Fixed-time outer synchronization of hybrid-coupled delayed complex networks via periodically semi-intermittent control

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Journal of the Franklin Institute 356 (2019) 6656–6677 www.elsevier.com/locate/jfranklin

Fixed-time outer synchronization of hybrid-coupled delayed complex networks via periodically semi-intermittent control Qintao Gan∗, Feng Xiao, Hui Sheng Shijiazhuang Campus, Army Engineering University, Shijiazhuang 050003, PR China Received 30 July 2018; received in revised form 8 December 2018; accepted 15 March 2019 Available online 27 June 2019

Abstract In this paper, the fixed-time synchronization between two delayed complex networks with hybrid couplings is investigated. The internal delay, transmission coupling delay and self-feedback coupling delay are all included in the network model. By introducing and proving a new and important differential equality, and utilizing periodically semi-intermittent control, some fixed-time synchronization criteria are derived in which the settling time function is bounded for any initial values. It is shown that the control rate, network size and node dimension heavily influence the estimating for the upper bound of the convergence time of synchronization state. Finally, numerical simulations are performed to show the feasibility and effectiveness of the control methodology by comparing with the corresponding finitetime synchronization problem. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Complex network is a useful modelling tool in understanding the dynamical behaviors of many natural and artificial systems, such as wireless communication networks, biological neural networks, power grids, the World Wide Web, Internet, social relation, traffic networks, scientific cooperation networks and disease transmission networks, and so on [7]. In the ∗

Corresponding author. E-mail address: [email protected] (Q. Gan).

https://doi.org/10.1016/j.jfranklin.2019.03.033 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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past decades, spurred by the discovery of the small-world and scale-free properties, complex networks have drawn increasing interest on modeling, analysis, control and applications. The models of complex networks have been traditionally formulated in relation to the time evolution and are govern by ordinary differential equations without delays. In fact, the inclusion of internal delay and coupling delay in such models make them more realistic by allowing to describe the effects of the finite speeds and spreading as well as traffic congestion for nodes’ behaviors and transmission exchange of information between nodes, respectively. The issues of integrating hybrid delays into the studies of dynamics and control for complex networks require more complicated analysis. Therefore, it is interesting to study this problem both in theory and in applications. Recently, the authors of [4,23,35] paid attentions on complex networks with different time-varying internal delay, transmission coupling delay from neighbors and self-feedback coupling delay transmission. Consequently, the results obtained in those papers are not only less conservative, but also effectually complement or improve the previously known results. Synchronization, as one of the most important collective behaviors in complex networks, has been extensively investigated [1,2,9,13,30,51,52]. Synchronization of complex networks can be divided into two major categories: inner synchronization and outer synchronization. The first one means that all the nodes in a complex network eventually approach to trajectory of a target node, and the second one exhibits the synchronization between two or more complex networks [20]. Furthermore, different from exponential synchronization and asymptotical synchronization, finite-time synchronization means that synchronization can be achieved in finite-time, and the settling time depends on the initial values [25,40]. It is well known that finite-time synchronization has demonstrated faster convergence rate, better robustness against uncertainties and disturbance rejection properties [3]. Therefore, many scientific and technical works have been joining the studies for finite-time synchronization of complex networks and a large body work has been reported in the literatures (see, for example [10,15,17,24,26,27,31,36] and the references therein). In many practical situations, the knowledge of initial values of complex networks can not be exactly known in advance, which limits the practical application scopes of finite-time synchronization. Recently, another kind of synchronization, called fixed-time synchronization have been introduced into complex networks for controlling and applications in efforts to overcome these drawbacks. Fixed-time synchronization means that synchronization can be achieved in a settling time which is bounded by a constant for any initial values. Up to now, many authors have studied different kinds of complex networks models and a significant body of work has been carried out. For example, the fixed-time synchronization problem of hybrid coupled networks with time-varying delays was investigated in [6]. In [12], the fixed-time synchronization of coupled neural networks with discontinuous was studied by designing a discontinuous control law and a high-precision estimation of the settling time was derived; Khanzadeh and Pourghoil [19] studied the fixed-time synchronization of complex dynamical networks with nonidentical nodes in the presence of bounded uncertainties and disturbances by using sliding mode control technique. the fixed-time cluster synchronization problem for complex networks with or without pinning control were discussed in [25]. Shi et al. [33] investigated the fixed-time outer synchronization of complex networks with noise coupling; fixed-time synchronization for a class of complex networks with synchronizing and desynchronizing impulses was considered by a unified method in [45]. Zhang et al. [48] developed a fixed-time control for a class of complex networks with nonidentical nodes

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and stochastic noise perturbations; fixed-time synchronization of multi-links complex networks was studied in [49]. In fact, due to the complexities of node dynamics and topological structure, complex networks are not always able to synchronize by themselves. Therefore, various effective control approaches such as feedback control [46], adaptive control [43], pinning control [25], impulsive control [1], intermittent control [4] and event-triggered control [32,34] have been reported to realize synchronization. It is worth noting that the works mentioned above focused on fixedtime synchronization via continuous control. Different from the continuous control approaches, the discontinuous control methods which include impulsive control, intermittent control and event-triggered control have attracted more and more interest because these control approaches are more economic and reduce the amount of the transmitted information. In [37], by using the algebraic graph theory and control theory, a novel second-order consensus protocol for multi-agent systems with synchronous intermittent information feedback and directed topology was proposed, and detailed analysis of the convergence of the states of the multiple agents steered by the presented protocol was performed. A kind of intermittent consensus protocols for second-order multi-agent systems with time delayed nonlinear dynamics and fixed directed communication topology was introduced and investigated in [38]. The difference between intermittent control and impulsive control is that intermittent control is activated during some time intervals (work intervals) and does not work during the other intervals (rest intervals), while impulsive control is activated only at some isolated time-points. Obviously, when rest intervals are zero, intermittent control reduces to continuous control, while work intervals are infinitely close to zero means that intermittent control becomes the impulsive control. The difference between intermittent control and event-triggered control is that intermittent control is based on the time-scheduled control, while event-triggered control is only updated when an event are triggered, and the triggering instants are based on a pre-designed condition. Recently, the finite-time synchronization problems for complex networks via intermittent control have been studied in [8,16,20,22,29,47]. In [21], the fixed-time consensus protocols for multi-agent systems with nonlinear uncertainties were presented based on event-triggered strategies which can significantly reduce control cost and the frequency of the controller updates. However, to the best of our knowledge, there are no results concerning the fixed-time synchronization methodology for complex networks by using intermittent control. The reasons or difficulties are listed in three aspects. The first reason is that it is difficult to design suitable intermittent controller to ensure fixed-time synchronization. The second reason is that it is lacking of the theory for fixed-time synchronization based on intermittent control. The third is that it is not easy to construct a Lyapunov functional to meet stringent technical requirements compared with asymptotic synchronization or exponential synchronization. Although the intermittent control has been applied to study asymptotic synchronization or exponential synchronization of complex networks in [4,5,28,41,50], it is difficult to deal with fixed-time synchronization of complex networks. Thus, it is urgent to develop some new analysis techniques and methods in investigating fixed-time synchronization of hybrid-coupled delayed complex networks via intermittent control. This situation motivates our present investigation. By introducing and proving a new and important differential inequality as a vital lemma, this paper is concerned with fixed-time outer synchronization for a class of hybrid-coupled delayed complex networks by using the periodically semi-intermittent control approach, in which both the internal delay, transmission coupling delay and self-feedback coupling delay are included in the networks model.

