Exponential synchronization for delayed coupled systems on networks via graph-theoretic method and periodically intermittent control

Journal Pre-proof Exponential synchronization for delayed coupled systems on networks via graph-theoretic method and periodically intermittent control Lei Zhang, Jing Liu

PII: DOI: Reference:

S0378-4371(19)32081-3 https://doi.org/10.1016/j.physa.2019.123733 PHYSA 123733

To appear in:

Physica A

Received date : 7 July 2019 Revised date : 24 October 2019 Please cite this article as: L. Zhang and J. Liu, Exponential synchronization for delayed coupled systems on networks via graph-theoretic method and periodically intermittent control, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123733. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

*Highlights (for review)

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Highlights 1. The exponential synchronization problem of delayed coupled systems on networks (DCSNs) under periodically intermittent control is considered.

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2. Lyapunov-type theorem and coefficients-type criterion

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are obtained by graph-theoretic approach and Lyapunov function method.

3.Numerical simulation are presented to show the

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effectiveness of the theoretical results.

*Manuscript Click here to view linked References

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Exponential synchronization for delayed coupled systems on networks via graph-theoretic method and periodically intermittent control✩ Lei Zhanga,∗, Jing Liub

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College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P.R. China b Economics and Management Institute, Xinjiang University, Urumqi 830046, P.R. China

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Abstract

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In this paper, we consider the exponential synchronization problem of delayed coupled systems on networks (DCSNs) under periodically intermittent control. Based on the graph-theoretic approach and Lyapunov function method, some easily verifiable synchronization criteria are derived in the form of Lyapunov-type theorem and coefficients-type criterion. As illustrations, the proposed theory is applied to research the exponential synchronization between two different delayed coupled oscillators on networks under periodically intermittent control. In the end, two numerical examples are presented to show the effectiveness of the theoretical results. Keywords: Exponential synchronization; Periodically intermittent control; Graph-theoretic approach; Delayed coupled systems; Delayed coupled oscillators

1. Introduction

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In recent decades, many researchers have devoted themselves to the coupled systems on networks (CSNs) due to their extensive applications in many domains such as the Internet, food webs, social organizations, power grids and others fields [1, 2, 3, 4, 5]. Time delay is always ubiquitous in many areas such as communication and traffic [6, 7, 8, 9]. Furthermore, it has been known that delayed coupling may modify dynamic behaviors of a system drastically, for example, stability and synchronization [10, 11, 12, 13]. Also, it changes over time as usual. Therefore, in order to simulate networks more practically, time-varying delay should be considered into CSNs. It is significant and practical to study the dynamical behaviors of delayed coupled systems on networks (DCSNs) because they play a vital role in practical applications. Lots of researchers have joined the study field with interest, for instance, Shi et al. [14] investigated the synchronization problem for complex networks with time-varying delay and without time-varying delay by utilizing suitable control strategy. Synchronization, as a typical kind of dynamic, is an important group behavior of networks and has drawn considerable attention since there exist many potential applications, including parallel recognition, secure communications, image processing, information science and so on [15, 16, 17, 18, 19, 20]. Carroll and Pecora in [21] proposed the synchronization between drive system and response system. After that, plenty of results regarding to synchronization have been derived, see for example [22, 23, 24]. More recently, synchronization in network or coupled oscillators has become an active field of researches. Hence, the problem of synchronization ✩ ∗

Supported by NSFC (No.61662079). Corresponding author. Email address: [email protected] (Lei Zhang)

Preprint submitted to Elsevier

October 23, 2019

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of delayed system has been an important research subject with the rapidly increasing research during the past few years. However, in many practical situations, some systems on networks cannot achieve synchronization directly. Therefore, abundant control techniques have been developed covering adaptive control [25], impulsive control [26], fuzzy control [27], observer-based control [28], intermittent control [29] and so on. It has been recognized that intermittent control has witnessed a growing interest since it can be used for a variety of purposes such as signal processing, manufacturing, transportation and communication. A significant benefit of intermittent control is that it has a nonzero control width, and can be easily implemented in practice. Hence, some synchronization conditions of coupled system by using intermittent control have been carried out [30, 31, 32, 33]. For instance, in [30], Hu et al. derived some stabilization criteria and synchronization conditions for a class of neural networks with time-varying delays by introducing multi-parameters and using the Lyapunov functional technique. In [34], the authors used Lyapunov stability theory and inequality techniques to analyze stochastic synchronization of coupled neural networks with intermittent control. In [35], Yang et al. considered exponential stabilization and synchronization for fuzzy model of memristive neural networks by utilizing the Lyapunov stability theory and periodically intermittent control. As we state above, when using the intermittent control to synchronize two different systems, the Lyapunov method provides a powerful tool to obtain the synchronization conditions. However, it is quite tough to construct a proper global Lyapunov function due to the intricate relations between the topological structure of networks and divergent dynamical properties of the coupled nodes. Fortunately, with the help of some results in graph theory, Li et al. [36, 37] put forward a graph-theoretic approach to construct the global Lyapunov function. They surprised to find that a network can be expressed as a weighted digraph (G, D), where G is a digraph with vertex set ℓ = {1, 2, · · · , l} and weight matrix D = (δkh )l×l . Each vertex k stands for an individual system called vertex system. Inter-connections or couplings among vertex systems are described by the edges of G, and the strength of the coupling is described by the weights of edges. In particular, an edge (h, k) from vertex h to k exists if and only if δkh > 0. They obtained some stability criteria by using the graph theory and further applied the theoretical results to coupled oscillators, epidemic models and predator-prey models. Following Li’s footstep, some novel results based on this approach were obtained (see [38, 39] and the references therein). As is shown in [36], the DCSNs can be described by a directed graph as well. However, to date, few results have been found in the literature combining graph-theoretic approach and periodically intermittent control to study the synchronization between two different DCSNs. Motivated by the above analysis, the objective of this paper is to investigate the exponential synchronization of DCSNs under periodically intermittent control and to use some graph-theoretical results to explore the relationship between synchronization and topology structure of the corresponding directed graph. The main contributions of this paper are listed as follows. Firstly, by using the graph theory and Lyapunov function method, two types of sufficient synchronization criteria are proposed, of which one is given in the form of Lyapunov functions and network topology, while another is given by means of coefficients of drive-response system. Secondly, some results in graph theory are effectively utilized to avoid the difficulty in directly finding the global Lyapunov function of DCSNs. Thirdly, periodically intermittent control was applied to synchronize delayed coupled oscillators on networks and some exponential synchronization conditions for them are also explored to demonstrate the applicability and feasibility of the derived criteria in this paper. The organization of this paper is as follows. In Section 2, some preliminaries and the model formulations are presented, respectively. In Section 3, based on the graph-theoretic approach and Lyapunov function method, some sufficient criteria for exponential synchronization between two different DCSNs are presented. An application on delayed coupled oscillators on networks is given in Section 4. In Section 5, we use two numerical examples to show the effectiveness of our theoretical results. Ultimately, a conclusion is drawn. 2

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2. Preliminaries and models In this section, some useful notations and a lemma on graph theory are presented in Subsection 2.1, and then we state the model formulations in Subsection 2.2.

