Synchronization of multi-stochastic-link complex networks via aperiodically intermittent control with two different switched periods

Physica A 509 (2018) 20–38

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Physica A journal homepage: www.elsevier.com/locate/physa

Synchronization of multi-stochastic-link complex networks via aperiodically intermittent control with two different switched periods✩ ∗

Mengzhuo Luo a,b,c , , Xinzhi Liu b , Shouming Zhong d , Jun Cheng e,f a

College of Science, Guilin University of Technology, Guilin, Guangxi, 541004, PR China Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 c Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin, Guangxi, 541004, PR China d School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, PR China e School of Science, Hubei University for Nationalities, Enshi, Hubei, 445000, PR China f College of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao, Shandong, 266061, PR China b

highlights • • • •

We model a general multi-stochastic-link complex networks with mixed time delays. Two aperiodically intermittent pinning control schemes are proposed. Control width and rates of control duration may be different in each switched period. Sufficient conditions are established based on Lyapunov stability theory.

article

info

Article history: Received 2 February 2018 Received in revised form 2 May 2018 Available online xxxx Keywords: Multi-stochastic-link complex networks Intermittent pinning control Two different switched periods Mixed delayed Exponential synchronization

a b s t r a c t This paper investigates the exponential synchronization problem for a class of multistochastic-link complex networks via novel aperiodically intermittent control approaches with two different switched periods, i.e. the control width and rates of control duration are different in each switched period. Firstly, we consider a general multi-stochastic-link model, it means that there is more than one edge between two nodes and the links among the nodes are perturbed by stochastic noises. The delay terms comprise both discrete and distributed delays. Furthermore, the multi-stochastic-link complex networks can be thought of consisting of many stochastic sub-networks with different time delays. Two new aperiodically intermittent control schemes are developed by virtue of the Lyapunov stability theory and pinning intermittent control techniques. Several novel and useful synchronization criteria are obtained, which guarantee global exponential synchronization of multi-stochastic-link complex networks in the mean square. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method. © 2018 Elsevier B.V. All rights reserved.

✩ This work was supported in part by the National Natural Science Foundation of China under Grants 11661028, 11661030, 61673308, 11502057, the Natural Science Foundation of Guangxi, PR China under Grant 2015GXNSFBA139005 and NSERC Canada. All of authors declare that there is no conflict of interest regarding the publication of this paper. ∗ Corresponding author at: College of Science, Guilin University of Technology, Guilin, Guangxi, 541004, PR China. E-mail addresses: [email protected] (M. Luo), [email protected] (X. Liu), [email protected] (S. Zhong), [email protected] (J. Cheng). https://doi.org/10.1016/j.physa.2018.05.145 0378-4371/© 2018 Elsevier B.V. All rights reserved.

M. Luo et al. / Physica A 509 (2018) 20–38

21

1. Introduction Nowadays, complex networks may be found everywhere in daily life, many systems in science and technology can be modeled as complex networks [1,2]. Therefore, in the past few years, there has been a substantial growth of research interest in the study of complex networks in various fields, which include mathematics, biology, physics, sociology etc. [3,4]. In general, a complex network usually consists of many sets of coupled interconnected nodes, and each node is a basic unit with specific contents and exhibiting dynamical behavior. Hence, the complexity nature of complex networks raises a series of important research problems. In particular, the synchronization phenomenon in complex dynamical networks occurs when all nodes have the same dynamical behavior under local protocols between neighbors of each node, which has become a hot topic in various fields [5–8]. Based on the requirements of practical problems, many synchronization patterns have been analyzed, such as complete synchronization [9,10], cluster synchronization [11], finite-time synchronization [12], etc.; on the other hand, synchronization phenomena has attracted widely attention in many hot research fields such as secure communication [13], image processing [14], and chemical and biological systems [15]. However, to the best of our knowledge, the most of existing results about synchronization problem of complex networks are based on two assumptions: one is that the edge between two nodes is only one; other assumption is that the links between nodes are deterministic or fixed. Unfortunately, in real world, such as communication networks, social networks and transport networks, etc. are not single link complex networks but are multi-link networks [16–18]. Multi-link means that there are more than one link between two nodes and each of the link has its own property, meantime, compared with ones with single-link, networks with multi-link are of the advantages of higher communication speed, lower communication cost, etc. Therefore, the research on the synchronization of networks with multi-link has important practice value. Furthermore, due to various reasons such as sudden failures, abrupt environmental changes, sensor aging and dynamical changes of the working conditions, the connectivity may be stochastic and time-varying [19]. Obviously, deterministic single link complex networks are a special case of multi-stochastic-link complex networks. However, few researchers focus on the multi-stochastic-link complex networks, and the synchronous analysis of multi-stochastic-link complex networks cannot be dealt with along the lines of aforementioned references. Therefore, it is necessary to further research in complex dynamical networks with multistochastic-link. It is well known that time delays in spreading information are ubiquitous in nature, technology, and society because of the finite speed of signal transmission over the links as well as network congestion [20,21]. The existence of time delays can make systems unstable and degrade performance [22,23]. Considerable attention has been devoted to time-varying delay systems due to their extensive application in practical problems containing circuit theory, complex dynamical networks, automatic control, etc. Generally speaking, there are two kinds of time delays: discrete time delays and distributed delays. While signal propagation is sometimes instantaneous and can be modeled with discrete delays, it may also be distributed during a certain time period so that distributed delays are incorporated into the model [24,25]. As a particular kind of time delays, the distributed delays have also received much research attention [26–28]. As is well known, in the case where the whole networks cannot synchronize by themselves, controllers should be designed and applied to force networks synchronization. Compared with continuous control strategies, discontinuous control strategies are more economical and can better simulate the real world, for example in impulsive control [29], sampledata control [30,31], edge snapping pinning control [32,33] and event-triggered scheduling algorithm [34,35]. Intermittent control is an important kind of discontinuous control in engineering fields, which has the advantage of reducing the amount of information required to be transmitted to achieve synchronization in complex networks. In this type of control strategy, each period usually contains two types of time, one being work time where the controller is activated, and the other one being rest time where the controller is off. Hence, intermittent control combined with pinning control can effectively reduce the control cost and have been extensively investigated by researchers in recent years. In [36], the author deals with the synchronization problem for a delayed complex network with hybrid-coupling via intermittent pinning control; in [37], the cluster synchronization problem of linearly coupled complex networks via aperiodically intermittent control is investigated; in [38], authors study the cluster synchronization of coupled genetic regulatory networks with time-varying delays via aperiodically adaptive intermittent control on some nodes, and two cases of delays are considered; in [39], authors investigate the exponential synchronization problem for linearly coupled networks with delay by pinning a simple aperiodically intermittent controller. It should be noted that in traditional intermittent control schemes, each control interval includes only one switched period, and each switched period possesses the same time width [40] and same rates of control duration [41]. This may be unreasonable and unavoidably restricts its scope of practical applications. Hence, aperiodically intermittent control with two switched periods usually uses nonidentical control width and rates of control duration, it may eliminate the imposed restriction on different switched periods and gives more flexibility to the designer. Therefore, base on real application and theoretical analysis, it is necessary to consider the synchronization problem for multi-stochastic-link complex networks under aperiodically intermittent pinning control with two different switched periods. Unfortunately, to the best of our knowledge, the results about aperiodically intermittent pinning control with two switched periods for multi-stochastic-link complex networks, especially the case of completely aperiodically has not yet appeared. Motivated by the above discussion, this paper aims to investigate the synchronization problem for a class of multistochastic-link complex networks via aperiodically intermittent pinning control with two different switched periods. The contribution of this paper can be summarized as follows: (1) The problem of synchronization for a class of generalized

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M. Luo et al. / Physica A 509 (2018) 20–38

multi-stochastic-link complex dynamical networks are investigated via aperiodically intermittent pinning control with two switched periods, and some novel results are developed which guarantee globally exponential synchronization in the mean square; (2) Compared with the periodically intermittent control studied in previous works, the proposed intermittent pinning control is completely aperiodic, it means that the control periods, rates of control duration, and control width in each switched period are allowed to be nonidentical; (3) By introduced an important Lemma 2.3 and some novel algorithms in this paper, we develop two different techniques to study the problem of synchronization, and some strict assumptions between the time delay, control width and non-control width in the original references have been removed. So, the conservativeness of our main conclusions are greatly improved. The organization of the remainder of this paper is described as follows. In Section 2, we give a brief account of the model and some mathematical preliminaries for subsequent use. In Section 3, we establish synchronization criteria for the proposed models with mixed delays and stochastic disturbance. We demonstrate the effectiveness of the theoretical results with two numerical examples in Section 4. Conclusions are finally drawn in Section 5. Notations: Throughout this letter Rn and Rn×m denote, respectively, the n-dimensioned Euclidean space and the set of all n × m real matrix. The notation X ≥ Y . (respectiveX > Y ) means that X and Y are symmetric matrices, and that X − Y is positive semi-definitive (respective positive definite). X + X T is denoted as He (X ) for simplicity. In is the n × n identity matrix. ∥·∥ is the Euclidean norm in (Rn . If A is a matrix, ) λmax (A) (respective λmin (A)) means the largest (respective smallest) eigenvalue of A. Moreover, let Ω , F , (Ft )t ≥0 , P be a complete probability space with a filtration. (Ft )t ≥0 satisfies the usual conditions (i.e, the filtration contains all P-null sets and is right continuous). E {·} stands for the mathematical expectation operator with respect to the given probability measure. Denote by L2F0 ([−τ , 0] : Rn ) the family of all F0 measurable C ([−τ , 0] : Rn )-valued random variables ϕ = {ϕ (s) : −τ ≤ s ≤ 0} such that sup−τ ≤s≤0 E ∥ϕ (s)∥2 < ∞. The asterisk * in a matrix is used to denote term that is induced by symmetry. Matrices, if not explicitly specified, are assumed to have appropriate dimensions. Sometimes, the arguments of function will be omitted in the analysis when no confusion can be arised. 2. Problem formulation and preliminaries Consider the following complex dynamical networks consisting of N identical notes with mixed time-varying delays:

