Finite-time synchronization of stochastic multi-links dynamical networks with Markovian switching topologies

Finite-time synchronization of stochastic multi-links dynamical networks with Markovian switching topologies

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Finite-time synchronization of stochastic multi-links dynamical networks with Markovian switching topologies Ying Guo, Bingdao Chen, Yongbao Wu PII: DOI: Reference:

S0016-0032(19)30834-8 https://doi.org/10.1016/j.jfranklin.2019.11.045 FI 4280

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Journal of the Franklin Institute

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24 March 2019 2 September 2019 17 November 2019

Please cite this article as: Ying Guo, Bingdao Chen, Yongbao Wu, Finite-time synchronization of stochastic multi-links dynamical networks with Markovian switching topologies, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.045

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Highlights    

The finite-time stabilization of stochastic coupled systems on networks with time-varying coupling structure are studied. Our investigation is based on Lyapunov method and Kirchhoff’s Matrix Tree Theorem. Some sufficient conditions and corollaries are proposed to guarantee the finite-time stability of systems. The main results are employed to stochastic coupled oscillators on networks with timevarying coupling structure.

The finite-time stabilization of stochastic coupled systems on networks with time-varying coupling structure are studied. Our investigation is based on Lyapunov method and Kirchhoff?s Matrix Tree Theorem. Some sufficient conditions and corollaries are proposed to guarantee the finite-time stability of systems. The main results are employed to stochastic coupled oscillators on networks with time-varying coupling structure.

Finite-time synchronization of stochastic multi-links dynamical networks with Markovian switching topologies Ying Guoa , Bingdao Chenb , Yongbao Wub,∗ a

b

Department of Mathematics, Qingdao University of Technology, Qingdao 266520, PR China Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, PR China

Abstract In this paper, finite-time synchronization of stochastic multi-links dynamical networks with Markovian switching topologies (SMDNM) via intermittent control is discussed. Different from previous literature, topological structure of multi-links complex networks is Markovian switching and each switching subnetwork is not required to contain a directed spanning tree or to be strongly connected. Meanwhile, a novel quantized aperiodically intermittent control is designed and the coupling function is nonlinear. Based on a new differential inequality and Kirchhoff’s Matrix Tree Theorem, a synchronization criterion is derived to ensure finite-time synchronization of SMDNM within a finite time which is closely related to the topology of the network. Moreover, we apply theoretical results to oscillators systems and a synchronization criterion is presented. Finally, a numerical example is provided to demonstrate the effectiveness and feasibility of the proposed methodology. Keywords: Finite-time synchronization; Markovian switching; Multi-links complex networks; Quantized aperiodically intermittent control; Graph theory

1. Introduction In the real world, scores of physical and social systems can be seen as complex dynamical networks (CDNs) such as World Wide Web, food webs, social organizations, ecological communications, communication networks and so forth [1, 2, 3, 4]. Recently, CDNs have attracted the attention of many scholars and interesting results have been obtained [5, 6, 7, 8, 9, 10]. We notice that complex networks in most of above literatures are considered to be single-link. However, in real life, multi-links complex networks are more practical. For example, for transportation networks, the transmission speed is different among airline networks, railway networks and highway networks. This means that multi-links complex networks need further researches. Moreover, some interesting results have been reported [11, 12, 13]. ∗

Corresponding author. Email address: [email protected] (Yongbao Wu)

Preprint submitted to Elsevier

November 27, 2019

Furthermore, to our best knowledge, the previous studies of CDNs mainly focused on deterministic systems. However, in the real world, many practical systems may be subjected to stochastic abrupt changes in their structures and parameters, such as components failures or repairs, changes of subsystem interconnections, sudden environmental changes and so on, which can be well simulated by the Markovian switching. In fact, complex networks with Markovian switching have been extensively researched and many results have been reported in literature [14, 15, 16, 17, 18, 19, 20]. The above literature mainly considers complex networks with Markov switching parameters or with Markov switching topologies. Moreover, in [17, 18, 19, 20], each of the switching subnetwork is required to contain a directed spanning tree or to be strongly connected, which may limit the applications of CDNs. Therefore, in order to make the results more general, this encourages the further research for the topological structure of switching subnetwork. Until now, many types of synchronization have been discovered including cluster synchronization [21], impulsive synchronization [22], asymptotic synchronization [23], finite-time synchronization and stability [24, 25, 26], exponential synchronization [27, 28, 29]. Thereinto, finite-time synchronization seems to be a potential one since it cater to results people desire that synchronization of numerous applications can be achieved in a finite time interval. Therefore, many scholars have incorporated the finite-time synchronization into the study of CDNs [11, 25, 26]. Most of the above literatures use state feedback control to study finite-time synchronization. Recently, intermittent control has attracted widespread attention owing to intermittent control can reduce the control cost and the amount of transmitted information [30, 31, 32]. Based on intermittent control, many results about finite-time synchronization have been reported [33, 34, 35]. In [33, 34], intermittent control is used to study finite-time synchronization of CDNs, but the Markov switching factor and multi-links are not considered. Most of existing results on finite-time synchronization were obtained under the framework of transmitted information of the signals, and few authors considered utilizing quantized controllers. Considering the advantages of intermittent control and quantitative control, it is necessary to use quantized aperiodically intermittent control to study finite-time synchronization. However, few authors use quantized aperiodically intermittent control to study finite-time synchronization for stochastic multi-links dynamical networks with Markovian switching topologies (SMDNM). Therefore, this is the motivation for our research. Motivated by the above discussions, the objective of this paper is to study the finite-time synchronization of stochastic multi-links dynamical networks with Markovian switching topologies via quantized aperiodically intermittent control. At fist, we integrate the stochastic disturbances, Markovian switching topologies and multilinks into CDNs to make model more practical. Inspired by reference [1], combining the Lyapunov method with Kirchhoff’s Matrix Tree Theorem, we present some sufficient conditions to guarantee the finite-time synchronization of SMDNM. Then, we apply theoretic results to the investigation of the finite-time synchronization of oscillators system. Compared to the relevant existing results, the main contributions of this paper are highlighted as follows. 1. The topology structure of multi-links complex networks is Markovian switching and each switching subnetwork is not required to contain a directed spanning tree or to be strongly connected. 2. By establishing a new differential inequality, combining Lyapunov method with graph theory, finite-time synchronization of SHMND via quantized aperiodically intermittent control is firstly studied. This paper is organized as follows. In Section 2, we give some preliminaries and a model description. Then, some sufficient conditions are given in Section 3. Subsequently, we apply our theoretic results to oscillators system in Section 4. Finally, a numerical example is provided in Section 5.

