Synchronization of complex networks with time-varying delay of unknown bound via delayed impulsive control

Journal Pre-proof Synchronization of complex networks with time-varying delay of unknown bound via delayed impulsive control Zhilu Xu, Xiaodi Li, Peiyong Duan

PII: DOI: Reference:

S0893-6080(20)30048-4 https://doi.org/10.1016/j.neunet.2020.02.003 NN 4401

To appear in:

Neural Networks

Received date : 21 October 2019 Revised date : 3 January 2020 Accepted date : 10 February 2020 Please cite this article as: Z. Xu, X. Li and P. Duan, Synchronization of complex networks with time-varying delay of unknown bound via delayed impulsive control. Neural Networks (2020), doi: https://doi.org/10.1016/j.neunet.2020.02.003. 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.

© 2020 Elsevier Ltd. All rights reserved.

Journal Pre-proof

Synchronization of complex networks with time-varying delay of unknown bound via delayed impulsive control I Zhilu Xua,b , Xiaodi Lia,b,∗, Peiyong Duanc,∗ a School

Abstract

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of Mathematics and Statistics, Shandong Normal University, Ji’nan, 250014, PR China for Control and Engineering Computation, Shandong Normal University, Ji’nan, 250014, PR China c School of Information Science and Engineering, Shandong Normal University, Ji’nan, 250014, PR China

b Center

The synchronization problem for complex networks with time-varying delays of unknown bound is investigated in this paper. From the impulsive control point of view, a novel delayed impulsive differential inequality is proposed, where the bounds of time-varying delays in continuous dynamic and discrete dynamic are both unknown. Based on the inequality, a class of delayed impulsive controllers is designed to achieve the synchronization of complex networks, where the restriction between impulses interval and time-varying delays is dropped. A numerical example is presented to illustrate the effectiveness of the obtained results. Keywords: Time-varying delay; Delayed impulsive control; Complex networks; Synchronization; Impulsive differential inequality. 1. Introduction

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It is well known that a complex network usually consists of a series of coupled interconnected nodes, where each node is a dynamical system. In the past decade, complex networks have been extensively studied due to their wide applications in fields of physics and biology, such as neural networks, power grids, world wide web, and so on [1–5]. In recent years, there has been rising interest in the research of complex networks involving time delays. This is prevailingly due to the fact that the time delay which always occurs in various plants is one of the causes of oscillation and instability. Many interesting results on delayed complex networks have been reported, see [2, 3, 6–9]. One of the significant aspects of investigating delayed complex networks is to consider the synchronization problems for complex networks. In the past few years, the synchronization problems for complex networks have attracted widespread attention due to its potential applications in various fields, such as secure communication, image process, biological systems, information science, and so on [6, 9, 10]. There are a large number of interesting results on synchronization of complex networks. [11] considered H∞ synchronization of complex networks; [12, 13] studied finite-time synchronization; [6, 14] investigated exponential synchronization. As we all known, due to the complex dynamical behaviors of each node and topological structure of a network, it is difficult that all the nodes achieve synchronization by themselves. Thus, to achieve synchronization of complex networks, there are many effective control I This

work is supported by National Natural Science Foundation of China (61673247), and the Research Fund for Distinguished Young Scholars (JQ201719). The paper has not been presented at any conference. ∗ Corresponding Author. Email address: [email protected] (X. Li), [email protected] (P. Duan). Submitted to Neural Networks

methods, for example, intermittent control [15], sampled-data control [16, 17], impulsive control [6], and so on. It is worth noting that impulsive control strategy is different from other control strategies. Impulsive control usually has a simple structure and is only activated at some discrete impulse instants to achieve the desired performance [18–22]. There are many interesting results for impulsive synchronization of complex networks [10, 11, 23, 24]. In [23], the pinning impulsive synchronization problem for a class of complex dynamical networks with time-varying delay was investigated. A novel impulsive synchronization criterion was proposed such that the restrictions on the size of time delays were relaxed. In [24], the authors established synchronization criteria for two fractionalorder complex networks by employing hybrid impulsive control strategies. The basic idea of impulsive synchronization is to transmit the state information of drive systems to response systems at discrete instants such that the synchronization is achieved. In practice, there are always measurement failures, loss of date, finite speeds of information transmission, and some other uncertainties at sampled instants so that the sampled state information of drive systems is not always available. Therefore, it is greatly significant to consider utilizing the state information of drive systems in recent history at every sample instant to achieve the impulsive synchronization. In recent years, the study of impulsive dynamical systems involving delayed impulses has received a great deal of attention, see [25–28]. [25] focused on the synchronization of linear complex networks via delayed impulses. [26] considered the exponential synchronization problem for complex networks with stochastic perturbations via delayed impulsive control. [27] paid attention on the double effect of time delays in impulses. However, most of them focused on the case that time delays are bounded or a January 2, 2020