⎧ m−1 N ∑ ∑ ) ( ⎪ dxi (t ) ⎪ ⎪ = f x t + C a¯ zij xj (t − τz (t )) − xi (t − τz (t )) ( ( )) ⎪ i ⎪ dt ⎪ ⎨ z =0 j=1,j̸ =i ∫ t ⎪ ⎪ + h (xi (s)) ds ⎪ ⎪ ⎪ t −d(t ) ⎪ ⎩ xi (t0 + s) = ϕi (s) s ∈ [t0 − τ , t0 ] , i = 1, 2, . . . , N

(1)

n n n where xi (t ) = (xi1 (t ) , xi2 (t ) , . . . , xin (t ))T (∈ R ) is the state vector of the ith node, f (·) , h (·) : R → R arez continuous denotes zth sub-network’s topological structure with a¯ ij = a¯ zji > nonlinear convex vector function. A¯ z = a¯ zij N ×N 0 (i ̸ = j) , z = 0, 1, . . . , m − 1 if there is a link between nodes i and j at time t, otherwise, a¯ zij = a¯ zji = 0. Constant C > 0 represents the coupling strength, while τz (t ) , d (t ) are the discrete-time and distributed time-varying delays respectively. We assume that τ0 (t ) = 0, 0 ≤ τz (t ) ≤ τz (z = 1, 2, . . . , m − 1) , 0 ≤ d (t ) ≤ d and τ = maxz {τz , d}. ϕi (·) denotes the initial condition of networks (1) for t ∈ [t0 − τ , t0 ]. To establish the main result of this paper, the following assumptions and definition are necessary.

Assumption 2.1. There exist positive constants Γ1 , Γ2 such that f (·) and h (·) in complex networks (1) satisfy the following Lipschitz conditions for any x, y ∈ Rn :

∥f (x) − f (y)∥ ≤ Γ1 ∥x − y∥

∥h (x) − h (y)∥ ≤ Γ2 ∥x − y∥

(2)

Remark 2.1. statement is also true: There exist matrices ( Notice that f (·) ) satisfies Assumption ( 2.1, then the following ) Q = diag q1 q2 · · · qn and 0 < H = diag h1 h2 · · · hn such that

(y − x)T H [f (y) − f (x) − Q (y − x)] ≤ −κ (y − x)T (y − x)

(3)

holds for some 0 < κ ∈ R and any x, y ∈ R . n

Assumption 2.2. In this paper, we assume that each entry a¯ zij of matrix A¯ z changes over time t in a stochastic way, i.e. matrix ( ) A¯ z can be expressed as Az (t ) = azij (t ) , and azij (t ) = azij + szij ω ˙ (t ) where

azij

is the mean of

azij

(t ),

(4) szij

is the standard deviation and ω (t ) is a Brownian motion.

M. Luo et al. / Physica A 509 (2018) 20–38

23

) ∑N z Let Lz (t ) = lij (t ) be the Laplacian matrix of Az (t ) with lzij (t ) = −azij (t ) if i ̸ = j, and lzii (t ) = j=1,j̸ =i aij (t ). Denote N × N ( ) ( ) ∑ N z z z Lz = lzij , where lzij = −azij if i ̸ = j and lzii = j=1,j̸ =i aij . Obviously, matrix L = lij N ×N is the Laplacian matrix of ( z ) N ×N ( z) ∑N z z z z A = aij . Let S = sij with sii = − j=1,j̸=i sij . Then, we can obtain that N ×N N ×N (z

˙ (t ) Lz (t ) = Lz + S z ω

(5)

Now the networks (1) can be rewritten into following multi-stochastic-link model form:

⎤ ⎡ ⎧ ∫ t m−1 N N ∑ ∑ ∑ ⎪ ⎪ z 0 ⎪ h (xi (s)) ds⎦ dt lij xj (t − τz (t )) + lij xj (t ) − C dxi (t ) = ⎣f (xi (t )) − C ⎪ ⎪ ⎪ t −d(t ) ⎪ z =1 j=1 j=1 ⎪ ⎨ ⎡ ⎤ m−1 N ∑ ∑ ⎪ ⎪ + ⎣−C szij xj (t − τz (t ))⎦ dω (t ) ⎪ ⎪ ⎪ ⎪ z =0 j=1 ⎪ ⎪ ⎩ xi (t0 + s) = ϕi (s) s ∈ [t0 − τ , t0 ] , i = 1, 2, . . . , N

(6)

Remark 2.2. As is well known, in the real world, many complex networks with multi-link, such as transportation networks, social networks and communication networks, etc. Hence, in this paper, the synchronization problem for a class of multilink complex networks with mixed delays (1) is investigated. However, the connection between nodes might involve with random behaviors due to various reasons, based on this consideration, in Ref. [19], authors discuss the synchronization problem of switched complex networks with stochastic link perturbations for the first time. Hence, based on the academic reasons and practical import in applications, a much more novel stochastic system model, such as multi-stochastic-link model, is needed to be further studied. Definition 2.1 ([19]). A network with N nodes, whose states are denoted xi (t ), i = 1, 2, . . . , N is said to be exponentially synchronized to be trajectory s (t ) in the mean square if there exist scalar constants α > 0 and M > 0 such that lim E ∥xi (t ) − s (t )∥2 ≤ M e−α t

{

}

t →+∞

i = 1, 2, . . . , N

∥ϕi (t ) − s (t )∥

sup t0 −τ ≤t ≤t0

∑N

Let s (t ) = N1 i=1 xi (t ) be the isolated node of networks (1) and d (s (t )) = matrices Lz and S z , we can have

⎡ 1 d (s (t )) = ⎣

N

+

[ =

N ∑

f (xi (t )) −

i=1

N ∫ 1 ∑

N

N 1 ∑

N

N

C

⎤

N ∑

N ∫ 1 ∑

⎡

N

i=1

t t −d(t )

∑N

i=1

d (xi (t )). Based on the definition of

) lzij

xj (t − τz (t ))

i=1

h (xi (s)) ds⎦ dt + ⎣−

f (xi (t )) +

i=1

(

z =0 j=1

t t −d(t )

i=1

m−1 N 1 ∑∑

1 N

(7)

m−1 N 1 ∑∑

N

z =0 j=1

( C

N ∑

) szij

⎤ xj (t − τz (t ))⎦ dω (t )

i=1

] h (xi (s)) ds dt

(8)

Next, we will introduce the aperiodically intermittent control scheme with two different switched periods in m∗ th control [ ]

∑m∗ −1 ∑m∗ −1 ∗ Ti , i=1 Ti + θm∗ 1 Tm∗ 1 is the control width in switched period Tm∗ 1 , E2m = i=1 ] [ ∑m∗ −1 ∑m∗ −1 ∗ m∗ −1 Ti + θm∗ 1 Tm∗ 1 , i=1 Ti + Tm∗ 1 is the non-control width in switched period Tm∗ 1 , E3m = Ti + Tm∗ 1 , i=1 i=1 ] [ ∑m∗ −1 ∑m∗ −1 m∗ Ti + Tm∗ 1 + θm∗ 2 Tm∗ 2 is the control width in switched period Tm∗ 2 , and E4 = Ti + Tm∗ 1 + θm∗ 2 Tm∗ 2 , i=1 i=1 ∑m∗ ] ∗ ∗ i=1 Ti is the non-control width in switched period Tm 2 . Particularly, if m = 1, let T0 = 0. As shown in Fig. 1, Tm∗ 1 and Tm∗ 2 are two switched appearing alternately in the m∗ th control period, if Tm∗ 1 > 0, then Tm∗ = Tm∗ 1 + Tm∗ 2 , θm∗ 1 (0 < θm∗ 1 < 1) and θm∗ 2 (0 < θm∗ 2 < 1) are called the rates of control duration in m∗ th control period, hence, θm∗ 1 Tm∗ 1 and θm∗ 2 Tm∗ 2 are called the control width in switched periods Tm∗ 1 and Tm∗ 2 . Similarly, (1 − θm∗ 1 ) Tm∗ 1 and (1 − θm∗ 2 ) Tm∗ 2 are called non-control width in switched periods Tm∗ 1 and Tm∗ 2 , respectively. ∗

period. We note that E1m

=

[∑

Remark 2.3. In this paper, the rates of control duration may be different, i.e θm∗ 1 ̸ = θm∗ 2 for any m∗ ∈ N+ . Without loss of generality, we also assume that the intermittent control scheme is aperiodically, it means that Tm∗ is different for any m∗ ∈ N+ , while θm∗ 1 , θm∗ 2 and Tm∗ are assumed to be the same in [41,42], so, our results are much more general than [41,42]. To reduce the number of controllers and save control cost effectively, pinning intermittent control scheme is introduced. In this paper, we assume that only the first node is pinned with a constant gain, which is defined as follows for the

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M. Luo et al. / Physica A 509 (2018) 20–38

Fig. 1. An intermittent control scheme with two different switched periods.

