√

N

X ̟ ω kGij Dk)exp( tki ) (exp(ℓTki ) − 1) (ℓ + σi∗ kDk + ℓ 2 j=1

√

ω (ℓ + σi∗ kDk + ℓ

N X j=1

kGij Dk) (exp(ℓtki +1 ) − exp(ℓtki )) .

√ If there exists ki such that Tki → 0, we have 0 < γ ≤ 0, which is invalid. At this point, one obtains Tki > 0, ki ∈ N, i = 1, 2, . . . , N . Thus, it is concluded that under the triggering condition (2.7), the system does not present Zeno behavior. The proof is completed. Remark 3.1 The time-varying formation of multi-agent systems with switching topologies has been studied in [29, 30]. Applying analogous analysis in Theorem 3.1 to the switching modes and combining with proper dwell time, the obtained results can be extended to deal with the cooperative control problems such as stationary or time-invariant (cluster) consensus and synchronization (to reach equilibria) even formation for systems with switching topologies. The cluster synchronization criterion in Theorem 3.1 is given by a linear matrix inequality. To facilitate finding reasonable gains in the event-triggered condition which ensures the synchronization, we will further discuss the relationships among the triggering gains µi , θi in the threshold, the feedback gain and coupling strength of the controlled GRNs such that the linear matrix inequality can be satisfied. Based on the method proposed in [18], we will give the ranges of reasonable feedback gain, intra-cluster coupling strength and the parameter δ (related to µ, θ) for synchronization. We introduce several notations which will be used in Theorem 3.2.

λl

m X

1 ¯ l > 0, 2℘((Pl G˜l ) ⊗ D) + max ε2l λ β 1≤l≤m l=1   ⌢ ⌢T = λmax (Pl G ll + G ll Pl ) ⊗ D ,

Θl = εl λl + λmax (Ψl ),

Ξ=

Ψl = Pl ⊗ (A + AT ) + χl + 2(m − 1) max{℘((Pl1 Gl1 l2 + GlT2 l1 Pl2 ) ⊗ D)}In(pl −pl−1 ) l2 6=l1

1¯ −1 ¯ T + Λ (3.19) l (Ipl −pl−1 ⊗ Γ) Λl , 2  
⌢ ⌢ ¯ l = λmax ((Pl G ll ) ⊗ D)T ((Pl G ll ) ⊗ D) , l = 1, 2, . . . , m. ¯ l = Pl ⊗ B + Ip −p ⊗ On ΓK T , λ where Λ 1 l l−1

Theorem 3.2 Consider network (2.6) with the event-triggered condition (2.7). For any given N ×N matrix G ∈ M2 with Gll3 ∈ RLl ×Ll3 (l, l3 = 1, 2, . . . , m) and positive constants σ ¯l (l = 1, 2, . . . , m), we construct the coupling matrix Gˆ with Gˆll =εl Gll and Gˆll3 =Gll3 (l3 6= l), and control gain σl = εl σ ¯l , where εl (l = 1, 2, . . . , m) is a positive constant. Suppose that Γ is an n1 × n1 positive definite diagonal matrix, Λ < 0 holds, if l , where δ is defined in (3.2) and Ψl , λl , Θl , Ξ are defined εl > − λmaxλl(Ψl ) (l = 1, 2, . . . , m), δ < min1≤l≤m ΘΘ l −Ξ in (3.19). Proof: ⌢

Denote G ll = Gll − diag(¯ σl , 0, . . . , 0). Then, one has Gˆll − Υl = εl G ll . By Lemma 2.1, we get ⌢

⌢

⌢T

Pl G ll + G ll Pl < 0, which leads to λl < 0. In view of the Schur Complement lemma, Λ < 0 is equivalent to −2IN ⊗ Γ < 0 and 1 Λ1 + Λ2 (IN ⊗ Γ)−1 ΛT2 < 0. 2 11

(3.20)