2

2. Preliminaries and model formulations Notations : We introduce some notations firstly. The notations L = {1, 2, · · · , N}, Z+ = {1, 2, · · · }, R≥0 = [0, +∞), S = {1, 2, · · · , s} and M = {1, 2, · · · , m} are used. And sign(·) represents the sign function. Let (Ω, F , F, P) be a complete probability space with a filtration F = {Ft }t≥0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all P-null sets). Denote B(t) as one-dimensional Brownian motion defined on the probability space (Ω, F , F, P). And E(·) represents the mathematical expectation with respect to the given probability measure P. Moreover, Rn stands for n-dimensional Euclidean space. If x ∈ Rn , then kxk is the Euclidean norm of x. Moreover, the superscript “T” stands for the transpose of a vector. The notation“Tr” denotes the trace for the matrix, and C 2,1 (Rn × S × R≥0 ; R≥0 ) is referred to the family of all nonnegative functions V(x, i, t) on Rn × S × R≥0 which are continuously twice differentiable in x and once in t. Matrix A = (akh )N×N is irreducible if and only if corresponding weighted digraph (G, A) is strongly connected [1]. Meanwhile, the P Laplacian matrix of (G, A) is defined as L = (lkh )N×N , where lkh = −akh for k , h and lkk = Nj,k, j=1 ak j . Let {r(t), t ≥ 0} be a right-continuous Markov chain on the probability space (Ω, F , F, P) taking values in the finite space S with generator Γ = (ri j ) s×s given by    if i , j, ri j ∆t + o(∆t), P{r(t + ∆t) = j|r(t) = i} =   1 + rii ∆t + o(∆t), if i = j,

where ∆t > 0, lim∆t→0 (o(∆t)/∆t) = 0, and ri j ≥ 0 is the transition rate from i to j if i , j, otherwise, rii = P − j,i ri j < 0. Next, we consider a multi-links stochastic complex dynamical network with Markovian switching topologies in the following:   m X N X   dxk (t) =  fk (xk (t), r(t), t) + bkhl (r(t))Gkhl (xk (t), xh (t), r(t), t) dt l=1 h=1

+ gk (xk (t), r(t), t)dB(t), t ≥ 0, k ∈ L,

(1)

where xk ∈ R p (p ∈ Z+ ) represents a state vector, fk , gk : R p × S × R≥0 → R p stand for continuous functions, (bkhl (i))N×N denotes the ith switching topology matrix of the lth subnetwork. Function Gkhl : R p ×R p ×S×R≥0 → R p represents coupling form. System (1) is regarded as a drive multi-links stochastic complex dynamical network, and the corresponding response multi-links stochastic complex dynamical networks can be written as follows:   m X N X   dyk (t) =  fk (yk (t), r(t), t) + bkhl (r(t))Gkhl (yk (t), yh (t), r(t), t) + uk (t) dt l=1 h=1

+ gk (yk (t), r(t), t)dB(t), t ≥ 0, k ∈ L,

where yk ∈ R p (p ∈ Z+ ) represents a state vector. Moreover, uk (t) is designed as ( − αk (i)ek (t) − γk (i)SIGN(Υ(ek (t)))|ek (t)|$ tn ≤ t < sn , uk (t) = 0, sn ≤ t < tn+1 ,

(2)

(3)

where αk (i), γk (i) are two positive constants and $ ∈ (0, 1), ek (t) = yk (t) − xk (t) for k ∈ L is an error vector, SIGN(Υ(ek (t))) = diag{sign(Υ(ek1 (t))), · · · , sign(Υ(ekN (t)))}, |ek (t)|$ = (|ek1 (t)|$ , · · · , |ekN (t)|$ )T , Υ(·) : R → Γˆ is 3

a logarithmic quantizer, where Γˆ = {±ωi : ωi = %i ω0 , i = 0, ±1, ±2, · · · } ∪ {0} with ω0 > 0. For any v ∈ R, Υ(v) is defined as  1 1  ωi < v ≤ 1−δ ωi , ω, if 1+δ    i 0, if v = 0, Υ(v) =  (4)    −Υ(−v), if v < 0, n −{s o n . in which δ = [(1 − %)/(1 + %)], % ∈ (0, 1). Let Ψ = supn ttn+1 n+1 −tn According to [33] and (4), we can see that quantization error satisfies the following condition Υ(v) − v = µv, ∃ µ ∈ [−δ, δ], ∀v ∈ R. From drive-response multi-links stochastic complex dynamical networks (1) and (2), we can get the following error networks:   m X N X   dek (t) =  fˇk (ek (t), r(t), t) + bkhl (r(t))Gˇ khl (ek (t), eh (t), r(t), t) + uk (t) dt l=1 h=1

+ gˇ k (ek (t), r(t), t)dB(t), t ≥ 0, k ∈ L,

(5)

where fˇk (ek (t), r(t), t) = fk (yk (t), r(t), t) − fk (xk (t), r(t), t), gˇ k (ek (t), r(t), t) = gk (yk (t), r(t), t) − gk (xk (t), r(t), t), Gˇ khl (ek (t), eh (t), r(t), t) = Gkhl (yk (t), yh (t), r(t), t) − Gkhl (xk (t), xh (t), r(t), t). Remark 1. Recently, complex dynamical networks have been considered by many scholars [34, 36, 37, 38]. It is worth noting that most literature focused on the model which has at most two coupling terms. However, in this paper, we consider stochastic complex networks with multiple coupling rather than single coupling or two coupling terms. Especially, the multi-links complex networks turn to single-link complex networks when m = 1, S = {1} (see [36, 37, 38]). However, a few scholars have considered the multi-coupling complex network, but most of their coupling structures are constant and the coupling function is linear. Different from above literature, the topology we considered is Markovian switching and the coupling function is nonlinear. In fact, the coupling function is nonlinear, which is more in line with the actual situations. For example, the secondorder Kuramoto oscillators model was considered in the literature [40] with a coupling function sin(xh − xk ), and the single-species model was considered in [41] with a coupling function xh − αkh xk . In order to study the finite-time synchronization of the drive-response multi-links stochastic complex dynamical networks (1) and (2), we give some useful lemmas, assumptions and essential definitions. Lemma 1. [42] Suppose that a1 , a2 , · · · , an are positive numbers. And z is a positive constant satisfying 0 < z < 1. Then the following inequality holds: (a1 + a2 + · · · + an )z ≤ az1 + az2 + · · · + azn . Assumption 1. For any k ∈ L, i ∈ S, there exist constants εk (i) and δk (i) such that eTk fˇk (ek , i, t) ≤ εk (i)kek k2 and kˇgk (ek , i, t)k2 ≤ δk (i)kek k2 . Assumption 2. For any k, h ∈ L, i ∈ S, there exists a positive constant Θ(l) kh (i) such that kGˇ khl (ek , eh , i, t)k ≤ Θ(l) kh (i)(kek k + keh k). 4

For the sake of simplicity, m = m1 + m2 + · · · + mN are used. And the definition of finite-time stability is as follows. Definition 1. [43] The error network (5) is said to be finite-time stable in probability, if the equation admits a unique solution for any initial data x0 ∈ R pN , denoted by x(t; x0 ), moreover, the following statements hold: (i) Finite-time attractiveness in probability: For every initial value x0 ∈ R pN \ {0}, the first hitting time τ x0 = inf{t; x(t; x0 ) = 0}, called the stochastic settling time, is finite almost surely, that is P{τ x0 < ∞} = 1; (ii) Stability in probability: for every pair of ε ∈ (0, 1) and r > 0, there exists δ = δ(ε, r) > 0 such that P{kx(t; x0 )k < r, ∀ t ≥ 0} ≥ 1 − ε, whenever kx0 k < δ. Definition 2. For Vk (ek , i, t) ∈ C 2,1 (R p × S × R≥0 ; R≥0 ), differential operator LVk (ek , i, t) with respect to the kth equation of error network (5) is defined by m X N X ∂Vk (ek , i, t) ∂Vk (ek , i, t)  ˇ + fk (ek (t), i, t) + bkhl (i)Gˇ khl (ek (t), eh (t), i, t) LVk (ek , i, t) , ∂t ∂ek l=1 h=1 # X s  1 " 2 ∂ Vk (ek , i, t) T gˇ k (ek , i, t) + γi j Vk (ek , j, t), +uk (t) + Tr gˇ k (ek , i, t) 2 ∂e2k j=1

where

    ∂Vk (ek , i, t)  ∂Vk (ek , i, t) ∂2 Vk (ek , i, t)  ∂2 Vk (ek , i, t)  ∂Vk (ek , i, t)  =  =  ,··· ,  and  . ( j)  ∂ek ∂e2k ∂xk(1) ∂xk(p) ∂e(i) k ∂ek p×p

