# Antisynchronization and Generalized Pinning Control of Multiweighted Coupled Complex-Valued Delayed Memristive Neural Networks

Authors: Limei Su; Yanli Huang
Journal: Discrete Dynamics in Nature and Society
 (2023)

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## Abstract

In this article, antisynchronization problem of multiweighted coupled complex-valued delayed memristive neural networks (MWCCVDMNNs) with and without coupling delays are investigated. First, via devising a suitable controller and constructing an appropriate Lyapunov functional, a criterion for ensuring antisynchronization of MWCCVDMNNs is derived. Second, we research the generalized pinning antisynchronization of MWCCVDMNNs by creating a generalized pinning controller to guarantee that the considered networks can accomplish antisynchronization. Similarly, several sufficient conditions guaranteeing the antisynchronization and generalized pinning antisynchronization of MWCCVDMNNs with coupling delays are also presented. Third, two numerical examples are provided to verify the correctness of the obtained antisynchronization results.


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## Body

## 1. Introduction

In recent decades, the applications of neural networks (NNs) in automatic control [1], pattern recognition [2], associative memory [3], and many other fields have attracted much attention. As a particular case of complex networks, coupled NNs (CNNs) are generated from a large number of single NN through complex interactions between different NNs. Enormous practical problems such as target recognition, edge detection, and image segmentation can be described and studied with the help of CNNs. Therefore, increasing interest has been dedicated to exploring the dynamic behaviors of CNNs, such as the passivity and synchronization of CNNs [4–8]. The authors in [4] obtained the global exponential synchronization and passivity conditions of CNNs under impulse control. Wang et al. [5] derived some adequate conditions to ensure the finite-time passivity of directed and undirected CNNs.Chua [9] first presented the concept of memristor in 1971. Numerous studies have shown that memristor shows the characteristic of pinched hysteresis, that is to say, there is a lag between the application and removal of a field and its succeeding effect, just like neurons in the human brain have [10, 11]. Due to this feature, the past dynamic history of memristor can be remembered, with a substantial similarity to the function of synapses. For this reason, many scholars have been trying to use memristors to simulate neurons to achieve brain-like computation, which leads to the development of memristive neural networks (MNNs) [12, 13]. An increasing number of scholars have recently investigated the dynamical behaviors of coupled MNNs (CMNNs) [14–16]. In [14], a distributed impulsive control strategy is employed to explore the multisynchronization problem of CMNNs. As discussed in [15], several synchronization conditions in finite-time and fixed-time for CMNNs were derived. Wu and Zeng [16] investigated a class of memristive recurrent NNs (MRNNs) and realized the exponential antisynchronization of the drive-response-based coupled MRNNs (CMRNNs).As a generalization of the real-valued NNs (RVNNs), the state, activation function, connection weight matrix, and external input of the complex-valued NNs (CVNNs) are all defined in the range of complex numbers. The CVNNs can solve many practical problems which cannot be solved by RVNNs. Thus, it is urgent to explore the dynamic properties of CVNNs, and some achievements have been made in recent years [17–19]. As discussed by Wei et al. [17], the antisynchronization and synchronization problems of complex-valued inertial NNs were studied, and some corresponding adequate conditions for synchronization and antisynchronization of complex-valued inertial NNs were established. The authors in [18] researched the multistability and robustness of CVNNs with delays and input perturbation. Moreover, Wei et al. [19] analyzed the matter of antisynchronization for CVNNs with time-varying delays and leakage delay and proposed several corresponding antisynchronization criteria. Furthermore, a large number of researchers have discussed the synchronization and passivity of complex-valued MNNs (CVMNNs) [20, 21]. In [20], the model of delayed CVMNNs is established for the first time, and some sufficient criteria for the passivity of the given CVMNNs are derived. Zhang et al. [21] discussed the global asymptotical synchronization of fractional-order CVMNNs with both parameter uncertainties and multiple time delays. To the best of the authors’ knowledge, there is no research result on the coupled CVMNNs (CCVMNNs) except that some conditions for ensuring passivity and synchronization of CCVMNNs were established in [22].In the works as mentioned above, the network models about CCVMNNs are coupled by single-weight. As a matter of fact, in the real world, it is more intuitive and convenient to describe some large-scale networks with multiweighted complex dynamical networks (MWCDNs), such as social networks, bus line networks, and communication networks. Hence, the multiweighted coupled neural networks (MWCNNs) as a special type of MWCDNs have received extensive attention [23, 24]. Wang et al. [23] obtained several novel criteria for ensuring finite-time passivity and synchronization of MWCNNs without and with coupling delays by proposing several novel concepts about passivity in finite-time and designing appropriate controllers. The authors derived some passivity and synchronization of MWCNNs by means of designing proportional-integral-derivative controllers [24]. As is known to all, stability is an essential theoretical problem in studying NNs. The introduction of time delays may lead to oscillations or instability or even chaos in the neural network system. Thus, multiweighted coupled complex-valued delayed MNNs (MWCCVDMNNs) is more worthy of being studied. Moreover, it is also important to discuss the MWCCVDMNNs with coupling delays. In [25, 26], the authors studied the multiweighted coupled delayed MNNs (MWCDMNNs) for event-triggered passivity and passivity-based synchronization, but the networks considered in these publications are all real-valued.It is generally known that antisynchronization of chaotic oscillators is a fascinating phenomenon. Antisynchronization is basically the same type of synchronization that Pecora and Carroll [27] first studied. The only discrepancy is that antisynchronization allows mutually symmetrical chaotic attractors to coexist. More precisely, antisynchronization refers to a phenomenon that the state vectors of the slave systems have the opposite signs but the same amplitude as those of the master system. Hence, the sum of two signals is anticipated to converge to 0 when antisynchronization occurs. So far, a wide range of methods have been proposed for the antisynchronization of chaotic systems, such as adaptive control, direct linear coupling, and nonlinear control. In [28], Ren et al. investigated the antisynchronization of chaotic systems and deduced some sufficient conditions for the given chaotic system to realize antisynchronization.In many practical situations, antisynchronization has gained widespread use in many areas including image processing, communication systems, and laser. Hence, antisynchronization, like synchronization, also plays a critical role in the study of the dynamic behaviors of CNNs [29, 30]. In [29], some finite-time antisynchronization conditions were derived for the MWCNNs. By integrating the definition of lag synchronization into the definition of decay synchronization, the concept of general decay lag antisynchronization was proposed and obtained several criteria for guaranteeing that multiweighted delayed CNNs with reaction-diffusion terms achieve decay lag antisynchronization [30].Different from other control methods, pinning control is a very effective control scheme in which only a small fraction of nodes are chosen to be controlled. So far, some scholars have investigated pinning synchronization control problem of the coupled delayed NNs (CDNNs) [31] and CMNNs [32]. In [31], several criteria were proposed to ensure the cluster synchronization of nonlinear CDNNs in both finite-time and fixed-time aspects based on the pinning control strategy. In a previous study [32], Yue et al. addressed the passivity and synchronization of CMNNs by making use of two effective pinning control strategies. However, the investigation on pinning antisynchronization of MWCNNs is very rare [33]. Hou et al. [33] analyzed the pinning antisynchronization of the MWCNNs. To the best of the authors’ knowledge, antisynchronization and pinning antisynchronization problems of MWCCVDMNNs have not been researched yet which motivates the study in this paper. Different from the classical pinning control method, what we study in this paper is the networks with discontinuous neurons; in this case, we develop the generalized antisynchronization pinning scheme for this kind of network.From what has been discussed above, this paper studies two classes of NNs: MWCCVDMNNs and MWCCVDMNNs with coupling delays. The main novelties of our work can be outlined as follows. (1) The network model of CCVDMNNs with multiweights is presented for the first time. (2) By virtue of employing appropriate Lyapunov functional and several inequality techniques, several adequate conditions are proposed to ensure the antisynchronization of MWCCVDMNNs with and without coupling delays. (3) Considering that MWCCVDMNNs are discontinuous neural networks, a novel pinning controller for this type of network is designed to realize the generalized pinning antisynchronization.


## 2. Preliminaries

### 2.1. Notations

LetℝN and ℂN be the N-dimensional real and complex vector space, respectively. The smallest and largest eigenvalue of the corresponding matrix are denoted by λm⋅ and λM⋅, respectively. ⊗ represents the Kronecker product. For any complex number a=aR+iaI, in which i=−1 is the imaginary unit, aI,aR∈ℝ are the imaginary part and real part of a, respectively. For any vector at∈ℂN, at=aHtat, where aHt represents the conjugate transposition of at. If the real and imaginary parts of at∈ℂN are denoted by aRt, aIt∈ℝN, then one has at=aRtTaRt+aItTaIt. Obviously, the vector norm in our paper is Euclidean norm, i.e., L2 norm. Note that the vector at is a complex vector, and the norm of at is the square of aHtat=aRtTaRt+aItTaIt.


### 2.2. Lemmas

Lemma 1.
(see [34]). For any real vectors α1, α2∈ℝn and matrix 0<Ξ∈ℝn×n,(1)2α1Tα2≤α1TΞα1+α2TΞ−1α2.Lemma 2.
(see [35]). Let A,B,C,andD be matrices with compatible dimensions and a∈ℝ. Then,(2)1aA⊗B=A⊗aB;2A⊗BT=AT⊗BT;3A⊗BC⊗D=AC⊗BD;4A+B⊗C=A⊗C+B⊗C.


## 2.1. Notations

LetℝN and ℂN be the N-dimensional real and complex vector space, respectively. The smallest and largest eigenvalue of the corresponding matrix are denoted by λm⋅ and λM⋅, respectively. ⊗ represents the Kronecker product. For any complex number a=aR+iaI, in which i=−1 is the imaginary unit, aI,aR∈ℝ are the imaginary part and real part of a, respectively. For any vector at∈ℂN, at=aHtat, where aHt represents the conjugate transposition of at. If the real and imaginary parts of at∈ℂN are denoted by aRt, aIt∈ℝN, then one has at=aRtTaRt+aItTaIt. Obviously, the vector norm in our paper is Euclidean norm, i.e., L2 norm. Note that the vector at is a complex vector, and the norm of at is the square of aHtat=aRtTaRt+aItTaIt.