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The organization of this paper is as follows: in Section 2, problem statement and preliminaries are presented; in Section 3, a periodically semi-intermittent controller is proposed to ensure fixed-time outer synchronization of the addressed hybrid-coupled delayed complex networks; numerical simulations will be given in Section 4 to demonstrate the effectiveness and feasibility of our theoretical results; we end this work with a conclusion in Section 5.

2. Modeling and preliminary Consider a class of complex networks consisting of N identical dynamical nodes with linearly non-delayed and delayed couplings, which can be described as follows [4]: x˙i (t ) = f (t, xi (t ), xi (t − τ1 (t ))) + c1

N 

ai j G1 (x j (t ) − xi (t ))

j =1, j =i N 

+ c2

bi j G2 (x j (t − τ2 (t )) − xi (t − τ3 (t ))), i = 1, 2, . . . , N,

(2.1)

j =1, j =i

where xi (t ) = (xi1 (t ), xi2 (t ), . . . , xin (t ))T ∈ Rn , f : R × Rn × Rn → Rn is a nonlinear vector function, cl (l = 1, 2) is the coupling strength, Gl ∈ Rn×n (l = 1, 2) is the inner matrix linking the coupled variables, τ 1 (t) ∈ [0, τ 1 ] is the internal delay occurring inside the dynamical node, τ 2 ∈ [0, τ 2 ] represent the transmission delay from node j to node i (i = j), τ 3 ∈ [0, τ 3 ] is the self-feedback delay, A = (ai j )N×N and B = (bi j )N×N denote the topological structure of the complex networks for non-delayed configuration and delayed one, respectively, in which, aij , bij > 0 if there is a connection from node i to node j (i = j), and ai j = bi j = 0 (i = j) otherwise, the diagonal elements of A and B are defined as aii = −

N  j =1, j =i

ai j , bii = −

N 

bi j , i = 1, 2, . . . , N.

j =1, j =i

The initial conditions of Eq. (2.1) are given as xi (s) = φi (s) ∈ C([−τ, 0], Rn ) (i = 1, 2, . . . , N ), where C([−τ, 0], Rn ) represents the set of all n-dimensional continuous differentiable functions defined on the interval [−τ, 0] and τ = max{τ1 , τ2 , τ3 }. Remark 1. Clearly, in this paper, the coupling configuration matrices A and B are not necessary to be identical, symmetric or irreducible, that is, the corresponding graphs generated by matrices A and B can be directed, weakly connected and even there is no rooted spanning directed tree. Furthermore, in order to realize the complete synchronization of system (2.1), the assumption b11 = b22 = · · · = bNN = −a < 0 must be imposed in [4,23], which has been abandoned in this paper. Obviously, network model (2.1) is a generalization of those considered in [4,23] and can describe many realistic complex networks better. Motivating applications include different reaction delays to the node’s own behavior and the behavior of its neighbors or computation delays in combination with transmission delays. As far as we know, many networks in the real world can be described by hybrid-coupled delayed complex network models, such as traffic flow models, communication networks, social networks, infectious disease spreads models, species development in balance models, and so on.

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System (2.1) can be rewritten as x˙i (t ) = f (t, xi (t ), xi (t − τ1 (t ))) + c1

N 

ai j G1 x j (t ) + c2

j=1

N 

bi j G2 x j (t − τ2 (t ))

j=1

− c2 bii G2 (xi (t − τ2 (t )) − xi (t − τ3 (t ))), i = 1, 2, . . . , N.

(2.2)

To make two hybrid-coupled complex networks achieve fixed-time synchronization, we refer to system (2.2) as the drive network, and the response network is given by y˙i (t ) = f (t, yi (t ), yi (t − τ1 (t ))) + c1

N 

ai j G1 y j (t ) + c2

j=1

N 

bi j G2 y j (t − τ2 (t ))

j=1

− c2 bii G2 (yi (t − τ2 (t )) − yi (t − τ3 (t ))) + ui (t ), i = 1, 2, . . . , N,

(2.3)

where yi (t ) = (yi1 (t ), yi2 (t ), . . . , yin (t ))T ∈ Rn , ui (t) is the control input of the ith node in the response network, and the other parameters have the same meanings with the corresponding parameters in the drive network (2.2). The initial conditions of Eq. (2.3) are given as yi (s) = ψi (s) ∈ C([−τ, 0], Rn ) (i = 1, 2, . . . , N ). Let ei (t ) = yi (t ) − xi (t ) be the error states for i = 1, 2, . . . , N . It is easy to see that e˙i (t ) = f¯(t, ei (t ), ei (t − τ1 (t ))) + c1

N  j=1

ai j G1 e j (t ) + c2

N 

bi j G2 e j (t − τ2 (t ))

j=1

− c2 bii G2 (ei (t − τ2 (t )) − ei (t − τ3 (t ))) − ui (t ), i = 1, 2, . . . , N,

(2.4)

where f¯(t, ei (t ), ei (t − τ1 (t ))) = f (t, yi (t ), yi (t − τ1 (t ))) − f (t, xi (t ), xi (t − τ1 (t ))). In this paper, the fixed-time outer synchronization is realized by using periodically semiintermittent control, which is designed in the following form: ⎧

p+1 3  t  ⎪ ξr ei (t ) T ⎪ 2 e (s) e (s)ds ⎪−ki ei (t ) − α i i t −τ (t ) 1 −σ ei (t )2 ⎪ r r ⎪ r=1 ⎪ ⎨

q+1 3  t  ξr ei (t ) T 2 ui (t ) = −β (2.5) e (s) e (s)ds i i 1−σr t −τr (t ) ei (t )2 ⎪ ⎪ r=1 ⎪ ⎪ ⎪ −αsign (ei (t ))|ei (t )| p − βsign (ei (t ))|ei (t )|q , mT ≤ t < (m + θ )T , ⎪ ⎩ −ki ei (t ), (m + θ )T ≤ t < (m + 1)T , where m = 0, 1, 2, . . . , ki and ξ r (r = 1, 2, 3) are positive constants denoting the control strengths, α and β are tunable constants, 0 < p < 1, q > 1, T > 0 denotes the control period, θ is the ratio of the control width to the control period called control rate. In order to obtain our main results, the following assumptions, definition and lemmas are necessary. Assumption 1 [4,5]. For the vector-valued function f : R × Rn × Rn → Rn , there exist positive constants η and ζ such that f satisfies  (y(t ) − x(t ))T f (t, y(t ), y(t − τ1 (t ))) − f (t, x(t ), x(t − τ1 (t ))) ≤ η(y(t ) − x(t ))T (y(t ) − x(t )) + ζ (y(t − τ1 (t )) − x(t − τ1 (t )))T (y(t − τ1 (t )) − x(t − τ1 (t ))) for any x(t ), y(t ) ∈ Rn and t ≥ 0.