Lemma 2. [44] Assume that a continuous, positive-definite function V(t) is defined on a neighborhood of the origin and satisfies the following differential inequality ˙ ≤ −qV η (t) − pV(t), ∀t ∈ u˜ \ {0}, V(t) where q > 0, 0 < η < 1, p > 0 are three constants. Then, for any given t0 , V(t) satisfies the following inequality V 1−η (t)exp{(1 − η)pt} ≤ V 1−η (t0 )exp{(1 − η)pt0 } + and

 q exp{(1 − η)pt0 } − exp{(1 − η)pt} , t0 ≤ t < t1 p

V(t) ≡ 0, ∀t ≥ t1 , with t1 given by t1 ≤

  ln 1 + qp V 1−η (0) p(1 − η)

,

for t0 = 0. Now, in order to get main results, a lemma is given as follows: 5

Lemma 3. Suppose that V(t) is a continuous and non-negative function when t ∈ [0, +∞) and satisfies the following condition  η   −αV (t) − p1 V(t), tn ≤ t < sn , m = 0, 1, 2, · · · , ˙ V(t) ≤  (6)   p2 V(t), sn ≤ t < tn+1 , m = 0, 1, 2, · · · , where α, p1 , p2 are positive constants and η ∈ (0, 1). If

(1 − Ψ)p1 − Ψp2 > 0, then the following inequality holds  α V 1−η (t) exp{(1 − η)p1 t} ≤ exp{(1 − η)(p1 + p2 )Ψt} V 1−η (0) − (exp{(1 − η)(1 − Ψ)p1 t} p1  exp{−(1 − η)p2 Ψt} − 1) .

Proof. Take M0 = V(0)1−η + α/p1 and W(t) = V 1−η (t) exp{(1 − η)p1 t}. Moreover, let Θ(t) = W(t) − M0 + exp{(1 − η)p1 t}α/p1 . Next, we will prove that Θ(t) ≤ 0, t ∈ [0, s0 ).

(7)

If it is not true, then there exists λ1 ∈ [0, s0 ] that satisfies ˙ 1 ) > 0, Θ(t) ≤ 0 when t ∈ [0, λ1 ). Θ(λ1 ) = 0, Θ(λ

(8)

According to (6) and (8), one has ˙ 1 ) = (1 − η)V −η (λ1 )V(λ ˙ 1 ) exp{(1 − η)p1 λ1 } + p1 (1 − η)V 1−η (λ1 ) exp{(1 − η)p1 λ1 } Θ(λ +α(1 − η) exp{(1 − η)p1 λ1 } ≤ (1 − η)V −η (λ1 )[−αV η (λ1 ) − p1 V(λ1 )] exp{(1 − η)p1 λ1 } + p1 (1 − η)V 1−η (λ1 ) exp{(1 − η)p1 λ1 } +α(1 − η) exp{(1 − η)p1 λ1 } = 0, ˙ 1 ) > 0 and implies that Θ(t) ≤ 0 holds for t ∈ [0, s0 ). which contradicts inequality Θ(λ Then we will prove that H(t) , W1 (t) − M0 + exp{(1 − η)p1 t} exp{−(1 − η)(p1 + p2 )(t − s0 )}α/p1 < 0, for all t ∈ [s0 , t1 ), where W1 (t) = V 1−η (t) exp{(1 − η)p1 t} exp{−(1 − η)(p1 + p2 )(t − s0 )}. Otherwise, there exists λ2 ∈ [s0 , t1 ] such that ˙ 2 ) ≥ 0, H(t) < 0 for all t ∈ [s0 , λ2 ). H(λ2 ) = 0, H(λ According to (6) and (9), one has ˙ 2) = W ˙ 1 (λ2 ) + α (1 − η)p1 exp{(1 − η)p1 λ2 } exp{−(1 − η)(p1 + p2 )(λ2 − s0 )} H(λ p1 α − (1 − η)(p1 + p2 ) exp{(1 − η)p1 λ2 } exp{−(1 − η)(p1 + p2 )(λ2 − s0 )} p1 6

(9)

≤ (1 − η)p2 V 1−η (λ2 ) exp{(1 − η)p1 λ2 } exp{−(1 − η)(p1 + p2 )(λ2 − s0 )} +V 1−η (λ2 )[(1 − η)p1 exp{(1 − η)p1 λ2 } exp{−(1 − η)(p1 + p2 )(λ2 − s0 )} −(1 − η)(p1 + p2 ) exp{−(1 − η)(p1 + p2 )(λ2 − s0 )}] α + (1 − η)p1 exp{(1 − η)p1 λ2 } exp{−(1 − η)(p1 + p2 )(λ2 − s0 )} p1 α − (1 − η)(p1 + p2 ) exp{(1 − η)p1 λ2 } exp{−(1 − η)(p1 + p2 )(λ2 − s0 )} p1 < 0, which contradicts inequality (9). Therefore, H(t) < 0 for all t ∈ [s0 , t1 ), that is " # α W(t) ≤ exp{(1 − η)(p1 + p2 )(t − s0 )} M0 − exp{(1 − η)p1 t} exp{−(1 − η)(p1 + p2 )(t − s0 )} p1 # " α exp{(1 − η)p1 t} exp{−(1 − η)(p1 + p2 )(t1 − s0 )} . (10) < exp{(1 − η)(p1 + p2 )(t1 − s0 )} M0 − p1 On the other hand, according to (7), for t ∈ [0, s0 ), one has α exp{(1 − η)p1 t} p1 α exp{(1 − η)p1 t} exp{−(1 − η)(p1 + p2 )(t1 − s0 )} ≤ M0 − p1 " # α ≤ exp{(1 − η)(p1 + p2 )(t1 − s0 )} M0 − exp{(1 − η)p1 t} exp{−(1 − η)(p1 + p2 )(t1 − s0 )} . (11) p1

W(t) ≤ M0 −

Therefore, combining (10) with (11), we get # " α exp{(1 − η)p1 t} exp{−(1 − η)(p1 + p2 )(t1 − s0 )} . W(t) ≤ exp{(1 − η)(p1 + p2 )(t1 − s0 )} M0 − p1 Similarly, we can prove that for all t ∈ [t1 , s1 ), one has " # α W(t) ≤ exp{(1 − η)(p1 + p2 )(t1 − s0 )} M0 − exp{(1 − η)p1 t} exp{−(1 − η)(p1 + p2 )(t1 − s0 )} p1 and for all t ∈ [s1 , t2 ), we obtain

 α W(t) ≤ exp{(1 − η)(p1 + p2 )(t1 − s0 + t − s1 )} M0 − exp{(1 − η)p1 t} p1  exp{−(1 − η)(p1 + p2 )(t1 − s0 + t − s1 )} .