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2.1. Preliminaries The following notations will be used throughout this paper. Let R be the set of all real numbers, Rn be the n−dimensional Euclidean space, R1+ be the set of all nonnegative real numbers and N+ = {0, 1, 2, · · · }. Also, let P Z+ = {1, 2, · · · }, ℓ = {1, 2, · · · , l}, m = li=1 mi for mi ∈ Z+ and | · | be the Euclidean norm in Rn . The superscript “T” stands for the transpose. In addition, C 1,1 (Rn ×R1+ ; R1+) represents for the family of all nonnegative functions V(x, t) on Rn × R1+ which are continuously differentiable in x and t, C([−τ, 0]; Rn ) is the space of continuous functions x : [−τ, 0] → Rn with norm kxk = sup−τ≤t≤0 |x(t)|. The Laplacian matrix of (G, D) is defined as P L = (pkh )l×l , where pkh = −δkh for k , h and pkh = j,k δk j for k = h. In order to state our main results clearly, let us present a useful lemma as follows.

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Lemma 1. [36] Suppose that l ≥ 2. Let ck denote the cofactor of the k-th diagonal element of L. And then the following identity holds: ck δkh Mkh (E k , E h ) =

X

W(Q)

Q∈Q

k,h=1

X

Mrs (E r , E s ),

(s,r)∈E(CQ )

where k, h ∈ ℓ and Mkh (E k , E h ) is an arbitrary function, Q is the set of all spanning unicyclic graphs of (G, D), W(Q) is the weight of Q, and CQ is the directed Q. Particularly, if (G, D) is strongly connected, then ck > 0 for k ∈ ℓ.

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2.2. Model formulations In this paper, the exponential synchronization for two different DCSNs will be researched. Letting the k-th system be represented by a digraph G with l vertices (l ≥ 2), we consider the following system: X X dxk (t) = bk (xk (t)) + ckh fh (xh (t)) + dkh gh (xh (t − τkh (t))) + Ik , t ≥ 0, k ∈ ℓ, dt h=1 h=1

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l

l

(1)

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where xk (t) ∈ Rmk corresponds to the state of the k-th vertex at time t; bk (·) : Rmk → Rmk is a continuously differentiable appropriate behaved function; fh (·) and gh (·): Rmh → Rmk are the continuously inner connecting functions, additionally, in neural networks, they denote the activation functions of the h-th unit ; τkh (t) ≥ 0 is the discrete time-varying delay along h-th vertex system from k-th vertex system; ckh and dkh are the connection strength and the time delay connection weight of the h-th vertex to the k-th vertex, respectively. Here ckh , dkh are defined as follows: if there is a connection from vertex k to vertex h, then the coupling ckh , 0, dkh , 0; otherwise, ckh = 0, dkh = 0. Ik represents an external input or bias. In order to observe the synchronization behavior of system (1), we design the response system as follows: X X dyk (t) = bk (yk (t)) + ckh fh (yh (t)) + dkh gh (yh (t − τkh (t))) + Ik + uk (t), dt h=1 h=1 l

l

3

(2)

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where for each k ∈ ℓ and t ≥ 0, uk (t) is a periodically intermittent control defined by ( Pl h=1 Qkh (yh (t) − xh (t)), nT ≤ t < (n + θ)T, uk (t) = 0, (n + θ)T ≤ t < (n + 1)T,

(3)

t ∈ [−τ, 0],

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x(t) = ϕ(t), y(t) = φ(t),

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in which n ∈ N+ , Qkh (k, h ∈ ℓ) are the constant control, T is the control period, and 0 < θ < 1 is the rate of control duration. Furthermore, we assume that system (1) and (2) possess different initial functions

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where ϕ(t) = (ϕT1 (t), ϕT2 (t), · · · , ϕTl (t))T ∈ C([−τ, 0], Rm ), φ(t) = (φT1 (t), φT2 (t), · · · , φTl (t))T ∈ C([−τ, 0], Rm ) and τ =maxk,h∈ℓ {τkh (t)}. In order to facilitate the reader, we give the architecture of drive-response systems (1) and (2) shown in Fig. 1. The solid lines represent the deterministic coupling among the nodes within a network, while dashed ones represent the coupling between the drive network and response network. 2

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5 Response system

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1 Drive system

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Fig. 1: Example of the structure for drive-response systems.

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Let ek (t) = xk (t) − yk (t) be the synchronization error. From the drive-response systems (1) and (2), we can know that the error system can be written as:  l l l X X X     ˜ ˜  Qkh eh (t), nT ≤ t < (n + θ)T, dkh g˜ h (eh (t − τkh (t))) + ckh fh (eh (t)) + e˙ k (t) = bk (ek (t)) +      h=1 h=1 h=1 (4)   l l  X X    ˜  ckh f˜h (eh (t)) + dkh g˜ h (eh (t − τkh (t))) (n + θ)T ≤ t < (n + 1)T,    e˙ k (t) = bk (ek (t)) + h=1

h=1

where b˜ k (ek (t)) = bk (yk (t)) − bk (xk (t)), f˜h (eh (t)) = fh (yh (t)) − fh (xh (t)) and g˜ h (eh (t − τkh (t))) = gh (yh (t − τkh (t))) − gh (xh (t − τkh (t))). Remark 1. As alluded to in the Introduction, synchronization of complex networks is an active field of researches since the synchronization mechanism can explain many natural phenomena well, including the synchronous information exchange in the Internet and worldwide web, and the synchronous transfer of digital or 4

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analog signals in communication networks. Usually, Lyapunov function is used to attain synchronization criteria. In [36], Li et al. offered a technique to construct the global Lyapunov function for coupled systems of ordinary differential equations on networks by using each vertex system’s Lyapunov function which can be obtained or be constructed easily. In fact, the practical coupled systems are very complex. To make progress, some simplifications have been made to avoid the issues of structural complexity and intensively study the system’s potentially formidable dynamics. For example, in nonlinear dynamics the simple and nearly identical dynamical systems are coupled together in simple, regular ways. Thus, in this article, we are interested in whether or not some graph theoretical results can overcome the difficulty of directly constructing an appropriate Lyapunov function for DCSNs and the synchronization of drive-response systems has a close relationship with topology property of the corresponding directed graph. We now begin to present the results derived in this paper. Before ending this section, we introduce the definition of exponential synchronization for drive-respond systems (1) and (2).