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Assumption 2. Time-delay functions τr (t ) : [0, +∞ ) → [0, +∞ ) (r = 1, 2, 3) are realvalued continuous and satisfy τ˙r (t ) ≤ σr < 1. Definition 1 [14]. The controlled response network (2.3) is said to achieve synchronization with the drive network (2.2) in finite time, if there exists a fixed settling time T˜ (e(0)) ≥ such that limt→T˜ (e(0)) ei (t ) = 0 and ei (t) ≡ 0 for t > T˜ (e(0)), i = 1, 2, . . . , N, where e(t ) = (e1 (t ), e2 (t ), . . . , eN (t ))T , e(0) = sup−τ ≤s≤0 e(s). The time

 T (e(0)) = inf T˜ (e(0)) ≥ 0 : ei (t ) ≡ 0for anyt > T˜ (e(0)), i = 1, 2, . . . , N is called the settling time of synchronization, which is dependent of the initial synchronization error e(0). Furthermore, if there exist a fixed-time Tf ≥ 0 which is independent of the initial synchronization error e(0), such that limt→Tf ei (t ) = 0 and ei (t) ≡ 0 for t > Tf , i = 1, 2, . . . , N, the controlled response network (2.3) is said to be fixed-time synchronized with the drive network (2.2). Lemma 1. If Z and Z˜ are real matrices with appropriate dimensions, then there exists a positive constant ζ such that 1 Z T Z˜ + Z˜ T Z ≤ ζ Z T Z + Z˜ T Z˜ . ζ Lemma 2 [18]. Let ai ≥ 0 (i = 1, 2, . . . , n), 0 < p < 1, q > 1. Then, the following inequalities hold: n n n n



   aip ≥ ai p , aiq ≥ n1−q ai q . i=1

i=1

i=1

i=1

Lemma 3 [22,29]. Suppose that function V(t) is continuous and non-negative when t ∈ [−τ, +∞ ) and satisfies the following conditions:  V˙ (t ) ≤ −αV η (t ), t ∈ [mT , (m + θ )T ), (2.6) V˙ (t ) ≤ 0, t ∈ [(m + θ )T , (m + 1)T ), where m = 0, 1, 2, . . . , 0 < η, θ < 1, α, T > 0, then the following inequality holds

 V 1−η (t ) ≤ sup V (s) 1−η − αθ (1 − η)t, 0 ≤ t ≤ Ts , −τ ≤s≤0

(2.7)

in which Ts is the settling time satisfies 

sup V (s) 1−η −τ ≤s≤0 Ts = . αθ (1 − η) In the following, we will give a very useful lemma to deal with the fixed-time synchronization for ordinary differential inequalities with periodically intermittent control. Lemma 4. Suppose that function V(t) is non-negative and satisfies the following conditions:  V˙ (t ) ≤ −αV q (t ) − βV p (t ), t ∈ [mT , (m + θ )T ), (2.8) V˙ (t ) ≤ 0, t ∈ [(m + θ )T , (m + 1)T ), where α > 0, β > 0, T > 0, 0 < p < 1, q > 1, 0 < θ < 1, m = 0, 1, 2, . . .. Then V(t) ≡ 0, if 1 1 t≥ + . αθ (q − 1) βθ (1 − p)

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Proof. Consider the following system for comparison purpose: ⎧ ⎧ ⎨−αW q (t ), W (t ) ≥ 1, ⎪ ⎪ ⎪ ⎪ ˙ t ∈ [mT , (m + θ )T ), ⎨W (t ) = −βW p (t ), 0 < W (t ) < 1, ⎩ 0, W (t ) = 0, ⎪ ⎪ W˙ (t ) = 0, t ∈ [(m + θ )T , (m + 1)T ), ⎪ ⎪ ⎩ W (0) = V (0).

(2.9)

Comparing Eq. (2.8) with Eq. (2.9), one can see that 0 ≤ V(t) ≤ W(t). Therefore, if there exists a time Tf > 0 such W(t) ≡ 0 for t > Tf , then V(t) ≡ 0 for t > Tf . Let U (t ) = W 1−q (t ) when W(t) ≥ 1. It is easy to see from Eq. (2.9) that U(0) → 0 when W (0) → +∞, and U(t) → 1 for W(t) → 1. Therefore, ⎧ t ∈ [mT , (m + θ )T ), 0 < U (t ) ≤ 1, ⎨U˙ (t ) = α(q − 1), (2.10) U˙ (t ) = 0, t ∈ [(m + θ )T , (m + 1)T ), ⎩ U (0) = U0 = W 1−q (0). Let U (t ) = W 1−p (t ) when 0 ≤ W(t) < 1. It follows from Eq. (2.9) that U(t) → 1 when W(t) → 1, and U(t) → 0 for W(t) → 0. Thus, ⎧ ⎨U˙ (t ) = −β(1 − p), t ∈ [mT , (m + θ )T ), 0 ≤ U (t ) < 1, (2.11) U˙ (t ) = 0, t ∈ [(m + θ )T , (m + 1)T ), ⎩ U (0) = 1. Consequently, the global stability within fixed time for the zero solution of Eq. (2.9) has been transformed to the two problems: (1) the solution of Eq. (2.10) approaches to 1 in a fixed time T1 ; (2) the solution of Eq. (2.11) approaches to 0 in a fixed time T2 . Hence, from any initial value W(0), W(t) → 0 in the fixed time T f = T1 + T2 . For t ∈ [mT , (m + θ )T ), it can be deduced from Eq. (2.10) that U (t ) = U (0) +

m−1   i=0

(i+θ )T

 α(q − 1)ds +

iT

t

α(q − 1)ds

mT

= U (0) + α(q − 1)[t − mT (1 − θ )].

(2.12)

It follows from the fact U(0) < 1 and limt→+∞ U (t ) = +∞ that there exists a time T1 such that limt→T1 U (t ) = 1 and 0 < U(t) < 1 for 0 < t < T1 . Consequently, it can be derived from Eq. (2.12) that U (0) + α(q − 1)[t − mT (1 − θ )] = 1, which means that α(q − 1)[t − mT (1 − θ )] ≤ 1.