Next, we use mathematical induction to prove the following two inequalities.   m  "  X    α   (tk − sk−1 ) W(t) ≤ exp  (1 − η)(p + p ) M − exp{(1 − η)p1 t}  1 2     0 p1 k=1 7

  m  #  X       exp  −(1 − η)(p + p ) (t − s ) , t ∈ [tn , sn ),  1 2  k k−1      

(12)

k=1

  m  "  X    α    exp{(1 − η)p1 t} W(t) ≤ exp  (1 − η)(p + p ) (t − s + t − s ) M0 −   1 2  k k−1 n      p1 k=1   m  #  X      (tk − sk−1 + t − sn ) exp  −(1 − η)(p + p ) , t ∈ [sn , tn+1 ).  1 2    

(13)

k=1

We assume that (12) and (13) are satisfied for any m ≤ n − 1, where n is a positive integer, that is, for any 0 ≤ l ≤ n − 1, when t ∈ [tl , sl ),   l  "  X    α   (tk − sk−1 ) exp{(1 − η)p1 t} W(t) ≤ exp  (1 − η)(p + p ) M −  1 2     0 p1 k=1   l  #  X       −(1 − η)(p + p ) exp  (t − s )    1 2  k k−1     k=1

and when t ∈ [sl , tl+1 ), we have   l  "  X    α   (tk − sk−1 + t − sl ) W(t) ≤ exp  exp{(1 − η)p1 t} (1 − η)(p + p ) M0 −  1 2     p1 k=1   l  #  X      (tk − sk−1 + t − sl ) exp  −(1 − η)(p + p )  1 2     k=1    l+1  "  X    α    exp{(1 − η)p1 t} (1 − η)(p + p ) ≤ exp  (t − s ) M0 −   1 2  k k−1      p 1 k=1   l+1  #  X      (tk − sk−1 ) exp  −(1 − η)(p + p )  1 2     k=1   n  "  X    α    exp{(1 − η)p1 t} < exp  (1 − η)(p1 + p2 )  (tk − sk−1 ) M −    0 p1  k=1   n  #  X      (tk − sk−1 ) exp  −(1 − η)(p + p ) .  1 2     k=1

Therefore, for any t ∈ [0, tn ), we have   n  "  X    α    M − W(t) ≤ exp  (1 − η)(p1 + p2 )  (tk − sk−1 ) exp{(1 − η)p1 t}    0 p1  k=1   n  #  X      (tk − sk−1 ) , exp  −(1 − η)(p + p )  1 2     k=1

8

(14)

it is easy to know that (14) holds for t ∈ [tn , sn ). And for t ∈ [sn , tn+1 ), we can verify that   n  "  X    α    W(t) ≤ exp  exp{(1 − η)p1 t} (1 − η)(p + p ) (t − s + t − s ) M0 −    1 2  k k−1 n     p 1 k=1   n  #  X      (tk − sk−1 + t − sn ) exp  −(1 − η)(p + p ) .  1 2     k=1

Thus according to mathematical induction, we can get that (12) and (13) are true for any positive integer m. As for any t ≥ 0, there is a nonnegative integer m that satisfies tn ≤ t < tn+1 , thus we can deduce the following estimations of W(t) for any t. When t ∈ [tn , sn ),   n  "   X   α   (tk − sk−1 ) W(t) ≤ exp  (1 − η)(p + p )  1 2   M0 − p1 exp{(1 − η)p1 t}   k=1   n  #   X       (t − s ) exp  −(1 − η)(p + p ) 1 2  k k−1      k=1   "  n    X tk − sk−1  α    (t − t ) exp{(1 − η)p1 t} M0 − = exp  (1 − η)(p + p )  k k−1 1 2     t − tk−1 p1 k=1 k   #  n   X tk − sk−1      exp  (t − t ) −(1 − η)(p + p )   k k−1  1 2      t − tk−1 k=1 k   n  "   X   α   (tk − tk−1 ) exp{(1 − η)p1 t} ≤ exp  (1 − η)(p + p )Ψ M0 −  1 2     p1 k=1   n  #  X       exp  −(1 − η)(p + p )Ψ (t − t )   1 2 k k−1       k=1 # " α exp{(1 − η)p1 t} exp {−(1 − η)(p1 + p2 )Ψt} . ≤ exp {(1 − η)(p1 + p2 )Ψt} M0 − p1 Similarly, when t ∈ [sn , tn+1 ), we can see that   n  "  X    α   (tk − sk−1 + t − sn ) W(t) ≤ exp  (1 − η)(p + p ) M − exp{(1 − η)p1 t}  1 2     0 p1 k=1   n  #  X       exp  −(1 − η)(p + p ) (t − s + t − s )   1 2  k k−1 n      k=1   n  "  X tk − sk−1    t − s α  n+1 n   ≤ exp  (1 − η)(p + p ) (t − t ) + (t − t ) M0 − exp{(1 − η)p1 t}  1 2 k k−1 n     t − tk−1 tn+1 − tn p1 k=1 k   n  #   X tk − sk−1   t − s  n+1 n   exp  −(1 − η)(p + p ) (t − t ) + (t − t )   1 2  k k−1 n      t − tk−1 tn+1 − tn k=1 k   n  "   X   α   (tk − tk−1 + t − tn ) ≤ exp  (1 − η)(p + p )Ψ  1 2   M0 − p1 exp{(1 − η)p1 t}   k=1 9

  n  #  X       exp  −(1 − η)(p + p )Ψ (t − t + t − t )   1 2 k k−1 n      k=1 # " α exp{(1 − η)p1 t} exp {−(1 − η)(p1 + p2 )Ψt} . = exp {(1 − η)(p1 + p2 )Ψt} M0 − p1 Thus from the definition of W(t), it is easy to get V

1−η

"

α α − exp{(1 − η)p1 t} (t) exp{(1 − η)p1 t} ≤ exp {(1 − η)(p1 + p2 )Ψt} V(0)1−η + p1 p1 # exp {−(1 − η)(p1 + p2 )Ψt} " α = exp {(1 − η)(p1 + p2 )Ψt} V(0)1−η − (exp{(1 − η)(1 − Ψ)p1 t} p1 # exp {−(1 − η)p2 Ψt} − 1) .

The proof is completed. Remark 2. In [44], scholars studied finite-time synchronization of neural networks by establishing differential ˙ ≤ −αV η (t) − p1 V(t) when lT ≤ t < lT + θT inequalities with periodically intermittent control, for example, V(t) ˙ ≤ p2 V(t) when lT + θT ≤ t < (l + 1)T , in which T > 0, 0 < θ < 1 are constants and l ∈ {0, 1, · · · }. Howand V(t) ever, different from literature [44], in our paper, we study finite-time synchronization of SMDNM by establishing ˙ ≤ −αV η (t) − p1 V(t) when differential inequalities with quantized aperiodically intermittent control, that is, V(t) ˙ ≤ p2 V(t) when sn ≤ t < tn+1 . Particularly, when aperiodically intermittent control becomes tn ≤ t < sn and V(t) periodically intermittent control, the conclusion in this article is still valid. Thus Lemma 3 is a generalization of literature [44]. 3. Finite-time synchronization control In this section, we provide some sufficient conditions to assure the finite-time synchronization of driveresponse complex networks (1) and (2) by Lyapunov method and Kirchhoff’s Matrix Tree Theorem. Theorem 1. Suppose that Assumptions 1, 2 are satisfied. The drive network (1) and the response network (2) can reach the finite-time synchronization, if (Qkh )N×N is irreducible, in which Qkh = maxi∈S,l∈M {qk (i)bkhl (i)Θ(l) kh (i)} where qk (i) are positive constants and the following inequality hold s m X N   X Qkh  X + ri j qk ( j) < 0. Λk (i) = qk (i) 2εk (i) − 2αk (i) + δk (i) + 3bkhl (i)Θ(l) (i) + kh q (i) k j=1 l=1 h=1

Moreover, the finite time interval can be estimated as   1−$ 2 ln 1 + ΛV (e(0),r(0),0) Ξ T ∗ = 1−$ , ((1 − Ψ)Λ − ΨΦ) 2 10

(15)