It is clear that −2IN ⊗Γ < 0. In what follows, we will prove that (3.20) holds. For ∀ z˜(t) = (˜ z1T (t), z˜2T (t), . . . , T T n z˜N (t)) , where z˜i (t) ∈ R , i = 1, 2, . . . , N , one has 1 z (t) z˜T (t)[Λ1 + Λ2 (IN ⊗ Γ)−1 ΛT2 ]˜ 2 m X ⌢ ⌢T 1¯ −1 ¯ T ¯ z¯˜lT (t)[Pl ⊗ (A + AT ) + εl (Pl G ll + G ll Pl ) ⊗ D + χl + ∆In(pl −pl−1 ) + Λ = ˜l (t) l (Ipl −pl−1 ⊗ Γ) Λl ]z 2 l=1 m X

+

z¯˜lT (t)[

m X

((Pl Gll3 + GlT3 l Pl3 ) ⊗ D)z¯˜l3 (t)],

(3.21)

l3 =1

l=1

l3 6=l

T where z¯˜l (t) = (˜ zpTl−1 +1 (t), z˜pTl−1 +2 (t), . . . , z˜pTl (t))T , l = 1, 2, . . . , m, and z¯˜l (t) is the transposition of z¯˜l (t). From m m P m m P P P T ℘((Pl Gll3 + GlT3 l Pl3 ) [ ((Pl Gll3 + GlT3 l Pl3 ) ⊗ D)z¯˜l3 (t)] ≤ z¯˜l (t)[ Lemma 2.2, it is easy to derive l=1 l3 =1

l3 =1

l=1

l3 6=l

⊗D)(z¯˜lT (t)z¯˜l (t) + z¯˜lT3 (t)z¯˜l3 (t))] ≤ 2(m − 1) max{℘((Pl1 Gl1 l2 + GlT2 l1 Pl2 ) ⊗ D)} l2 6=l1

becomes

m P

l=1

l3 6=l

zl (t). Then, (3.21) z¯lT (t)¯

  m X ⌢ ⌢T 1 z˜T (t)[Λ1 + Λ2 (IN ⊗ Γ)−1 ΛT2 ]˜ z¯˜lT (t) εl (Pl G ll + G ll Pl ) ⊗ D + Ψl + ∆In(pl −pl−1 ) z¯l (t). z (t) ≤ 2 l=1

The inequality εl λl + λmax (Ψl ) + ∆ < 0, l = 1, 2, . . . , m can ensure that z˜T (t)[Λ1 + 12 Λ2 (IN ⊗ Γ)−1 ΛT2 ]˜ z (t) < δΞ Note that ∆ = 1−δ . In the proof of Theorem 3.1,

(3.22)

0, which implies that (3.20) holds. we have required that δ < 1. So (3.22) is equivalent to

Θl + (Ξ − Θl )δ < 0, l = 1, 2, . . . , m. If Ξ < Θl , (3.23) is equivalent to δ >

Θl Θl −Ξ .

We note that

Θl Θl −Ξ

(3.23)

> 1. This contradicts with the premise

Θl Θl −Ξ . λmax (Ψl ) l εl > − λl ), which leads to ΘΘ < 1. Thus, we conclude l −Ξ Θl δ < min1≤l≤m Θl −Ξ , the inequality (3.23) holds. At this point,

δ < 1. It is easy to see that Ξ = Θl does not satisfy (3.23). If Ξ > Θl , (3.23) is equivalent to δ < Since δ > 0, we require that Θl < 0 (namely that when Θl < 0 (< Ξ) (l = 1, 2, . . . , m) and Λ < 0 is proved. This completes the proof.