(2.13)

If t ∈ [mT , (m + θ )T ), we have m ≤ t/T, then it follows from Eq. (2.13) that t≤

1 . αθ (q − 1)

(2.14)

Similarly, for t ∈ [(m + θ )T , (m + 1)T ), m  (i+θ )T  U (t ) = U (0) + α(q − 1)ds = U (0) + α(q − 1)(m + 1)θ T , i=0

iT

(2.15)

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and m + 1 > t/T , then we also have t≤

1 . αθ (q − 1)

(2.16)

Hence, for any t ∈ [0, +∞ ), we always have T1 =

1 . αθ (q − 1)

(2.17)

Next, we will estimate the time T2 such that U(t) of Eq. (2.10) approaches to 0 from 1. Similar to Eq. (2.12), for t ∈ [mT , (m + θ )T ), we can derive from Eq. (2.11) that U (t ) = U (0) −

m−1   i=0

(i+θ )T

 β(1 − p)ds −

iT

t

β(1 − p)ds

mT

= U (0) − β(1 − p)[t − mT (1 − θ )].

(2.18)

It can be found that U(t) is decreasing on [0, +∞ ). Let U (0) = 1 and U (t ) = 0, we have 0 = 1 − β(1 − p)[t − mT (1 − θ )] ≤ 1 − βθ (1 − p)t. Set h(t ) = 1 − βθ (1 − p)t. Since h(0) = 1 > 0, h(+∞ ) = −∞ < 0 and h˙ (t ) = −βθ (1 − p) < 0, there exists an unique T2 > 0 satisfying h(T2 ) = 0, and T2 =

1 . βθ (1 − p)

(2.19)

For t ∈ [(m + θ )T , (m + 1)T ), we have m  (i+θ )T  U (t ) = U (0) − β(1 − p)ds = U (0) − β(1 − p)(m + 1)θ T . i=0

(2.20)

iT

In a similar way, we can get T2 =

1 . βθ (1 − p)

(2.21)

Combining Eqs. (2.17) and (2.21), one has that, from any initial value W(0), W(t) → 0 in the fixed time T f = T1 + T2 . Therefore, V(t) ≡ 0 for t ≥ Tf . This completes the proof.  Remark 2. The controllers for realizing synchronization can be classified into two categories, namely, the continuous ones and discontinuous. In Refs. [6,19,25,33,48,49], the synchronization problems for complex networks via continuous controllers have been investigated. However, the existing methods can not be used to deal with the fixed-time synchronization problems via periodically intermittent control. Therefore, a new and important differential equality is introduced and proved here, based on which some fixed-time synchronization criteria will be derived in Section 3. 3. Synchronization analysis of complex networks In this section, we will discuss the fixed-time outer synchronization problem for the delayed complex networks with hybrid couplings and mixed time-varying delays under a periodically semi-intermittent controller. The main results are stated as follows.

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Theorem 1. Suppose that Assumptions 1 and 2 hold. If there exist positive constants η, ζ , ζ r , ξ r (r = 1, 2, 3) and ki , (i = 1, . . . , N ) such that εIN + c1 ρ0 A˜ − K ≤ 0, ζ − ξ21 ≤ 0, c2 − 2ζc22 bmin − ξ22 ≤ 0, 2ζ1 c2 − 2ζ3 bmin − ξ23 ≤ 0,

(i) (ii ) (iii ) (iv)

where IN is the N-dimensional identity matrix, K = diag(k1 , k2 , . . . , kN ), bmin = min (bii ) (i = 1, 2, . . . , N ), ρ min is the minimum eigenvalue of matrix (G1 + GT1 )/2, ρ0 = G1 , A˜ s = (A˜ + A˜ T )/2 with A˜ is a modified matrix of A by replacing the diagonal elements aii by (aii ρ min /ρ 0 ) (it is easy to see that A˜ is a symmetric matrix with nonnegative off-diagonal elements), and c 2 ζ1 1  ξr c2 (ζ2 + ζ3 ) λmax (BT B)λmax (GT2 G2 ) − λmax (GT2 G2 )bmin + η + , 2 2 2 r=1 1 − σr 3

ε=

then the drive network (2.2) and response network (2.3) can achieve fixed-time synchronization under the periodically semi-intermittent controller (2.5). The fixed settling time Tf can be estimated by Tf ≤

1 1 + . αθ (q − 1) β(N n) 1−q 2 θ (1 − p)

(3.1)

Proof. Let e(t ) = (eT1 (t ), eT2 (t ), . . . , eTN (t ))T and denote 1 T 1   ξr V (t ) = ei (t )ei (t ) + 2 i=1 2 i=1 r=1 1 − σr N

N

3



t

t −τr (t )

eTi (s)ei (s)ds.

(3.2)

When mT ≤ t < (m + θ )T (m = 0, 1, 2, . . . ), taking the derivative of V(t) with respect to time t along the solutions of Eq. (2.4), it follows from Assumptions 1 and 2 that V˙ (t ) ≤

N 

N N    eTi (t ) f¯(t , ei (t ), ei (t − τ1 (t ))) + c1 ai j G1 e j (t ) + c2 bi j G2 e j (t − τ2 (t ))

i=1

j=1

 − c2 bii G2 (ei (t − τ2 (t )) − ei (t − τ3 (t ))) −

N 

j=1

ki eTi (t )ei (t )

i=1

−α

N 

eTi (t )sign (ei (t ))|ei (t )| p − α

i=1

−β

N 

i=1 r=1

eTi (t )sign (ei (t ))|ei (t )|q − β

i=1

N  3   i=1 r=1

ξr 1 − σr ξr 1 − σr



t

t −τr (t )



t

t −τr (t )

eTi (s)ei (s)ds

3

N

p+1

eTi (s)ei (s)ds

1   ξr 1  T eTi (t )ei (t ) − ξr ei (t − τr (t ))ei (t − τr (t )). 2 i=1 r=1 1 − σr 2 i=1 r=1 N

+

N  3  

2

q+1 2

3

By virtue of Lemma 1, the following inequalities can be derived

(3.3)

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c2

N  N 

6665

bi j eTi (t )G2 e j (t − τ2 (t ))

i=1 j=1 N N  c2  T c 2 ζ1 λmax (BT B)λmax (GT2 G2 ) eTi (t )ei (t ) + e (t − τ2 (t ))ei (t − τ2 (t )), 2 2ζ1 i=1 i i=1

≤

− c2

N 

(3.4)

bii eTi (t )G2 ei (t − τ2 (t ))

i=1

c2

≤−

N  1 c2   T bii ζ2 ei (t )GT2 G2 ei (t ) + eTi (t − τ2 (t ))ei (t − τ2 (t )) 2 i=1 ζ2

≤−

N N  c 2 ζ2 c2  λmax (GT2 G2 ) bii eTi (t )ei (t ) − bii eTi (t − τ2 (t ))ei (t − τ2 (t )), 2 2ζ 2 i=1 i=1

N 

(3.5)

bii eTi (t )G2 ei (t − τ3 (t ))

i=1 N  1 c2   T ≤− bii ζ3 ei (t )GT2 G2 ei (t ) + eTi (t − τ3 (t ))ei (t − τ3 (t )) 2 i=1 ζ3

≤−

N N  c 2 ζ3 c2  λmax (GT2 G2 ) bii eTi (t )ei (t ) − bii eTi (t − τ3 (t ))ei (t − τ3 (t )), 2 2ζ 3 i=1 i=1