  1−$ where Λ = mink∈L,i∈S {Λk (i)}, Ξ = mink∈L,i∈S 2γk (i)(ck qk (i)) 2 , Φ = maxk∈L,i∈S {Φk (i)}, in which Φk (i) =   P PN P PN  Qkh qk (i) 2εk (i) + δk (i) + ml=1 h=1 (i) + + sj=1 ri j qk ( j), V(e(0), r(0), 0) = k=1 ck qk (r(0))kek (0)k2 3bkhl (i)Θ(l) kh qk (i) and ck is the cofactor of the kth diagonal element of the Laplacian matrix of (G, (Qkh )N×N ). PN Proof. Define function V(e, t, i) = k=1 ck Vk (ek , i, t), where Vk (ek , i, t) = qk (i)kek k2 , for any k ∈ L, i ∈ S. According to Assumptions 1 and 2, we obtain that   m X N X    LVk (ek (t), i, t) =2qk (i)eTk (t)  fˇk (ek (t), i, t) + bkhl (i)Gˇ khl (ek (t), eh (t), i, t) + uk (t) l=1 h=1

+ qk (i)kˆgk (ek (t), i, t)k2 +

s X

ri j qk ( j)kek (t)k2 .

j=1

When t ∈ [tn , sn ), applying Lemma 1 and (3), one can deduce   2eTk (t)uk (t) =2eTk (t) −αk (i)ek (t) − γk (i)sign(Υ(ek (t)))|ek (t)|$ p X ≤ − 2αk (i)kek (t)k2 − 2γk (i) |eki (t)|1+$ . i=1

On the basis of the above computation, we can get 2

LVk (ek (t), i, t) ≤qk (i)(2εk (i) − 2αk (i) + δk (i))kek (t)k + − 2qk (i)γk (i)

p X i=1

|eki (t)|1+$ +

s X

2qk (i)eTk (t)

m X N X

bkhl (i)Gˇ khl (ek (t), eh (t), i, t)

l=1 h=1

ri j qk ( j)kek (t)k2 .

(16)

j=1

By Young’s inequality kakc kbkd ≤

d c kakc+d + kbkc+d , c+d c+d

it yields that 2qk (i)eTk (t)

m X N X

bkhl (i)Gˇ khl (ek (t), eh (t), i, t)

l=1 h=1

≤2 ≤3

m X N X

l=1 h=1 m X N X l=1 h=1

qk (i)bkhl (i)Θ(l) kh (i)kek (t)k(kek (t)k + keh (t)k) 2 qk (i)bkhl (i)Θ(l) kh (i)kek (t)k

+

m X N X

2 qk (i)bkhl (i)Θ(l) kh (i)keh (t)k

l=1 h=1

m X N m X N X X   2 ≤ [3qk (i)bkhl (i)Θ(l) (i) + Q ]ke (t)k + Qkh keh (t)k2 − kek (t)k2 . kh k kh l=1 h=1

l=1 h=1

11

(17)

Combining (16) with (17), we derive that N  m X s    X Qkh  X (l) 3bkhl (i)Θkh (i) + ri j qk ( j) kek (t)k2 LVk (ek (t), i, t) ≤ qk (i) 2εk (i) − 2αk (i) + δk (i) + + qk (i) j=1 l=1 h=1

− 2qk (i)γk (i)

mk X i=1

1+$

|eki (t)|

+

m X N X l=1 h=1

  Qkh keh (t)k2 − kek (t)k2

N  m X s    X Qkh  X (l) ri j qk ( j) kek (t)k2 3bkhl (i)Θkh (i) + ≤ qk (i) 2εk (i) − 2αk (i) + δk (i) + + qk (i) j=1 l=1 h=1 m X N     1+$ X 2 2 Qkh keh (t)k2 − kek (t)k2 − 2qk (i)γk (i) keki (t)k + l=1 h=1

m X N   1+$ X   = − Λk (i)kek (t)k2 − 2qk (i)γk (i) keki (t)k2 2 + Qkh keh (t)k2 − kek (t)k2 ,

(18)

l=1 h=1

Therefore, according to (18) and the definition of function V(e, i, t) = LV(e(t), i, t) = ≤

N X k=1

N X k=1

ck LVk (ek (t), i, t)

PN

k=1 ck Vk (ek , i, t),

one has

  m X N X   1+$    2 2 2 2 2 ck −Λk (i)kek (t)k − 2qk (i)γk (i) keki (t)k + Qkh keh (t)k − kek (t)k  .

(19)

l=1 h=1

By using of Lemma 1, one has −

N X k=1

  1+$   1+$ 1−$ 2ck qk (i)γk (i) keki (t)k2 2 ≤ − min 2γk (i)(ck qk (i)) 2 V 2 (e(t), i, t). k∈L,i∈S

Moreover, according to the Theorem 2.2 in reference [1], we can gain that  m N  N X X    X 2 2 ck Qkh keh (t)k − kek (t)k  = 0.  k=1

(20)

(21)

l=1 h=1

Combining (19) and (20) with (21), we have LV(e(t), i, t) ≤ −

N X k=1

 1+$  1−$ ck Λk (i)kek (t)k2 − min 2γk (i)(ck qk (i)) 2 V 2 (e(t), i, t) k∈L,i∈S

  1+$ 1−$ ≤ − ΛV(e(t), i, t) − min 2γk (i)(ck qk (i)) 2 V 2 (e(t), i, t) k∈L,i∈S

= − ΛV(e(t), i, t) − ΞV

1+$ 2

(e(t), i, t).

Therefore, we can obtain ELV(e(t), r(t), t) ≤ ΛEV(e(t), r(t), t) − Ξ(EV(e(t), r(t), t)) 12

1+$ 2

.

(22)

Similarly, when t ∈ [sn , tn+1 ), according to Assumptions 1 and 2, we can get 2

2

LVk (ek (t), i, t) ≤2qk (i)εk (i)kek (t)k + qk (i)δk (i)kek (t)k + + 2qk (i)eTk (t)

m X N X

l=1 h=1

Therefore, one has LV(e(t), i, t) ≤

N X k=1

ck Φk (i)kek (t)k2 +

N X k=1

ck

ri j qk ( j)kek (t)k2

j=1

bkhl (i)Gˇ khl (ek (t), eh (t), i, t)

l=1 h=1 m X N X

≤Φk (i)kek (t)k2 +

s X

  Qkh keh (t)k2 − kek (t)k2 .

m X N X l=1 h=1

  Qkh keh (t)k2 − kek (t)k2 ≤ ΦV(e(t), i, t).