Theorem 3.1 and Theorem 3.2 are visualized and the demonstrations are straightforward. But to reduce the conservative of the condition on the triggering gains µ, θ (µi , θi ), we carry out further investigations. For convenience, we first give the following relations  Ppl 2 2 i=p θi kG˜li k  (p −p )γ 2 2 δ˜l l−1 +1  k¯ e (t)k ≤ k¯ z (t)k + (kz(t)k2 + ke(t)k2 ) + l 1−l−1 exp(−̟t),  l  1−δ˜l 1−δ˜l δ˜l  l 2 ˜ γexp(−̟t) (3.24) ke(t)k2 ≤ δkz(t)k +N , 1−δ˜  Ppl Ppl  ˜i k2 ˜i k2  2 2N G G θ θ k k i i  ˜ i=pl−1 +1 i=pl−1 +1 l l  k¯ el (t)k2 ≤ 1−δlδ˜ k¯ zl (t)k2 + kz(t)k2 + (pl − pl−1 + ) γexp(−̟t) , ˜ (1−δ)(1− δ˜ ) 1−δ˜ 1−δ˜ l

l

l

2 P Pp l

˜i where δ˜l = 4˜ µl (Nl − 1 + pl − pl−1 ) < 1, δ˜ = max1≤l≤m {δ˜l + 2θ( m l=1 i=pl−1 +1 Gl )} < 1, with µ ˜l = maxpl−1 +1≤i≤pl µi . Similar to the derivation of (3.10), by using (3.9), we get the first two inequalities in (3.24). Substituting the second inequality in (3.24) into the first one yields the third one. Replace m P el (t). Using the second inequality and the third ζl e¯Tl (t)¯ the item β1 max1≤l≤m ζl eT (t)e(t) in (3.11) with β1 l=1

one in (3.24), and combining with the fact δ˜l ≤ max1≤l≤m δ˜l , l = 1, 2, . . . , m, θi ≤ θ, i = 1, 2, . . . , N , 12

¯ one can derive an inequality analogous to (3.12) in which χ, λ ¯ and ℵ are replaced with χ, ¯ λ ¯¯ and ℵ, m m P P ˜ ζl δl ¯ = Nγ ˜ nN , ℵ where χ ¯l = χl + β(1− I is instead of χl , λ ¯¯ = ∆I 2℘((Pl G˜l ) ⊗ D) + (pl − δ˜ ) n(pl −pl−1 ) 1−δ˜ l

pl−1 +

2N

Ppl

i=pl−1

˜i 2 +1 θi kGl k

1−δ˜

γζl ) β(1− δ˜l )

˜ = > 0, with ∆

m δ˜ P 2℘((Pl G˜l ) ⊗ 1−δ˜ l=1

D)+

l=1 P P pl 2θ( m l=1 ζl i=p

2

l−1 +1

kG˜li k

˜ ˜ β(1−δ)(1−max 1≤l≤m δl )

l=1

)

. Following

˜ < 0, l = 1, 2, . . . , m, a similar argument to that of Theorem 3.2, we can prove Λ < 0 by demonstrating Ωl + ∆ ¯ δ˜ ε2 λ

l l l . It is easy to deduce that when εl > − λmaxλl(Ψl ) and which requires Ωl < 0, where Ωl = Θl + β(1− δ˜l ) Θl Θl δ˜l < ), the inequality Ωl < 0 holds. Combining with the defi(˜ µl < ¯ ¯ ε2 λ ε2 λ

Θl −

l l β

4(Θl −

l l )(N −1+p −p l l l−1 ) β

˜ ˜ and δ(< ˜ < 0, we derive θ < θ˜l , where θ˜l = (1 − nition of ∆ 1), and solving the inequality Ωl + ∆

2 pl m m P P P

˜i 2℘((Pl G˜l ) ⊗ D)+Ωl (1−max1≤l≤m δ˜l )][2(1−max1≤l≤m δ˜l )( max1≤l≤m δ˜l )[(max1≤l≤m δ˜l )

Gl )(Ωl

−

m P

l=1

2℘((Pl G˜l ) ⊗ D)) −

l=1 P P pl 2( m l=1 ζl i=p β

l=1 i=pl−1 +1

l−1 +1

kG˜li k

2

)