## 2.2. Lemmas

Lemma 1.
(see [34]). For any real vectors α1, α2∈ℝn and matrix 0<Ξ∈ℝn×n,(1)2α1Tα2≤α1TΞα1+α2TΞ−1α2.Lemma 2.
(see [35]). Let A,B,C,andD be matrices with compatible dimensions and a∈ℝ. Then,(2)1aA⊗B=A⊗aB;2A⊗BT=AT⊗BT;3A⊗BC⊗D=AC⊗BD;4A+B⊗C=A⊗C+B⊗C.


## 3. Antisynchronization and Generalized Pinning Antisynchronization of MWCCVDMNNs

### 3.1. Antisynchronization of MWCCVDMNNs

In this paper, a single model of CVDMNN is presented as follows:(3)a˙st=−dsast+∑h=1npshastyhaht−uht,s=1,2,…,n,where the complex-valued ast indicates the state variable associated with the s-th neuron; ds>0 represents the self-feedback coefficient; pshast stands for the weight of the synaptic connection of complex-valued memristor; the complex-valued nonlinear function yh⋅ denotes the activation function of the h-th neuron; uht is the time-varying delay with 0≤uht≤uh≤u=maxh=1,2,…,nuh and u˙ht≤ϱh<1.Let(4)ast=asRt+iasIt,pshast=pshRasRt+ipshIasIt,yhaht−uht=yhRahRt−uht+iyhIahIt−uht,where asIt,pshIasIt, yhIahIt−uht are the imaginary parts of ast,pshast,yhaht−uht and asRt,pshRasRt,yhRahRt−uht are the real parts of ast,pshast,yhaht−uht.According to the characteristics of memristor’s current and voltage, one has(5)pshRasRt=p^shR,asRt≤Λs,pˇshR,asRt>Λs,pshIasIt=p^shI,asIt≤Λs,pˇshI,asIt>Λs,where s,h∈1,2,…,n; p^shR,pˇshR,p^shI,pˇshI are constants; Γs>0 represents the threshold level. Let p˜shR=maxp^shR,pˇshR, p˜shI=maxp^shI,pˇshI, p¯shR=p^shR−pˇshR, p¯shI=p^shI−pˇshI, P¯R=p¯shRn×n, P¯I=p¯shIn×n, P˜R=diag∑h=1np˜1hR2,∑h=1np˜2hR2,…,∑h=1np˜nhR2, and P˜I=diag∑h=1np˜1hI2,∑h=1np˜2hI2,…,∑h=1np˜nhI2.In the following, we consider the MWCCVDMNNs made up ofN CVDMNNs as (3)(6)A˙δt=−DAδt+PAδtyAδt¯+cδt+∑ϵ=1η∑κ=1NlϵKδκϵHϵAκt,δ=1,2,…,N,in which the complex-valued vector Aδt=Aδ1t,Aδ2t,…,AδntT∈ℂn denotes the state vector for the δ-th node; 0<D=diagd1,d2,…,dn∈ℝn×n, Aδt¯=Aδ1t−u1t,Aδ2t−u2t,…,Aδnt−untT∈ℂn, yAδt¯=y1Aδ1t−u1t,y2Aδ2t−u2t, ⋯,ynAδnt−untT∈ℂn, and Hϵ∈ℝn×nϵ=1,2,…,η is an internally coupled matrix; PAδt=pshAδstn×n∈ℂn×n; cδt=cδRt+icδIt=cδ1t,cδ2t,…,cδntT∈ℂn represents the controller; ℝ∋lϵ>0ϵ=1,2,…,η symbols coupling strength for the ϵ-th coupling form; Kϵ=KδκϵN×N∈ℝN×Nϵ=1,2,…,η describes the outer coupling matrix, where ℝ∋Kδκϵ>0, if there is a connection from node ι to node κδ≠κ; otherwise, ℝ∋Kδκϵ=0.(7)Kδδϵ=−∑k=1k≠δNKδκϵ,δ=1,2,…,N.Remark 1.
CVNNs, as an extension of RVNNs, can solve some problems which cannot be dealt by RVNNs because their states, activation functions, and connection weights are all complex-valued. For instance, the detection of symmetric problem and the XOR problem cannot be modelled by a single real-valued neuron, but they can be achieved by a single complex-valued neuron with the orthogonal decision boundaries [36]. Moreover, it is more suitable to describe various physical phenomena (e.g., the phase progression and retardation, the superposition of fields, and the wave propagation) through complex numbers in reality, all of which can be drawn as the applications of CVNNs. Naturally, CVNNs have found widespread practical applications in physical systems dealing with quantum waves, ultrasonic, and electromagnetic waves. Therefore, it is very meaningful to consider the characteristic of complex-value when studying CMNNs.Then, separating network (6) into the following imaginary and real parts:(8)A˙δRt=−DAδRt+PRAδRtyRAιRt¯+∑ϵ=1η∑κ=1NlϵKδκϵHϵAκRt−PIAδItyIAδIt¯+cδRt,A˙δIt=−DAδIt+PRAδRtyIAδIt¯+∑i=1η∑κ=1NlϵKικϵHϵAκIt+PIAδItyRAδRt¯+cδIt,where yRAδRt¯=y1RAδ1Rt−u1t,y2RAδ2Rt−u2t,…,ynRAδnRt−untT, yIAδIt¯=y1IAδ1It−u1t,y2IAδ2It−u2t,…,ynIAδnIt−untT, AδRt=Aδ1Rt,Aδ2Rt,…,AδnRtT, PR⋅=pshR⋅n×n, AδIt=Aδ1It,Aδ2It,…,AδnItT, cδRt=cδ1Rt,cδ2Rt,…,cδnRtT, and PI⋅=pshI⋅n×n, cδIt=cδ1It,cδ2It,…,cδnItT.Assumption 1.
For allβ1,β2∈ℝ, there exist real numbers YδR,YδI,jδR, jδR>0 such that(9)yδR⋅≤YδR,yδI⋅≤YδI,yδRβ1+yδRβ2≤jδRβ1+β2,yδIβ1+yδIβ2≤jδIβ1+β2.For the network (6), assume that A∗t=A∗1t,A∗2t,…,A∗ntT∈Cn∈ℂn is an arbitrary solution, then(10)A˙∗t=−DA∗t+PA∗tyA∗t.Here,A∗t=A∗Rt+iA∗It. Then, (10) can be expressed by separating it into the following two real systems:(11)A˙∗Rt=−DA∗Rt+PRA∗RtyRA∗Rt−PIA∗ItyIA∗It,A˙∗It=−DA∗It+PRA∗RtyIA∗It+PIA∗ItyRA∗Rt.Letzδt=Aδt+A∗t, then(12)z˙δt=−Dzδt+PAδtyAδt¯+cδt+PA∗tyA∗t+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκt,where zδt=zδ1t,zδ2t,…,zδntT, δ=1,2,…,N.Then, the decomposition form of (12) can be expressed as(13)z˙δRt=−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+cδRt+PIAδIt−PIA∗ItyIA∗It+∑ϵ=1η∑κ=1NlϵKικϵHϵzκRt,z˙δIt=−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+cδIt−PIAδIt−PIA∗ItyRA∗Rt+∑ϵ=1η∑κ=1NlϵKϵκϵHϵzκIt,where zδRt=zδ1Rt,zδ2Rt,…,zδnRtT, zδIt=zδ1It,zδ2It,…,zδnItT, zδRt¯=zδ1Rt−u1t,zδ2Rt−u2t,…,zδnRt−untT, zδIt¯=zδ1It−u1t,zδ2It−u2t,…,zδnIt−untT, GRzδRt¯=yRAδRt¯+yRA∗Rt, and GIzδIt¯=yIAδIt¯+yIA∗It.Definition 1.