Fig. 2. Random disturb signal ω (t ).

Fig. 3. The chaotic attractor of the complex network (40) with initial φ = (0.01, 0.55)T , T11 = T12 = 2.2 s.

networks (1): ui (t ) =

{ −∆ (xi (t ) − s (t )) 0

∗

∗

t ∈ E1m ∪ E3m ∗ ∗ t ∈ E2m ∪ E4m

(9)

M. Luo et al. / Physica A 509 (2018) 20–38

25

Fig. 4. The synchronization errors curve with C = 1 and multi-link index m = 3 for Example 1.

Fig. 5. Parameter feasible field.

Fig. 6. The state trajectories of errors systems for Example 1.

where ∆ = diag g 0 behaviors of ei (t ) are

(

···

)

0

N ×N

, g > 0. For any node i = 1, 2, . . . , N, denote ei (t ) = xi (t ) − s (t ), then the dynamical

26

M. Luo et al. / Physica A 509 (2018) 20–38

Fig. 7. The synchronization errors curve with C = 1 and multi-link index m = 3 for Example 2.

Fig. 8. The state trajectories of errors systems for Example 2.

∗

∗

if t ∈ E1m ∪ E3m

⎡ dei (t ) = ⎣f (xi (t )) +

t

∫

t −d(t )

N

−

∗

1 ∑ N

f (xi (t )) −

i=1

h (xi (s)) ds + C

N ∑

˜¯l e (t ) − C ij j

1 N

∑∫ i=1

⎤

t t −d(t )

lzij ej (t − τz (t ))

z =1 j=1

j=1 N

m−1 N ∑ ∑

⎡

h (xi (s)) ds⎦ dt + ⎣−C

m−1 N ∑ ∑

⎤ szij ej (t − τz (t ))⎦ dω (t )

(10)

z =0 j=1

∗

if t ∈ E2m ∪ E4m

⎡ dei (t ) = ⎣f (xi (t )) +

∫

t

t −d(t )

N

−

1 ∑ N

i=1

f (xi (t )) −

h (xi (s)) ds + C

N ∑

¯lij ej (t ) − C

j=1

1 N

N

∑∫ i=1

t t −d(t )

m−1 N ∑ ∑

lzij ej (t − τz (t ))

z =1 j=1

⎤

⎡

h (xi (s)) ds⎦ dt + ⎣−C

m−1 N ∑ ∑ z =0 j=1

⎤ szij ej

(t − τz (t ))⎦ dω (t )

(11)

M. Luo et al. / Physica A 509 (2018) 20–38

( )

where L˜¯ = ˜¯lij

27

{ 0 ( ) −lii − g i = j = 1 , L¯ = ¯lij N ×N = −L0 . = 0 −lij other

N ×N

Before proceeding further, we will introduce the following lemmas which will be played an important role in the derivation of our main results. Lemma 2.1 ([43]). For any constant matrix M > 0, any scalars a and b with a < b, and a vector function x (t ) : [a, b] → Rn such that the integrals concerned are well defined, then the following inequality holds:

]T

b

[∫

b

[∫

x (s) ds

]

x (s) ds ≤ (b − a)

M a

a

b

∫

xT (s) Mx (s) ds

(12)

a

Lemma 2.2 ([44]). Suppose a N × N matrix A satisfies aij ≥ 0, i ̸ = j and aii = −

)T

∑N

j=1,j̸ =i T

aij . For the new matrix A˜ = A − ∆,

there exists a vector ξ = ξ1 ξ2 · · · ξN , ξi > 0 such that ξ A = 0. Then Ξ A˜ + A˜ Ξ < 0, where Ξ = diag (ξ ). In the following, for convenience of calculation, we assume that max {ξi } = 1.

(

∗

∗

∗

∗

T

Lemma 2.3. Let W1m (t ), W2m (t ), W3m (t ), W4m (t ) be piecewise functions as defined by the following: ∗ W1m

[

m∗ −1

(t ) = h exp γ (1 − θm∗ 1 )

∑

] ∗

t ∈ E1m

Ti

i=1

(m∗ −1 ∑

[ ( ∗ W2m

(t ) = h exp γ

t − θm∗ 1

))] ∗

t ∈ E2m

Ti + Tm∗ 1

i=1

(m∗ −1 ∑

[ ∗ W3m

(t ) = h exp γ (1 − θm∗ 2 )

)] ∗

t ∈ E3m

Ti + Tm∗ 1

i=1 m∗

[ ( ∗ W4m

(t ) = h exp γ

t − θm∗ 2

∑

)] t ∈ E4m

Ti

∗

i=1 ∗

∗

∗

where h > 1, γ > 0, θ11 ≥ θ12 ≥ θ21 ≥ θ22 ≥ · · · ≥ θm∗ 1 ≥ θm∗ 2 ≥ · · ·, and let θ = minm∗ ∈N+ θm∗ ,1 , θm∗ ,2 . Ti , E1m , E2m , E3m , ∗ E4m are defined as above. Then

}

{

∗

· · · ≤ W1m

(

∗

t1m

)

∗

≤ W2m

(

∗

t2m

)

∗

< W3m

(

∗

t3m

)

∗

≤ W4m

(

∗

t4m

)

∗ +1

< W1m

(

∗ +1

t1m

)

≤ ···

i.e. the piecewise functions are increasing functions. ∗

∗

∗

Proof. For any t2m ∈ E2m , we can have t2m ≥ ∗ t2m

− θm∗ 1

(m∗ −1 ∑

≥

i=1

⇒

∗ W2m

(

∗ t2m

)

i=1

Ti + θm∗ 1 Tm∗ 1 , which implies that

m∗ −1

) Ti + Tm∗ 1

∑m∗ −1

∑

Ti + θm∗ 1 Tm∗ 1 − θm∗ 1

(m∗ −1 ∑

i=1

= h exp γ

Ti + Tm∗ 1

= (1 − θm∗ 1 )

i=1

[ ( ∗ t2m

m∗ −1

)

− θm∗ 1

(m∗ −1 ∑

Ti

i=1

))]

m∗ −1

[ ≥ h exp γ (1 − θm∗ 1 )

Ti + Tm∗ 1

∑

∑

i=1

] Ti

∗

= W1m

i=1

m∗ −1 ∗

t2m <

∵

∑

Ti + Tm∗ 1

i=1 ∗ t2m

⇒

− θm∗ 1

(m∗ −1 ∑

) Ti + Tm∗ 1

i=1

< (1 − θm∗ 2 )

(m∗ −1 ∑

) Ti + Tm∗ 1

i=1

[ (∑ ∗ )] ∗ ( ∗) m −1 < h exp γ (1 − θm∗ 2 ) Ti + Tm∗ 1 = W3m t3m i=1 ∑m∗ −1 ∗ Next, for any t4m ≥ Ti + Tm∗ 1 + θm∗ 2 Tm∗ 2 i=1 ∗

then, W2m

(

∗

t2m

)

∗

⇒

∗ t4m

− θm∗ 2

m ∑ i=1

m∗ −1

Ti ≥

∑ i=1

∗

Ti + Tm∗ 1 + θm∗ 2 Tm∗ 2 − θm∗ 2

m ∑ i=1

Ti = (1 − θm∗ 2 )

(m∗ −1 ∑ i=1

) Ti + Tm∗ 1

(

∗

t1m

)

28

M. Luo et al. / Physica A 509 (2018) 20–38

∗ W4m

⇒

(

∗ t4m

)

∗

[ ( ∗ t4m

= h exp γ

∗

∗

∵

t4m <

m ∑

− θm∗ 2

∗

Ti ⇒ t4m − θm∗ 2

i=1

∗

≥ W3m

Ti

i=1 ∗

∗

m ∑

)]

m ∑

Ti <

i=1

m ∑

(

∗

t3m

)