]−1 , l = 1, 2, . . . , m. When θ < min1≤l≤m θ˜l , the inequality

˜ < 0 holds for l = 1, 2, . . . , m. Thus, Λ < 0 is proved. Ωl + ∆ ˜ = Ωl + ∆ ˜ = 0 and Φ2 (δ) = Θl + ∆ = 0 by δ˜∗ and δ∗ , respectively. It is clear Denote the solutions of Φ1 (δ) ∗ ∗ ˜ ˜ < 0 and Φ2 (δ) < 0, respectively. Note that that δ and δ are the upper bounds of the solutions of Φ1 (δ) P Ppl Pm P pl i 2 ˜i 2 ˜ G 2θ m 2θ ζ k k ˜ l=1 l=1 l i=pl−1 +1 kGl k i=pl−1 +1 l δmax max1≤l≤m δ˜l ζl δ˜l 1≤l≤m ζl ≤ (max ζ )[ + ] = + . 1≤l≤m l β(1−max ˜ ˜ ˜ δ˜ ) δ˜ ) δ˜ ) β(1−δ˜ ) β(1−δ)(1−max β(1−δ)(1−max β(1−δ) l

1≤l≤m l

1≤l≤m l

1≤l≤m l

a δ˜∗ Ξ . Since the function 1−a is increasing, we get δ˜∗ ≥ δ∗ . Denote the Then, Φ2 (δ∗ ) = Φ1 (δ˜∗ ) ≤ Θl + 1− δ˜∗ values of µ, θ derived from δ˜ and δ by µ ˜∗ , θ˜∗ and µ∗ , θ ∗ , respectively. From the definition of δ˜ and δ, it is concluded that fixing other parameters and µi , i = 1, 2, . . . , N , we have θ˜∗ ≥ θ ∗ ; fixing other parameters and θi , i = 1, 2, . . . , N , one has µ ˜∗ ≥ µ∗ . At this point, the following less conservative is presented.

Lemma 3.1 For the constructed coupling matrix and control gains in Theorem 3.2, if εl > − λmaxλl(Ψl ) Θl (l = 1, 2, . . . , m), µ ˜l < and θ < min1≤l≤m θ˜l , then Λ < 0 holds. 2¯ 4(Θl −

ε λl l )(Nl −1+pl −pl−1 ) β

Under the presented triggering condition with the above conditions on µ ˜l and θ (µi , θi ), the cluster synchronization can still be ensured with the upper bound of ̟ being modified with the related items that are derived in the above analysis. Remark 3.2 Under the presented technique, the triggering q gains µ, θ (µi , θi , i = 1, 2, . . . , N ) may be small i.e., less than 1. Yet, for the threshold in (2.7), we have µi φi (tki , tkj1 ) + θi ϕi (tki , tkj2 ) + γexp(−̟t) ≥ q q q q q q q µi γ µi θi θi ̟t exp(− ). When µ and θ are far less than 1, and φ (t , t ) + ϕ (t , t ) + i i i i k k k k i j1 i j2 3 3 3 2 3 3 may not be too small. In other words, even µi , θi are small, their real effect on the triggering strategy is still normal [23]. Besides, we can also adjust γ and ̟ to get large threshold to avoid frequently updating the events. Theorem 3.1 shows that the coupled GRNs are cluster synchronized under the presented strategy. The local stability and oscillatory dynamics of a GRN have been studied in [31]. It only concerns the isolated model, and the studies are carried out by mean of eigenvalue analysis and qualitative discussions (numerical simulations and experiments). In the following, we analytically show that the presented event-triggered method can also stabilize an isolated GRN(including the GRN in [31]). Denote the state variable of a single GRN of (2.6) (or (2.3) with control) by X(t). The controlled GRN is described by ˙ X(t) = AX(t) + Bf (X(t)) + L − σ ˜ (X(tk ) − Y ),

13

(3.25)

where X(t) = (X1 (t), X2 (t), . . . , Xn (t))T is the state variable, σ ˜ > 0 is the control gain, and Y = (Y1 , Y2 , . . . , Yn )T is an equilibrium. Define the measurement error as e˜(t) = X(tk ) − X(t), where tk is the event instant. Based on (2.7), we design a reduced triggering condition p (3.26) k˜ e(t)k < γexp(−̟t).