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Definition 1. [31] Drive-respond systems (1) and (2) are said to be exponentially synchronized, if there exist M ≥ 1 and λ > 0, such that |y(t) − x(t)| ≤ Mkφ − ϕke−λt

for any t ≥ 0. Here λ is called the degree of exponential synchronization. 3. Synchronization Analysis

The main objective of this section is to consider the exponential synchronization between the drive-response systems (1) and (2) under periodically intermittent control (3). Some sufficient laws for exponential synchronization are established by combining graph theory and Lyapunov function method. Firstly, we present a theorem in the form of Lyapunov functions.

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3.1. Lyapunov-type theorem Before stating the main results, a definition about the k-th vertex-Lyapunov function and two necessary hypotheses are given to arrive at our results.

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Definition 2. For p ≥ 2 and k, h ∈ ℓ, we assume that the following assumptions are satisfied for system (4), A1. There exist positive constants αk and βk , such that αk |ek (t)| p ≤ Vk (ek (t), t) ≤ βk |ek (t)| p .

(5)

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A2. There exist constants σ ˜ k > 0, function Mkh (E k , E h ) ∈ C(R pmk × R pmh ; R) and δkh ≥ 0, k, h ∈ ℓ satisfying,  l l X X    p p   −σk |ek (t)| + δkh |ek (t − τhk (t))| + δkh Mkh (E k , E h ), nT ≤ t < (n + θ)T,    dVk (ek (t), t)   h=1 h=1 ≤  l l  X X dt    p p  δkh Mkh (E k , E h ), (n + θ)T ≤ t < (n + 1)T, δkh |ek (t − τhk (t))| +   −σ˜ k |ek (t)| +  h=1

h=1

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where σk = σ˜ k − ςk and ςk = pQkk + (p − 1) h=1,h,k |Qkh |. A3. Along each directed cycle C of weighted digraph (G, D), for all E k ∈ R pmk , E h ∈ R pmh , there is X Mkh (E k , E h ) ≤ 0. (h,k)∈E(C)

  Then function Vk (ek (t), t) ∈ C 1,1 Rmk × R1+ ; R1+ is called the k-th vertex-Lyapunov function of system (4). 5

(6)

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We are now ready to present our main results in the following theorem.

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Theorem 1. Suppose that digraph (G, D) is strongly connected and system (4) admits the vertex-Lyapunov function Vk (ek (t), t) for any k ∈ ℓ. Exponential synchronization between drive-response systems (1) and (2) can be achieved under intermittent control (3) provided that θ > 1 − ε/ζ, where ζ = maxk∈ℓ {|ςk /βk |} and ε satisfies ( ) ( Pl ) σk h=1 δkh ε − min + max eετ ≤ 0. (7) k∈ℓ k∈ℓ βk αk

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P Proof. First, we set V(e, t) = lk=1 ck Vk (ek , t), where ck is the cofactor of the k-th diagonal element of the Laplacian matrix of (G, D). Noting that (G, D) is strongly connected, we have ck > 0 for any k ∈ ℓ by Lemma P 1. Denote β = lk=1 ck βk and 1− 2p   l  2p  X   min{ck αk } . α =  ck αk  k∈ℓ k=1

V(e(t), t) ≤ and

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Thus, from (5) we have

  l  X ck βk |ek (t)| ≤  ck βk  |e(t)| p = β|e(t)| p

V(e(t), t) ≥

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k=1 l X

ck αk |ek (t)| p

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  l X  2p   ck αk   P = ci αi |ek (t)|2  l c α j=1 j j i=1 k=1  2p  l l X  X ck αk ≥ |ek (t)|2  ci αi  Pl j=1 c j α j k=1 i=1 p 1− 2   l  2p  X   min{ck αk } |e(t)| p ≥  ck αk  k∈ℓ

k=1

Hence,

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=α|e(t)| p .

α|e(t)| p ≤ V(e(t), t) ≤ β|e(t)| p .

(8)

In the sequel, we divide the proof into the following three steps for convenience of the reader. Step 1. Prove P(t) = eεt V(e(t), t) − hβ̟ < 0, for all t ∈ [−τ, θT ), where h > 1 is a constant and ̟ = sup−τ≤s≤0 |φ(s) − ϕ(s)| p . Firstly, for t ∈ [−τ, 0], it is easy to check that P(t) ≤ eεt β|e(t)| p − hβ̟ < 0. 6

(9)

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Next, we prove that for t ∈ [0, θT ):

P(t) < 0.

(10)

P(t0 ) = 0, D+ P(t0 ) ≥ 0,

(11)

and P(t) < 0,

for t ∈ [0, t0 ),

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If not, there exists a t0 ∈ [0, θT ) such that

(12)

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where D+ denotes Dini derivative. From the condition A2 and Lemma 1 we can derive that D+ P(t0 ) =εeεt0 V(e(t0 ), t0) + eεt0

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ck V˙ k (ek (t0 ), t0 )

k=1

k=1

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  l l l l   X X X X     p p εV(e(t ), t ) − ≤eεt0  c σ |e (t )| + c δ |e (t − τ (t ))| + c δ M (E , E ) 0 0 k k k 0 k kh k 0 hk 0 k kh kh k h      k=1 k,h=1 k=1 h=1   l l   X X X X    εt0  p p =e  ), t ) − εV(e(t c σ |e (t )| + c δ |e (t − τ (t ))| + W(Q) M (E , E ) .  0 0 k k k 0 k kh k 0 hk 0 kh k h     k,h=1