The network (6) is said to be antisynchronized if(14)limt⟶+∞Aδt+A∗t=0,for allδ=1,2,…,N.
In this paper, the authors construct the following controller for the MWCCVDMNNs (6):(15)cδRt=−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I,cδIt=−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R,where δ=1,2,…,N; ℝ∋vδR>0 and ℝ∋vδI>0; Y¯R=Y1R,Y2R,…,YnRT; ΘδR=diagvδ1R,vδ2R,…,vδnR∈ℝn×n and ΘδI=diagvδ1I, vδ2I,…,vδnI∈ℝn×n are the positive definite control gain matrices; sign zδRt=diag (sign zδ1Rt, sign zδ2Rt,…, sign zδnRt); Y¯I=Y1I,Y2I,…,YnIT; and sign zδIt=diag (sign zδ1It, sign zδ2It,…, sign zδnIt).
For convenience, some symbols are denoted as follows:(16)zRt=z1RtT,z2RtT,…,zNRtTT,zIt=z1ItT,z2ItT,…,zNItTT,zRt¯=z1Rt¯T,z2Rt¯T,…,zNRt¯TT,zIt¯=z1It¯T,z2It¯T,…,zNIt¯TT,JR=diagj1R2,j2R2,…,jnR2,JI=diagj1I2,j2I2,…,jnI2,Λ=diag11−ϱ1,11−ϱ2,…,11−ϱn.Theorem 1.
The antisynchronization of network (6) can be realized under the controller (15) if the following conditions hold:(17)Φ1R<0andΦ1I<0,where Φ1R=IN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵT,Φ1I=IN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵT; ΘR=diagΘ1R,Θ2R,…,ΘNR∈ℝnN×nN, and ΘI=diagΘ1I,Θ2I,…,ΘNI∈ℝnN×nN.Proof.
Consider the Lyapunov function as(18)V1t=∑δ=1NzδRtTzδRt+2∑δ=1N∑h=1n∫t−uhttjhRzδhRε21−ϱhdε+2∑δ=1N∑h=1n∫t−uhttjhIzδhIε21−ϱhdε+∑δ=1NzδItTzδIt.
Then, one can yield(19)V˙1t≤2∑δ=1NzδRtT−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯+∑ϵ=1η∑κ=1N​lϵKικϵHϵzκRt−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I−PRAδRt−PRA∗RtyRA∗Rt+PIAδIt−PIA∗ItyIA∗It+2∑δ=1NzδItT−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯+∑ϵ=1η∑κ=1N​lϵKικϵHϵzκIt−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R−PRAδRt−PRA∗RtyIA∗It−PIAδIt−PIA∗ItyRA∗Rt+2zRtTIN⊗JRΛzRt−2zRt¯TIN⊗JRzRt¯+2zItTIN⊗JIΛzIt−2zIt¯TIN⊗JIzIt¯.
From Assumption1, we can deduce(20)2∑δ=1NzδRtTPRAδRtGRzδRt¯≤2∑δ=1N∑s=1n∑h=1nzδsRtp˜shRyhRAδhRt¯+yhRA∗hRt≤∑δ=1N∑s=1n∑h=1nzδsRt2p˜shR2+∑δ=1N∑h=1njhR2zδhRt¯2=zRtTIN⊗P˜RzRt+zRt¯TIN⊗JRzRt¯and(21)−2∑δ=1NzδRtTPIAδItGIzδIt¯≤zRtTIN⊗P˜IzRt+zIt¯TIN⊗JIzIt¯,(22)2∑δ=1NzδItTPRAδRtGIzδIt¯≤zItTIN⊗P˜RzIt+zIt¯TIN⊗JIzIt¯,(23)2∑δ=1NzδItTPIAδItGRzδRt¯≤zItTIN⊗P˜IzIt+zRt¯TIN⊗JRzRt¯.
In addition,(24)2∑ϵ=1η∑δ=1N∑κ=1NlϵKδκϵzδRtTHϵzκRt=zRtT∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵTzRt.
Similarly,(25)2∑ϵ=1η∑δ=1N∑κ=1NlϵKικϵzδItTHδzκIt=zItT∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵTzIt.
Furthermore,(26)2∑δ=1NzδRtTPRAδRt−PRA∗RtyRA∗Rt=2∑δ=1N∑h=1n∑s=1nzδsRtpshRAδsRt−pshRA∗sRtyhRA∗hRt≤2∑δ=1N∑h=1n∑s=1nzδsRtp^shR−p⌣shRYhR=2∑δ=1NzδRtTP¯RY¯R.
Likewise,(27)2∑δ=1NzδRtTPIAδIt−PIA∗ItyIA∗It≤2∑δ=1NzδRtTP¯IY¯I,(28)2∑δ=1NzδItTPRAδRt−PRA∗RtyIA∗It≤2∑δ=1NzδItTP¯RY¯I,(29)2∑δ=1NzδItTPIAδIt−PIA∗ItyRA∗Rt≤2∑δ=1NzδItTP¯IY¯R.
It then follows from (19)–(29) that(30)V˙1t≤zRtTIN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵTzRt+zItTIN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵTzIt≤α1zt2,where α1=maxλMΦ1R,λMΦ1I<0.
From (30) and the construction of V1t, one can obtain that V1t is a bounded function that does not increase. Therefore, V1t converges to a nonnegative constant when t⟶+∞. In addition, according to (30), we can get(31)zt2≤V˙1tα1.
From (31), one can infer that limt⟶+∞∫0tzε2dε exists and is a nonnegative real number. Furthermore,(32)0≤limt⟶+∞∑δ=1N∑h=1n∫t−uhtt2jhRzδhRε21−ϱhdε≤limt⟶+∞∫t−utzRεTIN⊗2JRΛzRεdε≤λMIN⊗2JRΛlimt⟶+∞∫t−utzε2dε=0.
Similarly,(33)0≤limt⟶+∞∑δ=1N∑h=1n∫t−uhtt2jhIzδhIε21−ϱhdε=0.
From (32) and (33), we know that limt⟶+∞∑δ=1NzδRtTzδRt+zδItTzδIt exists and is a nonnegative real number. In the next, we will prove limt⟶+∞∑δ=1NzδRtTzδRt+zδItTzδIt=0. If this is not true, we have(34)limt⟶+∞∑δ=1NzδRtTzδRt+zδItTzδIt=θ>0.
In other words, there exists a positive real constantϖ^ such that(35)∑δ=1NzδRtTzδRt+zδItTzδIt>θ2fort≥ϖ^.
Hence,(36)zt2>θ2,whent≥ϖ^.
From (30) and (36), we can obtain(37)V˙1t<α1θ2,t≥ϖ^.
Then, according to (37), one gets(38)−V1ϖ^≤V1+∞−V1ϖ^=∫ϖ^+∞V˙1tdt<∫ϖ^+∞α1θ2dt=−∞,which leads to contradiction. Therefore,(39)limt⟶+∞∑δ=1NzδRtTzδRt+zδItTzδIt=0.
Then, we have(40)limt⟶+∞zt=0,which means that the network (6) achieves antisynchronization.