(23)

Combining (22) with (23), we get that  1+$   tn ≤ t < sn ,  − ΛEV(e(t), r(t), t) − ΞE(V(e(t), r(t), t)) 2 , ELV(e(t), r(t), t) ≤    ΦEV(e(t), r(t), t), sn ≤ t < tn+1 ,

Therefore, according to Lemma 3, one has ) ( )" ( 1−η 1 − $ 1 − $ Λt ≤ exp (Λ + Φ)Ψt V 1−η (e(0), r(0), 0) (EV(e(t), r(t), t)) 2 exp 2 2 ( ! ) !#
1−$ Ξ − exp (1 − Ψ)Λ − ΨΦ t − 1 , Λ 2 Moreover, according to Lemma 2, estimation of the finite time is given as follows:   1−$ ΛV 2 (e(0),r(0),0) ln 1 + Ξ = T ∗. t ≤ 1−$ ((1 − Ψ)Λ − ΨΦ) 2

It means that systems (1) and (2) are finite-time synchronization. This completes the proof. Remark 3. It is well-known that Lyapunov method is an important approach for investigating the stability and synchronization of networks. However, the difficulty is how to construct a proper Lyapunov function. In Theorem PN 1, we utilize a systematic method to construct a Lyapunov function, namely V(e, t, i) = k=1 ck Vk (ek , t, i), which overcomes this challenge and simplifies the complex analysis of the study of finite-time synchronization. Remark 4. In fact, complex dynamical networks with Markovian switching topologies have been considered by many scholars in [17, 18, 19, 20]. For example, in [17], each switching subnetwork of complex networks is required to be strongly connected. In [18, 19, 20], the topological structure of every switching subnetwork needs to contain a directed spanning tree. Different from previous related literature, from Theorem 1, we can see that each switching subnetwork is not required to contain a directed spanning tree or to be strongly connected. Therefore, the results obtained in this article are the extension of previous literature. 13

Remark 5. We can clearly see that quantized aperiodically intermittent control plays a key role in the finite-time synchronization of stochastic complex dynamical networks from Theorem 1. In fact, In fact, (15) in Theorem 1, if the coefficient αk (i) of controllers uk (t) is enough large, then the finite-time synchronization can be achieved.   1−$   2 (e(0),r(0),0)  ln1+ ΛV  Ξ

∂ ∂ ∂ ∗ T ∗ < 0, ∂Φ T ∗ < 0, ∂Ξ T > 0. It means that Remark 6. Let T ∗ = 1−$ ((1−Ψ)Λ−ΨΦ) , it is easy to calculate that ∂Λ 2 ∗ T is a strictly monotone decreasing function for variable Λ and variable Φ, while the function T ∗ is monotone increasing for variable Ξ. Thus, with the parameter Λ or Φ getting larger, the convergence time will be shorter. With the parameter Ξ getting smaller, the convergence time will be shorter.

4. An application to oscillators systems In recent years, oscillators systems have attracted wide attention [9, 45]. In view of it, we apply our main results to stochastic multi-links oscillators systems with Markovian switching topologies in this section. The model can be described by x¨k (t) + φk (r(t)) x˙k (t) + xk (t) +

m X N X l=1 h=1

˙ t ≥ 0, k ∈ L, bkhl (r(t))Gkhl (xk (t), xh (t), r(t), t) = hk (xk (t), r(t), t) B(t),

(24)

where φk : S → R≥0 is the damping coefficient, Gkhl : R × R × S × R≥0 → R represents the coupling form, and hk : R × S × R≥0 → R stands for the perturbation intensity to the kth vertex, (bkhl (i))N×N denotes the ith switching topology matrix of the lth subnetwork. Let yk (t) = x˙k (t) + k xk (t), where k > 0, then system (24) is transformed into the following:     dxk (t) = h yk (t) − k xk (t) dt,     i     dyk (t) = (k − φk (r(t)))yk (t) + k φk (r(t)) − k2 − 1 xk (t) dt t ≥ 0, k ∈ L. (25)   m P N  P    − bkhl (r(t))Gkhl (xk (t), xh (t), r(t), t)dt + hk (xk (t), r(t), t)dB(t),  l=1 h=1

Set Xk (t) = (xk (t), yk (t)) , Xh (t) = (xh (t), yh (t)) , G˜ khl (Xk (t), Xh (t), r(t), t) = T

Fk (Xk (t), r(t), t) =

T

! 0 , −Gkhl (xk (t), xh (t), r(t), t)

! yk (t) − k xk (t)  , h˜ k (Xk (t), r(t), t) = (k − φk (r(t)))yk (t) + k φk (r(t)) − k2 − 1 xk (t)

! 0 . hk (xk (t), r(t), t)

Then, we can rewrite system (35) as   m X N X   dXk (t) = Fk (Xk (t), r(t), t) + bkhl (r(t))G˜ khl (Xk (t), Xh (t), r(t), t) dt + h˜ k (Xk (t), r(t), t)dB(t), t ≥ 0, k ∈ L.(26) l=1 h=1

System (26) is regarded as a drive stochastic multi-links oscillators systems, and the corresponding response stochastic multi-links oscillators systems can be given as follows:   m X N X   dYk (t) = Fk (Yk (t), r(t), t) + bkhl (r(t))G˜ khl (Yk (t), Yh (t), r(t), t) + Uk (t) dt l=1 h=1

14

+ h˜ k (Yk (t), r(t), t)dB(t), t ≥ 0, k ∈ L, where Yk (t) = ( xˆk (t), yˆ k (t))T and quantized aperiodically intermittent control can be given as ( − αk (i)Ek (t) − γk (i)SIGN(Υ(Ek (t)))|ek (t)|$ tn ≤ t < sn , Uk (t) = 0, sn ≤ t < tn+1 ,

(27)

(28)

where αk (i), γk (i) are two positive constants and $ ∈ (0, 1), Ek (t) = Yk (t) − Xk (t) = ( xˆk (t) − xk (t), yˆ k (t) − yk (t))T =  (ek (t), eˆ k (t))T , SIGN(Ek (t)) = diag sign (ek (t)) , sign (ˆek (t)) , |Ek (t)|$ = (|ek (t)|$ , |ˆek (t)|$ )T and other parameters are similarly defined in Section 2. Therefore, combining system (35) with system (27), we can get the following error system   m X N X   dEk (t) = Fˆ k (Ek (t), r(t), t) + bkhl (r(t))Gˆ khl (Ek (t), Eh (t), r(t), t) + Uk (t) dt (29) l=1 h=1 + hˆ k (Ek (t), r(t), t)dB(t), t ≥ 0, k ∈ L, where Fˆ k (Ek (t), r(t), t) = Fk (Yk (t), r(t), t) − Fk (Xk (t), r(t), t), Gˆ khl (Ek (t), Eh (t), r(t), t) = G˜ khl (Yk (t), Yh (t), r(t), t) − G˜ khl (Xk (t), Xh (t), r(t), t), hˆ k (Ek (t), r(t), t) = h˜ k (Yk (t), r(t), t) − h˜ k (Xk (t), r(t), t). Next, a criterion about drive-response systems (26) and (27) is given. Theorem 2. The drive system (26) and the response system (27) can reach the finite-time synchronization, if the following conditions hold for any k, h ∈ L. (C1 ) There are positive constants Θ(l) kh (i) and δk (i) such that kGkhl (yk , yh , i, t) − Gkhl (xk , xh , i, t)k ≤ Θ(l) kh (i)(kyk − xk k + kyh − xh k), khk (yk , i, t) − hk (xk , i, t)k2 ≤ δk (i)kyk − xk k2 .

(C2 ) The following inequalities hold s m X N   X Qkh  X (l) + ri j qk ( j) < 0, Λk (i) = qk (i) 2εk (i) − 2αk (i) + δk (i) + 3bkhl (i)Θkh (i) + qk (i) j=1 l=1 h=1

n o where εk (i) = max −k + 12 k φk (i) − k2 , −φk (i) + k + 12 k φk (i) − k2 . Moreover, the finite time interval can be estimated as   1−$ 2 ln 1 + ΛV (e(0),r(0),0) Ξ T ∗ = 1−$ , ((1 − Ψ)Λ − ΨΦ) 2   1−$ 2 where Λ = mink∈L,i∈S {Λk (i)}, Ξ = mink∈L,i∈S 2γk (i)(ck qk (i)) , Φ = maxk∈L,i∈S {Φk (i)}, in which Φk (i) =    P PN Ps PN Qkh 2 qk (i) 2εk (i)+δk (i)+ ml=1 h=1 3bkhl (i)Θ(l) k=1 ck qk (r(0))kek (0)k , kh (i)+ qk (i) + j=1 ri j qk ( j) and V(e(0), r(0), 0) = ck is the cofactor of the kth diagonal element of the Laplacian matrix of (G, (Qkh )N×N ), (Qkh )N×N is irreducible and Qkh = maxi∈S,l∈M {qk (i)bkhl (i)Θ(l) kh (i)}, qk (i) is a positive constant. 15

Proof. According to condition (C1 ), we know that   EkT Fˆ k (Ek , i, t) = k φk (i) − k2 ek eˆ k (t) − k e2k + (k − φk (i))ˆe2k ! ! 1 1 2 2 2 2 ≤ −k + k φk (i) − k ek + −φk (i) + k + k φk (i) − k eˆ k 2 2 2 ≤ εk (i)kEk k .