Q∈Q

(h,k)∈E(C)

According to W(Q) > 0, condition A1, (6), (11) and (12), we get

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D+ P(t0 )   l l   X X   σ δ   k kh (e (e ) εV(e(t ), t ) − c V V (t ), t (t − τ (t )), t − τ (t )) + c ≤eεt0  0 0 k k k 0 k k 0 0 hk 0 0 hk 0  k     β α k k k=1 k,h=1  l  ( )!    σk X δkh   εt0 εt0 = ε − min e V (e(t0 − τhk (t0 )), t0 − τhk (t0 )) e V (e(t0 ), t0 ) + max      k∈ℓ k∈ℓ βk αk  h=1 ( )! ( Pl ) σk h=1 δkh < ε − min hβ̟ + max eετ hβ̟ k∈ℓ k∈ℓ βk αk ( ) ( Pl ) ! σk h=1 δkh = − −ε + min − max eετ hβ̟. k∈ℓ k∈ℓ βk αk

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It follows from (7) that D+ P(t0 ) < 0, which leads to a contradiction with D+ P(t0 ) ≥ 0. Hence inequality (10) holds. Step 2. Prove Q(t) = eεt V(e(t), t) − hβ̟eζ(t−θT ) < 0, for all t ∈ [θT, T ). Were the argument false, there would exist a t1 ∈ [θT, T ) such that Q(t1 ) = 0, D+ Q(t1 ) ≥ 0, and Q(t) < 0,

for t ∈ [θT, t1 ). 7

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For k, h ∈ ℓ, if θT ≤ t1 − τhk (t1 ) ≤ t1 , we get εt1

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ck Vk (ek (t1 − τhk (t1 )), t1 − τhk (t1 )) = eεt1 V (e(t1 − τhk (t1 )), t1 − τhk (t1 )) < eετ hβ̟eζ(t1 −θT ) ,

l X k=1

ck Vk (ek (t1 − τhk (t1 )), t1 − τhk (t1 ))
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eεt1

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and if −τ ≤ t1 − τhk (t1 ) ≤ θT , from Step 1, we have

Hence, for any k ∈ ℓ, we always have εt1

e

l X k=1

ck Vk (ek (t1 − τhk (t1 )), t1 − τhk (t1 )) < eετ hβ̟eζ(t1 −θT ) .

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Then, the following inequalities hold D+ Q(t1 )

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ζ(t1 −θT ) ˙ =εeεt1 V(e(t1 ), t1 ) + eεt1 V(e(t 1 ), t1 ) − ζhβ̟e   l l   X X    p p − ζhβ̟eζ(t1 −θT ) εV(e(t ), t ) − c σ ˜ |e (t )| + c δ |e (t − τ (t ))| ≤eεt1   1 1 k k k 1 k kh k 1 hk 1     k=1 k,h=1  l  l     X X  + eεt1  c δ M (E , E ) k kh kh k h      k=1 h=1   l l   X X   δ σ ˜   kh k (e (e ) V V (t ), t (t − τ (t )), t − τ (t )) + c ≤eεt1  εV(e(t ), t ) − c  k k k 1 1 k 1 hk 1 1 hk 1 k 1 1 k     βk αk k,h=1 k=1

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− ζhβ̟eζ(t1 −θT ) ( Pl ) ( )! σ ˜k h=1 δkh ζ(t1 −θT ) hβ̟e + max eετ hβ̟eζ(t1 −θT ) − ζhβ̟eζ(t1 −θT ) < ε − min k∈ℓ k∈ℓ βk αk ( ( ) ( Pl ) ) ! σ ˜ k ςk ςk h=1 δkh ζ(t1 −θT ) ετ = − −ε + min + max − − max e hβ̟e hM0 eζ(t1 −θT ) − ζhβ̟eζ(t1 −θT ) k∈ℓ k∈ℓ k∈ℓ βk βk αk βk ( ) ( Pl ) ! ( ) ςk σk h=1 δkh ζ(t1 −θT ) ετ + max − max e hβ̟e hβ̟eζ(t1 −θT ) − ζhβ̟eζ(t1 −θT ) . = − −ε + min k∈ℓ k∈ℓ k∈ℓ βk αk βk

From Assumption 2, we can obtain that D+ Q(t1 ) < 0, which contradicts D+ Q(t1 ) ≥ 0. Hence Q(t) < 0, for all t ∈ [θT, T ). According to Step 1 that, for t ∈ [−τ, θT ), eεt V(e(t), t) < hβ̟ < hβ̟eζ(1−θ)T . From Step 2 that for t ∈ [θT, T ), eεt V(e(t), t) < hβ̟eζ(t−θT ) < hβ̟eζ(1−θ)T . 8

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So, for all t ∈ [−τ, T ), we have

eεt V(e(t), t) < hβ̟eζ(1−θ)T .

Similar to the proof of (10), we can get that

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eεt V(e(t), t) < hβ̟eζ(1−θ)T , for T ≤ t < (1 + θ)T . Analogous to Step 2, we have

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eεt V(e(t), t) < hβ̟eζ(1−θ)T eζ[t−(1+θ)T ] = hβ̟eζ(t−2θT ) , for (1 + θ)T ≤ t < 2T .

and for (n + θ)T ≤ t < (n + 1)T ,

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Step 3. Apply mathematical induction method to certificate the following estimates are true for any integer n. For nT ≤ t < (n + θ)T , eεt V(e(t), t) < hβ̟enζ(1−θ)T , (13)

eεt V(e(t), t) < hβ̟enζ(1−θ)T eζ[t−(n+θ)T ] = hβ̟eζ[t−(n+1)θT ] .

(14)

Assume that (13) and (14) are true for any n ≤ a − 1, where a > 0 is a integer. Then, for any integer b satisfying 0 ≤ b ≤ a − 1, when bT ≤ t < (b + θ)T , eεt V(e(t), t) < hβ̟ebζ(1−θ)T < hβ̟eaζ(1−θ)T , and when (b + θ)T ≤ t < (b + 1)T ,

al

eεt V(e(t), t) < hβ̟eζ[t−(b+1)θT ] < hβ̟eζ(b+1)(1−θ)T ] ≤ hβ̟eaζ(1−θ)T . Combining it with (9), for any t ∈ [−τ, aT ), we have

urn

eεt V(e(t), t) < hβ̟eaζ(1−θ)T .