˜ n, ˜ be an analogous matrix to Λ in which Λ1 , Λ2 and −2IN ⊗ Γ are replaced with A + AT − (2˜ Let Λ σ − β)I
T B + On1 ΓK and −2Γ, respectively. Using similar analysis to that in Theorem 3.1 and Theorem 3.2, ˜ n+ we deduce that when σ ˜ > λmax2(W) , the GRN (3.25) is exponentially stable, where W = A + AT + βI

T T T 1 −1 ˜ On1 ΓK )Γ (B + On1 ΓK ) , β > 0, 2 (B +

Lemma 3.2 Suppose that Γ is an n1 × n1 positive definite diagonal matrix. If σ ˜ > λmax2(W) , the network (3.25) is exponentially stable under the event-triggered condition (3.26) with ̟ < min{˜ κ, 2ℓ}, where κ ˜ = ˜ Furthermore, the Zeno behavior does not occur in the network. −λmax (Λ).

4

Numerical simulations

In this section, numerical examples are presented to validate the correctness of our theoretical results. A synthetic GRN analogous to that in Escherichia coli which is proposed in [31] is considered to be a single network in our coupled GRNs. The variables represent the concentrations of mRNAs (like lacl, tetR or cl) and proteins. The graph of the genetic regulatory networks are given in Figure 1. For clearness, the genetic expression processes of gene 1 are explicitly shown, and the genetic expression of other genes which are similar to that of gene 1 are omitted. The upper part shows the three-nodes networks, and such networks are contained in cells which are displayed in the lower part. The cells transmit the gene products ( e.g. proteins) to interact with each other. The parameters in the single system (2.3) are       ˜ −I3 O3 W1 L A= , B= , L= , 0.2I3 −0.2I3 O3 03 

 0 0 −4.8 W1 =  −4.8 0 0 , 0 4.8 0



 4.8 ˜ =  4.8  , L 0

v2 f = (f˜1 , f˜2 , f˜3 )T , f˜i1 (v) = 1+v 2 , i1 = 1, 2, 3, where O3 is a 3 × 3 zero matrix and 03 is a 3-dimension vector. The system has three positive equilibria

y1 = (0.2183, 4.5817, 4.5817, 0.2183, 4.5817, 4.5817)T , y2 = (3.2951, 0.4048, 0.6758, 3.2951, 0.4048, 0.6758)T , y3 = (4.5817, 0.2183, 0.2183, 4.5817, 0.2183, 0.2183)T , which satisfy (2.5). We regard y1 , y2 and y3 as desired synchronization states of three clusters, respectively. By calculations, we derive K = 0.65I3 . The coupled and controlled GRNs are given as follows:  N P    x˙ i (t) = Axi (t) + Bf (xi (t)) + L + Gij xj (tkj ) − σ¯i (xi (tki ) − y¯i ),    j=1 i = 1, 5, 9   N P    Gij xj (tkj ), i = 2, 3, 4, 6, 7, 8, 10, 11, 12.  x˙ i (t) = Axi (t) + Bf (xi (t)) + L + j=1

14

Transcription Transcription factor Translation DNA

cis-regulatory RNA DNA sequence protease

mRNA u3 Protein v3 Protein v1

mRNA u1

Degradation

mRNA u2 Protein v2

Protein

Protein

Protein Cell

Cell

Figure 1: The sketch of genetic regulatory networks. The grey thick lines ⊣ and → represent repression and activation of transcriptions, respectively. 