Journal Pre-proof 2. Preliminaries

constant. In fact, in many practical applications, time delays are variant and even, there may not exist a priori condition on the bound of time delays. In such cases, it is necessary and important to investigate the synchronization problem of complex networks subject to time-varying delays whose bound is unknown. Recently, [29] considered the stability problem for delayed neural networks with time-varying delays of unknown bound and proposed a new concept of stability, namely, µ-stability. The basic idea of µ-stability is to describe the relation between stability and exponential stability. In other words, µ-stability can be regarded as the generalization of Lyapunov stability. Moreover, µ-stability can be applied to the case involving timevarying delays of unknown bound due to the fact that µ function can establish the relationship between historical value and current value. Based on the such idea, lots of interesting results on time-varying delays of unknown bound and µ-stability have been reported, see [30–34]. In [31], a class of recurrent neural networks with time-varying delays of unknown bound was considered, where it was derived that the neural networks have at least 3n equilibrium points and two thirds of them are µ-stable. In [33], authors considered µ-stability of cellular neural networks with mixed delays, where impulsive disturbances were involved. In [34], the µ-synchronization criteria of impulsive complex-valued neural networks were derived, where the proposed conditions were dependent on the sizes of mixed time-varying delays. It is noted that although there are many results on the stability and synchronization control involving unbounded time-varying delays in the literature, such as those in [29–34], most of them ignore the effect of impulses or only focused on the case involving impulsive disturbances. Moreover, the possible of time-varying delays in impulses was excluded in [29–34]. Even, there is no available result on the synchronization of complex networks involving time-varying delays whose bounds are unknown via the delayed impulsive control. Therefore, investigating the delayed impulsive synchronization of complex networks with time-varying delays, where the bounds of time-varying delays are both unknown, still remains as a significant and open problem. Motivated by the above discussion, in this paper, the synchronization problem of complex networks with time-varying delays whose bounds are unknown is investigated. Based on a novel impulsive differential inequality, some sufficient conditions are derived by using Lyapunov method. A class of delayed impulsive controllers is designed such that every node in response systems can be synchronized with corresponding node in drive systems. It is worth noting that the bounds of time-varying delays in impulses and systems are both unknown. Moreover, our results can be applied to the case that the restriction between impulses interval and time-varying delays is dropped. The rest of the paper is organized as follows: In Section 2, some fundamental definitions, necessary assumptions, and a novel impulsive differential inequality are given. In Section 3, some synchronization criteria are presented. In Section 4, a numerical example is given to illustrate the effectiveness of the derived results. The summary of this paper is given in Section 5.

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Notations. Let R (R+ ) denote a set of real numbers (nonnegative real numbers), Z+ a set of positive integer numbers, and Rn (Rn×n ) the n−dimensional (n × n−dimensional) real spaces equipped with the Euclidean norm | · |. I represents the identity matrix with appropriate dimensions. Matrix A is a symmetric and positive definite matrix, which is equivalent to A > 0. A < 0 (A ≤ 0) denotes that A is a symmetric and negative definite (semi-definite) matrix. ∗ denotes the symmetric block in one symmetric matrix. For matrix A, let λmax (A) and λmin (A) denote the maximum and minimum eigenvalue of matrix A, respectively. Λ = {1, 2, · · · , N}. For any set H1 ⊆ R, set H2 ⊆ Rm , 1 ≤ m ≤ n, C1 (H1 , H2 ) = {ψ : H1 → H2 , ψ is continuously differentiable}, PC(H1 , H2 ) = {ψ : H1 → H2 is continuous almost everywhere except at some finite points t in which ψ(t+ ) and ψ(t− ) exist and moreover, ψ(t+ ) = ψ(t)}. For any τ > 0, define set PCBτ = {φ ∈ PC([−τ, 0], Rn ), φ is bounded}. For any φ ∈ PCBτ , the norm of φ is defined by ∥φ∥ = sup−τ≤s≤0 |φ(s)|. When τ = ∞, the interval [−τ, 0] is understood to be replaced by (−∞, 0]. Define set ζ = {µ(t) ∈ C1 (R+ , [1, ∞]) : µ(t) is nondecreasing on [0, ∞) and µ(t) → ∞ as t → ∞}. Let α ∨ β denote the maximum value of α and β. ⊗ denotes as the Kronecker product.

Consider the following complex networks with time-varying delays: N ∑ x˙ i (t) = f (xi (t), xi (t − τ(t))) + c bij Γx j (t) j=1

+ c˜

N ∑

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j=1

di j Γx j (t − τ(t)), t > 0, i ∈ Λ,

xi (s) =ϕi (s), s ∈ [−τ, 0], i ∈ Λ,

where xi (·) = [xi1 (·), xi2 (·), · · · , xin (·)]T ∈ Rn is the state variable of the ith node; f : Rn × Rn → Rn is a nonlinear vector value function. τ(t) is time-varying delay whose bound may be unknown or unmeasured. Assume that it satisfies 0 ≤ τ(t) ≤ τ, τ ≤ ∞. ϕi ∈ PCBτ is the initial value of ith node; c and c˜ represent non-delayed and delayed coupling strength, respectively; Γ = (γi j ) ∈ Rn×n is an inner-coupling matrix; bi j and dij are the weight or coupling strength between the ith node and the jth node. If there is a connection from node j to node i ( j , i), then bi j (dij ) > 0, otherwise, bi j (dij ) = 0. ∑ ∑ Specially, bii = − Nj=1,j,i bij (dii = − Nj=1,j,i dij ). Based on the construct of drive-response system, we take system (1) as drive system. Then the dynamic of the corresponding response system is described by y˙ i (t) = f (yi (t), yi (t − τ(t))) + c + c˜