(3.6)

and c1

N  N 

ai j eTi (t )G1 e j (t )

i=1 j=1

= c1

N N  

ai j eTi (t )G1 e j (t ) + c1

i=1 j =1, j =i

≤ c1

N 

N 

N  i=1

ai j ρ0 ei (t )e j (t ) + c1

i=1 j =1, j =i

= c1 ρ0

N  N 

aii eTi (t )G1 ei (t ) N 

aii ρmin eTi (t )ei (t )

i=1

a˜i j ei (t )e j (t )

i=1 j=1

= c1 ρ0

N 

eTi (t )A˜ ei (t ),

(3.7)

i=1

where a˜i j = ai j when i = j and a˜ii = aii ρmin /ρ0 for i, j = 1, 2, . . . , N . Furthermore, it follows from Assumption 1 and Lemma 2 that N  i=1

eTi (t ) f¯(t, ei (t ), ei (t

− τ1 (t ))) ≤ η

N  i=1

eTi (t )ei (t )

+ζ

N  i=1

eTi (t − τ1 (t ))ei (t − τ1 (t )), (3.8)

6666

−α

Q. Gan, F. Xiao and H. Sheng / Journal of the Franklin Institute 356 (2019) 6656–6677

N 

eTi (t )sign (ei (t ))|ei (t )| p = −α

i=1

N  n 

|ei j (t )|1+ p ≤ −α

i=1 j=1

N 

eTi (t )ei (t )

1+ p 2

,

(3.9)

i=1

and −β

N 

eTi (t )sign (ei (t ))|ei (t )|q = −β

i=1

N  n 

|ei j (t )|1+q ≤ −β(N n)

i=1 j=1

1−q 2

N 

eTi (t )ei (t )

1+q 2

.

i=1

(3.10) Substituting Eqs. (3.4)–(3.10) to Eq. (3.3), combining with Lemma 2, for mT ≤ t < (m + θ )T (m = 0, 1, 2, . . . ), we obtain 

c 2 ζ1 c2 (ζ2 + ζ3 ) λmax (BT B)λmax (GT2 G2 ) − λmax (GT2 G2 )bmin + c1 ρ0 A˜ + η 2 2  3 N N  ξ1  T 1  ξr + −K eTi (t )ei (t ) + ζ − e (t − τ1 (t ))ei (t − τ1 (t )) 2 r=1 1 − σr 2 i=1 i i=1

V˙ (t ) ≤

N c c2 ξ2  T 2 + − bmin − e (t − τ2 (t ))ei (t − τ2 (t )) 2ζ1 2ζ2 2 i=1 i N c ξ3  T 2 bmin + e (t − τ3 (t ))ei (t − τ3 (t )) 2ζ3 2 i=1 i  t N N  3 

1+ p

p+1   ξr −α eTi (t )ei (t ) 2 − α eTi (s)ei (s)ds 2 1 − σr t −τr (t ) i=1 i=1 r=1  t N N  3  

1+q

q+1  1−q ξr − β(N n) 2 eTi (t )ei (t ) 2 − β eTi (s)Pei (s)ds 2 1 − σr t −τr (t ) i=1 i=1 r=1  t N N  3 

1+ p

p+1   ξr T 2 ≤ −α ei (t )ei (t ) −α eTi (s)ei (s)ds 2 1 − σr t −τr (t ) i=1 i=1 r=1  t N N 3  

1+q

q+1  1−q ξr T 2 2 −β(N n) ei (t )ei (t ) −β eTi (s)ei (s)ds 2 1 − σr t −τr (t ) i=1 i=1 r=1    t N N  3  p+1 ξr ≤ −α eTi (t )ei (t ) + eTi (s)ei (s)ds 2 1 − σ r t −τr (t ) i=1 i=1 r=1   N N 3   ξr  t 1−q q+1 − β(N n) 2 eTi (t )ei (t ) + eTi (s)ei (s)ds 2 1 − σr t −τr (t ) i=1 i=1 r=1

−

= −αV

p+1 2

(t ) − β(N n)

1−q 2

V

q+1 2

(t ).

Similarly, when (m + θ )T ≤ t < (m + 1)T (m = 0, 1, 2, . . . ), we can get that

(3.11)

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c 2 ζ1 c2 (ζ2 + ζ3 ) λmax (BT B)λmax (GT2 G2 ) − λmax (GT2 G2 )bmin + c1 ρ0 A˜ 2 2  3 N 1  ξr +η+ −K eTi (t )ei (t ) 2 r=1 1 − σr i=1

V˙ (t ) ≤

≤ 0.

(3.12)

According to Lemma 4, for any initial value V(0) > 0, there exists a fixed time Tf , which is independent on V(0), such that limt→Tf V (t ) = 0 and V(t) ≡ 0 for t > Tf . Since e(t)2 ≤ V(t)/λmin (P), one has limt→Tf ei (t ) = 0 and ei (t) ≡ 0 for t > Tf . It follows from Definition 1 that the drive network (2.2) and response network (2.3) can achieve outer synchronization under the periodically semi-intermittent controller (2.5), and the fixed settling time is estimated by Eq. (3.1). The proof of Theorem 1 is completed.  Remark 3. By introducing a novel and important inequality in Lemma 4, the sufficient conditions on fixed-time outer synchronization via periodically semi-intermittent control are obtained, which reduces the control cost. Our results have been shown to be the generalization and improvement of the existing results reported recently in the literatures [6,33,48]. Remark 4. In the proof process of Theorem 1, many inequalities have been utilized, which lead to the conservativeness of the synchronization criteria. Evidently, there is an interesting open problem concerning the fixed-time outer synchronization conditions in terms of linear matrix inequalities for hybrid-coupled delayed complex networks by using intermittent control. This will become our future investigative direction. Remark 5. Similar to [16,42], the controller (2.5) is called periodically semi-intermittent controller, which is somewhat conservative than the classical periodically intermittent controller. It is worth mentioning that the strong inequality condition V˙ (t ) ≤ 0 in Lemma 4 is indispensable when t ∈ [(m + θ )T , (m + 1)T ) (m = 0, 1, 2, . . . ), which means that networks (2.2) and (2.4) should be asymptotical synchronization when t ∈ [(m + θ )T , (m + 1)T ). Since complex networks are not always able to synchronize by themselves, some control inputs should be imposed on nodes when t ∈ [(m + θ )T , (m + 1)T ). This strong inequality condition limits the applications of Lemma 4 and the designing of intermittent controller. To overcome these difficulties, some conservative restrictions on vector function f were required in [29,47], which can guarantee that networks can be self-synchronization without control inputs. In [20], a novel differential inequality was established, in which the derivative of the Lyapunov function V(t) ≤ γ V(t) when no controllers are added into networks, some new and useful finite-time synchronization criteria were then derived via periodically intermittent control. Hence, a natural problem is whether a periodically intermittent controller with no control in rest time can be also designed based on these new schemes to ensure fixed-time synchronization of complex networks. It is our intention in the future to reduce the possible conservatism for the sake of broader applications of the intermittent control technique. Remark 6. The uncertain positive constants η, ζ , ζ r , ξ r (r = 1, 2, 3) and ki (i = 1, . . . , N ) make Theorem 1 flexible in realistic applications. However, inappropriate choosing of these parameters may enlarge the values of the polynomials on the left side of the inequalities in Theorem 1, and then make the results conservative. Therefore, it is important to determine appropriate parameters such that the inequalities in Theorem 1 is less conservative. The detailed algorithm for determining these parameters in the inequalities is given as follows:

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(1) The positive constants η and ζ can be computed by Assumption 1. It is easy to show that the smaller constants η and ζ are, the less conservative of Theorem 1 is. (2) Inspired by Cai et al. [4,39,44], and the extreme value theory of multivariate function,  it is easy to verify that a good choice of parameters ζ r (r = 1, 2, 3) is ζ1 =

1/ λmax (BT B)λmax (GT2 G2 ) and ζ2 = ζ3 = 1/ λmax (GT2 G2 ), respectively. (3) It is easy to show that the larger control strengths ki and ξ r (i = 1, . . . , N, r = 1, 2, 3) are, the less conservative of Theorem 1 is. However, larger control strengths means higher control cost. Therefore, we need to choose the smallest when choosing the appropriate control strengths, as well as satisfying the inequalities in Theorem 1 as smaller as possible. To realize the finite-time synchronization of networks (2.2) and (2.4), the following periodically semi-intermittent controller can be designed as ⎧

p+1 3  t  ⎪ ξr ei (t ) T ⎪ 2 e (s) e (s)ds ⎨−ki ei (t ) − α i 1−σr t −τr (t ) i ei (t )2 r=1 ui (t ) = (3.13) ⎪ −αsign (ei (t ))|ei (t )| p , mT ≤ t < (m + θ )T , ⎪ ⎩ −ki ei (t ), (m + θ )T ≤ t < (m + 1)T , where the parameters involved in Eq. (3.13) have the same meanings with the corresponding parameters in Eq. (2.5). Based on Lemma 3, following the similar line of the proof for Theorem 1, the finite-time synchronization criteria can be easily derived. Theorem 2. Suppose that Assumptions 1 and 2 hold. If there exist positive constants η, ζ , ζ r , ξ r (r = 1, 2, 3) and ki , (i = 1, . . . , N ) such that (i) (ii ) (iii ) (iv)

εIN + c1 ρ0 A˜ − K ≤ 0, ζ − ξ21 ≤ 0, c2 − 2ζc22 bmin − ξ22 ≤ 0, 2ζ1 − 2ζc23 bmin − ξ23 ≤ 0,

where IN is the N-dimensional identity matrix, K = diag(k1 , k2 , . . . , kN ), bmin = min (bii ) (i = 1, 2, . . . , N ), ρ min is the minimum eigenvalue of matrix (G1 + GT1 )/2, ρ0 = G1 , A˜ s = (A˜ + A˜ T )/2 with A˜ is a modified matrix of A by replacing the diagonal elements aii by (ρ min aii /ρ 0 ), and c 2 ζ1 1  ξr c2 (ζ2 + ζ3 ) ε= λmax (BT B)λmax (GT2 G2 ) − λmax (GT2 G2 )bmin + η + , 2 2 2 r=1 1 − σr 3

then the drive network (2.2) and response network (2.3) can achieve finite-time synchronization under the periodically semi-intermittent controller (3.13). The settling time Ts can be estimated by

 sup V (s) 1−p −τ ≤s≤0 Ts ≤ . (3.14) αθ (1 − p)

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Suppose θ = 1, the periodically semi-intermittent controller becomes a general continuous controller. Based on Theorem 1, the following result can be obtained easily. Corollary 1. Suppose that Assumptions 1 and 2 hold. If there exist positive constants η, ζ , ζ r , ξ r (r = 1, 2, 3) and ki , (i = 1, . . . , N ) such that εIN + c1 ρ0 A˜ − K ≤ 0, ζ − ξ21 ≤ 0, c2 − 2ζc22 bmin − ξ22 ≤ 0, 2ζ1 c2 − 2ζ3 bmin − ξ23 ≤ 0,

(i) (ii ) (iii ) (iv)

where IN is the N-dimensional identity matrix, K = diag(k1 , k2 , . . . , kN ), bmin = min (bii ) (i = 1, 2, . . . , N ), ρ min is the minimum eigenvalue of matrix (G1 + GT1 )/2, ρ0 = G1 , A˜ s = (A˜ + A˜ T )/2 with A˜ is a modified matrix of A by replacing the diagonal elements aii by (ρ min aii /ρ 0 ), and c 2 ζ1 1  ξr c2 (ζ2 + ζ3 ) λmax (BT B)λmax (GT2 G2 ) − λmax (GT2 G2 )bmin + η + , 2 2 2 r=1 1 − σr 3

ε=

then the drive network (2.2) and response network (2.3) can achieve fixed-time synchronization under the controller  t 3 

p+1 e (t )  ξr i ui (t ) = −ki ei (t ) − α eTi (s)ei (s)ds 2 − αsign (ei (t ))|ei (t )| p 2 1 − σ  e (t )  r i t −τr (t ) r=1  3 

q+1 e (t ) t  ξr i −β eTi (s)ei (s)ds 2 − βsign (ei (t ))|ei (t )|q . 2 1 − σ  e (t )  r i t −τ (t ) r r=1 The fixed settling time Tf can be estimated by Tf ≤

1 1 + . α(q − 1) β(N n) 1−q 2 (1 − p)

(3.15)

Suppose τr (t ) = 0 (r = 1, 2, 3), complex network (2.2) can be simplified as follows: xi (t )

= f (t , xi (t )) + c1

N 

ai j G1 x j (t ), i = 1, 2, . . . , N.

(3.16)

j=1

Corresponding, the response network can be designed as y˙i (t ) = f (t , yi (t )) + c1

N 

ai j G1 y j (t ) + ui (t ), i = 1, 2, . . . , N,

(3.17)

j=1

where ui (t) is a periodically semi-intermittent control designed as  −ki ei (t ) − αsign (ei (t ))|ei (t )| p − βsign (ei (t ))|ei (t )|q , mT ≤ t < (m + θ )T , ui (t ) = −ki ei (t ), (m + θ )T ≤ t < (m + 1)T . (3.18) For the complex networks without delays and hybrid couplings (3.16) and (3.17), the following corollary can be derived based on Theorem 1.