 ε1 G11 G12 G13 Based on the technique in Theorem 3.2, we give the coupling matrix Gˆ =  G21 ε2 G22 G23 , G31 G ε3 G33     32 −5 0 0 5 −6 6 0 0  7 −9 2  0  0   , G22 = G33 = 0.1 ·  7 −9 2  , G12 = G13 = where G11 = 0.1 ·   0  2 8 −8 0  0 −8 6  3 0 4 −7 10 0 0 −10   1 −1 0 0  0 1 −1 0   , G = G23 = 1.5G12 , G31 = G32 = 1.6G12 . It is computed that δ = 20µ+1.8592θ. 0.1·  0 0 1 −1  21 −1 0 0 1 Take σ ¯1 = 0.8, σ ¯2 = σ ¯3 = 0.9, P1 = diag(0.1, 0.1, 0.1, 0.1), P2 = P3 = diag(0.1, 0.1, 0.08, 0.07), which satisfy Pl (Gll − Υl ) + (Gll − Υl )T Pl < 0, where Υl = diag(¯ σl , 0, . . . , 0), l = 1, 2, 3. Given Γ = I3 and β = 0.2. By computations, we obtain λmax (Ψ1 ) = 2.8335, λmax (Ψ2 ) = 3.1935, λmax (Ψ3 ) = 3.2655, λ1 = −0.0327 and λ2 = λ3 = −0.043. Let εl = 90, which satisfies εl > − λmaxλl(Ψl ) , l = 1, 2, 3. Then, one has Ξ = 1667.5. From Theorem 3.2, δ is required to satisfy δ < 6.5663 × 10−5 . Let µi = 2.88 × 10−6 , θi = 4.2 × 10−6 , i = 1, 2, . . . , 12. We get δ = 6.5409 × 10−5 < 6.5663 × 10−5 . By simple calculations, one derives κ = 11.07 and ℓ = 4.1406. Let ̟ = 6 < min{κ, 2ℓ}, γ = 1. The synchronization condition in Theorem 3.1 is ensured. The initial values are xi (0) = (0.3 + 0.014i, 0.5 + 0.014i, 0.7 + 0.014i, 0.9 + 0.014i, 1.1 + 0.014i, 1.3 + 0.014i), i = 1, 2, . . . , 12. The step size in the simulations is 0.001, and the total simulation time is 2.5 s. Figure 2(a), 2(b) and 2(c) show the evolution trajectories of xi1 (t), xi2 (t) and xi3 (t) (mRNAs), respectively, i = 1, 2, . . . , 12. For saving space, trajectories of other state elements (proteins) are omitted. It should be pointed out that the initial values do not affect the finial convergence. We see that undergoing a period of evolution, each cluster synchronizes to the desired equilibrium. The GRNs (2.6) achieves desired cluster synchronization. At this point, the presented triggering condition is validated, and the technique to find triggering gains is also certified. Figure 3 depicts the event time instants of all nodes. We observe that the event time instants are sporadic not at every simulation time-instant. Using Algorithm 1, the triggering times of all nodes in the simulation are counted: [135, 122, 109, 80, 152, 91, 97, 111, 152, 85, 86, 130]. Correspondingly, the triggering frequency of all nodes, namely the ratio of event triggering times to total iteration times are [5.4%, 4.88%, 4.36%, 3.2%, 6.08%, 3.64%, 3.88%, 4.44%, 6.08%, 3.4%, 3.44%, 5.2%]. The average trigger15

5.5

4.4

4.4 States xi2(t), i = 1,2, … ,12

States xi1(t), i = 1,2, … ,12

5.5

3.3 Cluster 1 Cluster 2 Cluster 3 2.2

1.1

0

3.3 Cluster 1 Cluster 2 Cluster 3 2.2

1.1

0

0.5

1

1.5

2

0

2.5

0

0.5

1

Time(s)

1.5

2

2.5

Time(s)

(a)

(b) 5.5

States xi3(t), i = 1,2, … ,12

4.4

3.3 Cluster 1 Cluster 2 Cluster 3 2.2

1.1

0

0

0.5

1

1.5

2

2.5

Time(s)

(c)

Figure 2: (a) Evolution trajectories of xi1 (t) for all nodes. (b) Evolution trajectories of xi2 (t) for all nodes. (c) Evolution trajectories of xi3 (t) for all nodes. ing frequency is about 4.5%. It is shown that under the proposed triggering condition (2.7), the events are not frequently updated, which means that the energy-consumption are low. Figure 4(a) shows the control N P inputs u ¯¯i1 := −σ¯i (xi1 (tki ) − y¯i1 ), i = 1, 5, 9. Figure 4(b) shows the coupling states ri1 (t) := Gij xj1 (tkj ) j=1