N ∑ j=1

N ∑

bij Γy j (t)

j=1

dij Γy j (t − τ(t)) + ui (t), t > 0, i ∈ Λ,

yi (s) =ψi (s), s ∈ [−τ, 0], i ∈ Λ, 2

(1)

(2)

Journal Pre-proof where yi (·) = [yi1 (·), yi2 (·), · · · , yin (·)]T ∈ Rn is the state variable of the ith node for response system (2); ψi ∈ PCBτ is the initial value of ith node; ui (·) is the control input of ith node. Note that impulsive control as a class of synchronization control schemes has some particular advantages. For example, in communication security systems, it can be combined with conventional cryptographic techniques to achieve the process of encrypting information such that encrypted information is more securely transmitted across the public channel. However, timedelay phenomenons are inevitable due to the finite speeds of the information transmission. In such process, it is hard to immediately complete the sampling, processing, and transferring impulses information at some discrete moments, which leads to the sampling delays in impulse instants, see Figure 1. Therefore, the impulses acting on response systems will possess a delay term representing the sampling delays. Such kind of impulses is said to be delayed impulses whose impulsive transients depend on the states in recent history. In this sense, a class of delayed impulsive controller can be designed as ∞ ∑ ui (t) = [Kei (t − ςn ) − ei (t)]δ(t − tn ), i ∈ Λ, (3)

Drive systems

Sensor

yi(t)

+

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n=1

Response systems

i

Figure 1: Delayed impulsive control loop.

when µ(t) = (1+ρt)γ , ρ > 0, γ > 0, the µ-synchronization becomes power synchronization; when µ(t) = ln(e + ρt), ρ > 0, the µ-synchronization becomes log synchronization. Especially, when µ(t) = exp(λt), λ > 0, the µ-synchronization becomes exponential synchronization. We will give the corresponding synchronization criteria for concerned complex networks in Section 3. In this paper, we make the following assumptions: Assumption 1. There exist three nonnegative constants µ1 ≥ 1, µ2 ≥ 1, and µ3 ≥ 1 such that function µ(t) ∈ ζ satisfies the following inequalities:

Jou

where F(ei (t), ei (t − τ(t))) = f (ei (t) + xi (t), ei (t − τ(t)) + xi (t − τ(t))) − f (xi (t), xi (t − τ(t))). We assume that the solution of (4) is right continuous, i.e., ei (tn ) = ei (t+n ), n ∈ Z+ . The synchronization problem between response system (2) and drive system (1) is reduced to find a control gain matrix K such that error system (4) is µ-stable.

µ(tn ) µ(t) µ(tn ) ≤ µ1 , ∗ ≤ µ2 , ∗ ≤ µ3 , µ(tn−1 ) µ (t − τ(t)) µ (tn − ςn )

where

{

∗

n ∈ Z+ , µ (t) =

µ(t), 1,

t ≥ 0, t < 0.

Assumption 2. There exist two nonnegative constants l1 and l2 such that the nonlinear item f (u, v) satisfies the following inequality: ¯ v)| ¯ ≤ l1 |u − u| ¯ + l2 |v − v|, ¯ | f (u, v) − f (u, ¯ v¯ ∈ Rn . for any u, v, u, Lemma 1. ([23]) For any real vectors x, y ∈ Rn and P ∈ Rn×n , there exists an n × n matrix Q > 0 such that:

Definition 1. Response system (2) is said to be globally µsynchronized with drive system (1), namely error system (4) is said to be globally µ-stable, if there exist a function µ ∈ ζ and a scalar M > 0 such that |ei (t)| ≤

Delays

Impulsive generator

rna

i

ei(tn)

ui

where ei (·) = yi (·) − xi (·) is the synchronization error of the ith node, ςn denotes the time delay when impulse sampling takes place. Assume that it satisfies 0 ≤ ςn ≤ ς, n ∈ Z+ , ς ≤ ∞. Impulse time sequences {tn , n ∈ Z+ } satisfy 0 = t0 < t1 < · · · < tn → ∞ as n → ∞ and we denote such kind of impulse time sequences by set F0 for latter use. For any T > 0, let F (T) denote the set of all impulse time sequences in F0 satisfying tn − tn−1 ≤ T for any n ∈ Z+ . Furthermore, one may obtain the following error system:  N ∑     ˙ e (t) = F(e (t), e (t − τ(t))) + c bij Γe j (t) i i i     j=1   N  ∑   + c˜ dij Γe j (t − τ(t)), t , tn , (4)     j=1    ei (t) = Kei (t− − ςn ), t = tn , n ∈ Z+ ,       e (s) = ψ (s) − ϕ (s), s ∈ [−(τ ∨ ς), 0], i ∈ Λ, i