### 3.2. Generalized Pinning Antisynchronization of MWCCVDMNNs

In the following, we generalize the traditional pinning control method. One part of the controller we designed controls all the nodes, and the other part controls the firstm nodes of network (6). Then, the MWCCVDMNNs with corresponding pinning controller is described by(41)A˙δt=−DAδt+PAδtyAδt¯+vδt+∑ϵ=1η∑κ=1NlϵKικϵHϵAκt,δ=1,2,…,N,where vδt=vδRt+ivδIt∈ℂn is the generalized pinning adaptive controller and 0<D∈ℝn×n, Aδt∈ℂn, PAδt∈ℂn×n, yAδt¯∈ℂn, ℝ∋lϵ>0, Kϵ∈ℝN×N, and Hϵ∈ℝn×n have the same meanings as in Section 3.1. Without loss of generality, we select the first m nodes and pin them with the following generalized pinning adaptive controller designed as(42)vδt=cδt+c^δt,δ=1,2,…,N,where cδt=cδRt+icδIt, c^δt=c^δRt+ic^δIt, in which cδRt and cδIt are given as in (15) for δ=1,2,…,N, and(43)c^δRt=−∑ϵ=1ηlϵρδϵtRHϵzδRt,δ=1,2,…,m,0,δ=m+1,m+2,…,N,c^δIt=−∑ϵ=1ηlϵρδϵtIHϵzδIt,δ=1,2,…,m,0,δ=m+1,m+2,…,N,with(44)ρ˙δϵtR=ζδϵzδRtTHϵ+HϵT2zδRt,ρ˙δϵtI=ζδϵzδItTHϵ+HϵT2zδIt,when δ=1,2,…,m, where ℝ∋lϵ>0, zδRt=AδRt+A∗Rt, zδIt=AδIt+A∗It, zδRt=zδ1Rt,zδ2Rt,…,zδnRtT, zδIt=zδ1It,zδ2It,…,zδnItT, Hϵ∈ℝn×n, Hϵ+HϵT>0, 1≤m<N, and ℝ∋ζδϵ>0; ℝ∋ρδϵ0R>0; and ℝ∋ρδϵ0I>0 for δ=1,2,…,m.Similar to Section 3.1, the error vectorzδt=Aδt+A∗t can be governed by(45)z˙δt=−Dzδt+PAδtfAδt¯+vδt+PA∗tyA∗t+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκt,δ=1,2,…,N.In order to get the desired result, we separate networks (45) into the following equivalent imaginary and real parts:(46)z˙δRt=−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκRt+PIAδIt−PIA∗ItyIA∗It+vδRt,z˙δIt=−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκIt−PIAδIt−PIA∗ItyRA∗Rt+vδIt,where(47)vδRt=−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I−∑ϵ=1ηlϵρδϵtRHϵzδRt,vδIt=−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R−∑ϵ=1ηlϵρδϵtIHϵzδIt,in which ρ˙δεtR=ζδεzδRtTHε+HεT/2zδRt, ρ˙δεtI=ζδεzδItTHε+HεT/2zδIt for δ=1,2,…,m, and ρδϵtR=ρδϵtI≡0 for δ=m+1,m+2,…,N.Theorem 2.
If there are matrices0<ρ^ϵR=diagρ^1ϵR,ρ^2ϵR,…,ρ^mϵR,0,…,0∈ℝN×N and 0<ρ^ϵI=diagρ^1ϵI,ρ^2ϵI,…,ρ^mϵI,0,…,0∈ℝN×N, ϵ=1,2,…,u, such that(48)Φ2R<0andΦ2I<0,where Φ2R=IN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ−ρ^ϵR⊗Hϵ+KϵT−ρ^ϵR⊗HϵT,Φ2I=IN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ε=1ηlεKε−ρ^εI⊗Hε+KεT−ρ^εI⊗HεT; ΘR=diagΘ1R,Θ2R,…,ΘNR∈ℝnN×nN, ΘI=diagΘ1I,Θ2I,…,ΘNI∈ℝnN×nN; ρ^δϵR>0 and ρ^δϵI>0 for δ=1,2,…,m, then the network (41) is pinning adaptive antisynchronized.Proof.
Take the Lyapunov functional for network (45) as follows:(49)V2t=∑ϵ=1η∑δ=1NlϵζδϵρδϵtR−ρ^δϵR2+∑ϵ=1η∑δ=1NlϵζδϵρδϵtI−ρ^δϵI2+∑δ=1NzδRtTzδRt+∑δ=1NzδItTzδIt+2∑δ=1N∑h=1η∫t−uhttjhRzδhRε21−ϱhdε+2∑δ=1N∑h=1η∫t−uhttjhIzδhIε21−ϱhdε.
Then, we have(50)V˙2t≤2∑δ=1NzδRtT−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκRt−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I−∑ϵ=1ηlϵρδϵtRHϵzδRt+PIAδIt−PIA∗ItyIA∗It+2∑δ=1NzδItT−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+∑ϵ=1η∑κ=1N​lϵKδκϵHϵzκIt−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R−∑ϵ=1ηlϵρδϵtIHϵzδIt−PIAδIt−PIA∗ItyRA∗Rt+∑ϵ=1η∑δ=1mlϵρδϵtR−ρ^δϵRzδRtTHϵ+HϵTzδRt+∑ϵ=1η∑δ=1m​lϵρδϵtI−ρ^δϵIzδItTHϵ+HϵTzδIt+2zRtTIN⊗JRΛzRt−2zRt¯TIN⊗JRzRt¯+2zItTIN⊗JIΛzIt−2zIt¯TIN⊗JIzIt¯.
Moreover, we can derive that(51)−2∑δ=1N∑ϵ=1ηlϵρδϵtRzδRtTHϵzδRt+∑ϵ=1η∑δ=1mlϵρδϵtR−ρ^δϵRzδRtTHϵ+HϵTzδRt=−2∑δ=1N∑ϵ=1η​lϵρδϵtRzδRtTHϵzδRt+∑ϵ=1η∑δ=1m​lϵρδϵtRzδRtTHϵ+HϵTzδRt−∑ϵ=1η∑δ=1mlϵρ^δϵRzδRtTHϵ+HϵTzδRt=−∑ϵ=1η∑δ=1mlϵρ^δϵRzδRtTHϵ+HϵTzδRt=−zRtT∑ϵ=1ηlϵρ^ϵR⊗Hϵ+HϵTzRt.
Similarly,(52)−2∑δ=1N∑ϵ=1ηlϵρδϵtIzδItTHϵzδIt+∑ϵ=1η∑δ=1mlϵρδϵtI−ρ^δϵIzδItTHϵ+HϵTzδIt=−∑ϵ=1η∑δ=1mlϵρ^δϵIzδItTHϵ+HϵTzδIt=−zItT∑ϵ=1ηlϵρ^ϵI⊗Hϵ+HϵTzIt.
Thus, it is easy to get that by (20)–(29) and (50)–(52)(53)V˙2t≤zRtTIN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ−ρ^ϵR⊗Hϵ+KϵT−ρ^ϵR⊗HϵTzRt+zItTIN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ϵ=1ηlϵKϵ−ρ^ϵI⊗Hϵ+KϵT−ρ^ϵI⊗HϵTzIt≤α2zt2,where α2=maxλMΦ2R,λMΦ2I<0.
From (49) and (53), we can get V2t is nonincreasing, and any term of V2t is bounded. Therefore, limt⟶+∞V2t≥0, ρδϵtR and ρδϵtI are bounded. According to (44), we can get ργϵtR and ρδϵtI are monotonically increasing; thus, ρδϵtR and ρδϵtI converge to a finite nonnegative value, which means both the limitation of ∑ϵ=1η∑δ=1mlϵ/ζδϵρδϵtR−ρ^δϵ2 and ∑ϵ=1η∑δ=1mlϵ/ζδϵρδϵtI−ρ^δϵ2 exist as well as limt⟶+∞∑ϵ=1η∑δ=1mlϵ/ζδϵρδϵtR−ρ^δr2≥0 and limt⟶+∞∑ϵ=1η∑δ=1mlϵ/ζδrρδϵtI−ρ^δϵ2≥0. Thus, we have limt→+∞∑δ=1N∑h=1n∫t−uhtt2jhRxδhRε2/1−ϱhdε+∫t−uhtt2jhIzδhIε2/1−ϱhdε+∑δ=1NzδRtTzδRt+zδItTzδItexist and is a nonnegative real number. Then, by virtue of the similar proof method as in Theorem 1, we can obtain that limt⟶+∞zt=0. Therefore, the network (41) achieves antisynchronization under the generalized pinning adaptive controller (47).