(15)

On the other hand, when aT ≤ t < (a + θ)T , we obtain (15) holds. And when (a + θ)T ≤ t < (a + 1)T , we can verify that eεt V(e(t), t) < hβ̟eaζ(1−θ)T eζ[t−(a+θ)T ] = hβ̟eζ[t−(a+1)θT ] .

Jo

Here the proof is analogous to that of (10) and Step 2, respectively. Therefore, it follows from mathematical induction method that inequalities (13) and (14) hold for any integer n > 0. Noting that for any t ≥ 0, there exists an integer n such that nT ≤ t < (n + θ)T . When nT ≤ t < (n + θ)T , and n ≤ t/T , we have eεt V(e(t), t) < hβ̟enζ(1−θ)T ≤ hβ̟eζ(1−θ)t . Moreover, when (n + θ)T ≤ t < (n + 1)T , and n + 1 > t/T , we get

eεt V(e(t), t) < hβ̟eζ[t−(n+1)θT ] < hβ̟eζ(1−θ)t . Finally, let h → 1, according to (8), it is readily seen that eεt α|e(t)| p ≤ eεt V(e(t), t) ≤ β̟eζ(1−θ)t . 9

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Hence, we get the following estimate |y(t) − x(t)| = |e(t)| ≤ 1 p

 β  1p α

1

1

̟ p e− p [ε−ζ(1−θ)]t = Mkφ − ϕke−λt ,

of

for any t ≥ 0, where λ = ε−ζ(1−θ) , and M = ( αβ ) ≥ 1. Therefore, the exponential synchronization between drivep response systems (1) and (2) can be achieved via periodically intermittent control (3). The proof is therefore complete.

p ro

Remark 2. Theorem 1 shows that the rate of control duration θ can be obtained as θ > 1 − ε/ζ if the parameter ε and control strength Qkh are chosen properly. In fact, this provides an available way to design the time interval of periodically intermittent control. Its validity is verified by numerical tests in Section 5.

Pr e-

Remark 3. Conditions A1 and A2 in Definition 2 are applicable to the exponential synchronization of coupled systems in some applications, which can show that the decay of pth moment of the solution is exponential. In practice, one needs only let ( ) ( Pl ) σk h=1 δkh min > max (16) k∈ℓ k∈ℓ βk αk then there exists a scalar ε fulfills condition (7) in Theorem 1. In fact, estimate (16) is a restriction on the coupling matrix (δkh )l×l . We know from condition A2 that δkh ≥ 0, so the sum of δkh (h = 1, 2, · · · , l) can not be sufficiently large to ensure the vertex-Lyapunov function V˙ k (ek (t), t) ≤ 0. In other words, the restriction of coupling matrix provides a valuable guarantee for the stability of error system (4). The validity of the technique is presented in Theorem 2. In what follows, based on some results in graph theory, two corollaries can be obtained. Suppose that (G, D) is balanced, we can readily get X 1X W(Q) [Mkh (E k , E h ) + Mhk (E h , E k )] . 2 Q∈Q (h,k)∈E(C)

al

l X

ck δkh Mkh (E k , E h ) =

k,h=1

X

urn

Thus, (6) can be replaced by

(h,k)∈E(C)

[Mkh (E k , E h) + Mhk (E h , E k )] ≤ 0.

(17)

Consequently, we obtain an easier stability criterion as follows. Corollary 1. The conclusion of Theorem 1 holds if (6) is replaced by (17) provided that (G, D) is balanced.

Jo

Furthermore, if for every Mkh (E k , E h) there exist functions Pk (E k ) and Ph (E h ), such that Mkh (E k , E h) ≤ Pk (E k ) − Ph (E h ),

(18)

then (6) can be readily verified. Thus, one more corollary is derived below. Corollary 2. The result of Theorem 1 holds if (6) is replaced by (18). The results obtained above have provided some powerful synchronization laws for drive-response systems (1) and (2). So, a natural question arises: how to find some appropriate functions to check the availability of Theorem 1. Answering this question is the aim of Theorem 2. The detail how to find such functions Mkh (E k , E h) to verify conditions A2 and A3 in Definition 2 will be shown in the following Theorem 2. 10

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3.2. Coefficients-type criterion In this subsection, based on Lyapunov-type theorem, a coefficients-type criterion is stated as follows.

of

Theorem 2. Suppose that digraph (G, D) is strongly connected and the following conditions hold, B1. There exist positive constants ηk , Ah and Bh, such that (19)

| fh (yh ) − fh (xh )| ≤ Ah |yh − xh |,

(20)

p ro

(yk − xk )T (bk (yk ) − bk (xk )) ≤ −ηk |yk − xk |2 ,

|gh (yh ) − gh (xh )| ≤ Bh |yh − xh |. B2. It holds

(21)

Pr e-

   l  l l l    X X X       X   min  pη − (p − 1) |c |A − (p − 1) |d |B − ς − δ > max δ ,    k kh h kh h k kh kh       k∈ℓ  k∈ℓ  h=1

where p ≥ 2 and

δkh ,

h=1

(

h=1

h=1

maxk∈ℓ {|ckh |Ah + |Qkh |, |dkh |Bh } , h , k, 0, h = k.

Then the exponential synchronization can be realized between systems (1) and (2) by periodically intermittent control (3). Proof. For any k ∈ ℓ, denote

al

Vk (ek , t) = |ek (t)| p ,

dVk (ek (t), t) dt

urn

which satisfies condition A1 obviously. If nT ≤ t < (n + θ)T , differentiating with respect to time, we can get  

 =p|ek (t)| p−2 eTk (t) b˜ k (ek (t))

+

l X

ckh f˜h (eh (t)) +

h=1

l X h=1

dkh g˜ h (eh (t − τkh (t))) +

≤p|ek (t)| p−2 eTk (t) (bk (yk (t)) − bk (xk (t))) + p|ek (t)| p−2 |eTk (t)|

Jo

l X

+ p|ek (t)| p−2 |eTk (t)|

+ p|ek (t)| p−2 |eTk (t)|

l X h=1

l X h=1

  Qkh eh (t)

|ckh || fh (yh (t)) − fh (xh (t))|

|dkh ||gh (yh (t − τkh (t))) − gh (xh (t − τkh (t)))| + p|ek (t)| p−2 eTk (t)Qkk ek (t)

h=1 l X

h=1,k,k

|Qkh ||eh (t)|.