-

xi(t)

2xT Py ≤ xT PQ−1 PT x + yT Qy. Lemma 2. Assume that function f (t) ∈ PC(R, R+ ) satisfies the following inequality:  +    D f (t) ≤ α f (t) + β f (t − τ(t)), t , tn , (5)    f (t) ≤ κn f (t− − ςn ), t = tn , n ∈ Z+ ,

M , t ≥ 0, i ∈ Λ. µ(t)

Remark 1. Definition 1 introduces µ-synchronization which includes a class of dynamical behaviors between asymptotic synchronization and exponential synchronization. For example, 3

where D+ denotes the upper right-hand Dini derivative; α ∈ R, β ∈ R+ , and κn ∈ R+ , n ∈ Z+ . Under Assumption 1, if

Journal Pre-proof there exist constants δ > 1, T > 0, and function µ(t) ∈ ζ such that ln µ1 + (|α| + βδµ2 )T < ln δ, (6) δµ3 κn ≤ 1, n ∈ Z+ . (7)

It follows from the condition (7) that F(tm ) = µ(tm ) f (tm ) ≤

sup

Suppose that (10) does not hold, then there exists t∗ = inf{t ∈ (tm , tm+1 ), F(t) > δµ(0) f (0)} such that F(t∗ ) = δµ(0) f (0) and F(t) < δµ(0) f (0) for t < t∗ . Note that t∗ > tm and F(tm ) ≤ µ(0) f (0). Furthermore, there exists t∗ = sup{t ∈ [tm , t∗ ), F(t) ≤ µ(0) f (0)}, which implies that F(t∗ ) = µ(0) f (0) and F(t) ≥ µ(0) f (0) for t ≥ t∗ . Then one may derive that F(s) ≤ δµ(0) f (0) ≤ δF(t) for s ∈ [−(τ ∨ ς), t∗ ] and t ∈ [t∗ , t∗ ]. Therefore, for t ∈ [t∗ , t∗ ],

(8)

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over the class F (T), where f (0) :=

f (s).

−(τ∨ς)≤s≤0

P ROOF. To show (8), we construct an auxiliary function F(t) = µ∗ (t) f (t). Now we only need to show that

F(t) ≤ δµ(0) f (0), t ∈ [tn−1 , tn ), ∀n ∈ Z+ . For t < 0, we have F(t) = f (t) ≤ µ(0) f (0). Let n = 1. We first show that

− ςm )

≤ δµ3 κm µ(0) f (0) ≤ µ(0) f (0).

Then the solution of the inequality (5) satisfies f (t) ≤ δµ(0) f (0)/µ(t), t ≥ 0,

κm µ(tm ) F(t− ∗ µ (tm − ςm ) m

F(t) ≤ δµ(0) f (0), t ∈ [0, t1 ).

˙ f (t) + µ(t)D+ f (t) D+ F(t) = µ(t) ˙ βµ(t) µ(t) F(t) + αF(t) + ∗ F(t − τ(t)) ≤ µ(t) µ (t − τ(t)) ˙ µ(t) δβµ(t) ≤ F(t) + αF(t) + ∗ F(t). µ(t) µ (t − τ(t))

(9)

Note that F(0) = µ(0) f (0) ≤ µ(0) f (0) < δµ(0) f (0). Suppose that (9) does not hold, then there exists t∗ = inf{t ∈ (0, t1 ), F(t) > δµ(0) f (0)} such that F(t∗ ) = δµ(0) f (0) and F(t) < δµ(0) f (0) for t < t∗ . Note that t∗ > 0 and F(0) ≤ µ(0) f (0). Furthermore, there exists t∗ = sup{t ∈ [0, t∗ ), F(t) ≤ µ(0) f (0)}, which implies that F(t∗ ) = µ(0) f (0) and F(t) ≥ µ(0) f (0) for t ≥ t∗ . Therefore, F(s) ≤ δµ(0) f (0) ≤ δF(t) for s ∈ [−(τ ∨ ς), t∗ ] and t ∈ [t∗ , t∗ ]. For t ∈ [t∗ , t∗ ], we have

By integrating both sides of above inequality from t∗ to t∗ , we can get that ∫

ln δ =

∫

≤

dt ≤ F(t)

t2

˙ µ(t) + |α| + δµ2 βdt µ(t)

t1

t∗

t∗

˙ µ(t) δβµ(t) +α+ ∗ dt µ(t) µ (t − τ(t))

≤ ln µ1 + (|α| + βδµ2 )T.