∗

Ti − θ(m∗ +1)1

i=1

m ∑

Ti

i=1

[ ( ( ∗ ) [ ( ∗ ) ∑m∗ ] ∑ ∗ )] m∗ +1 < h exp γ 1 − θ(m∗ +1)1 t m +1 . = h exp γ t4m − θm∗ 2 m i=1 Ti i=1 Ti = W1 ( ∗ ) ( ∗ ) ( ∗ ) (1 ∗ ) ∗ ∗ ∗ ∗ By the similar method, we can obtain that W1m +1 t1m +1 ≤ W2m +1 t2m +1 < W3m +1 t3m +1 ≤ W4m +1 t4m +1 . Hence, for the (m∗ + 1) th control period, the statement is also true. The proof is complete. □ ∗

Then, one has that W4m

(

∗

t4m

)

3. Main results In this section, we shall analyze the issue of synchronization for the multi-stochastic-link complex networks (6) and the isolated node s (t ) under the aperiodically pinning intermittent controller (9) with two different switched periods. Then, we will derive the synchronization criteria for the controlled networks in the following two cases. case I: τ0 (t ) = 0, 0 ≤ τz (t ) ≤ τz (z = 1, 2, . . . , m − 1) , 0 ≤ d (t ) ≤ d and τ = maxz {τz , d}, θ11 ≥ θ12 ≥ θ21 ≥ θ22 ≥

· · · ≥ θm∗ 1 ≥ θm∗ 2 ≥ · · · case II: τ0 (t ) = 0, 0 ≤ τz (t ) ≤ τz (z = 1, 2, . . . , m − 1) , 0 ≤ d (t ) ≤ d and τ = maxz {τz , d}, 0 ≤ τ˙z (t ) ≤ τ˙z < 1, where τ˙z > 0 Theorem 3.1. Suppose that Assumptions 2.1 and 2.2 hold, for the positive vector ξ = ξ1 We denote that

ξ2

(

((

υ1 = −2C λmax

Ξ L˜¯

)s )

− υ3 > 0

υ2z =

C 2 (λmax S z )2 + C min {ξi }

···

ξN

)T

defined in Lemma 2.2.

(z = 1, 2, . . . , m − 1)

( ) ( z 2) ∑ −1 λmax Ξ Q + QT Ξ + Ξ 2 + C 2 S 0 Ξ S 0 + C m − 2κ z =1 λmax (L ) >0 min {ξi } (( )s ) γ = υ1 + υ3 = −2C λmax Ξ L˜¯ >0 υ3 =

υ4 =

Γ22 min {ξi }

and for positive vectors υ1 , υ2 , υ3 , υ4 , γ such that the following conditions hold

υ1 >

m−1 ∑

υ2z + υ4 d

ϑ − γ (1 − θ) > 0

(13)

z =1

υ

υ

where ϑ > 0 is the unique solution of equation ϑ − υ1 + z =1 υ2z eϑτz + ϑ4 eϑ d − ϑ4 = 0, then multi-stochastic-link complex networks with isolated node s (t ) can achieve globally exponential synchronization in the mean square under the case I.

∑m−1

Proof. Choose the following Lyapunov functional candidate for synchronization error system (10) and (11) as V1 (t ) = eT (t ) (Ξ ⊗ In ) e (t ) =

N ∑

ξi eTi (t ) ei (t )

(14)

i=1

)T

where e (t ) = eT1 (t ) eT2 (t ) · · · eTN (t ) Let L be the weak infinitesimal generator of V (t ) along the trajectory of the error system (10). By the Itô’s differential ∗ ∗ formula, one has that, for t ∈ E1m ∪ E3m , m∗ = 1, 2, . . .

(

L V1 (t ) = 2e (t ) (Ξ ⊗ In ) T

−C

m−1 ∑ (

(

F (x (t )) +

∫

t t −d(t )

L ⊗ In e (t − τz (t )) − z

)

z =1

+ C2

m−1 ∑

(

(

)

H (x (s)) ds + C L˜¯ ⊗ In e (t )

1 N

1TN

)

⊗ In F (x (t )) −

(

1 N

1TN

⊗ In

)

t

)∫

t −d(t )

H (x (s)) ds

eT (t − τz (t )) S z Ξ S z ⊗ In e (t − τz (t ))

(

)

(15)

z =0

where F (x (t )) =

[

1N = 1



1

···  N

( T f (x (t )) ]T 1

1



f T (x2 (t ))

···

)T

f T (xN (t ))

H (x (t )) =

(

hT (x1 (t ))

hT (x2 (t ))

···

)T

hT (xN (t )) ,

M. Luo et al. / Physica A 509 (2018) 20–38

29

From the Assumption 2.1, it can be seen that

(

(

2e (t ) (Ξ ⊗ In ) F (x (t )) − T

=2

N ∑

1

1TN

N

)

⊗ In F (x (t ))

)

( ξ

T i ei

N 1 ∑

(t ) f (xi (t )) − f (s (t )) + f (s (t )) −

N

i=1

≤2

N ∑

ξi eTi (t ) Qei (t ) − 2κ

N ∑

i=1

2e (t ) (Ξ ⊗ In ) N ∑

f (xi (t ))

i=1

eTi (t ) ei (t )

i=1

(∫

T

≤

)

t t −d(t )

ξi2 eTi (t ) ei (t ) +

H (x (s)) ds −

N ∫ ∑ i=1

i=1

(

1 N

1TN

t

)∫ ⊗ In

)

t −d(t )

t

(h (xi (θ)) − h (s (θ)))T dθ

H (x (s)) ds t

∫

(h (xi (θ)) − h (s (θ))) dθ t −d

t −d

∫ t ∫ t ( ) ( ) Γ22 eT (s) e (s) ds ≤ eT (t ) Ξ 2 ⊗ In e (t ) + ≤ eT (t ) Ξ 2 ⊗ In e (t ) + Γ22 V1 (s) ds min {ξi } t −d(t ) t −d(t ) ( ( )) (( )s ) (( )s ) 2eT (t ) (Ξ ⊗ In ) C L˜¯ ⊗ In e (t ) ≤ 2CeT (t ) Ξ L˜¯ ⊗ In e (t ) ≤ 2C λmax Ξ L˜¯ V1 (t ) ( ) m−1 m−1 ∑ ∑ ( z ) (( ) ) 2eT (t ) (Ξ ⊗ In ) −C L ⊗ In e (t − τz (t )) = 2C eT (t ) −Ξ Lz ⊗ In e (t − τz (t )) z =1

≤C

m−1 ∑

λmax

z =1

(( ) ) z 2

eT (t ) e (t ) + C

L

z =1

m−1 ∑

eT (t − τz (t )) e (t − τz (t ))

z =1

Then, LV1 (t ) ≤ −υ1 V1 (t ) +

m−1 ∑

υ2z V1 (t − τz (t )) + υ4

t

∫

t −d(t )

z =1

V1 (s) ds

∗

∗

t ∈ E1m ∪ E3m

(16)

Using the similar method, we can seen that the following inequality is true based on the fact that λmax L V1 (t ) ≤ υ3 V1 (t ) +

m−1 ∑

υ2z V1 (t − τz (t )) + υ4

z =1

∫

t

t −d(t )

V1 (s) ds

∗

(( ) ) s Ξ L¯ = 0:

∗

t ∈ E2m ∪ E4m

(17)

Next, we will show that error system (10) and (11) are global exponential synchronization in the mean square. Let t0 = 0, M = sup−τ ≤s≤0 V1 (s), W (t ) = eϑ t V1 (t ) and we denote Q11 (t ) = W (t ) − hM. Obviously, it has that for any 1 1 t ∈ [−τ , 0], inequality( Q)11 (t ) < 0 (is true. In the next step, we will[ show that ) ) Q1 (t ) < 0 also holds for t ∈ E1 , otherwise, 1 1 1 1 1 1 1 1 ˙ ∃t1 ∈ E1 such that Q1 t1 = 0, Q1 t1 ≥ 0 and Q1 (t ) < 0 for t ∈ −τ , t1 .