## 3.1. Antisynchronization of MWCCVDMNNs

In this paper, a single model of CVDMNN is presented as follows:(3)a˙st=−dsast+∑h=1npshastyhaht−uht,s=1,2,…,n,where the complex-valued ast indicates the state variable associated with the s-th neuron; ds>0 represents the self-feedback coefficient; pshast stands for the weight of the synaptic connection of complex-valued memristor; the complex-valued nonlinear function yh⋅ denotes the activation function of the h-th neuron; uht is the time-varying delay with 0≤uht≤uh≤u=maxh=1,2,…,nuh and u˙ht≤ϱh<1.Let(4)ast=asRt+iasIt,pshast=pshRasRt+ipshIasIt,yhaht−uht=yhRahRt−uht+iyhIahIt−uht,where asIt,pshIasIt, yhIahIt−uht are the imaginary parts of ast,pshast,yhaht−uht and asRt,pshRasRt,yhRahRt−uht are the real parts of ast,pshast,yhaht−uht.According to the characteristics of memristor’s current and voltage, one has(5)pshRasRt=p^shR,asRt≤Λs,pˇshR,asRt>Λs,pshIasIt=p^shI,asIt≤Λs,pˇshI,asIt>Λs,where s,h∈1,2,…,n; p^shR,pˇshR,p^shI,pˇshI are constants; Γs>0 represents the threshold level. Let p˜shR=maxp^shR,pˇshR, p˜shI=maxp^shI,pˇshI, p¯shR=p^shR−pˇshR, p¯shI=p^shI−pˇshI, P¯R=p¯shRn×n, P¯I=p¯shIn×n, P˜R=diag∑h=1np˜1hR2,∑h=1np˜2hR2,…,∑h=1np˜nhR2, and P˜I=diag∑h=1np˜1hI2,∑h=1np˜2hI2,…,∑h=1np˜nhI2.In the following, we consider the MWCCVDMNNs made up ofN CVDMNNs as (3)(6)A˙δt=−DAδt+PAδtyAδt¯+cδt+∑ϵ=1η∑κ=1NlϵKδκϵHϵAκt,δ=1,2,…,N,in which the complex-valued vector Aδt=Aδ1t,Aδ2t,…,AδntT∈ℂn denotes the state vector for the δ-th node; 0<D=diagd1,d2,…,dn∈ℝn×n, Aδt¯=Aδ1t−u1t,Aδ2t−u2t,…,Aδnt−untT∈ℂn, yAδt¯=y1Aδ1t−u1t,y2Aδ2t−u2t, ⋯,ynAδnt−untT∈ℂn, and Hϵ∈ℝn×nϵ=1,2,…,η is an internally coupled matrix; PAδt=pshAδstn×n∈ℂn×n; cδt=cδRt+icδIt=cδ1t,cδ2t,…,cδntT∈ℂn represents the controller; ℝ∋lϵ>0ϵ=1,2,…,η symbols coupling strength for the ϵ-th coupling form; Kϵ=KδκϵN×N∈ℝN×Nϵ=1,2,…,η describes the outer coupling matrix, where ℝ∋Kδκϵ>0, if there is a connection from node ι to node κδ≠κ; otherwise, ℝ∋Kδκϵ=0.(7)Kδδϵ=−∑k=1k≠δNKδκϵ,δ=1,2,…,N.Remark 1.
CVNNs, as an extension of RVNNs, can solve some problems which cannot be dealt by RVNNs because their states, activation functions, and connection weights are all complex-valued. For instance, the detection of symmetric problem and the XOR problem cannot be modelled by a single real-valued neuron, but they can be achieved by a single complex-valued neuron with the orthogonal decision boundaries [36]. Moreover, it is more suitable to describe various physical phenomena (e.g., the phase progression and retardation, the superposition of fields, and the wave propagation) through complex numbers in reality, all of which can be drawn as the applications of CVNNs. Naturally, CVNNs have found widespread practical applications in physical systems dealing with quantum waves, ultrasonic, and electromagnetic waves. Therefore, it is very meaningful to consider the characteristic of complex-value when studying CMNNs.Then, separating network (6) into the following imaginary and real parts:(8)A˙δRt=−DAδRt+PRAδRtyRAιRt¯+∑ϵ=1η∑κ=1NlϵKδκϵHϵAκRt−PIAδItyIAδIt¯+cδRt,A˙δIt=−DAδIt+PRAδRtyIAδIt¯+∑i=1η∑κ=1NlϵKικϵHϵAκIt+PIAδItyRAδRt¯+cδIt,where yRAδRt¯=y1RAδ1Rt−u1t,y2RAδ2Rt−u2t,…,ynRAδnRt−untT, yIAδIt¯=y1IAδ1It−u1t,y2IAδ2It−u2t,…,ynIAδnIt−untT, AδRt=Aδ1Rt,Aδ2Rt,…,AδnRtT, PR⋅=pshR⋅n×n, AδIt=Aδ1It,Aδ2It,…,AδnItT, cδRt=cδ1Rt,cδ2Rt,…,cδnRtT, and PI⋅=pshI⋅n×n, cδIt=cδ1It,cδ2It,…,cδnItT.Assumption 1.
For allβ1,β2∈ℝ, there exist real numbers YδR,YδI,jδR, jδR>0 such that(9)yδR⋅≤YδR,yδI⋅≤YδI,yδRβ1+yδRβ2≤jδRβ1+β2,yδIβ1+yδIβ2≤jδIβ1+β2.For the network (6), assume that A∗t=A∗1t,A∗2t,…,A∗ntT∈Cn∈ℂn is an arbitrary solution, then(10)A˙∗t=−DA∗t+PA∗tyA∗t.Here,A∗t=A∗Rt+iA∗It. Then, (10) can be expressed by separating it into the following two real systems:(11)A˙∗Rt=−DA∗Rt+PRA∗RtyRA∗Rt−PIA∗ItyIA∗It,A˙∗It=−DA∗It+PRA∗RtyIA∗It+PIA∗ItyRA∗Rt.Letzδt=Aδt+A∗t, then(12)z˙δt=−Dzδt+PAδtyAδt¯+cδt+PA∗tyA∗t+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκt,where zδt=zδ1t,zδ2t,…,zδntT, δ=1,2,…,N.Then, the decomposition form of (12) can be expressed as(13)z˙δRt=−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+cδRt+PIAδIt−PIA∗ItyIA∗It+∑ϵ=1η∑κ=1NlϵKικϵHϵzκRt,z˙δIt=−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+cδIt−PIAδIt−PIA∗ItyRA∗Rt+∑ϵ=1η∑κ=1NlϵKϵκϵHϵzκIt,where zδRt=zδ1Rt,zδ2Rt,…,zδnRtT, zδIt=zδ1It,zδ2It,…,zδnItT, zδRt¯=zδ1Rt−u1t,zδ2Rt−u2t,…,zδnRt−untT, zδIt¯=zδ1It−u1t,zδ2It−u2t,…,zδnIt−untT, GRzδRt¯=yRAδRt¯+yRA∗Rt, and GIzδIt¯=yIAδIt¯+yIA∗It.Definition 1.
The network (6) is said to be antisynchronized if(14)limt⟶+∞Aδt+A∗t=0,for allδ=1,2,…,N.
In this paper, the authors construct the following controller for the MWCCVDMNNs (6):(15)cδRt=−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I,cδIt=−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R,where δ=1,2,…,N; ℝ∋vδR>0 and ℝ∋vδI>0; Y¯R=Y1R,Y2R,…,YnRT; ΘδR=diagvδ1R,vδ2R,…,vδnR∈ℝn×n and ΘδI=diagvδ1I, vδ2I,…,vδnI∈ℝn×n are the positive definite control gain matrices; sign zδRt=diag (sign zδ1Rt, sign zδ2Rt,…, sign zδnRt); Y¯I=Y1I,Y2I,…,YnIT; and sign zδIt=diag (sign zδ1It, sign zδ2It,…, sign zδnIt).
For convenience, some symbols are denoted as follows:(16)zRt=z1RtT,z2RtT,…,zNRtTT,zIt=z1ItT,z2ItT,…,zNItTT,zRt¯=z1Rt¯T,z2Rt¯T,…,zNRt¯TT,zIt¯=z1It¯T,z2It¯T,…,zNIt¯TT,JR=diagj1R2,j2R2,…,jnR2,JI=diagj1I2,j2I2,…,jnI2,Λ=diag11−ϱ1,11−ϱ2,…,11−ϱn.Theorem 1.
The antisynchronization of network (6) can be realized under the controller (15) if the following conditions hold:(17)Φ1R<0andΦ1I<0,where Φ1R=IN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵT,Φ1I=IN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵT; ΘR=diagΘ1R,Θ2R,…,ΘNR∈ℝnN×nN, and ΘI=diagΘ1I,Θ2I,…,ΘNI∈ℝnN×nN.Proof.
Consider the Lyapunov function as(18)V1t=∑δ=1NzδRtTzδRt+2∑δ=1N∑h=1n∫t−uhttjhRzδhRε21−ϱhdε+2∑δ=1N∑h=1n∫t−uhttjhIzδhIε21−ϱhdε+∑δ=1NzδItTzδIt.
Then, one can yield(19)V˙1t≤2∑δ=1NzδRtT−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯+∑ϵ=1η∑κ=1N​lϵKικϵHϵzκRt−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I−PRAδRt−PRA∗RtyRA∗Rt+PIAδIt−PIA∗ItyIA∗It+2∑δ=1NzδItT−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯+∑ϵ=1η∑κ=1N​lϵKικϵHϵzκIt−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R−PRAδRt−PRA∗RtyIA∗It−PIAδIt−PIA∗ItyRA∗Rt+2zRtTIN⊗JRΛzRt−2zRt¯TIN⊗JRzRt¯+2zItTIN⊗JIΛzIt−2zIt¯TIN⊗JIzIt¯.
From Assumption1, we can deduce(20)2∑δ=1NzδRtTPRAδRtGRzδRt¯≤2∑δ=1N∑s=1n∑h=1nzδsRtp˜shRyhRAδhRt¯+yhRA∗hRt≤∑δ=1N∑s=1n∑h=1nzδsRt2p˜shR2+∑δ=1N∑h=1njhR2zδhRt¯2=zRtTIN⊗P˜RzRt+zRt¯TIN⊗JRzRt¯and(21)−2∑δ=1NzδRtTPIAδItGIzδIt¯≤zRtTIN⊗P˜IzRt+zIt¯TIN⊗JIzIt¯,(22)2∑δ=1NzδItTPRAδRtGIzδIt¯≤zItTIN⊗P˜RzIt+zIt¯TIN⊗JIzIt¯,(23)2∑δ=1NzδItTPIAδItGRzδRt¯≤zItTIN⊗P˜IzIt+zRt¯TIN⊗JRzRt¯.
In addition,(24)2∑ϵ=1η∑δ=1N∑κ=1NlϵKδκϵzδRtTHϵzκRt=zRtT∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵTzRt.
Similarly,(25)2∑ϵ=1η∑δ=1N∑κ=1NlϵKικϵzδItTHδzκIt=zItT∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵTzIt.
Furthermore,(26)2∑δ=1NzδRtTPRAδRt−PRA∗RtyRA∗Rt=2∑δ=1N∑h=1n∑s=1nzδsRtpshRAδsRt−pshRA∗sRtyhRA∗hRt≤2∑δ=1N∑h=1n∑s=1nzδsRtp^shR−p⌣shRYhR=2∑δ=1NzδRtTP¯RY¯R.
Likewise,(27)2∑δ=1NzδRtTPIAδIt−PIA∗ItyIA∗It≤2∑δ=1NzδRtTP¯IY¯I,(28)2∑δ=1NzδItTPRAδRt−PRA∗RtyIA∗It≤2∑δ=1NzδItTP¯RY¯I,(29)2∑δ=1NzδItTPIAδIt−PIA∗ItyRA∗Rt≤2∑δ=1NzδItTP¯IY¯R.
It then follows from (19)–(29) that(30)V˙1t≤zRtTIN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵTzRt+zItTIN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ϵ=1ηlϵKϵ⊗Hϵ+KϵT⊗HϵTzIt≤α1zt2,where α1=maxλMΦ1R,λMΦ1I<0.
From (30) and the construction of V1t, one can obtain that V1t is a bounded function that does not increase. Therefore, V1t converges to a nonnegative constant when t⟶+∞. In addition, according to (30), we can get(31)zt2≤V˙1tα1.
From (31), one can infer that limt⟶+∞∫0tzε2dε exists and is a nonnegative real number. Furthermore,(32)0≤limt⟶+∞∑δ=1N∑h=1n∫t−uhtt2jhRzδhRε21−ϱhdε≤limt⟶+∞∫t−utzRεTIN⊗2JRΛzRεdε≤λMIN⊗2JRΛlimt⟶+∞∫t−utzε2dε=0.
Similarly,(33)0≤limt⟶+∞∑δ=1N∑h=1n∫t−uhtt2jhIzδhIε21−ϱhdε=0.
From (32) and (33), we know that limt⟶+∞∑δ=1NzδRtTzδRt+zδItTzδIt exists and is a nonnegative real number. In the next, we will prove limt⟶+∞∑δ=1NzδRtTzδRt+zδItTzδIt=0. If this is not true, we have(34)limt⟶+∞∑δ=1NzδRtTzδRt+zδItTzδIt=θ>0.
In other words, there exists a positive real constantϖ^ such that(35)∑δ=1NzδRtTzδRt+zδItTzδIt>θ2fort≥ϖ^.
Hence,(36)zt2>θ2,whent≥ϖ^.
From (30) and (36), we can obtain(37)V˙1t<α1θ2,t≥ϖ^.
Then, according to (37), one gets(38)−V1ϖ^≤V1+∞−V1ϖ^=∫ϖ^+∞V˙1tdt<∫ϖ^+∞α1θ2dt=−∞,which leads to contradiction. Therefore,(39)limt⟶+∞∑δ=1NzδRtTzδRt+zδItTzδIt=0.
Then, we have(40)limt⟶+∞zt=0,which means that the network (6) achieves antisynchronization.