(22)

Applying the following inequality (see [40], p.52) |a| p |b|q ≤

p q |a| p+q + |b| p+q , p+q p+q 11

(23)

Journal Pre-proof

it yields

≤ p|ek (t)| p−1 l X

=p

h=1,h,k

l X

h=1,h,k

|Qkh ||eh (t)|

|Qkh ||eh (t)|

|Qkh ||ek (t)| p−1 |eh (t)|

h=1,h,k

l X

≤ (p − 1)

l X

h=1,h,k

|Qkh ||ek (t)| +

|eTk (t)|

≤ p|ek (t)|

p−1

≤ (p − 1)

l X

h=1,h,k

|Qkh ||eh (t)| p .

Pr e-

p|ek (t)|

l X

p

Together with (20) and (23), we have p−2

of

|eTk (t)|

p ro

p|ek (t)|

p−2

h=1

l X h=1

l X h=1

(24)

|ckh || fh(yh (t)) − fh (xh (t))|

|ckh |Ah |eh (t)| p

|ckh |Ah |ek (t)| +

l X h=1

|ckh |Ah |eh (t)| p .

(25)

|eTk (t)|

l X h=1

l X

urn

p|ek (t)|

p−2

al

From (21) and (23), one obtains

≤ p|ek (t)| =p

l X h=1

p−1

|dkh |Bh|eh (t − τkh (t))|

|dkh |Bh |ek (t)| p−1 |eh (t − τkh (t))|

≤ (p − 1)

Jo

h=1

|dkh ||gh (yh (t − τkh (t))) − gh (xh (t − τkh (t)))|

l X h=1

p

|dkh |Bh |ek (t)| +

l X h=1

12

|dkh |Bh|eh (t − τkh (t))| p .

(26)

Journal Pre-proof

Submitting (19) and (24)-(26) into (22), one sees that

h=1

+

l X h=1

|ckh |Ah |eh (t)| p +

of

dVk (ek , t)   dt l l l X X X   |ckh |Ah + (p − 1) |dkh |Bh + pQkk + (p − 1) |Qkh | |ek (t)| p ≤ −ηk p + (p − 1) h=1

l X

h=1,h,k

|Qkh ||eh (t)| p +

h=1,h,k

l X h=1

|dkh |Bh |eh (t − τkh (t))| p

h=1

h=1 l X

δkh |eh (t)| p +

= − σk |ek (t)| p + = − σk |ek (t)| p +

l X h=1

l X h=1

h=1

h=1

δkh |eh (t − τkh (t))| p −

δkh |ek (t − τhk (t))| p +

δkh |ek (t − τhk (t))| p +

in which σk = ηk p − (p − 1)

l X

h=1

l X h=1

δkh |ek (t − τhk (t))| p +

Pr e-

+

l X

p ro

  l l l l X X X X   p   δkh |ek (t)| p |ckh |Ah − (p − 1) |dkh |Bh − ςk − δkh  |ek (t)| − ≤ − ηk p − (p − 1)

h=1

l X

l X h=1

h=1

δkh |ek (t − τhk (t))| p

δkh [|eh (t)| p + |eh (t − τkh (t))| p − (|ek (t)| p + |ek (t − τhk (t))| p )] δkh Mkh (E k , E h),

h=1

l X h=1

|ckh |Ah − (p − 1)

l X h=1

|dkh |Bh − ςk −

l X

δkh ,

h=1

Jo

urn

al

Mkh (E k , E h) = (|eh (t)| p + |eh (t − τkh (t))| p ) − (|ek (t)| p + |ek (t − τhk (t))| p ) .

13

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Similarly, for (n + θ)T ≤ t < (n + 1)T , we have  

 =p|ek (t)| p−2 eTk (t) b˜ k (ek (t))

+

l X

ckh f˜h (eh (t)) +

h=1

l X h=1

  dkh g˜ h (eh (t − τkh (t)))

of

dVk (ek (t), t) dt

  l l l X X X   p   |ckh |Ah |eh (t)| p |ckh |Ah + (p − 1) |dkh |Bh  |ek (t)| + ≤ −ηk p + (p − 1) h=1

h=1

h=1

h=1

|dkh |Bh |eh (t − τkh (t))| p

p ro

+

l X

  l l l l X X X X   p   δkh |ek (t)| p |ckh |Ah − (p − 1) |dkh |Bh − δkh  |ek (t)| − ≤ − ηk p − (p − 1) h=1

h=1 l X

δkh |eh (t)| p +

=−σ ˜ k |ek (t)| p +

l X h=1

h=1

h=1

δkh |eh (t − τkh (t))| p −

δkh |ek (t − τhk (t))| p +

in which

h=1

l X

δkh |ek (t − τhk (t))| p +

Pr e-

+

l X

σ ˜ k = ηk p − (p − 1)

l X h=1

l X

h=1

h=1

l X

δkh |(ek (t − τhk (t)))| p

l X

δkh .

h=1

δkh Mkh (E k , E h ),

h=1

|ckh |Ah − (p − 1)

l X h=1

|dkh |Bh −

h=1

h=1

h=1

h=1

urn

h=1

al

Hence, condition A2 satisfies. From B2, there exists a scalar ε satisfying    l  l l l    X X X       X   ετ ε − min  pη − (p − 1) |c |A − (p − 1) |d |B − ς − δ + max e ≤ 0. δ    k kh h kh h k kh kh         k∈ℓ k∈ℓ

Thus, according to Theorem 1, we can conclude that the exponential synchronization between system (1) and (2) can achieve. This completes the proof.