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Corollary 2. Suppose that Assumptions 1 and 2 hold. If there exist positive constants η, ki , (i = 1, . . . , N ) and positive definite diagonal matrix P and  such that ηIN + c1 ρ0 A˜ − K ≤ 0, where IN is the N-dimensional identity matrix, K = diag(k1 , k2 , . . . , kN ), ρ min is the minimum eigenvalue of matrix (G1 + GT1 )/2, ρ0 = G1 , A˜ s = (A˜ + A˜ T )/2 with A˜ is a modified matrix of A by replacing the diagonal elements aii by (ρ min aii /ρ 0 ), then the drive network (3.16) and response network (3.17) can achieve fixed-time synchronization under the periodically semiintermittent controller (3.18). The fixed settling time Tf can be estimated by Tf ≤

1 1 + . αθ (q − 1) β(N n) 1−q 2 θ (1 − p)

(3.19)

4. Numerical simulations In this section, in order to illustrate the effectiveness of the periodically semi-intermittent controller in fixed-time synchronization obtained above, we first consider an undirected complex network with 200 dynamical nodes. The node dynamics is chosen as: x˙(t ) = Dx(t ) + f1 (x(t )) + f2 (x(t − τ1 (t ))), where ⎡

−10 D = ⎣ 28 0

10 −1 0

(4.1)

⎤ 0 0 ⎦, f1 (x(t ))=(0, −x1 x3 , x1 x2 )T , f2 (x(t − τ1 (t )))=(0, 6x2 (t −1), 0)T . −8/3

System (4.1) with above coefficients exhibit a chaotic behavior as shown in Fig. 1. To verify the feasibility and effectiveness of the proposed fixed-time synchronization scheme in Theorem 1, we take A as the coupling matrix of BA scale-free network [4] with m0 = m1 = 5, where m0 is the number of original nodes in the network and m1 is the number of new edges added to the network at each time. For brevity, taking B = 0.1A, G1 = diag(1.1, 1, 1), G2 = diag(1, 1, 1), c1 = c2 = 1, τ2 (t ) = et /(1 + et ) and τ3 (t ) = 0.02| sin (t )|, and choosing T = 0.2, θ = 0.4, p = 0.5, q = 1.5, α = β = 10, ξr = 5 (r = 1, 2, 3), ζ1 = ζ2 = 2, ζ3 = 1, ki = 100 (i = 1, 2, . . . , 200) in the periodically semi-intermittent controller (2.5). By simple calculation, we can get bmin = −4.7, λmax (A˜ s ) = 0.1924, λmax (BT B) = 23.4167, λmax (GT2 G2 ) = 1, ρ0 = 1.1, ρmin = 1, η = 3.5064, ζ = 0.1, σ1 = 0, σ2 = 0.25 and σ3 = 0.02. Hence, all conditions in Theorem 1 are satisfied. From Theorem 1, the drive network (2.2) and response network (2.3) can achieve fixed-time synchronization under the periodically semi-intermittent controller (2.5) as shown in Fig. 2, and the fixed settling time can be estimated as Tf ≤ 2.97. Next, we will show the numerical simulations on the fixed-time synchronization of directed network. As we all know, the biggest difference between undirected and directed networks

Q. Gan, F. Xiao and H. Sheng / Journal of the Franklin Institute 356 (2019) 6656–6677

6671

80

60

X3

40

20

0 40 20

40 20

0

0

-20

-20

X2

-40

-40

X1

Fig. 1. Chaotic attractor of system (4.1) with initial values (0.2, 0.4, 0.8)T . 6 3 4

2

2

e i2 (t)

e i1 (t)

1

0

0

-1 -2 -2 -4 -3 -6 0

1.5

3

4.5

6

7.5

9

0

1.5

3

t

4.5

6

7.5

9

6

7.5

9

t 2.5

5

4

2

3

E(t)

e i3 (t)

1.5 2

1 1 0.5

0

-1

0 0

1.5

3

4.5

t

6

7.5

9

0

1.5

3

4.5

t

Fig. 2. Trajectories of the synchronization errors and the total synchronization error for undirected network.

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is their outer coupling matrices. Therefore, the different outer coupling matrices of directed network A and B are chosen as follows: ⎡

−5 ⎢0.72 ⎢ ⎢ 0.3 ⎢ ⎢ 0.8 ⎢ ⎢0.38 A = I20  ⎢ ⎢ 0.5 ⎢ ⎢0.68 ⎢ ⎢0.42 ⎢ ⎣0.64 0.46

0.4 −5 0.3 0.6 0.52 0.7 0.56 0.62 0.44 0.66

0.8 0.48 −4 0.32 0.5 0.7 0.64 0.46 0.72 0.38

0.4 0.48 0.54 −5 0.58 0.74 0.52 0.54 0.48 0.62

0.6 0.72 0.42 0.64 −4 0.6 0.48 0.54 0.76 0.44

0.6 0.56 0.58 0.6 0.58 −6 0.36 0.42 0.32 0.78

0.64 0.64 0.5 0.64 0.34 0.74 −5 0.3 0.48 0.62

0.56 0.4 0.56 0.4 0.3 0.82 0.76 −4 0.36 0.74

0.6 0.64 0.14 0.52 0.22 0.7 0.28 0.4 −5 1.3

⎤ 0.4 0.36⎥ ⎥ 0.66⎥ ⎥ 0.48⎥ ⎥ 0.58⎥ ⎥ 0.5 ⎥ ⎥ 0.72⎥ ⎥ 0.3 ⎥ ⎥ 0.8 ⎦ −6

and ⎡

−1.1 ⎢ 0.18 ⎢ ⎢ 0.13 ⎢ ⎢ 0.3 ⎢ ⎢ 0.12 B = I20  ⎢ ⎢ 0.15 ⎢ ⎢ 0.17 ⎢ ⎢ 0 ⎢ ⎣ 0.1 0

0 −1.3 0.14 0 0.18 0 0.27 0.38 0 0.16

0.3 0.24 −1.0 0.29 0 0 0.16 0.2 0.27 0.12

0.1 0 0.16 −1.6 0.15 0.16 0.1 0 0.21 0.18

0.16 0.2 0 0 −1.2 0.15 0.12 0.32 0.35 0.14

0.14 0.12 0.17 0.31 0.29 −1.0 0 0.12 0.1 0.13

0.2 0.16 0 0.1 0.16 0.12 −1.1 0.1 0.19 0.1

0.1 0.1 0.3 0.1 0.1 0.18 0.15 −1.4 0 0.27

0 0.15 0 0.2 0.2 0.14 0 0.15 −1.5 0

⎤ 0.1 0.15 ⎥ ⎥ 0.1 ⎥ ⎥ 0.3 ⎥ ⎥ 0 ⎥ ⎥. 0.1 ⎥ ⎥ 0.13 ⎥ ⎥ 0.13 ⎥ ⎥ 0.28 ⎦ −1.1

By plain calculation, it is easy to get bmin = −1,λmax (A˜ s ) = 0.4663, λmax (BT B) = 3.4956, λmax (GT2 G2 ) = 1,ρ0 = 1.1, ρmin = 1, η = 3.5064, ζ = 0.1, σ1 = 0, σ2 = 0.5 and σ3 = 0.06. Therefore, all conditions in Theorem 1 are satisfied, the drive network (2.2) and response network (2.3) can achieve fixed-time synchronization under the periodically semi-intermittent controller (2.5) as shown in Fig. 3, and the fixed settling time can be estimated by Tf ≤ 2.15. The numerical simulations clearly verify the effectiveness and correctness of the developed periodically semi-intermittent controller to the fixed-time synchronization between two delayed complex networks with hybrid couplings. It is shown from inequality (3.19) that the periodically semi-intermittent control rate θ , network size N and node dimension n heavily influence the estimating for the upper bound of the convergence time of synchronization state. Fig. 4 shows that the more nodes of network, the slower the convergence rate of network is. Fig. 5 describes that the larger periodically semi-intermittent control rate, the faster the convergence rate of network is. Comparing with the corresponding finite-time synchronization, fixed-time synchronization can be achieved in a settling time, which is bounded and independent of the initial states. Fig. 6 shows the comparisons of finite-time synchronization and fixed-time synchronization, which proved that the convergence rate of fixed-time synchronization is faster than the rate of finite-time synchronization. Therefore, compare with finite-time control strategy, fixed-time control scheme shows more effectiveness and superiority.