˙ f (t) + µ(t)D+ f (t) D+ F(t) = µ(t) ˙ µ(t) βµ(t) ≤ F(t) + αF(t) + ∗ F(t − τ(t)) µ(t) µ (t − τ(t)) ˙ µ(t) δβµ(t) ≤ F(t) + αF(t) + ∗ F(t). µ(t) µ (t − τ(t))

rna

It follows from condition (6) that

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Integrating both sides of above inequality from t∗ to t∗ , one may derive that ∫ t∗ ∫ t∗ ˙ µ(t) δβµ(t) dt ln δ = ≤ +α+ ∗ dt µ (t − τ(t)) t∗ F(t) t∗ µ(t) ∫ t1 ˙ µ(t) ≤ + |α| + δµ2 βdt t0 µ(t) ≤ ln µ1 + (|α| + βδµ2 )T. It follows from condition (6) that

ln δ ≤ ln µ1 + (|α| + βδµ2 )T < ln δ,

which is a contradiction. Therefore, (9) holds. Now we assume that there exists m ∈ Z+ such that F(t) ≤ δµ(0) f (0), t ∈ [tk−1 , tk ), k = 1, 2, · · · , m. Next we shall show that F(t) ≤ δµ(0) f (0), t ∈ [tm , tm+1 ).

t∗

∫

t∗

(10) 4

ln δ ≤ ln µ1 + (|α| + βδµ2 )T < ln δ, which is a contradiction. Then we derive that (10) holds. Therefore, we come to a conclude via mathematical induction that (8) holds. The proof is completed.  Remark 2. If the parameters of inequality (5) satisfy α < 0, β > 0, −α > β > 0, τ(t) ≤ τ < ∞, ςn = 0, κn = 1, n ∈ Z+ , then the inequality (5) becomes well-known Halanay inequality which has been generalized to various forms by many authors. In [18], authors developed Halanay inequality and obtained a class of impulsive differential inequality for the class of systems with impulsive disturbance. [19, 22] studied the exponential stabilization for impulsive systems and established a class of novel impulsive differential inequality from impulsive control point of view. However, most of existing results are only valid for systems with bounded time-varying delays or constant time delays. In the case that the bound of time-varying delay is unknown or there is no priori condition on the bound of time delays, those results become invalid. Lemma 2 provides some effective conditions which can be used to solve the case that time-varying delays are unbounded or their bounds are unknown.

Journal Pre-proof 3. Main results

It follows from condition (11) that ( )T ( ) e(t) Θ c˜D ⊗ PΓ + D V(t) ≤ e(t − τ(t)) ∗ IN ⊗ L2 QL2 − βIN ⊗ P ( ) e(t) + αV(t) + βV(t − τ(t)) e(t − τ(t))

In this section, we will present some sufficient conditions and design a class of delayed impulsive controllers to guarantee the µ-synchronization of complex networks with non-delay coupling and delayed coupling.

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≤αV(t) + βV(t − τ(t)),

Theorem 1. Under Assumption 1 and Assumption 2, if there exist function µ(t) ∈ ζ, some positive constants α, β, T, δ > 1, n × n matrices G, P > 0, and n × n diagonal matrix Q > 0 such that condition (6) and the following inequalities hold:  c˜D ⊗ PΓ  Ξ IN ⊗ P  ∗ −I ⊗ Q 0  N  ∗ ∗ IN ⊗ L2 QL2 − βIN ⊗ P ( ) − δµ1 3 P G ≤ 0, ∗ −P

    ≤ 0, 

where Θ = cB ⊗ PΓ + c(B ⊗ PΓ)T + IN ⊗ L1 QL1 + IN ⊗ PQ−1 P − αIN ⊗P, e(·) = [e1 (·), e2 (·), · · · , eN (·)]T . When t = tn , n ∈ Z+ , one may derive that V(tn ) =

(11)

N ∑

eTi (tn )Pei (tn )

i=1

=

(12)

N ∑ i=1

[Kei (t−n − ςn )]T P[Kei (t−n − ςn )].

where Ξ = cB ⊗ PΓ + c(B ⊗ PΓ)T + IN ⊗ L1 QL1 − αIN ⊗ P, B = (bi j ) ∈ RN×N , and D = (dij ) ∈ RN×N . Then response system (2) is said to be globally µ-synchronized with drive system (1) under the control gain matrix K = GT P−1 over the class F (T).

Based on Lemma 1, one may derive that condition (12) is equivalent to the following inequality:

P ROOF. We consider the following Lyapunov function:

which implies that

V(t) =

N ∑

eTi (t)Pei (t).

i=1

KT PK −

V(tn ) ≤

eTi (t)Pe˙i (t)

i=1

=2

N ∑ i=1

+ c˜

eTi P[F(ei (t), ei (t − τ(t))) + c

N ∑ j=1

V(t) ≤ δµ(0)

N ∑

bij Γe j (t)

j=1

dij Γe j (t − τ(t))].

Jou

Based on Assumption 2 and Lemma 1, there exists a diagonal matrix Q > 0 such that 2

N ∑ i=1

≤

N ∑ i=1

eTi (t)PF(ei (t), ei (t − τ(t)))

[eTi (t)PQ−1 Pei (t)

+ FT (ei (t), ei (t − τ(t)))QF(ei (t), ei (t − τ(t)))]

≤

N ∑

i=1

It follows from Lemma 2 that

rna

N ∑

N 1 ∑ T − ei (tn − ςn )Pei (t−n − ςn ) δµ3

=V(t−n − ςn )/δµ3 .