Jo

Remark 4. Condition B1 is the condition for the connecting functions. Condition B2 seems to be complex since the model is a network of coupled systems, where there are interactions among l vertex dynamical systems. Condition B2 shows that if both the strength of coupling and control are small, the synchronization between systems (1) and (2) is achieved. In addition, the new criteria derived in Theorem 2 are helpful to design some synchronization coupled networks by adjusting the coupling matrix. 4. An application on delayed coupled oscillators on networks In this section, we will apply the main results to discuss the synchronization for two coupled oscillators on networks. The synchronization of coupled oscillators of large populations is a hot topic in various scientific disciplines such as mechanics, electrical engineering, and biological systems. In these applications, an simple oscillator model x¨(t) + ε x˙(t) + ̺x(t) = 0, t ≥ 0, 14

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l X

c˜ kh Mh (xh (t)) +

h=1

l X

d˜kh Nh (xh (t − τkh (t)) = 1, t ≥ 0, k ∈ ℓ,

p ro

x¨k (t) + εk x˙k (t) + ̺k xk (t) +

of

is one of the commonest examples. Here ε > 0 is the damping coefficient. Recently, some coupled oscillators on networks have also been studied as models of self-excited systems. Then, the synchronization of coupled oscillators will be investigated. Since there naturally exists a consequence of finite information transmission and processing speeds among the units in many large-scale networks, time-delays are always taken into account in the realistic modeling, such as the time-delayed coupling in biological neural networks, epidemiological models and electrical power grids. These coupled oscillators are modeled on a digraph G which has l (l ≥ 2) vertices. Firstly, coupled oscillators are described by

h=1

(27)

Pr e-

in which xk (t) ∈ R is the state variable of the k-th dynamical node at time t, Mh , Nh : R → R is the form of influence from vertex h to vertex k. c˜kh , d˜kh represents the intensity of influence from vertex h to vertex k. If there is a connection from vertex k to vertex h, then the coupling c˜ kh , 0, d˜kh , 0; otherwise, c˜ kh = 0, d˜kh = 0. Also, we denote that x¯k (t) = x˙k (t) + ξk xk (t), where ξk > 0 and then we can get the system as follows: ( x˙k (t) = x¯k (t) − ξk xk (t), P P (28) x¯˙k (t) = (ξk − εk ) x¯k (t) − lh=1 c˜ kh Mh (xh (t)) − lh=1 d˜kh Nh (xh (t − τkh (t))) + (ξk εk − ξk2 − ̺k )xk (t) + 1.

urn

al

In order to realize synchronization between two DCONs, we refer to system (28) as the drive system, and then we design the response system as following: ( y˙k (t) = y¯ k (t) − ξk yk (t), P P y˙¯k (t) = (ξk − εk )¯yk (t) − lh=1 c˜kh Mh (yh (t)) − lh=1 d˜kh Nh (yh (t − τkh (t))) + (ξk εk − ξk2 − ̺k )yk (t) − uk (t) + 1, (29) where uk (t) is defined by (3). Next, in order to facilitate further study, we design that Xk = (xk , x¯k )T , Yk = (yk , y¯ k )T , Uk (t) = (0, uk )T , Hh (Xk (t)) = (0, −Mh (xh (t)))T , Gh (Xh (t − τkh (t))) = (0, −Nh (xh (t − τkh (t))))T , Zk (t) = Yk −Xk = (yk − xk , y¯ k − x¯k )T ,  T (ek , e¯ k )T , F k (Xk (t)) = x¯k (t) − ξk xk (t), (ξk − εk ) x¯k (t) + (ξk εk − ξk2 − ̺k )xk (t) + 1 . Then we can rewrite the drive system (28) as X X dXk (t) = F k (Xk (t)) + c˜ kh Hh (Xh (t)) + d˜khGh (Xh (t − τkh (t))), t ≥ 0, k ∈ ℓ. dt h=1 h=1 l

l

(30)

Similarly, the response system (29) can also be represented by X X dYk (t) = F k (Yk (t)) + c˜kh Hh (Yh (t)) + d˜khGh (Yh (t − τkh (t))) + Uk (t), t ≥ 0, k ∈ ℓ. dt h=1 h=1

Jo

l

l

(31)

Then the synchronization error system can be written as X X dZk (t) = F k (Zk (t)) + c˜kh Hh (Zh (t)) + d˜khGh (Zh (t − τkh (t))) + Uk (t). dt h=1 h=1 l

l

(32)

In what follows, an easily verifiable exponential synchronization result is stated for drive-response systems (30) and (31). 15

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Theorem 3. The systems (30) and (31) can achieve exponential synchronization if for any k, h ∈ ℓ, the following conditions hold. C1. There exist positive constants A¯ h and B¯ h such that |Mh (yh ) − Mh (xh )| ≤ A¯ h |yh − xh |, |Nh (yh ) − Nh (xh )| ≤ B¯ h|yh − xh |.

h=1

h=1

of

¯ is strongly connected and there is C2. Digraph (G, D)     l l l l     X X X X         ˜ ¯ ¯ ¯ ¯ min  pγ − (p − 1) |˜ c | A − (p − 1) | d | B − ς − , δ δ > max  k kh h kh h k kh  kh        k∈ℓ  k∈ℓ  h=1

h=1

|ξ ε −ξ2 −̺ +1| − k k 2k k

|ξ ε −ξ2 −̺ +1| − k k 2k k

p ro

+ ξk > 0, ςk = where D¯ = (δ¯ kh )l×l , p ≥ 2, γk = min{̟k , ςk }, ̟k = k∈ℓ n o ( ¯ h + |Qkh |, |d˜kh | B¯ h , h , k, max |˜ c | A k∈ℓ kh δ¯ kh , 0, h = k,

+ εk − ξk > 0 and

Proof. Firstly, for arbitrary Xk = (xk , x¯k )T and Yk = (yk , y¯ k )T , we can get that

Pr e-

(Yk − Xk )T (F k (Yk ) − F k (Xk ))

= ek e¯k − ξk e2k + (ξk − εk )¯ek + (ξk εk − ξk2 − ̺k )ek e¯ k

= −ξk e2k + (ξk − εk )¯ek + (ξk εk − ξk2 − ̺k + 1)ek e¯k ! ! |ξk εk − ξk2 − ̺k + 1| |ξk εk − ξk2 − ̺k + 1| 2 ≤ − ξk ek + − εk + ξk e¯ 2k 2 2 ≤ −γk e2k − γk e¯ 2k = −γk |Zk |2 ,

al

Together with condition C1, which implies that the condition B1 holds. From condition C2 that   l   l l l     X X X     X    δ − max pη − (p − 1) |c |A − (p − 1) |d |B − ς − δ min     kh k kh h kh h k kh         k∈ℓ k∈ℓ h=1 h=1 h=1 h=1     l l l l    X X X  X        ˜kh | B¯ h − ςk − ¯ h − (p − 1) ¯ kh  − max  δ¯ kh  = min  pγ − (p − 1) |˜ c | A | d δ  k kh      k∈ℓ   k∈ℓ  > 0,

urn

h=1

h=1

h=1

h=1

where ηk = ξ2k , Ah = A¯ h and Bh = B¯ h . Hence, for any k ∈ ℓ, condition B2 holds. So, all conditions in Theorem 2 are fulfilled, which implies that systems (30) and (31) can realize exponential synchronization via periodically intermittent control (3).