˙ 11 t11 = ϑ eϑ t1 V1 t11 + eϑ t1 L V1 t11 Q 1

( )

( )

1

(

( ))

( ≤ ϑ W t1 + e

( 1)

ϑ t11

−υ1 V1 t1 + υ

( 1)

1 2 V1

t11

(

− τ1 t1

( 1 ))

+ ··· + υ

m−1 V1 2

t11

(

− τm−1 t1

( 1 ))

+ υ4

∫

)

t11

( )

t11 −d t11

V1 (s) ds

( υ4 ) ( 1 ) υ4 ≤ ϑ − υ1 + υ21 eϑτ1 + · · · + υ2m−1 eϑτm−1 + eϑ d − W t1 = 0 ϑ ϑ ( ) This leads to a contradiction with Q˙ 11 t11 ≥ 0, hence, Q11 (t ) < 0 is true for t ∈ E11 . 1 1 Now, based on the (Lemma 2.3, we ) ( )will prove that Q2 (t ) = W ([t ) − hM ) exp [γ (t − θ11 T11 )] < 0 for t ∈ E2 . Otherwise, ∃t21 ∈ E21 such that Q12 t21 = 0, Q˙ 12 t21 ≥ 0 and Q12 (t ) < 0 for t ∈ −τ , t21 ( ( )) ( ) ( ) ( ) 1 1 ˙ 12 t21 = ϑ eϑ t2 V1 t21 + eϑ t2 L V1 t21 − γ W t21 Q ( ( ) ( ) ( ( )) ( ( )) 1 ≤ ϑ W t21 + eϑ t2 υ3 V1 t21 + υ21 V1 t21 − τ1 t21 + · · · + υ2m−1 V1 t21 − τm−1 t21 ∫ +υ4

)

t21

( )

t21 −d t21

V1 (s) ds

( ) − γ W t21

30

M. Luo et al. / Physica A 509 (2018) 20–38

( ) ( ) υ4 υ4 ≤ ϑ + υ3 + υ21 eϑτ1 + · · · + υ2m−1 eϑτm−1 + eϑ d − − γ W t21 = 0 ϑ ϑ ( ) Obviously, this leads to a contradiction with Q˙ 12 t21 ≥ 0, hence, Q12 (t ) < 0 is true for t ∈ E21 . ( ) exp [γ (1 − θ12 ) T11 ] < 0 for t ∈ E31 . Otherwise, ∃t31 ∈ E31 such that Q13 t31 = 0, We will prove that Q13 (t ) = W ([t ) − hM ) ( ) ˙ 13 t31 ≥ 0 and Q13 (t ) < 0 for t ∈ −τ , t31 Q ( ) ( ) ( ( )) 1 1 Q˙ 31 t31 = ϑ eϑ t3 V1 t31 + eϑ t3 L V1 t31 ( ( )) ( ( )) ( ( ) ( 1) ϑ t31 −υ1 V1 t31 + υ21 V1 t31 − τ1 t31 + · · · + υ2m−1 V1 t31 − τm−1 t31 ≤ ϑ W t3 + e +υ4

)

t31

∫

( )

t31 −d t31

V1 (s) ds

( ) ( ) υ4 υ4 ≤ ϑ − υ1 + υ21 eϑτ1 + · · · + υ2m−1 eϑτm−1 + eϑ d − − γ W t31 = 0 ϑ ϑ ( 1) 1 ˙ Hence, this leads to a contradiction with Q3 t3 ≥ 0, so, Q13 (t ) < 0 is true for t ∈ E31 . ( ) Furthermore, we will show that [Q41 (t ) = W (t )− hM exp [γ (t − θ12 T1 )] < 0. Otherwise, ∃t41 ∈ E41 such that Q14 t41 = 0, ) ) ( ˙ 14 t41 ≥ 0 and Q14 (t ) < 0 for t ∈ −τ , t41 Q ( ) ( ) ( ( )) ( ) 1 1 ˙ 14 t41 = ϑ eϑ t4 V1 t41 + eϑ t4 L V1 t41 − γ W t41 Q ( ( )) ( ( )) ( ( 1) ( ) ϑ t41 ≤ ϑ W t4 + e υ3 V1 t41 + υ21 V1 t41 − τ1 t41 + · · · + υ2m−1 V1 t41 − τm−1 t41 +υ4

)

t41

∫

( )

t41 −d t41

( ) − γ W t41

V1 (s) ds

( ) ( ) υ4 υ4 ≤ ϑ + υ3 + υ21 eϑτ1 + · · · + υ2m−1 eϑτm−1 + eϑ d − − γ W t41 = 0 ϑ ϑ ( 1) 1 Obviously, this also leads to a contradiction with Q˙ 4 t4 ≥ 0, hence, it has that Q14 (t ) < 0 is true for t ∈ E41 . Similarly, we can prove the following results are also true in the (m∗ − 1)th control period: [ ∗ −2 ] ( ) m∑ ∗ W (t ) < hM exp γ 1 − θ(m∗ −1),1 Ti t ∈ E1m −1 i=1

[ ( W (t ) < hM exp γ

t − θ(m∗ −1),1

(m∗ −2 ∑

))] ∗ −1

t ∈ E2m

Ti + T(m∗ −1),1

i=1

[ W (t ) < hM exp γ 1 − θ(m∗ −1),2

(

( ∗ −2 ) m∑

)] ∗ −1

t ∈ E3m

Ti + T(m∗ −1),1

i=1 m∗ −1

[ ( W (t ) < hM exp γ

t − θ(m∗ −1),2

∑

)] t ∈ E4m

Ti

∗ −1

(18)

i=1

[

Then, form the Lemma 2.3, for any t ∈ −τ ,

i=1

m∗ −1

[ ( W (t ) < hM exp γ

∑m∗ −1 )

t − θ(m∗ −1),2

∑

Ti , we can have

)]

∗

[ < hM exp γ 1 − θm∗ ,1

(

Ti

i=1

−1 ) m∑

] Ti

i=1

Hence, in the (m∗ ) th control period, we will continue to prove that ∗

[ ∗ Q1m

(t ) = W (t ) − hM exp γ 1 − θm∗ ,1

(

−1 ) m∑

] Ti

<0

∗

t ∈ E1m

i=1 ∗ t1m

Otherwise, ∃ ∗

˙m Q 1

(

∗

t1m

)

∈

∗ E1m ϑ t11

= ϑe

such that

(

∗

V1 t1m

)

∗ ∗ Qm t1m 1 ∗ ϑ t1m

+e

(

[ ) ∗ ( ∗) ∗ m∗ = 0, Q˙ m t m ≥ 0 and Qm 1 (t ) < 0 for t ∈ −τ , t1 ( ( m∗ ))1 1 )

L V1 t1

(19)

M. Luo et al. / Physica A 509 (2018) 20–38

31

≤ ϑ − υ1 + υ21 eϑτ1 + · · · + υ2m−1 eϑτm−1 + υϑ4 eϑ d − ϑ W t1 = 0 ∗ ( m∗ ) ∗ m∗ Hence, this leads to a contradiction with Q˙ m t1 ≥ 0, then, Qm 1 1 (t ) < 0 is true for t ∈ E1 . υ4 )

(

(

) m∗

Based on the similar proof, we can also obtain that

[ ( W (t ) < hM exp γ

t − θm∗ ,1

(m∗ −1 ∑

))] ∗

t ∈ E2m

Ti + Tm∗ ,1

i=1

[ W (t ) < hM exp γ 1 − θm∗ ,2

(

)

(m∗ −1 ∑

)] ∗

t ∈ E3m

Ti + Tm∗ ,1

i=1 ∗

[ ( W (t ) < hM exp γ

t − θm∗ ,2

m ∑

)] ∗

t ∈ E4m

Ti

(20)

i=1 ∗

∗

∗

∗

∑m∗ −1

∑m∗

Combining with above discussion, for any t, ∃m∗ ∈ N+ , one has t ∈ E1m ∪ E2m ∪ E3m ∪ E4m , i.e. i=1 Ti ≤ t < i=1 Ti . ) ) ∑m∗ −1 ( ( ∑m∗ −1 ∗ ∗ ,1 t ≤ (1 − θ) t, then, the following ∗ ,1 T ≤ 1 − θ While t ∈ E1m , it is equivalent to T ≤ t ⇒ 1 − θ i m i m i=1 i=1 inequality is true: W (t ) < hM exp [γ (1 − θ) t] m∗

while t ∈ E2

i.e. t <

∑m∗ −1 i=1

∗

t ∈ E1m

(21)

Ti + Tm∗ ,1 ⇒ t − θm∗ ,1

(∑

m∗ −1 i=1

Ti + Tm∗ ,1

)

∗

< 1 − θm∗ ,1 t ≤ (1 − θ) t, that is , for t ∈ E2m ,

)

(

∗

∗

(21) is also true. In the same way, for t ∈ E3m ∪ E4m , we can deduce that (21) holds for any t ≥ 0, which is equivalent to E {V1 (t )} < hM exp [− (ϑ − γ (1 − θ)) t]

(22)

So, from the conditions of Theorem 3.1 and Definition 2.1, it can be derived that multi-stochastic-link complex networks (6) is exponentially synchronization to the trajectory s (t ) in the mean square. This proof is complete. □ Remark 3.1. In this paper, we discuss the problem of exponentially synchronization for multi-stochastic-link complex networks via aperiodically intermittent pinning control with two different switched periods. Particularly, in order to obtain more general results, we assume the rates of control duration and the control width in each switched period are different. Hence, it has brought great challenges to our research work under these assumptions. Remark 3.2. Obviously, Lemma 2.3 plays a very important role in analyzing our main results, and the monotonicity of the rates of control duration is introduced, which may cause a certain conservatism. But, in order to avoid this restriction, we will continue to develop the following Theorem 3.2 to improve our results based on Case II. Theorem 3.2. Suppose that Assumptions 2.1 and 2.2 hold, if for any given positive constants σ1 and σ2 , then multi-stochastic-link system (6) and isolated node s (t ) are synchronized under the case II if there exist positive-definite matrices Ξ , Qz , Q¯, R , Pz and P¯ , such that for (z = 1, 2, . . . , m − 1)

{ } Γ22 IN < min e−σ1 d Q¯, e−σ2 d P¯ ⎡ Υ1 ⎢ ⎢∗ ⎢ ⎢ ⎢ ⎢∗ ⎢ ⎣ ∗

(( ) ) C − Ξ L1 ⊗ In ( ) C 2 S 1 Ξ S 1 ⊗ In − (1 − τ˙1 ) e−σ1 τ1 (Q1 ⊗ In )

⎡ Υ2 ⎢ ⎢∗ ⎢ ⎢ ⎢ ⎢∗ ⎢ ⎣ ∗

σ1 − ··· ···

∗

..