When t , tn , n ∈ Z+ , we calculate the derivative of V(t) along the trajectory of error system (4): D+ V(t) =2

1 P ≤ 0, δµ3

[eTi (t)PQ−1 Pei (t) + eTi (t)L1 QL1 ei (t)

i=1

+ eTi (t − τ(t))L2 QL2 ei (t − τ(t))]. 5

sup

−(τ∨ς)≤s≤0

V(s)/µ(t), t ≥ 0,

which implies that |e(t)|2 ≤

δµ(0)λmax (P)∥e(0)∥2 . λmin (P)µ(t)

Therefore, response system (2) is globally µ-synchronized with drive system (1). The proof is completed.  Remark 3. The µ-synchronization between response system (2) and drive system (1) is substantially equivalent to the µstability of error system (4). In the past decades, the µstability for neural networks with time-varying delays of unknown bound has been extensively studied. For example, [29, 31] addressed the µ-stability for neural networks with timevarying delays of unknown bound. However, the above results cannot be applied to delayed systems subject to the effect of impulses. Recently, [30, 33, 34] considered the influence of impulses and established µ-stability criteria for concerned systems, where [34] handled with µ-synchronization and [30, 33] handled with the global µ-stability. However, due to D+ V < 0 in [30, 33, 34], the effect of impulses is only studied from the impulsive disturbance point of view. In other words, they cannot be applied to the cases that the plant is unstable. Moreover,

Journal Pre-proof ) 1 − δ[1−ln(1−ω)] P G (ii) ≤ 0. ∗ −P Then response system (2) is said to be globally log synchronized with drive system (1) under the control gain matrix K = GT P−1 over the class F (T). (

the upper bound of time delays in [34] is strictly restricted. In this paper, Theorem 1 proposes a sufficient condition from the impulsive control point of view, which can be applied to the synchronization control of complex dynamical networks subject to unbounded time-varying delays or time-varying delays whose bounds are unmeasured or unknown. Moreover, it is an LMI-based criterion, which can be easily verified via LMIMatlab toolbox.

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P ROOF. Similarly, we verify Assumption 1. For any t ≥ 0, n ∈ Z+ ,

Remark 4. Recently, refs [25–27] considered the effect of delayed impulses. However, the exact information of time delays in impulses is priori needed. Moreover, the relationship between impulsive interval and the upper bound of time delays is artificially limited. In this paper, Theorem 1 can be used to the case that time-varying delay in impulses may be unbounded or its bound is unknown or unmeasured. Moreover, it can be applied to the case that the time delays in impulses are independent of the impulse signals.

≤1+

ln(1 +

ρ(tn −tn−1 ) e+ρtn−1 )

≤ 1 + ln(1 + ρT/e), ln(e + ρtn−1 ) µ(t) ln(e + ρt) 1 = = , µ∗ (t − τ(t)) γ ln(e + ρt) γ

e+ρt

n ln e+ρ(1−ω)t ln(e + ρtn ) µ(tn ) n = =1+ ∗ µ (tn − ςn ) ln[e + ρ(tn − ωtn )] ln[e + ρ(1 − ω)tn ]

Especially, if we further strengthen the restrictions for timevarying delays and the function µ(t), then the following synchronization criteria can be derived.

≤ 1 − ln(1 − ω).

Let µ1 = 1 + ln(1 + ρT/e), µ2 = γ1 , and µ3 = 1 − ln(1 − ω). Similar to the proof procedure of Theorem 1, one may derive Corollary 2. 

Corollary 1. (Global power synchronization) Suppose that µ(t) = (1 + ρt)γ , τ(t) = ω1 t, and ςn = ω2 tn , n ∈ Z+ , where ρ > 0, γ > 0, and 0 ≤ ωi < 1, i = 1, 2. Under Assumption 2 and condition (11), if there exist constants δ > 1, T > 0 such that (i) γ ln(1 + ρT) + αT + δβT/(1 − ω1 )γ < ln δ, ( (1−ω )γ ) − δ2 P G (ii) ≤ 0. ∗ −P Then response system (2) is said to be globally power synchronized with drive system (1) under the control gain matrix K = GT P−1 over the class F (T).