## 3.2. Generalized Pinning Antisynchronization of MWCCVDMNNs

In the following, we generalize the traditional pinning control method. One part of the controller we designed controls all the nodes, and the other part controls the firstm nodes of network (6). Then, the MWCCVDMNNs with corresponding pinning controller is described by(41)A˙δt=−DAδt+PAδtyAδt¯+vδt+∑ϵ=1η∑κ=1NlϵKικϵHϵAκt,δ=1,2,…,N,where vδt=vδRt+ivδIt∈ℂn is the generalized pinning adaptive controller and 0<D∈ℝn×n, Aδt∈ℂn, PAδt∈ℂn×n, yAδt¯∈ℂn, ℝ∋lϵ>0, Kϵ∈ℝN×N, and Hϵ∈ℝn×n have the same meanings as in Section 3.1. Without loss of generality, we select the first m nodes and pin them with the following generalized pinning adaptive controller designed as(42)vδt=cδt+c^δt,δ=1,2,…,N,where cδt=cδRt+icδIt, c^δt=c^δRt+ic^δIt, in which cδRt and cδIt are given as in (15) for δ=1,2,…,N, and(43)c^δRt=−∑ϵ=1ηlϵρδϵtRHϵzδRt,δ=1,2,…,m,0,δ=m+1,m+2,…,N,c^δIt=−∑ϵ=1ηlϵρδϵtIHϵzδIt,δ=1,2,…,m,0,δ=m+1,m+2,…,N,with(44)ρ˙δϵtR=ζδϵzδRtTHϵ+HϵT2zδRt,ρ˙δϵtI=ζδϵzδItTHϵ+HϵT2zδIt,when δ=1,2,…,m, where ℝ∋lϵ>0, zδRt=AδRt+A∗Rt, zδIt=AδIt+A∗It, zδRt=zδ1Rt,zδ2Rt,…,zδnRtT, zδIt=zδ1It,zδ2It,…,zδnItT, Hϵ∈ℝn×n, Hϵ+HϵT>0, 1≤m<N, and ℝ∋ζδϵ>0; ℝ∋ρδϵ0R>0; and ℝ∋ρδϵ0I>0 for δ=1,2,…,m.Similar to Section 3.1, the error vectorzδt=Aδt+A∗t can be governed by(45)z˙δt=−Dzδt+PAδtfAδt¯+vδt+PA∗tyA∗t+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκt,δ=1,2,…,N.In order to get the desired result, we separate networks (45) into the following equivalent imaginary and real parts:(46)z˙δRt=−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκRt+PIAδIt−PIA∗ItyIA∗It+vδRt,z˙δIt=−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκIt−PIAδIt−PIA∗ItyRA∗Rt+vδIt,where(47)vδRt=−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I−∑ϵ=1ηlϵρδϵtRHϵzδRt,vδIt=−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R−∑ϵ=1ηlϵρδϵtIHϵzδIt,in which ρ˙δεtR=ζδεzδRtTHε+HεT/2zδRt, ρ˙δεtI=ζδεzδItTHε+HεT/2zδIt for δ=1,2,…,m, and ρδϵtR=ρδϵtI≡0 for δ=m+1,m+2,…,N.Theorem 2.
If there are matrices0<ρ^ϵR=diagρ^1ϵR,ρ^2ϵR,…,ρ^mϵR,0,…,0∈ℝN×N and 0<ρ^ϵI=diagρ^1ϵI,ρ^2ϵI,…,ρ^mϵI,0,…,0∈ℝN×N, ϵ=1,2,…,u, such that(48)Φ2R<0andΦ2I<0,where Φ2R=IN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ−ρ^ϵR⊗Hϵ+KϵT−ρ^ϵR⊗HϵT,Φ2I=IN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ε=1ηlεKε−ρ^εI⊗Hε+KεT−ρ^εI⊗HεT; ΘR=diagΘ1R,Θ2R,…,ΘNR∈ℝnN×nN, ΘI=diagΘ1I,Θ2I,…,ΘNI∈ℝnN×nN; ρ^δϵR>0 and ρ^δϵI>0 for δ=1,2,…,m, then the network (41) is pinning adaptive antisynchronized.Proof.
Take the Lyapunov functional for network (45) as follows:(49)V2t=∑ϵ=1η∑δ=1NlϵζδϵρδϵtR−ρ^δϵR2+∑ϵ=1η∑δ=1NlϵζδϵρδϵtI−ρ^δϵI2+∑δ=1NzδRtTzδRt+∑δ=1NzδItTzδIt+2∑δ=1N∑h=1η∫t−uhttjhRzδhRε21−ϱhdε+2∑δ=1N∑h=1η∫t−uhttjhIzδhIε21−ϱhdε.
Then, we have(50)V˙2t≤2∑δ=1NzδRtT−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκRt−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I−∑ϵ=1ηlϵρδϵtRHϵzδRt+PIAδIt−PIA∗ItyIA∗It+2∑δ=1NzδItT−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+∑ϵ=1η∑κ=1N​lϵKδκϵHϵzκIt−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R−∑ϵ=1ηlϵρδϵtIHϵzδIt−PIAδIt−PIA∗ItyRA∗Rt+∑ϵ=1η∑δ=1mlϵρδϵtR−ρ^δϵRzδRtTHϵ+HϵTzδRt+∑ϵ=1η∑δ=1m​lϵρδϵtI−ρ^δϵIzδItTHϵ+HϵTzδIt+2zRtTIN⊗JRΛzRt−2zRt¯TIN⊗JRzRt¯+2zItTIN⊗JIΛzIt−2zIt¯TIN⊗JIzIt¯.
Moreover, we can derive that(51)−2∑δ=1N∑ϵ=1ηlϵρδϵtRzδRtTHϵzδRt+∑ϵ=1η∑δ=1mlϵρδϵtR−ρ^δϵRzδRtTHϵ+HϵTzδRt=−2∑δ=1N∑ϵ=1η​lϵρδϵtRzδRtTHϵzδRt+∑ϵ=1η∑δ=1m​lϵρδϵtRzδRtTHϵ+HϵTzδRt−∑ϵ=1η∑δ=1mlϵρ^δϵRzδRtTHϵ+HϵTzδRt=−∑ϵ=1η∑δ=1mlϵρ^δϵRzδRtTHϵ+HϵTzδRt=−zRtT∑ϵ=1ηlϵρ^ϵR⊗Hϵ+HϵTzRt.
Similarly,(52)−2∑δ=1N∑ϵ=1ηlϵρδϵtIzδItTHϵzδIt+∑ϵ=1η∑δ=1mlϵρδϵtI−ρ^δϵIzδItTHϵ+HϵTzδIt=−∑ϵ=1η∑δ=1mlϵρ^δϵIzδItTHϵ+HϵTzδIt=−zItT∑ϵ=1ηlϵρ^ϵI⊗Hϵ+HϵTzIt.
Thus, it is easy to get that by (20)–(29) and (50)–(52)(53)V˙2t≤zRtTIN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ−ρ^ϵR⊗Hϵ+KϵT−ρ^ϵR⊗HϵTzRt+zItTIN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ϵ=1ηlϵKϵ−ρ^ϵI⊗Hϵ+KϵT−ρ^ϵI⊗HϵTzIt≤α2zt2,where α2=maxλMΦ2R,λMΦ2I<0.
From (49) and (53), we can get V2t is nonincreasing, and any term of V2t is bounded. Therefore, limt⟶+∞V2t≥0, ρδϵtR and ρδϵtI are bounded. According to (44), we can get ργϵtR and ρδϵtI are monotonically increasing; thus, ρδϵtR and ρδϵtI converge to a finite nonnegative value, which means both the limitation of ∑ϵ=1η∑δ=1mlϵ/ζδϵρδϵtR−ρ^δϵ2 and ∑ϵ=1η∑δ=1mlϵ/ζδϵρδϵtI−ρ^δϵ2 exist as well as limt⟶+∞∑ϵ=1η∑δ=1mlϵ/ζδϵρδϵtR−ρ^δr2≥0 and limt⟶+∞∑ϵ=1η∑δ=1mlϵ/ζδrρδϵtI−ρ^δϵ2≥0. Thus, we have limt→+∞∑δ=1N∑h=1n∫t−uhtt2jhRxδhRε2/1−ϱhdε+∫t−uhtt2jhIzδhIε2/1−ϱhdε+∑δ=1NzδRtTzδRt+zδItTzδItexist and is a nonnegative real number. Then, by virtue of the similar proof method as in Theorem 1, we can obtain that limt⟶+∞zt=0. Therefore, the network (41) achieves antisynchronization under the generalized pinning adaptive controller (47).


## 4. Antisynchronization and Pinning Antisynchronization of MWCCVDMNNs with Coupling Delays

### 4.1. Antisynchronization of MWCCVDMNNs with Coupling Delays

In this section, the MWCCVDMNNs with coupling delays is considered as follows:(54)A˙δt=−DAδt+PAδtyAδt¯+cδt+∑ϵ=1η∑κ=1NlϵKδκϵHϵAκt^,δ=1,2,…,N,where D,Aδt, PAδt,yAδt¯, cδt,lϵ,Kϵ,Hϵ have the similar meanings as in Section 3.1; Aκt^=Aκ1t−w1t,Aκ2t−w2t,…,Aκnt−wntT∈ℝn; wϵtϵ=1,2,…,η denote the coupling delay with 0≤wϵt≤wϵ≤w=maxϵ=1,2,…,nwϵ and w˙ϵt≤ϱ¯ϵ<1.Letzδt=Aδt+A∗t, then(55)z˙δt=−Dzδt+PAδtyAδt¯+cδt+PA∗tyY∗t+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκt^,δ=1,2,…,N.Then, by separating the imaginary and real parts, system (55) can be shown as(56)z˙δRt=−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκRt^+PIAδIt−PIA∗ItyIA∗It+cδRt,z˙δIt=−DzδI+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκIt^−PIAδIt−PIA∗ItyRA∗Rt+cδIt,where zδRt, zδIt, zδRt¯, zδIt¯, GRzδRt¯, and GIzδIt¯ have the same definitions as in Section 3.1; zκRt^=zκ1Rt−w1t,zκ2Rt−w2t,…,zκnRt−wntT and zκIt^=zκ1It−w1t,zκ2It−w2t,…,zκnIt−wntT.Theorem 3.
If there are some matrices0<Ψϵ=diagb1ϵ,b2ϵ,…,bnNϵ∈ℝnN×nN, ϵ=1,2,…,η, such that(57)Φ3R<0andΦ3I<0,where Φ3R=IN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ε=1ηlεKε⊗HεΨε−1KεT⊗HεT+1/1−ϱ¯εΨε; Φ3I=IN⊗−2D+P˜R+P˜I+2JRΛ−2ΘI+∑ε=1ηlεKε⊗HεΨε−1KεT⊗HεT+1/1−ϱ¯εΨε; ΘR=diagΘ1R,Θ2R,…,ΘNR∈ℝnN×nN; and ΘI=diagΘ1I,Θ2I,…,ΘNI∈ℝnN×nN. Then, the network (54) achieves antisynchronized under the controller (15).Proof.
The Lyapunov function is chosen as(58)V3t=∑δ=1NzδRtTzδRt+2∑δ=1N∑h=1n∫t−uhttjhRzδhRε21−ϱhdε+2∑δ=1N∑h=1n∫t−uhttjhIzδhIε21−ϱhdε+∑δ=1NzδItTzδIt+∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzRεTΨϵzRεdε+∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzIεTΨϵzIεdε.
Then, we have(59)V˙3t≤2∑δ=1NzδRtT−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯+∑ϵ=1η∑κ=1NlϵKικϵHϵzκRt^−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I−PRAδRt−PRA∗RtyRA∗Rt+PIAδIt−PIA∗ItyIA∗It+2∑δ=1NzδItT−AzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκIt^−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R−PRAδRt−PRA∗RtyIA∗It−PIAδIt−PIA∗ItyRA∗Rt+2zRtTIN⊗JRΛzRt−2zRt¯TIN⊗JRzRt¯+2zItTIN⊗JIΛzIt−2zIt¯TIN⊗JIzIt¯+∑ϵ=1ηlϵ1−ϱ¯ϵzRtTΨϵzRt−∑ϵ=1ηlϵzRt^TΨrzRt^+∑ϵ=1ηlϵ1−ϱ¯ϵzItTΨϵzIt−∑ϵ=1ηlϵzIt^TΨϵzIt^.
Obviously,(60)2∑δ=1N∑ϵ=1η∑κ=1NlϵKικϵzδRtTHϵzκRt^=2∑ϵ=1ηlϵzRtTKϵ⊗HϵzRt^≤∑ϵ=1ηlϵzRtTKϵ⊗HϵΨϵ−1KϵT⊗HϵTzRt+∑ϵ=1ηlϵzRt^TΨϵzRt^.
Similarly,(61)2∑δ=1N∑ϵ=1η∑κ=1NlϵKικϵzδItTHϵzκIt^≤∑ϵ=1ηlϵzItTKϵ⊗HϵΨϵ−1KϵT⊗HϵTzIt+∑ϵ=1ηlϵzIt^TΨϵzIt^.
By (20)–(30), (26)–(29), and (60)-(61), one obtains(62)V˙3t≤zRtTIN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT+11−ϱ¯ϵΨϵzRt+zItTIN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT+11−ϱ¯ϵΨϵzIt≤α3zt2,where α3=maxλMΦ3R,λMΦ3I<0. Similar to the deduction of (31), we can obtain that limt⟶+∞∫0txε2dε exists and is a real nonnegative number. In addition,(63)0≤∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzRεTΨϵzRεdε≤∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wtzRεTΨϵzRεdε≤∑ϵ=1ηlϵ1−ϱ¯ϵλMΨϵ∫t−wtzε2dε=0.
Similarly,(64)0≤∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzIεTΨϵzIεdε=0.
From (32)-(33) and (63)-(64), one can obtain that limt⟶+∞∑δ=1NzδRtTzδRt+zδItTzδIt exists and is a nonnegative real number. Then, based on the method for proving Theorem 1, one can easily obtain limt⟶+∞zt=0. Thus, under the controller (15), the network (54) is antisynchronized.