∗

∗

(( ) ) C −RL1 ⊗ In ( ) C 2 S 1 RS 1 ⊗ In − (1 − τ˙1 ) e−σ2 τ1 (P1 ⊗ In )

···

.

···

∗

..

∗

∗

.

ln (µν) I1

− L (σ1 + σ2 ) > 0 C

(( ) ) −Ξ Lm−1 ⊗ In

(23)

⎤

⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ 0 ⎥ ( ) 2 m−1 m−1 ⎦ C S ΞS ⊗ In − (1 − τ˙m−1 ) e−σ1 τm−1 (Qm−1 ⊗ In ) 0

C

((

) ) −RLm−1 ⊗ In

(24)

⎤

⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ 0 ⎥ ( ) 2 m−1 m−1 ⎦ C S RS ⊗ In −σ2 τm−1 − (1 − τ˙m−1 ) e (Pm−1 ⊗ In ) 0

(25)

32

M. Luo et al. / Physica A 509 (2018) 20–38

where

( Υ1 =

ΞQ + Q Ξ + Ξ + T

2

m−1 ∑

( (( )s )) Qz + dQ¯ + C S Ξ S + σ1 − κ + 2C λmax Ξ L˜¯ IN 2

0

0

) ⊗ In

z =1

( Υ2 =

T

2

RQ + Q R + R +

m−1 ∑

) Pz + dP¯ + C S RS − σ2 R − κ IN 2

0

0

⊗ In

z =1

Proof. Choose the following Lyapunov functional candidate for synchronization error system (10) and (11) as V2 (t ) = e (t ) (Ξ ⊗ In ) e (t ) + T

m−1 ∫ ∑

0

∫

t

e

+ −d

−σ1 (t −s) T

e (s) Q¯ ⊗ In e (s) dsdθ

(

)

m−1 ∫ ∑

∫

t

∗

(26)

eσ2 (t −s) eT (s) (Pz ⊗ In ) e (s) ds

eσ2 (t −s) eT (s) P¯ ⊗ In e (s) dsdθ

∗

∗

t ∈ E2m ∪ E4m

)

(

+ −d

t t −τz (t )

z =1 0

∗

t ∈ E1m ∪ E3m

t +θ

V3 (t ) = eT (t ) (R ⊗ In ) e (t ) +

∫

e−σ1 (t −s) eT (s) (Qz ⊗ In ) e (s) ds

t −τz (t )

z =1

∫

t

(27)

t +θ ∗

∗

By the Itô’s differential formula, the stochastic differential of V2 (t ) along the trajectory of error system (10), for t ∈ E1m ∪ E3m , one has that L V2 (t ) ≤ eT (t ) Υ1 e (t ) +

N ∑

Γ22

t −d(t )

i=1

+ 2C

m−1 ∑

eT (t )

((

t

∫

eT (s) e (s) ds − e−σ1 d

) ) −Ξ Lz ⊗ In e (t − τz (t )) −

m−1 ∑

∫

t

eT (s) Q¯ ⊗ In e (s) ds

(

t −d(t )

)

(1 − τ˙z ) e−σ1 τz eT (t − τz (t )) (Qz ⊗ In ) e (t − τz (t ))

z =1

z =1

− σ 1 V2 ( t ) + C 2

m−1 ∑

eT (t − τz (t )) S z Ξ S z ⊗ In e (t − τz (t ))

)

(

(28)

z =1

Then, from the conditions of Theorem 3.2, it has that L V2 (t ) ≤ −σ1 V2 (t )

∗

∗

t ∈ E1m ∪ E3m

(29)

In the same way, we can have L V3 (t ) ≤ eT (t ) Υ2 e (t ) +

N ∑

Γ22

t

∫

i=1

t −d(t )

∑

∫

t

eT (s) P¯ ⊗ In e (s) ds

(

t −d(t )

)

m−1

m−1

+ 2C

eT (s) e (s) ds − e−σ2 d

eT (t )

((

∑ ) ) −RLz ⊗ In e (t − τz (t )) − (1 − τ˙z ) e−σ2 τz eT (t − τz (t )) (Pz ⊗ In ) e (t − τz (t ))

z =1

+ σ2 V3 (t ) + C 2

z =1 m−1 ∑

eT (t − τz (t )) S z RS z ⊗ In e (t − τz (t ))

(

)

(30)

z =1

Similarly, it can be seen that the following inequality is also true L V3 (t ) ≤ σ2 V3 (t )

∗

∗

t ∈ E2m ∪ E4m

(31)

Following, we will analyze the problem of synchronization in five steps. Step I: By simple calculation, there exist two positive constants µ and ν such that V3 (t ) ≤ µV2 (t )

V2 (t ) ≤ νV3 (t )

where

( )} λmax (R ) (σ1 +σ2 )τz λmax (Pz ) (σ1 +σ2 )d λmax P¯ ( ) ,e ,e µ = max z λmini (ξi ) λmin (Qz ) λmin Q¯ {

(32)

M. Luo et al. / Physica A 509 (2018) 20–38

{

33

( )} λmax (Qz ) λmax Q¯ ( ) , ν = max , (33) z λmin (R ) λmin (Pz ) λmin P¯ ( ∑ ∗ ) ) [∑ ∗ −1 ∑m∗ −1 ∗ −σ1 t − m Ti m −1 i=1 ∗ ∗ t ≤ e T + θ T , and from the (29), one has that V T , Step II: If t ∈ E1m = ( ) i m 1 m 1 2 i i=1 i=1 ) (∑ ∗ m −1 Ti . Obviously, based on the (32), we can have V2 i=1 (m∗ −1 ) (m∗ −1 ) ∑ ∑ V2 Ti ≤ νV3 Ti 1

i=1

≤ νe

i=1

σ2

(∑ ∗ m −1 i=1

Ti −

∑m∗ −2 i=1

Ti −T(m∗ −1)1 −θ(m∗ −1)2 T(m∗ −1)2

)

V3

(m∗ −2 ∑

) Ti + T(m∗ −1)1 + θ(m∗ −1)2 T(m∗ −1)2

i=1

σ2

)

((

1−θ(m∗ −1)2 T(m∗ −1)2

≤ µνe

)

V2

(m∗ −2 ∑

) Ti + T(m∗ −1)1 + θ(m∗ −1)2 T(m∗ −1)2

i=1

σ2

((

)

)

1−θ(m∗ −1)2 T(m∗ −1)2 −σ1 θ(m∗ −1)2 T(m∗ −1)2

≤ µνe

V2

(m∗ −2 ∑

) Ti + T(m∗ −1)1

i=1

Furthermore, V2

(m∗ −2 ∑

(m∗ −2 ∑

) Ti +

T(m∗ −1)1

≤ νV3

i=1

≤ νe

) Ti +

T(m∗ −1)1

i=1

σ2

(∑ ∗ m −2 i=1

∑m∗ −2

Ti +T(m∗ −1)1 −

Ti −θ(m∗ −1)1 T(m∗ −1)1

i=1

)

V3

(m∗ −2 ∑

) Ti + θ(m∗ −1)1 T(m∗ −1)1

i=1

σ2

((

)

1−θ(m∗ −1)1 T(m∗ −1)1

≤ µνe

)

V2

(m∗ −2 ∑

) Ti + θ(m∗ −1)1 T(m∗ −1)1

i=1

σ2

((

≤ µνe

)

1−θ(m∗ −1)1 T(m∗ −1)1

)

e

−σ1 θ(m∗ −1)1 T(m∗ −1)1

V2

(m∗ −2 ) ∑ Ti

i=1

In summary, V2

(m∗ −1 ) ∑ Ti

σ2

≤ (µν)2 e

((

)

(

)