of all nodes. When the cluster synchronization is realized, u ¯¯i1 and ri1 (t) evolve to zero. It is shown that the control inputs and coupling states are both piecewise constants due to the event-triggered mechanism. The control inputs and coupling states are fixed on the time intervals of two events and only updated at the event time instants. This helps reducing the implementation cost. Fix the values of other parameters, in view of Lemma 3.1, we require that µ ˜1 < 4.3887 × 10−6 , µ ˜2 < −5 −5 −6 2.0313 × 10 and µ ˜3 < 1.8151 × 10 . Taking µi = 3.87 q × 10 , i = 1, 2, . . . , 12, which indicates that the real coefficient of the first item in (2.7) is not less than µ3i = 0.00114, we derive θ˜1 = 4.2853 × 10−6 , θ˜2 = 1.8443 × 10−4 and θ˜3 = q 1.6018 × 10−4 . Let θi = 4.2 × 10−6 , we see that the real coefficient of the second item in (2.7) is not less than θ3i = 0.00118, i = 1, 2, . . . , 12. Using the presented way in Theorem 3.2, when q θi = 4.2 × 10−6 , µ is required to satisfy µ < 2.8927 × 10−6 (< 3.87 × 10−6 ), which implies µ3 < 0.000982. We calculate the ratio (0.00114 − 0.000982)/0.000982 ≈ 16.09%. Although the value of µ does not change 16

12

10

All nodes

8

6

4

2

0

0

0.5

1

1.5

2

2.5

Time(s)

Figure 3: Event time instants of all nodes.

350

200

300 20

Coupling states of all nodes

Control inputs of controlled nodes

150 250

10 200 0 150

−10

100

−20 0.2

u ¯11 (t) u ¯21 (t) u ¯31 (t) 0.3

0.4

50

r11 (t) r21 (t) r31 (t) r41 (t) r51 (t) r61 (t) r71 (t) r81 (t) r91 (t) r10,1 (t) r11,1 (t) r12,1 (t)

10 100

0 −10

50

0.5

0.6

0.7

0

−50 0 −50

0

0.5

1

1.5

2

−100

2.5

Time(s)

0

0.5

1

1.5

2

2.5

Time(s)

(a)

(b)

Figure 4: (a) Control inputs of controlled nodes 1, 5, 9. (b) Coupling states of all nodes. too much, the real effect of µ on the threshold of the triggering condition increases. Besides, one derives κ = 11.12. With the same parameter values as those derived from Theorem 3.2 except µi and θi , the state evolution trajectories can be shown. Nevertheless, since the value of µi and θi are not enlarged too much, the trajectories have no great difference with those in Figure 2. For saving space, they are omitted. The pinning control contributes to the realization of cluster synchronization of GRNs (2.6). To verify this point, we show the state evolution trajectories of GRNs (2.6) without pinning control. All parameter values are the same as those in Figure 2 except that σ¯i = 0, i = 1, 5, 9. Figure 5 displays the evolution trajectories of xi1 (t), xi2 (t) and xi3 (t), i = 1, 2, . . . , 12. We see that when pinning control is absent, undergoing a long-time evolution, the nodes realize synchronization under the presented triggering strategy. However, not all clusters are synchronized to the correspondingly desired equilibria. According to the definition of cluster synchronization of GRNs (2.6), we say that cluster synchronization is not realized. It illustrates that although the intra-cluster coupling strength is much larger than the extra-cluster coupling strength, pinning control is still necessary for GRNs (2.6) to realize desired cluster synchronization. In the following, we give an example for the controlled single GRN (3.25) whose specific model (repressilator) is proposed in [31]. We carry out comparisons between the presented event-triggered method with

17

1.5

5.5

States xi2(t), i = 1,2, … ,12

States xi1(t), i = 1,2, … ,12

4.4

1 Cluster 1 Cluster 2 Cluster 3 0.5

3.3 Cluster 1 Cluster 2 Cluster 3 2.2

1.1

0

0

10

20

30

40

0

50

0

10

20

Time(s)

30

40

50

Time(s)

(a)