### 4.2. Generalized Pinning Antisynchronization of MWCCVDMNNs with Coupling Delays

In this section, we add generalized pinning adaptive controller as in Section3.2 to the MWCCVDMNNs with coupling delays, then the network (54) under the pinning adaptive controller can be expressed as follows:(65)A˙δt=−DAδt+PAδtyAδt¯+vδt+∑ϵ=1η∑κ=1NlϵKδκϵHϵAκt^,δ=1,2,…,N,where vδt=vδRt+ivδIt∈ℂn is the generalized pinning adaptive controller as given in (42), 0<D∈ℝn×n, Aδt∈ℂn, PAδt∈ℂn×n, yAδt¯∈ℂn, ℝ∋lϵ>0, Kϵ∈ℝN×N, Hϵ∈ℝn×n, and Aκt^∈ℝn have the same meanings as in Section 4.1.Similarly, the error vectorzδt=Aδt+A∗t can be governed by equations as follows:(66)z˙δt=−Dzδt+PAδtyAδt¯+vδt+PA∗tyA∗t+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκt^,δ=1,2,…,N.Separating (66) into the following imaginary and real parts:(67)z˙δRt=−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+vδRt+PIAδIt−PIA∗ItyIA∗It+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκRt^,z˙δIt=−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+vδIt−PIAδIt−PIA∗ItyRA∗Rt+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκIt^,where vδRt and vδIt are given as in (47).Theorem 4.
The network (65) is pinning adaptive antisynchronization if there are matrices 0<ρ^ϵR=diagρ^1ϵR,ρ^2ϵR,…,ρ^mϵR,0,…,0∈ℝN×N, 0<ρ^ϵI=diagρ^1ϵI,ρ^2ϵI,…,ρ^mϵI,0,…,0∈ℝN×N, and 0<Ψϵ=diagb1ϵ,b2ϵ,…,bnNϵ∈ℝnN×nN, ϵ=1,2,…,η, satisfying(68)Φ4R<0andΦ4I<0,where Φ4R=IN⊗−2D+P˜R+P˜I+2JRΛ−2ΥR+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT−ρ^ϵR⊗Hϵ+HϵT+1/1−ϱ¯ϵΨϵ; Φ4I=IN⊗−2D+P˜R+P˜I+2JIΛ−2ΥI+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT−ρ^ϵR⊗Hϵ+HϵT+1/1−ϱ¯ϵΨϵ; ΘR=diagΘ1R,Θ2R,…,ΘNR∈ℝnN×nN, ΘI=diagΘ1I,Θ2I,…,ΘNI∈ℝnN×nN; ρ^δrR>0 and ρ^δrI>0 for δ=1,2,…,m.Proof.
A proper Lyapunov functional for (66) is constructed as follows:(69)V4t=∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzRεTΨϵzRεdε+∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzIεTΨϵzIεdε+∑ϵ=1η∑δ=1NlϵζδϵρδϵtR−ρ^δϵR2+∑ϵ=1η∑δ=1NlϵζδϵρδϵtI−ρ^δrI2+2∑δ=1N∑h=1n∫t−uhttjhRzδhRε21−ϱhdε+2∑δ=1N∑h=1n∫t−uhttjhIzδhIε21−ϱhdε+∑δ=1NzδRtTzδRt+∑δ=1NzδItTzδIt.
Taking the derivative ofV4t, we have(70)V˙4t≤2∑δ=1NzδRtT−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+∑ϵ=1η∑κ=1NlϵKικϵHϵzκRt^−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I−∑ϵ=1ηlϵρδϵtRHϵzδRt+PIAδIt−PIA∗ItyIA∗It+2∑δ=1NzδItT−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+∑ϵ=1η∑κ=1NlϵKικϵHϵzκIt^−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R−∑ϵ=1ηlrρδϵtIHϵzδIt−PIAδIt−PIA∗ItyRA∗Rt+∑ϵ=1η∑δ=1mlϵρδϵtR−ρ^δϵRzδRtTHϵ+HϵTzδRt+∑ϵ=1η∑δ=1mlϵρδϵtI−ρ^δϵIzδItTHϵ+HϵTzδIt+2zRtTIN⊗JRΛzRt−2zRt¯TIN⊗JRzRt¯+2zItTIN⊗JIΛzIt−2zIt¯TIN⊗JIzIt¯+∑ϵ=1ηlϵ1−ϱ¯ϵzRtTΨϵzRt−∑ϵ=1ηlϵzRt^TΨϵzRt^+∑ϵ=1ηlϵ1−ϱ¯ϵzItTΨϵzIt−∑ϵ=1ηlϵzIt^TΨϵzIt^.
By (20)–(23), (26)–(29), (51)-(52), (60)-(61), and (70), we have(71)V˙4t≤zRtTIN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT−ρ^ϵR⊗Hϵ+HϵT+11−ϱ¯ϵΨϵzRt+zItTIN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT−ρ^ϵI⊗Hϵ+HϵT+11−ϱ¯ϵΨϵzIt≤α4zt2,where α4=maxλMΦ4R,λMΦ4I<0. By combining the proofs of Theorems 2 with 3, we can obtain limt⟶+∞zt=0. Thus, the network (65) is antisynchronized under the controller (47). □