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Corollary 3. (Global exponential synchronization) Suppose that µ(t) = exp(λt), 0 ≤ τ(t) ≤ τ, 0 ≤ ςn ≤ ς, where λ > 0, τ < ∞, ς < ∞. Under Assumption 2 and condition (11), if there exist constants δ > 1, T > 0 such that

P ROOF. In fact, we just verify Assumption 1. For any t ≥ 0, n ∈ Z+ , µ(tn ) 1 + ρtn γ 1 + ρ(tn − tn−1 ) + ρtn−1 γ =( ) =[ ] µ(tn−1 ) 1 + ρtn−1 1 + ρtn−1 γ

e+ρt

n ln(e + ρtn−1 ) + ln e+ρtn−1 ln(e + ρtn ) µ(tn ) = = µ(tn−1 ) ln(e + ρtn−1 ) ln(e + ρtn−1 )

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≤ (1 + ρT) , µ(t) 1 + ρt 1 =( )γ ≤ , µ∗ (t − τ(t)) 1 + ρ(1 − ω1 )t (1 − ω1 )γ 1 + ρtn µ(tn ) 1 =( )γ ≤ . µ∗ (tn − ςn ) 1 + ρ(1 − ω2 )tn (1 − ω2 )γ

Let µ1 = (1 + ρT)γ , µ2 = 1/(1 − ω1 )γ , and µ3 = 1/(1 − ω2 )γ . Similar to the proof procedure of Theorem 1, we can obtain that response system (2) is globally power synchronized with drive system (1). 

(i) λT + αT + δβ exp(λτ)T < ln δ, ) ( 1 − δ exp{λς} P G ≤ 0, (ii) ∗ −P

then response system (2) is said to be globally exponentially synchronized with drive system (1) under the control gain matrix K = GT P−1 over the class F (T). P ROOF. Here, we also verify Assumption 1. For any t ≥ 0, n ∈ Z+ , µ(tn ) exp(λtn ) = = exp[λ(tn − tn−1 )] ≤ exp(λT), µ(tn−1 ) exp(λtn−1 ) µ(t) exp(λt) = ≤ exp(λτ), ∗ µ (t − τ(t)) exp[λ(t − τ(t))] µ(tn ) exp(λtn ) = ≤ exp(λς). ∗ µ (tn − ςn ) exp[λ(tn − ςn )] Let µ1 = exp(λT), µ2 = exp(λτ), µ3 = exp(λς). Similar to the proof procedure of Theorem 1, one may derive Corollary 3. 

Corollary 2. (Global log synchronization) Suppose that µ(t) = ln(e + ρt), τ(t) = t + [e − (t + e)γ ]/ρ, and ςn = ωtn , n ∈ Z+ , where ρ > 0, 0 < γ ≤ 1, and 0 ≤ ω < 1. Under Assumption 2 and condition (11), if there exist constants δ > 1, T > 0 such that (i) ln[1 + ln(1 + ρT/e)] + αT + δβT/γ < ln δ,

4. Examples In this section, a numerical example is presented to show the validity of our results. 6

Journal Pre-proof Example 1. Consider drive system (1) and response system (2) with N = 4, n = 2, f (u, v) = A1 tanh(u) + A2 tanh(v), c = 0.5, c˜ = 1, τ(t) = 0.2t, Γ = I, where matrices A1 and A2 are given by: [ ] [ ] 1 1 1 −0.5 A1 = , A2 = . −1 1 0 1

ϯ

20

|e4(t)|

150

|e |, i=1,2,3,4

ei2, i=1,2,3,4

|e3(t)|

0 −20

i

−40 100 −60 400 200

50

0 ei1, i=1,2,3,4 0

c

ϭ

ϰ

|e2(t)|

0

20

40

60

80

100

t

10

−200 −400

|e2(t)|

0

t 100

3

|e4(t)|

7

ϯ

80

60

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|e (t)|

8

40

20

d

|e1(t)|

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6

6 5

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•

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−2

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−4

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−6 6

4

1 0

0

5

10

e

Ϯ

15

20

t

4

2 ei1, i=1,2,3,4 0

−4

t 20 15 10 5 0

|e1(t)| |e2(t)| |e3(t)|

8

|e4(t)|

7

Figure 2: The network topologies.

−2

10

9

|ei|, i=1,2,3,4

Ϯ

40

1

|ei|, i=1,2,3,4

ϰ

b |e (t)|

200

ei2, i=1,2,3,4

ď

ϭ

250

lP repro of

Ă

a

6 5 4 3 2 1

The network structure is shown in Figure 2. Figure 2 (a) corresponds to the non-delayed coupling matrix B and Figure 2 (b) corresponds to the time-varying delayed coupling matrix D. The matrices B and D are given by:     1 0  0 1   −1 0  −1 0  1 −1 0   0  0    0 −1 1 B =   , D =  . 1 −2 0  0 −1 0   1  1     1 0 1 −2 0 1 0 −1

0

0

20

40

60

80

t 100

Figure 3: (a) The trajectories of the synchronization errors |ei (t)| without control; (b) Phase plots of error systems (4) without control; (c) The trajectories of the

synchronization errors |ei (t)| with delayed impulse control; (d) Phase plots of error

systems (4) with delayed impulse control; (e) The trajectories of the synchronization errors |ei (t)| with delayed impulses, where the gain matrix is K∗ = K − 0.2I.

drive system (1).