(b) 5.5

States xi3(t), i = 1,2, … ,12

4.4

3.3 Cluster 1 Cluster 2 Cluster 3 2.2

1.1

0

0

10

20

30

40

50

Time(s)

(c)

Figure 5: (a) Evolution trajectories of xi1 (t) for all nodes without pinning control. (b) Evolution trajectories of xi2 (t) for all nodes without pinning control. (c) Evolution trajectories of xi3 (t) for all nodes without pinning control.

the reduced triggering condition (3.26) and the method in [31]. The parameters of the model are given as follows:           1 0 0 −6.2 ˜ −I3 O3 W1 L ˜ = 6.2062  1  . 0 , L A= , B= , L= , W1 =  −6.2 0 I3 −I3 O3 03 1 0 6.2 0

Figure 6 (a) shows the evolution trajectory of the state X4 (t) according to the analysis in [31]. We see that the network is unstable and it presents oscillations. Taking β˜ = 0.2, under the parameters satisfying Lemma 3.2: σ ˜ = 10, ̟ = 0.15, γ = 1, Figure 6 (b) displays the stable trajectory. It illustrates that under the presented method, the repressilator [31] is stabilized to the equilibrium Y = 1.6588I6 . But the techniques used in [31] only give the regions of stability and oscillation, they can not be directly utilized to stabilize even synchronize GRNs.

18

5

2.6

4

State X4(t)

State X4(t)

2 3

2

1.4 1

0

0

40

80 Time(s)

120

160

0.8

0

20

40

60

Time(s)

(a)

(b)

Figure 6: (a) Evolution trajectories of X4 (t) in [31]. (b) Evolution trajectories of X4 (t) that is derived by using the presented event-based method.

5

Conclusions

In this paper, the cluster synchronization of coupled GRNs with a directed topology has been investigated by utilizing the event-based strategy and pinning control. Taking advantage of the feature of cluster synchronization, we have designed the event-triggered condition which involves the intra-cluster information and extra-cluster information in individual ways to ensure the synchronization. This scheme has reduced utilizing the connection matrix. Under the proposed event-triggered condition, the cluster synchronization criterion has been given by a linear matrix inequality. It is shown that the continuous transmission of information and the Zeno phenomenon have been averted in the network. The explicit conditions on gains in the threshold of the triggering condition have been derived. The stability issue of a single GRN has also been addressed under the reduced triggering condition. The correctness of the theoretical results has been verified by numerical examples. The cluster synchronization of coupled GRNs with diffusion and stochastic coupling via the event-triggered strategies will be considered in future works.

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[26] L. Li, D.W.C. Ho, J. Cao, J. Lu, Pinning cluster synchronization in an array of coupled neural networks under event-based mechanism, Neural Netw. 76 (2016) 1–12. [27] A.T. Kwon, H.H. Hoos, R. Ng, Inference of transcriptional regulation relationships from gene expression data, Bioinformatics 19 (2003) 905–912. [28] Q. Li, B. Shen, Y. Liu, F.E. Alsaadi, Event-triggered H∞ state estimation for discrete-time stochastic genetic regulatory networks with Markovian jumping parameters and time-varying delays, Neurocomputing 174 (2016) 912–920. [29] X. Dong, G. Hu, Time-varying formation control for general linear multi-agent systems with switching directed topologies, Automatica 73 (2016) 47–55. [30] X. Dong, Y. Zhou, Z. Ren, Y. Zhong, Time-varying formation tracking for second-order multi-agent systems subjected to switching topologies with application to quadrotor formation flying, IEEE Trans. Ind. Electron. (2016) doi:10.1109/TIE.2016.2593656. [31] M.B. Elowitz, S. Leibler, A synthetic oscillatory network of transcriptional regulators, Nature 403 (2000) 335–338.

21

Highlights     

Cluster synchronization of GRNs with directed coupling is studied.

A triggering condition using the neighbors’ states and a decay function is proposed.

The states of intra-cluster and extra-cluster nodes are involved in individual ways. The synchronization criteria are obtained via the event-based strategy.

The continuous transmission of information and the Zeno behavior are avoided.