(30)

From conditions (C1 ), (C2 ) and (30), we can conclude that Assumptions 1 and 2 are satisfied. Thus, according to Theorem 1, we can get that the drive-response systems (26) and (27) are synchronized in a finite time. This completes the proof. Remark 7. Recently, many scholars have studied coupled oscillators [2, 17, 38, 39]. For example, in [38], the constant coupling structure was discussed. Meanwhile, the coupling structure with Markovian switching is considered in [2, 17] and it is assumed that the topology of each switching subnetwork is strongly connected. Different from these results, we consider the case when the coupling structure is Markovian switching and the topology of each switching subnetwork is not required to contain a directed spanning tree or to be strongly. Thus, this condition is weakened in this paper. Remark 8. The dynamic of oscillators systems is a hot topic. Thereinto, many scholars investigated the exponential synchronization and asymptotic synchronization, which are defined on infinite time [2, 17, 38]. However, from perspective of engineering, the study of finite-time synchronization is more necessary. In this paper, we firstly investigate the finite-time synchronization of stochastic multi-links oscillator with Markovian switching topologies by designing a quantized aperiodically intermittent controller. 5. Numerical test Example 1. In this section, we give an example to illustrate the effectiveness of the theoretical results obtained in this paper. Let m = 3, N = 9, S = {1, 2}. To begin with, it is assumed that Markov chain r(t) takes values in S = {1, 2} with generator ! −2 2 Γ = (ri j )2×2 = . 1 −1 Then consider a coupled oscillators network with 9 oscillators and three different kinds of weights (that is m = 3) at state 1 and state 2. Moreover, a coupled oscillators network is given in Figure 1. Meanwhile, the coupling weights of corresponding switching subnetworks are as follows. a121 (1) = 1.20 × 10−4 , a311 (1) = 1.23 × 10−4 , a241 (1) = 1.33 × 10−4 , a641 (1) = 1.63 × 10−4 , a691 (1) = 1.34 × 10−4 , a791 (1) = 1.10 × 10−4 , a851 (1) = 1.60 × 10−4 , a351 (1) = 1.37 × 10−4 , a541 (1) = 1.06 × 10−4 , a212 (1) = 1.30 × 10−4 , a622 (1) = 1.10 × 10−4 , a162 (1) = 1.23 × 10−4 ,

a242 (1) = 1.32 × 10−4 , a452 (1) = 1.21 × 10−4 , a892 (1) = 1.35 × 10−4 , a532 (1) = 1.29 × 10−4 , a132 (1) = 1.43 × 10−4 , a182 (1) = 1.25 × 10−4 , a712 (1) = 1.30 × 10−4 , a673 (1) = 1.51 × 10−4 ,

a263 (1) = 1.56 × 10−4 , a463 (1) = 1.54 × 10−4 , a943 (1) = 1.35 × 10−4 , a953 (1) = 1.12 × 10−4 , a783 (1) = 1.11 × 10−4 , a383 (1) = 1.30 × 10−4 ,

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Figure 1: A coupled oscillators network with 3 links at state 1 and state 2, respectively.

and a611 (2) = 1.21 × 10−4 , a261 (2) = 1.43 × 10−4 , a811 (2) = 1.30 × 10−4 , a971 (2) = 1.23 × 10−4 ,

a491 (2) = 1.24 × 10−4 , a951 (2) = 1.17 × 10−4 , a351 (2) = 1.64 × 10−4 , a132 (2) = 1.77 × 10−4 , a832 (2) = 1.16 × 10−4 , a982 (2) = 1.32 × 10−4 , a582 (2) = 1.14 × 10−4 , a962 (2) = 1.43 × 10−4 ,

a762 (2) = 1.32 × 10−4 , a942 (2) = 1.31 × 10−4 , a422 (2) = 1.31 × 10−4 , a213 (2) = 1.79 × 10−4 ,

a263 (2) = 1.43 × 10−4 , a763 (2) = 1.35 × 10−4 , a793 (2) = 1.31 × 10−4 , a593 (2) = 1.58 × 10−4 , a543 (2) = 1.56 × 10−4 , a383 (2) = 1.55 × 10−4 , a533 (2) = 1.37 × 10−4 ,

and other weights are 0. Meanwhile, the functions and parameters are given as follows. Let φk (1) = 0.02, φk (2) = 0.01, hk (xk , 1, t) = 0.15 sin xk , hk (xk , 2, t) = 0.5 sin xk , k = 0.05, Gkh1 ( x˜k , x˜h , i, t) = 0.04 sin( x˜h − x˜k ), Gkh2 ( x˜k , x˜h , i, t) = 0.05 sin( x˜h − x˜k ), Gkh3 ( x˜k , x˜h , i, t) = 0.1 sin( x˜h − x˜k ). Moreover, we choose $ = 8/9, αk (i) = 12.5, γk (i) = 5.6.

17

By a simple calculation, we can get that s m X N   X Qkh  X (l) ri j qk ( j) < 0. + qk (i) 2εk (i) − 2αk (i) + δk (i) + 3bkhl (i)Θkh (i) + qk (i) j=1 l=1 h=1

Therefore, all conditions in Theorem 2 are verified, which means the drive system (26) and the response system (27) can reach the finite-time synchronization. And corresponding numerical simulations are shown in Figures 6, 3, 4 and 5, which shows the effectiveness and feasibility of theoretical results. 200

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Figure 3: The sample path figure of system (27) when N = 9.

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Figure 5: The sample path figure of error system (29) when N = 9.

Example 2. Recently, the second-order Kuramoto model has received extensive attention [46, 47]. Peron et al. investigated second-order Kuramoto oscillators in [47] as follows: N X d2 θk (t) dθk (t) = −α + Ωk + λ akh sin(θh (t) − θk (t)), t ≥ 0, k ∈ L, dt2 dt h=1

(31)

where α is the dissipating parameter, λ is the coupling strength, Ωk is the natural frequency of oscillator k, akh ≥ 0 represents coupling weight constant between the hth node and the kth node. 18

Now, we make a transformation of θ˜k (t) = θ˙k (t) + ηk θk (t), where ηk > 0, then system (31) could be converted to  dθk (t) = [θ˜k (t) − ηk θk (t)]dt,     # " N  X   ˜ ˜  akh sin(θh (t) − θk (t)) dt, t ≥ 0, k ∈ L.   dθk (t) = (−α + ηk )(θk (t) − ηk θk (t)) + Ωk + λ h=1