Jo

Remark 5. As an application of Theorem 2, the exponential synchronization between two different DCONs is considered. Some sufficient conditions ensuring the exponential synchronization of DCONs are presented in Theorem 3. In engineering practice, there inevitably exists collision or vibration when the machinery parts contact, in particular, which have a big impact on the design of precision machine parts. As we know, the complicated mechanical system is used to be seen as a mass-spring-damper system to model system’s dynamics. In fact, the mass-spring-damper system is a typical example of oscillator model. In [41], Zhang et al. considered the stability of stochastic coupled oscillators with time-varying delays by graph-theoretic approach. Differing from the previous results, this paper first investigates the synchronization between two different DCONs via periodically intermittent control. So, our result endows DCONs more applicability in physics and other fields. In the next section, we will use an example to check the validity of the obtained results. 16

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5. Numerical tests

p ro

of

In this section, we present a detailed numerical examples to illustrate the effectiveness of our analytical results. Example. Let l = 4. We consider the delayed coupled oscillators on networks (30) as a drive system and system (31) as a response system. It follows from Section 4 that system (32) is the corresponding synchronization error system. We assume that Mh (xh (t)) = 0.1xh (t), Nh (xh (t − τkh (t))) = tanh(xh (t − τkh (t))), τkh (t) = 0.5(sin(t) + 1), the coefficient εk = 0.1, k, h = 1, 2, 3, 4 and ξ1 = ξ2 = ξ3 = 0.06, ξ4 = 0.07. In addition, other parameters are assigned in Table 1. Table 1: The values of c˜ kh and d˜kh

d˜kh 1 2 3 4

1 2 3 4 0 0.01 0 0.1 0.3 0 0.01 0.3 0 0.22 0 0.43 0.1 0.63 0.231 0

1 2 3 4 0 0.2 0.015 0.015 0.03 0 0.35 0.1 0.055 0.02 0 0.1 -0.01 0.06 0.01 0

Pr e-

c˜kh 1 2 3 4

The periodically intermittent controller uk (t) is designed as ( Pl h=1 Qkh (yh (t) − xh (t)), 5n ≤ t < 5(n + 0.9), uk (t) = 0, 5(n + 0.9) ≤ t < 5(n + 1),

(33)

urn

al

where n ∈ N+ , Q11 = −0.8, Q12 = 0.002, Q13 = 0.00003, Q14 = 0.000014, Q21 = 0.000011, Q22 = −0.9, Q23 = 0.0003, Q24 = 0.00004, Q31 = 0.0001, Q32 = 0.00012, Q33 = −0.7, Q34 = 0.00009, Q41 = 0.00001, Q42 = 0.0001, Q43 = 0.00003, Q44 = −0.9. Let A¯ h = 0.1 and B¯ h = 1 (h = 1, 2, 3, 4), thus, according to condition C2, we can get the weighted matrix of ¯ digraph (G, D)   0 0.2000 0.0150 0.0150    0.0300 0 0.3500 0.1000  (δ¯ kh )4×4 =  . 0 0.1000   0.0550 0.0220  0.0100 0.0630 0.0231 0

Jo

Specially, choose p = 2. By some simple calculations, we can know that conditions C1 and C2 in Theorem 3 are all fulfilled. What’s more, ε = 0.28 and ζ = 1.8 can be got by computation. Hence, it follows from Theorem 1 that θ > 0.844. Selecting θ = 0.9 and T = 5, it can be concluded from Theorem 3 that the drive-response systems (30) and (31) can realize exponential synchronization under periodically intermittent control (33). Additionally, the simulation results are shown in Figs. 2-4. Fig. 2 is the paths of the state vector Xk (k = 1, 2, 3, 4) in system (30) with the initial values (x1 (t) = 5.01, x¯1(t) = −2.3, x2 (t) = 4.02, x¯2 (t) = 4.02, x3(t) = −5.2, x¯3(t) = 6.01, x4(t) = 2.5, x¯4 (t) = 8 for t ∈ [−1, 0]). The response of the state vector Yk (k = 1, 2, 3, 4) in system (31) with the initial values (y1 (t) = 3, y¯ 1 (t) = −1, y2 (t) = 4, y¯ 2 (t) = 3, y3(t) = −4.8, y¯ 3 (t) = 6, y4 (t) = 0.5, y¯ 4(t) = 7 for t ∈ [−1, 0]) is described in Fig. 3. Fig. 4 is the simulation results on synchronization error Zk (k = 1, 2, 3, 4) of system (32). In fact, these numerical simulations support our theoretical results.

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Fig. 2: The paths of the solution to drive system (30).

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−4 −6 −8

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Fig. 3: The paths of the solution to response system (31).

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Fig. 4: The paths of the solution to synchronization error system (32).

6. Conclusions

References

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In this paper, the exponential synchronization of coupled systems with time delays has been investigated via periodically intermittent control technique. The synchronization analysis problem for DCSNs has been addressed by using graph-theoretic approach and Lyapunov function method. Several novel synchronization criteria have been derived. It should be especially pointed out that Lyapunov function for the general DCSNs has been developed in graph theory, which means the difficulty of constructing Lyapunov function directly is overcome. Furthermore, as a special case, the synchronization analysis problem for coupled oscillators on networks with time delay has been also discussed to illustrate the applicability of the theoretical outcomes. The method proposed in this paper builds a platform for the study of exponential synchronization of DCSNs under periodically intermittent control. These synchronization criteria also provide some new insight for the DCSNs from the view of topology property of networks. This is the first time for graph theory to study the exponential synchronization of DCSNs under periodically intermittent control. However, this paper focuses only the exponential synchronization problem of DCSNs. We expect that the coupled systems with periodically intermittent control can gain many valuable applications. In future work, we will consider to extend the proposed method for solving other systems, e.g. stochastic coupled systems, complex networks with distributed delay and so on.

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*Declaration of Interest Statement

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Dear editor

The authors do not have any conflict of interest.

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Mr. Lei Zhang

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Xinjiang University