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As indicated in Figure 3 (a) and (b), we know that response system (2) cannot be synchronized with drive system (1) without any control inputs.√ Based on Corollary 1, we consider the case, where µ(t) = 1 + 2t, ςn = tn /6, n ∈ Z+ . Choose δ = 1.75, α = 7.5, β = 1.3. Let impulse time sequences {tn } satisfy tn − tn−1 ≤ 0.05, n ∈ Z+ , i.e., T = 0.05. Then one may derive that µ1 = 1.0488, µ2 = 1.1180, µ3 = 1.0954, and (6) holds. And using the Matlab LMI toolbox, it is easy to verify that LMIs (11) and (12) hold and derive the following feasible solutions: [ ] [ ] 0.2254 −0.0408 0.0721 0 P= , Q= , ∗ 0.2130 ∗ 0.0721 [ ] 0.1500 0.0239 G= . 0.0238 −0.1489

In addition, if the time-varying√delay τ(t) is not linear but nonlinear, such as τ(t) = t + 5[e − t + e]. Then, it is adequately shown that response system (2) cannot achieve synchronization with drive system (1) without control, see Figure 4 (a) and (b). It follows from Corollary 2 that we choose µ(t) = ln(e + t/5), α = 8.5, β = 1.3, δ = 2, T = 0.05, ςn = tn /5, n ∈ Z+ . One may derive that µ1 = 1.0037, µ2 = 2, µ3 = 1.2231 such that (6), LMIs (11), and (12) hold. By utilizing the Matlab LMI toolbox, the following feasible solutions can be derived: [ ] [ ] 0.3385 0.0263 0.0905 0 , Q= , P= ∗ 0.2162 ∗ 0.0905 [ ] 0.2019 0.0278 G= . 0.0190 0.1350 Then impulsive gain K is designed as follows: [ ] 0.5920 0.0078 K= . 0.0568 0.6236

Then delayed impulsive gain is designed as follows: [ ] 0.7104 −0.0215 K= . 0.2483 −0.7030

For simulations, we take tn − tn−1 = 0.05, n ∈ Z+ . As indicated in Figure 4 (c) and (d), we know that response system (2) become synchronized with drive system (1) under such class of delayed impulsive control. Similarly, if we change the gain matrix and simulate the case that the gain matrix becomes K∗ = K + 0.2I, which is against Corollary 2, then we know that response system (2) cannot be synchronized with drive system (1) as indicated in Figure 4 (e).

For simulations, we take tn − tn−1 = 0.05, n ∈ Z+ . As indicated in Figure 3 (c) and (d), we know that response system (2) become synchronized with drive system (1) under such class of delayed impulsive control. However, if we make a slight change on the gain matrix K∗ = K − 0.2I, then it is easy to check that this case is against Corollary 1. In the case, it is shown in Figure 3 (e) that response system (2) cannot be synchronized with 7

Journal Pre-proof

a 1000

|e1(t)|

900

|e2(t)|

800

|e3(t)|

700

|e4(t)|

b

400

ei2, i=1,2,3,4

|ei|, i=1,2,3,4

600

600 500 400

200 t 0

100

300 −200 1000

200

80 60 500

100

c

0

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|e2(t)|

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ei2, i=1,2,3,4

|ei|, i=1,2,3,4

d

|e1(t)|

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1

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0 1

−4 0

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30 t

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|e2(t)| |e3(t)|

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|ei|, i=1,2,3,4

t

|e1(t)|

9

6 5 4 3 2 1 0

0

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6 −6 100

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10

0

40

0 ei1, i=1,2,3,4

t

lP repro of

0

t

0

10

20

30

40

50

60

Figure 4: (a) The trajectories of the synchronization errors |ei (t)| without control; (b) Phase plots of error systems (4) without control; (c) The trajectories of the

synchronization errors |ei (t)| with delayed impulsive control; (d) Phase plots of error

systems (4) with delayed impulses; (e) The trajectories of the synchronization errors |ei (t)| with the delayed impulses, where the gain matrix is K∗ = K + 0.2I.

5. Conclusion

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In this paper, delayed impulsive synchronization problems for complex networks with time-varying delays of unknown bound have been investigated. Based on a class of novel delayed impulsive differential inequality and the Lyapunov function method, a class of delayed impulsive controllers have been proposed to guarantee the synchronization of complex networks with time-varying delays of unknown bound. Some different µ-synchronization criteria have been obtained. A numerical example has been given to illustrate the effectiveness of the theoretical results. In the near future, we shall consider finitetime synchronization of complex networks via delayed impulsive control. References

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[1] Faydasicok, O., & Arik, S. (2013). A new upper bound for the norm of interval matrices with application to robust stability analysis of delayed neural networks. Neural Networks, 44, 64-71. [2] Selvaraj, P., Kwon, M., & Sakthivel, R. (2019). Disturbance and uncertainty rejection performance for fractional-order complex dynamical networks. Neural Networks, 112, 73-84. [3] Ali, S., Usha, M., Orman, Z., & Arik, S. (2019). Improved result on state estimation for complex dynamical networks with time varying delays and stochastic sampling via sampled-data control. Neural Networks, 114, 2837. [4] Guan, Z., & Zhang, H. (2008). Stabilization of complex network with hybrid impulsive and switching control. Chaos, Solitons and Fractals, 37, 1372-1382.

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Conflict of interest statement

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All authors decare that there is no any conflict of interest.