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8 10 6 8 4 2

e i2 (t)

e i1 (t)

6

4

0 -2

2

-4 0 -6 -2 0

0.5

1

1.5

2

2.5

3

0

3.5

0.5

1

1.5

2

2.5

3

3.5

t

t 9 10

8 6

E(t)

e i3 (t)

6

4 3 2

0 0

-2 0

0.5

1

1.5

2

2.5

3

3.5

t

t

Fig. 3. Trajectories of the synchronization errors and the total synchronization error for directed network.

Undirected network

Directed network

3.5

10

N=100 N=200 N=400

3

N=100 N=200 N=400

8

6

2

E(t)

E(t)

2.5

1.5

4

1 2 0.5 0

0 0

0.5

1

1.5

2

2.5

t

3

3.5

4

t

Fig. 4. Trajectories of the total synchronization error when θ = 0.4 and N = 100, N = 200, N = 400, respectively.

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Fig. 5. Trajectories of the total synchronization errors when N = 200 and θ = 0.4, 0.6, 0.8, respectively.

Undirected network

Directed network

24 3.5

Fixed-time total error Finite-time total error

Fixed-time total error Finite-time total error

21

3 18 2.5

E(t)

E(t)

15 2 1.5

12 9

1

6

0.5

3

0 0

0.5

1

1.5

2

2.5

t

3

3.5

4

0 0

1.5

3

4.5

6

7.5

9

10.5

12

13.

5

t

Fig. 6. Trajectories of the total synchronization errors with fixed-time control (β = 10) and finite-time control (β = 0) when N = 200 and θ = 0.4, respectively.

5. Conclusion In this paper, we have studied the fixed-time synchronization problem between two complex networks with hybrid coupling. By introducing and proving a new and important differential equality, and utilizing periodically semi-intermittent control, some fixed-time synchronization criteria have been derived in which the settling time function is bounded for any initial values. It is worth noting that control rate, network size and node dimension have played important roles in estimating the settling time of fixed-time synchronization. Moreover, our results show that the synchronization criteria depend on the ratio of control width to control period, but not control width or control period. For the practical applications, we can thus randomly choose the control period for achieving fixed-time synchronization. These results improve and generalize previously known results and will bring convenience in understanding the influence of network structure and consequently finding the effective way to improve network performance. Some remarks and numerical simulations have been used to demonstrate the effectiveness of the obtained results.

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In fact, the limitation of periodicity for the intermittent control is quite restricted and may not be realistic in many real applications, in particular, when uncertainties are taken into the consideration. Compared to the periodically intermittent control, a more general intermittent control technique, called the aperiodically intermittent control, is more flexible, and its application scope is wider [11,23]. The issues of integrating pinning control strategy (only a small fraction of nodes are controlled) and aperiodically intermittent control approach into the research on synchronization of complex networks for further reducing of the control cost. Therefore, it is interesting to study this problem both in theory and applications. However, to the best of our knowledge, there are no results concerning the fixed-time synchronization schemes for complex networks via pinning aperiodically intermittent control. This is an interesting problem and will become our future investigative direction. Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant 61305076. References [1] M.S. Ali, J. Yogambigai, Finite-time robust stochastic synchronization of uncertain Markovian complex dynamical networks with mixed-varying delays and reaction-diffusion terms via impulsive control, J. Frankl. Inst. 354 (2017) 2415–2436. [2] A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks, Physics Reports 469 (2008) 93–153. [3] S. Bhat, D. Bernstein, Finite-time stability of homogeneous system, in: Proceeding of the American Control Conference, 1997, pp. 2513–2514. Albuquerque, New Mexico [4] S. Cai, P. Zhou, Z. Liu, Pinning synchronization of hybrid-coupled directed delayed dynamical network via intermittent control, Chaos 24 (2014) 033102. [5] S. Cai, Q. Jia, Z. Liu, Cluster synchronization for directed heterogeneous dynamical networks via decentralized adaptive intermittent pinning control, Nonlinear Dyn. 82 (2015) 689–702. [6] C. Chen, L. Li, H. Peng, J. Kurths, Y. Yang, Fixed-time synchronization of hybrid coupled networks with time-varying delays, Chaos Solitons Fract. 108 (2018) 49–56. [7] G. Chen, X. Wang, X. Li, Introduction to Complex Networks: Models, Structures and Dynamics, Science Press, Beijing, 2015. January [8] L. Cheng, Y. Yang, L. Li, X. Sui, Finite-time hybrid projective synchronization of the drive-response complex networks with distributed-delay via adaptive intermittent control, Physica A 500 (2018) 273–286. [9] F. Dörfler, F. Bullo, Synchronization in complex networks of phase oscillators: a survey, Automatica 50 (2014) 1539–1564. [10] J. Feng, N. Li, Y. Zhao, C. Xu, J. Wang, Finite-time synchronization analysis for general complex dynamical networks with hybrid couplings and time-varying delays, Nonlinear Dyn. 88 (2017) 2723–2733. [11] Q. Gan, Exponential synchronization of generalized neural networks with mixed time-varying delays and reaction-diffusion terms via aperiodically intermittent control, Chaos 27 (2017) 013113. [12] C. Hu, J. Yu, Z. Chen, H. Jiang, T. Huang, Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks, Neural Netw. 89 (2017) 74–83. [13] C. Hu, J. Yu, H. Jiang, Z. Teng, Exponential synchronization of complex networks with finite distributed delays coupling, IEEE Trans. Neural Netw. 22 (2011) 1999–2010. [14] G. Ji, C. Hu, J. Yu, H. Jiang, Finite-time and fixed-time synchronization of discontinuous complex networks: a unified control framework design, J. Frankl. Inst. 355 (2018) 4665–4685. [15] S. Jiang, X. Lu, C. Xie, S. Cai, Adaptive finite-time control for overlapping cluster synchronization in coupled complex networks, Neurocomputing 266 (2017) 188–195. [16] T. Jing, F. Chen, X. Zhang, Finite-time lag synchronization of time-varying delayed complex networks via periodically intermittent control and sliding mode control, Neurocomputing 199 (2016) 178–184.

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