Next, taking stochastic perturbations, Markovian switching topologies and multi-links factors into account, we can get the following system:   dθk (t) = [θ˜k (t) − ηk θk (t)]dt + δk (r(t))θk (t)dB(t),    # "  N m X  X   ˜ (32) akhl (r(t)) sin(θh (t) − θk (t)) dt dθk (t) = (−α(r(t)) + ηk )(θ˜k (t) − ηk θk (t)) + Ωk + λ(r(t))      h=1 l=1    + δk (r(t))θ˜k (t)dB(t), t ≥ 0, k ∈ L,

where B(t) is one-dimensional Brownian motion. Then, we can rewrite system (32) as   m X N X   dXk (t) = Fk (Xk (t), r(t), t) + akhl (r(t))G˜ khl (Xk (t), Xh (t), r(t), t) dt + h˜ k (Xk (t), r(t), t)dB(t), t ≥ 0, k ∈ L,(33) l=1 h=1

where Fk (Xk (t), r(t), t) = (θ˜k (t)−ηk θk (t), (−α(r(t))+ηk )(θ˜k (t)−ηk θk (t))+Ωk )T , h˜ k (Xk (t), r(t), t) = (δk (r(t))θk (t), δk (r(t))θ˜k (t))T and G˜ khl (Xk (t), Xh (t), r(t), t) = (0, λ(r(t)) sin(θh (t) − θk (t)))T . System (33) is regarded as a drive system, and the corresponding response system can be given as follows:   m X N X   dYk (t) = Fk (Yk (t), r(t), t) + akhl (r(t))G˜ khl (Yk (t), Yh (t), r(t), t) + Uk (t) dt l=1 h=1

+ h˜ k (Yk (t), r(t), t)dB(t), t ≥ 0, k ∈ L,

(34)

where Yk (t) = ( xˆk (t), yˆ k (t))T and quantized aperiodically intermittent control can be given as ( − αk (i)Ek (t) − γk (i)SIGN(Υ(Ek (t)))|ek (t)|$ tn ≤ t < sn , Uk (t) = 0, sn ≤ t < tn+1 , where αk (i), γk (i) are two positive constants and $ ∈ (0, 1), Ek (t) = Yk (t) − Xk (t) = ( xˆk (t) − θk (t), yˆ k (t) − θ˜k (t))T =  (ek (t), eˆ k (t))T , SIGN(Ek (t)) = diag sign (ek (t)) , sign (ˆek (t)) , |Ek (t)|$ = (|ek (t)|$ , |ˆek (t)|$ )T and other parameters are similarly defined in Section 2. Therefore, combining system (33) with system (34), we can get the following error system   m X N X   dEk (t) = Fˆ k (Ek (t), r(t), t) + akhl (r(t))Gˆ khl (Ek (t), Eh (t), r(t), t) + Uk (t) dt (35) l=1 h=1 + hˆ k (Ek (t), r(t), t)dB(t), t ≥ 0, k ∈ L, where Fˆ k (Ek (t), r(t), t) = Fk (Yk (t), r(t), t) − Fk (Xk (t), r(t), t), Gˆ khl (Ek (t), Eh (t), r(t), t) = G˜ khl (Yk (t), Yh (t), r(t), t) − G˜ khl (Xk (t), Xh (t), r(t), t), hˆ k (Ek (t), r(t), t) = h˜ k (Yk (t), r(t), t) − h˜ k (Xk (t), r(t), t). 19

Now, some numerical simulations are given. Let m = 2, N = 4, S = {1, 2}, α(1) = 1, α(2) = 1.2, Ωk = 0.2, λ(1) = 0.4, λ(2) = 0.3. It is assumed that Markov chain r(t) takes values in S = {1, 2} with generator ! −1 1 Γ = (ri j )2×2 = . 2 −2 Moreover, a coupled network is given in Figure 6. Meanwhile, the coupling weights are as follows: a411 (1) = 0.2 × 10−3 , a311 (1) = 1.5 × 10−3 , a241 (1) = 1.3 × 10−3 , a122 (1) = 1.6 × 10−3 , a342 (1) = 0.4 × 10−3 ,

a121 (2) = 1.5 × 10−3 , a241 (2) = 0.5 × 10−3 , a132 (2) = 0.7 × 10−3 , a342 (2) = 0.8 × 10−3 ,

The computer simulation of the Markov chain r(t) is shown in Figure 7. The sample path figures of systems (33)

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and (34) are shown in Figures 8 and 9. Meanwhile, the sample path figure of error system (35) is shown in Figure 10. From the simulation results, it can be seen that systems (33) and (34) achieve finite-time synchronization. Conclusions This paper discussed finite-time synchronization of stochastic multi-links dynamical networks with Markovian switching topologies. Based on quantized aperiodically intermittent control, Kirchhoff’s Matrix Tree Theorem and a new differential inequality, a synchronization criterion is derived to ensure finite-time synchronization of SMDNM within a finite time which is closely related to the topological structure of the network. Moreover, we apply theoretical results to oscillators systems and second-order Kuramoto model. Meanwhile, a synchronization criterion is presented. In the future, we consider the effect of time delay on the finite-time synchronization of stochastic multi-links dynamical networks with Markovian switching topologies. 20

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Figure 8: The sample path figure of system (33) when N = 4.

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Figure 10: The sample path figure of error system (35) when N = 4.

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Statement of Contribution The finite-time synchronization of stochastic multi-links dynamical networks with Markovian switching topologies via intermittent control is investigated in this paper. Topological structure of multi-links complex networks is Markovian switching and each switching subnetwork is not required to contain a directed spanning tree or to be strongly connected. From the applications? point of view, to illustrate this view, we apply our novel approach to research stochastic coupled oscillators. Finally, we provide a numerical example to demonstrate the effectiveness and feasibility of the theoretical results. Acknowledgements The authors really appreciate the valuable comments of the editors and reviewers. This work was supported by grants from Natural Science Foundation of Shandong Province of China (Nos. ZR2017MA008, ZR2017BA007, ZR2018MA005, ZR2018MA020 and ZR2018MA023), Project of Shandong Province Higher Educational Science and Technology Program of China ( No. J18KA218 and J16LI09), Innovation Technology Funding Project in Harbin Institute of Technology (No. HIT.NSRIF.201703), Key Project of Science and Technology of Weihai (No. 2014DXGJMS08). References References [1] MY. Li, Z. Shuai, Global-stability problem for coupled systems of differential equations on networks. J Differ Equ., 2010;248;1-20. [2] C. Sowmiya, R. Raja, Q. Zhu, G. Rajchakit, Further mean-square asymptotic stability of impulsive discrete-time stochastic BAM neural networks with Markovian jumping and multiple time-varying delays. J. Frankl Inst.-Eng. Appl. Math., 2019;356(1); 561-591. [3] M. Liu, X. He, J. Yu, Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays. Nonlinear Anal.-Hybrid Syst., 2018;28;87-104. [4] Y. Guo, W. Zhao, X. Ding, Input-to-state stability for stochastic multi-group models with multi-dispersal and time-varying delay, Appl. Math. Comput., 2018;290;507-520. [5] X. Yang, J. Lu, Finite-time synchronization of coupled networks with Markovian topology and impulsive effects. IEEE Trans. Autom. Control, 2016;61(8);2256-2261. [6] S. Li, H. Su, X. Ding, Synchronized stationary distribution of hybrid stochastic coupled systems with applications to coupled oscillators and a Chua’s circuits network, J. Frankl Inst.-Eng. Appl. Math., 2018;355; 8743-8765. [7] H. Shen, J. Park, Z. Wu, Finite-time synchronization control for uncertain Markov jump neural networks with input constraints. Nonlinear Dyn., 2014;77(4);1709-1720. [8] R. Manivannan, R. Samidurai, Q. Zhu, Further improved results on stability and dissipativity analysis of static impulsive neural networks with interval time-varying delays. J. Frankl Inst.-Eng. Appl. Math., 2017;354(14);6312-6340. 22

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