1−θ(m∗ −1)1 T(m∗ −1)1 + 1−θ(m∗ −1)2 T(m∗ −1)2

)

i=1

( ) −σ1 θ(m∗ −1)1 T(m∗ −1)1 +θ(m∗ −1)2 T(m∗ −1)2

×e

V2

(m∗ −2 ) ∑ Ti

i=1

Now, based on the mathematical induction, the following inequality is true V2

(m∗ −1 ) ∑ Ti

∗

≤ (µν)2m e

∑m∗ −1

{σ2 ((1−θi1 )Ti1 +(1−θi2 )Ti2 )−σ1 (θi1 Ti1 +θi2 Ti2 )}

∑m∗ −1

{(σ1 +σ2 )((1−θi1 )Ti1 +(1−θi2 )Ti2 )−σ1 Ti }

i=1

V2 (0)

i=1 ∗

= (µν)2m e [

≤e

i=1

ln(µν) I1 +L(σ1 +σ2 )−σ1

]∑ ∗ m −1 T i=1

i

V2 (0) ≤ e

[

V2 (0)

ln(µν) I1 +L(σ1 +σ2 ) t −σ1

]

∑m∗ −1 i =1

Ti

V2 (0)

Taking mathematical expectation, it is easy to see that the following inequality is true E {V2 (t )} ≤ e

[ ] (µν) − σ1 − lnI −L(σ1 +σ2 ) t

1 E {V2 (0)} } 1−θm∗ 1 )Tm∗ 1 +(1−θm∗ 2 )Tm∗ 2 ( where L = supm∗ ∈N+ , I1 = infm∗ ∈N+ {Tm∗ 1 , Tm∗ 2 } Tm∗ [∑ ∗ ) ∑ ∗ m −1 m∗ −1 Step III: If t ∈ E2m = Ti + θm∗ 1 Tm∗ 1 , i=1 Ti + Tm∗ 1 , then i=1 (m∗ −1 ) ( ∑ ∗ ) ∑ −1 σ2 t − m T −θ T ∗ ∗ i m 1 m 1 i=1 V3 ( t ) ≤ e V3 Ti + θm∗ 1 Tm∗ 1

{

i=1

(34)

34

M. Luo et al. / Physica A 509 (2018) 20–38

holds. Next, we will continue to analyze the term of V3

V3

(m∗ −1 ∑

) Ti + θm∗ 1 Tm∗ 1

≤ µV2

(m∗ −1 ∑

(∑ ∗ m −1 i=1

)

Ti + θm∗ 1 Tm∗ 1 . From the (29), (31) and (32), one has

) −σ1

Ti + θm∗ 1 Tm∗ 1

≤ µe

(∑ ∗ ∑ ∗ −1 ) m −1 Ti +θm∗ 1 Tm∗ 1 − m Ti i=1 i=1

V2

≤ µe

−σ1

Ti

i=1

i=1

i=1

(m∗ −1 ) ∑

(∑ ∗ m −1

Ti +θm∗ 1 Tm∗ 1

i=1

]∑ ∗ m −1

ln(µν) I1 +L(σ1 +σ2 )

) [

e

i=1

Ti

V2 (0)

and, V3 (t ) ≤ µeσ2 t e

−σ2

[

= µeσ2 t e

(∑ ∗ ) (∑ ∗ ) [ ][∑ ∗ ] ln(µν) m −1 m −1 m −1 Ti +θm∗ 1 Tm∗ 1 −σ1 Ti +θm∗ 1 Tm∗ 1 +L(σ1 +σ2 ) Ti +θm∗ 1 Tm∗ 1 −S I i=1 i=1 i=1

e

(µν) −σ2 −σ1 + lnI +L(σ1 +σ2 ) 1

e

][∑ ∗ m −1 i=1

Ti +θm∗ 1 Tm∗ 1

]

1

[ ] (µν) − lnI +L(σ1 +σ2 ) S

e

1

V2 (0)

V2 (0)

(35)

where S = infm∗ ∈N {θm∗ 1 Tm∗ 1 , θm∗ 2 Tm∗ 2 } Next, we should discuss the (35) in two cases: ln(µν) + L (σ1 + σ2 ) > 0, from the (35), we can have (1): If −σ2 − σ1 + I 1

E {V3 (t )} ≤ µe

[ ] [ ] (µν) (µν) − lnI +L(σ1 +σ2 ) S − σ1 − lnI −L(σ1 +σ2 ) t

e

1

1

E {V2 (0)}

(36)

( ) + L (σ1 + σ2 ) < 0, let I2 = supm∗ ∈N {Tm∗ 1 , Tm∗ 2 }, one has (2): If −σ2 − σ1 + I 1 S + I2 − I2 ≥ t + S − I2 , and ln µν

E {V3 (t )} ≤ µeσ2 t e

[

[ ] ] ln(µν) − I +L(σ1 +σ2 ) S (µν) −σ2 −σ1 + lnI +L(σ1 +σ2 ) (t +S−I2 ) 1

e

1

(µν) I2 (I2 −S−LI2 )(σ1 +σ2 )− lnI 1

= µe In summary, for t ∈

∗ E2m ,

E {V3 (t )} ≤ Σ1 e

where Σ1 = sup µe

1

[ ] (µν) − lnI +L(σ1 +σ2 ) S 1

∗

Step IV: If for t ∈ E3m = V2 (t ) ≤ e

1

i=1

Ti + θm∗ 1 Tm∗ 1 ≥

∑m∗ −1 i=1

Ti +

E {V2 (0)}

E {V2 (0)}

we obtain that

[ ] (µν) − σ1 − lnI −L(σ1 +σ2 ) t

{

e

[ ] (µν) − σ1 − lnI −L(σ1 +σ2 ) t

∑m∗ −1

[∑

m∗ −1 i=1

, µe

E {V2 (0)}

(µν) I2 (I2 −S−LI2 )(σ1 +σ2 )− lnI 1

Ti + Tm∗ 1 ,

( (∑ ∗ )) m −1 −σ1 t − Ti +Tm∗ 1 i=1

V2

(37)

}

(m∗ −1 ∑

∑m∗ −1 i=1

)

Ti + Tm∗ 1 + θm∗ 2 Tm∗ 2 , then

) Ti + Tm∗ 1

i=1

and, V2

(m∗ −1 ∑

) Ti + Tm∗ 1

≤ νV3

(m∗ −1 ∑

) Ti + Tm∗ 1

≤ νe

σ2 ((1−θm∗ 1 )Tm∗ 1 )

V3

(m∗ −1 ∑

i=1

i=1

) Ti + θm∗ 1 Tm∗ 1

i=1

∗

hence, for t ∈ E3m

E {V2 (t )} ≤ µνe−σ1 t eσ2 ((1−θm∗ 1 Tm∗ 1 )) e [

×e

]∑ ∗ m −1

ln(µν) I1 +L(σ1 +σ2 )

i=1

[ ] (µν) − σ1 − lnI −L(σ1 +σ2 ) t

≤ Σe

1

where Σ = µνe(σ1 +σ2 )(I[2 −S) . ∗

Step V: If t ∈ E4m = V3 (t ) ≤ e

∑m∗ −1 i=1

σ1

Ti

(∑ ∗ m −1 i=1

Ti +Tm∗ 1

)

e

−σ1

(∑ ∗ m −1 i=1

Ti +θm∗ 1 Tm∗ 1

E {V2 (0)}

E {V2 (0)}

(38)

Ti + Tm∗ 1 + θm∗ 2 Tm∗ 2 ,

( ∑ ∗ ) σ2 t − im=1−1 Ti −Tm∗ 1 −θm∗ 2 Tm∗ 2

V3

(m∗ −1 ∑

∑m∗

i=1

)

Ti , then

) Ti + Tm∗ 1 + θm∗ 2 Tm∗ 2

i=1

and, V3

(m∗ −1 ∑ i=1

) Ti + Tm∗ 1 + θm∗ 2 Tm∗ 2

)

≤ µ V2

(m∗ −1 ∑ i=1

) Ti + Tm∗ 1 + θm∗ 2 Tm∗ 2

M. Luo et al. / Physica A 509 (2018) 20–38

≤ µe−σ1 θm∗ 2 Tm∗ 2 V2

(m∗ −1 ∑

35

) Ti + Tm∗ 1

i=1

so, ( ∑ ∗ ) [ ](∑ ∗ ) (µν) −1 m −1 σ2 t − m Ti −Tm∗ 1 −θm∗ 2 Tm∗ 2 −σ θ ∗ T ∗ −σ1 + lnI +L(σ1 +σ2 ) Ti +Tm∗ 1 i =1 i=1 1 m 2 m 2 1

V3 ( t ) ≤ µ Σ e

e

e

[ [ ] ][∑ ∗ ] (µν) (µν) m −1 − lnI +L(σ1 +σ2 ) S σ2 t −σ1 −σ2 + lnI +L(σ1 +σ2 ) Ti +Tm∗ 1 +θm∗ 2 Tm∗ 2 i=1 1 1

≤ µΣ e

e

e

V2 (0)

V2 (0)

Along the similar line of (37), we can obtain that [ ] (µν) − σ1 − lnI −L(σ1 +σ2 ) t

E {V3 (t )} ≤ Σ2 e

{

where Σ2 = sup µΣ e

1

[ ] (µν) − lnI +L(σ1 +σ2 ) S 1

E {V2 (0)}

(39)

(µν) I2 (I2 −S−LI2 )(σ1 +σ2 )− lnI

, µΣ e

1

} .

( ) Hence, when t ∈ ∪ ∪ ∪ if σ1 − I − L (σ1 + σ2 ) > 0 holds, one has limt →+∞ E {V (t )} = 0. Then, 1 from Definition 2.1, we can find that the conclusion of synchronization can be achieved. This proof is complete. □ ∗ E1m

∗ E2m

∗ E3m

∗ E4m ,

ln µν

Remark 3.3. Based on the purpose of cost control, we hope that the work widths θm∗ 1 Tm∗ 1 and θm∗ 2 Tm∗ 2 are small and the rest width L is large enough in each switched period if control period width is fixed. So, from condition (23) in Theorem 3.2, we can adjust parameters σ1 and σ2 appropriately to guarantee both (23) holds and L is large enough. Remark 3.4. The present work is necessary and nontrivial extension on the previous works of pinning intermittent control on complex networks. Firstly, most of the previous works had been concentrated on the networks with deterministic or fixed link, single switched period, identical rates of control duration and control width in every switched period; Secondly, the main features in this paper are focused on extending the conclusion of intermittent control schemes with single switched period to the case of two switched periods via Lemma 2.3, on the other hand, by using novel iterative algorithm in Theorem 3.2, we obtain a unified sufficient condition (23), regardless in the work time or in the rest time. Hence, our results are easy to simulate and achieve. Remark 3.5. Inspired by Ref. [38], a novel mathematical iteration method is adopted to deal with the problem of synchronization via intermittent control. Based on this approach, we can find that there exist no special restriction on rates of control duration such as θm∗ 1 and θm∗ 2 are monotonic and some strict assumptions between the time delay, control width and non-control width in the original references have been removed. Hence, the new proposed results here are very easy to verify and also complement. Furthermore, following an analogous line to Theorems 3.1 and 3.2, we can investigate the synchronization problem of the aperiodically adaptive intermittent control with l different periods for stochastic networks, stochastic complex-valued switch networks or sampled-data communication networks, and all these interesting topics may be covered in our future works.

4. Examples In this section, two numerical simulations are given to demonstrate the effectiveness of proposing synchronization schemes for synchronizing the multi-stochastic-link complex networks (6) to be trajectory s (t ) with aperiodically pinning intermittent controller (9). In order to show the effectiveness of the theoretical results, we consider a 3-node error system (10) and (11) to verify the effectiveness of Theorems 3.1 and 3.2 with discrete-time and distributed delay. Example 1. Consider three links of complex networks as follows:

⎡ ⎤ ⎧ ∫ t 3 2 3 ∑ ∑ ∑ ⎪ ⎪ ⎪ dxi (t ) = ⎣f (xi (t )) − C l0ij xj (t ) − C lzij xj (t − τz (t )) + h (xi (s)) ds⎦ dt ⎪ ⎪ ⎪ t −d(t ) ⎪ j=1 z =1 j=1 ⎪ ⎨ ⎡ ⎤ 2 3 ∑ ∑ ⎪ ⎪ + ⎣−C szij xj (t − τz (t ))⎦ dω (t ) ⎪ ⎪ ⎪ ⎪ z =0 j=1 ⎪ ⎪ ⎩ xi (t ) = ϕi (·) t ∈ [−τ , 0] , i = 1, 2, 3

(40)

where f (x (t )) = [−2.6x1 − 2.2 (|x1 + 1| − |x1 − 1|) + 3x2 , x1 − x2 + x3 , −x3 + 1]T , h (x (t )) = 0.2f (x (t )). Moreover, simple calculations show that κ = 0.5, Q = 0, Γ2 = 0.8, Brownian motion ω (t ) is depicted in Fig. 2. Next, we assume

36

M. Luo et al. / Physica A 509 (2018) 20–38

that the associated Laplacian matrices given by

(

−0.5

L¯ =

0 0.5

0 −0.2 0.2

0.5 0.2 −0.7

)

( 1

L =

0.1 −0.1 0

−0.1 0.2 −0.1

0 −0.1 0.1

)

( 2

L =

0.2 −0.2 0

−0.2 0.5 −0.3

0 −0.3 0.3

) (41)

and S 0 = −0.2L¯, S 1 = 0.2L1 , S 2 = 0.2L2 , then, we can show that the chaotic (behavior of )T stochastic complex network (40) with the given initial data in Fig. 3. Let τ1 = τ2 = d = 0.001, θ = 0.39, ξ = 1 1 1 , obviously, ξ T L¯ = 0 is true. Since L˜¯ = L¯ − diag 3

(

0

((

0 , and one has that γ = −2C λmax

)

Ξ L˜¯

)s )

= 7.1740 if C = 1.

For the above parameters, according to our analysis in Theorem 3.1, we can have υ1 = 6.4478, υ21 = 1.0036,

∑2 z = 1.0234, υ3 = 0.7262, υ4 = 0.6400. Furthermore, we also have υ1 = 6.4478 > 2.0276 = z =1 υ2 + υ4 d, ∑m−1 z ϑτz υ4 υ4 ϑ d and solve equation ϑ − υ1 + + ϑ e − ϑ = 0 by MATLAB LMI Toolbox yields: ϑ = 2.2468, therefore, z =1 υ2 e ϑ = 4.4112 > 4.3761 = γ (1 − θ) holds, which implies (13) is true. Now, according to Theorem 3.1, the global exponentially υ

2 2

synchronization in mean square can be achieved, and the synchronization error curves and the state trajectories of errors systems are shown in Figs. 4 and 6 respectively. Based on the above discussion, inequality (13) should play a important role in Theorem 3.1 for achieving synchronization, that is, a reasonable analysis of the relationship between the ϑ and θ is very crucial, which derived the red area of Fig. 5, in general, we can obtain suitable parameter ϑ by adjusting τz and d in the ∑min −1 υ υ equation ϑ − υ1 + z =1 υ2z eϑτz + ϑ4 eϑ d − ϑ4 = 0, then, we can also obtain the appropriate interval of θ from (13) by the simple calculation. Example 2. For convenience, in this example, we will consider the same model and appropriate parameters as that in the et Example 1 with the delay τ (t ) under the case II. We consider time-varying delay τ1 (t ) = τ2 (t ) = d (t ) = 0.001 1+ , et therefore, τ1 = τ2 = d = τ˙1 = τ˙2 = 0.001. In addition, we select σ1 = 1.5, σ2 = 0.3, I1 = 2, simultaneously, we select that the associated Laplacian matrices given by

−0.1 0.1 L¯ =

(

0

0.1 −0.2 0.1

0 0.1 −0.1

)

( 1

L =

0.2 −0.1 −0.1

−0.1 0.1

−0.1

0

)

0 0.1

( 2

L =

0.2 0 −0.2

0 0.5 −0.5

) −0.2 −0.5 0.7

(42)

In order to facilitate the analysis of numerical results, we choose the diagonal matrix as our feasible solution, then, by using Matlab Toolbox, we obtain that

( Q¯ =

11.8475 0 0

( Q2 =

( P¯ =

0 0 11.8469

)

2.4085 0 0

0 2.4278 0

0 0 2.4187

0.1869 0 0

0 0.1756 0

0 0 0.1822

( P1 =

0 11.8457 0

11.2801 0 0

0 11.2507 0

( Q1 =

)

( R=

)

0.2363 0 0

( P2 =

2.3875 0 0

0.1955 0 0

0 2.2923 0

0 0.1281 0 0 0.2109 0

0 0 2.35333

)

0 0 0.1910

)

0 0 0.2008

)

0 0 11.2722

)

then, we calculate µ = 0.9540, ν = 13.5962. According to Theorem 3.2, we only need L < 0.1215, condition (23) will be satisfied. Fig. 7 gives the evolutions of synchronization error variables with 3 agents and Fig. 8 shows the trajectory of error system with pinning intermittent controller (9). It can be shown from the simulation results that the stochastic complex networks (6) can synchronize with the isolate node s (t ) under the conditions of Theorem 3.2. Remark 4.1. In this section, the effectiveness of our main results are demonstrated by two numerical examples. By using simulation algorithms, it can be seen that the feasibility solutions of our results are easy to obtain. Firstly, inequality (13) plays an important role in Theorem 3.1, we are able to choose appropriate system parameters and the range of θm∗ 1 , θm∗ 2 so that an appropriate unique positive solution ϑ will be found to guarantee the (13) is true, and the feasible domain of (13) are also showed in Fig. 5; Secondly, in order to obtain synchronization in Theorem 3.2, we need to make sure the inequality (23) holds. According to Example 2, we can find that although the iteration process in Theorem 3.2 is relatively complicated, (23) is still easily to satisfy by adjusted σ1 , σ2 , control width and non-control width. 5. Conclusions We have investigated, in this paper, the problem of exponential synchronization for a class of multi-stochastic-link complex networks with mixed delays. By applied aperiodically pinning intermittent control schemes and two novel

M. Luo et al. / Physica A 509 (2018) 20–38

37

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