## 4.1. Antisynchronization of MWCCVDMNNs with Coupling Delays

In this section, the MWCCVDMNNs with coupling delays is considered as follows:(54)A˙δt=−DAδt+PAδtyAδt¯+cδt+∑ϵ=1η∑κ=1NlϵKδκϵHϵAκt^,δ=1,2,…,N,where D,Aδt, PAδt,yAδt¯, cδt,lϵ,Kϵ,Hϵ have the similar meanings as in Section 3.1; Aκt^=Aκ1t−w1t,Aκ2t−w2t,…,Aκnt−wntT∈ℝn; wϵtϵ=1,2,…,η denote the coupling delay with 0≤wϵt≤wϵ≤w=maxϵ=1,2,…,nwϵ and w˙ϵt≤ϱ¯ϵ<1.Letzδt=Aδt+A∗t, then(55)z˙δt=−Dzδt+PAδtyAδt¯+cδt+PA∗tyY∗t+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκt^,δ=1,2,…,N.Then, by separating the imaginary and real parts, system (55) can be shown as(56)z˙δRt=−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκRt^+PIAδIt−PIA∗ItyIA∗It+cδRt,z˙δIt=−DzδI+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκIt^−PIAδIt−PIA∗ItyRA∗Rt+cδIt,where zδRt, zδIt, zδRt¯, zδIt¯, GRzδRt¯, and GIzδIt¯ have the same definitions as in Section 3.1; zκRt^=zκ1Rt−w1t,zκ2Rt−w2t,…,zκnRt−wntT and zκIt^=zκ1It−w1t,zκ2It−w2t,…,zκnIt−wntT.Theorem 3.
If there are some matrices0<Ψϵ=diagb1ϵ,b2ϵ,…,bnNϵ∈ℝnN×nN, ϵ=1,2,…,η, such that(57)Φ3R<0andΦ3I<0,where Φ3R=IN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ε=1ηlεKε⊗HεΨε−1KεT⊗HεT+1/1−ϱ¯εΨε; Φ3I=IN⊗−2D+P˜R+P˜I+2JRΛ−2ΘI+∑ε=1ηlεKε⊗HεΨε−1KεT⊗HεT+1/1−ϱ¯εΨε; ΘR=diagΘ1R,Θ2R,…,ΘNR∈ℝnN×nN; and ΘI=diagΘ1I,Θ2I,…,ΘNI∈ℝnN×nN. Then, the network (54) achieves antisynchronized under the controller (15).Proof.
The Lyapunov function is chosen as(58)V3t=∑δ=1NzδRtTzδRt+2∑δ=1N∑h=1n∫t−uhttjhRzδhRε21−ϱhdε+2∑δ=1N∑h=1n∫t−uhttjhIzδhIε21−ϱhdε+∑δ=1NzδItTzδIt+∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzRεTΨϵzRεdε+∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzIεTΨϵzIεdε.
Then, we have(59)V˙3t≤2∑δ=1NzδRtT−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯+∑ϵ=1η∑κ=1NlϵKικϵHϵzκRt^−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I−PRAδRt−PRA∗RtyRA∗Rt+PIAδIt−PIA∗ItyIA∗It+2∑δ=1NzδItT−AzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκIt^−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R−PRAδRt−PRA∗RtyIA∗It−PIAδIt−PIA∗ItyRA∗Rt+2zRtTIN⊗JRΛzRt−2zRt¯TIN⊗JRzRt¯+2zItTIN⊗JIΛzIt−2zIt¯TIN⊗JIzIt¯+∑ϵ=1ηlϵ1−ϱ¯ϵzRtTΨϵzRt−∑ϵ=1ηlϵzRt^TΨrzRt^+∑ϵ=1ηlϵ1−ϱ¯ϵzItTΨϵzIt−∑ϵ=1ηlϵzIt^TΨϵzIt^.
Obviously,(60)2∑δ=1N∑ϵ=1η∑κ=1NlϵKικϵzδRtTHϵzκRt^=2∑ϵ=1ηlϵzRtTKϵ⊗HϵzRt^≤∑ϵ=1ηlϵzRtTKϵ⊗HϵΨϵ−1KϵT⊗HϵTzRt+∑ϵ=1ηlϵzRt^TΨϵzRt^.
Similarly,(61)2∑δ=1N∑ϵ=1η∑κ=1NlϵKικϵzδItTHϵzκIt^≤∑ϵ=1ηlϵzItTKϵ⊗HϵΨϵ−1KϵT⊗HϵTzIt+∑ϵ=1ηlϵzIt^TΨϵzIt^.
By (20)–(30), (26)–(29), and (60)-(61), one obtains(62)V˙3t≤zRtTIN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT+11−ϱ¯ϵΨϵzRt+zItTIN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT+11−ϱ¯ϵΨϵzIt≤α3zt2,where α3=maxλMΦ3R,λMΦ3I<0. Similar to the deduction of (31), we can obtain that limt⟶+∞∫0txε2dε exists and is a real nonnegative number. In addition,(63)0≤∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzRεTΨϵzRεdε≤∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wtzRεTΨϵzRεdε≤∑ϵ=1ηlϵ1−ϱ¯ϵλMΨϵ∫t−wtzε2dε=0.
Similarly,(64)0≤∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzIεTΨϵzIεdε=0.
From (32)-(33) and (63)-(64), one can obtain that limt⟶+∞∑δ=1NzδRtTzδRt+zδItTzδIt exists and is a nonnegative real number. Then, based on the method for proving Theorem 1, one can easily obtain limt⟶+∞zt=0. Thus, under the controller (15), the network (54) is antisynchronized.


## 4.2. Generalized Pinning Antisynchronization of MWCCVDMNNs with Coupling Delays

In this section, we add generalized pinning adaptive controller as in Section3.2 to the MWCCVDMNNs with coupling delays, then the network (54) under the pinning adaptive controller can be expressed as follows:(65)A˙δt=−DAδt+PAδtyAδt¯+vδt+∑ϵ=1η∑κ=1NlϵKδκϵHϵAκt^,δ=1,2,…,N,where vδt=vδRt+ivδIt∈ℂn is the generalized pinning adaptive controller as given in (42), 0<D∈ℝn×n, Aδt∈ℂn, PAδt∈ℂn×n, yAδt¯∈ℂn, ℝ∋lϵ>0, Kϵ∈ℝN×N, Hϵ∈ℝn×n, and Aκt^∈ℝn have the same meanings as in Section 4.1.Similarly, the error vectorzδt=Aδt+A∗t can be governed by equations as follows:(66)z˙δt=−Dzδt+PAδtyAδt¯+vδt+PA∗tyA∗t+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκt^,δ=1,2,…,N.Separating (66) into the following imaginary and real parts:(67)z˙δRt=−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+vδRt+PIAδIt−PIA∗ItyIA∗It+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκRt^,z˙δIt=−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+vδIt−PIAδIt−PIA∗ItyRA∗Rt+∑ϵ=1η∑κ=1NlϵKδκϵHϵzκIt^,where vδRt and vδIt are given as in (47).Theorem 4.
The network (65) is pinning adaptive antisynchronization if there are matrices 0<ρ^ϵR=diagρ^1ϵR,ρ^2ϵR,…,ρ^mϵR,0,…,0∈ℝN×N, 0<ρ^ϵI=diagρ^1ϵI,ρ^2ϵI,…,ρ^mϵI,0,…,0∈ℝN×N, and 0<Ψϵ=diagb1ϵ,b2ϵ,…,bnNϵ∈ℝnN×nN, ϵ=1,2,…,η, satisfying(68)Φ4R<0andΦ4I<0,where Φ4R=IN⊗−2D+P˜R+P˜I+2JRΛ−2ΥR+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT−ρ^ϵR⊗Hϵ+HϵT+1/1−ϱ¯ϵΨϵ; Φ4I=IN⊗−2D+P˜R+P˜I+2JIΛ−2ΥI+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT−ρ^ϵR⊗Hϵ+HϵT+1/1−ϱ¯ϵΨϵ; ΘR=diagΘ1R,Θ2R,…,ΘNR∈ℝnN×nN, ΘI=diagΘ1I,Θ2I,…,ΘNI∈ℝnN×nN; ρ^δrR>0 and ρ^δrI>0 for δ=1,2,…,m.Proof.
A proper Lyapunov functional for (66) is constructed as follows:(69)V4t=∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzRεTΨϵzRεdε+∑ϵ=1ηlϵ1−ϱ¯ϵ∫t−wϵttzIεTΨϵzIεdε+∑ϵ=1η∑δ=1NlϵζδϵρδϵtR−ρ^δϵR2+∑ϵ=1η∑δ=1NlϵζδϵρδϵtI−ρ^δrI2+2∑δ=1N∑h=1n∫t−uhttjhRzδhRε21−ϱhdε+2∑δ=1N∑h=1n∫t−uhttjhIzδhIε21−ϱhdε+∑δ=1NzδRtTzδRt+∑δ=1NzδItTzδIt.
Taking the derivative ofV4t, we have(70)V˙4t≤2∑δ=1NzδRtT−DzδRt+PRAδRtGRzδRt¯−PIAδItGIzδIt¯−PRAδRt−PRA∗RtyRA∗Rt+∑ϵ=1η∑κ=1NlϵKικϵHϵzκRt^−ΘδRzδRt−signzδRtP¯RY¯R+P¯IY¯I−∑ϵ=1ηlϵρδϵtRHϵzδRt+PIAδIt−PIA∗ItyIA∗It+2∑δ=1NzδItT−DzδIt+PRAδRtGIzδIt¯+PIAδItGRzδRt¯−PRAδRt−PRA∗RtyIA∗It+∑ϵ=1η∑κ=1NlϵKικϵHϵzκIt^−ΘδIzδIt−signzδItP¯RY¯I+P¯IY¯R−∑ϵ=1ηlrρδϵtIHϵzδIt−PIAδIt−PIA∗ItyRA∗Rt+∑ϵ=1η∑δ=1mlϵρδϵtR−ρ^δϵRzδRtTHϵ+HϵTzδRt+∑ϵ=1η∑δ=1mlϵρδϵtI−ρ^δϵIzδItTHϵ+HϵTzδIt+2zRtTIN⊗JRΛzRt−2zRt¯TIN⊗JRzRt¯+2zItTIN⊗JIΛzIt−2zIt¯TIN⊗JIzIt¯+∑ϵ=1ηlϵ1−ϱ¯ϵzRtTΨϵzRt−∑ϵ=1ηlϵzRt^TΨϵzRt^+∑ϵ=1ηlϵ1−ϱ¯ϵzItTΨϵzIt−∑ϵ=1ηlϵzIt^TΨϵzIt^.
By (20)–(23), (26)–(29), (51)-(52), (60)-(61), and (70), we have(71)V˙4t≤zRtTIN⊗−2D+P˜R+P˜I+2JRΛ−2ΘR+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT−ρ^ϵR⊗Hϵ+HϵT+11−ϱ¯ϵΨϵzRt+zItTIN⊗−2D+P˜R+P˜I+2JIΛ−2ΘI+∑ϵ=1ηlϵKϵ⊗HϵΨϵ−1KϵT⊗HϵT−ρ^ϵI⊗Hϵ+HϵT+11−ϱ¯ϵΨϵzIt≤α4zt2,where α4=maxλMΦ4R,λMΦ4I<0. By combining the proofs of Theorems 2 with 3, we can obtain limt⟶+∞zt=0. Thus, the network (65) is antisynchronized under the controller (47). □


## 5. Numerical Examples

Example 1.
Consider the MWCCVDMNNs illustrated by(72)A˙δt=−DAδt+PAδtyAδt¯+cδt+l1∑κ=16Kδκ1H1Aκt+l2∑κ=16Kδκ2H2Aκt+l3∑κ=16Kδκ3H3Aκt,whereδ=1,2,…,6, yiRμ=yiIμ=μ+1−μ−1/5i=1,2,3, D=diag0.4,0.8,0.5, l1=0.1, l2=0.2, l3=0.4, uht=1−1/2+hx−t, u=1, ϱh=1/2+h, h=1,2,3, and the matricesHϵ, Kϵ=Kικϵ6×6ϵ=1,2,3, and the elements in the matrices PAδtδ=1,2,…,6 are selected as follows:(73)H1=0.40.10.20.10.40.20.40.20.5,H2=0.40.30.40.20.70.30.40.90.6,H3=0.40.30.20.20.80.40.21.10.7,p11Raδ1Rt=0.42,aδ1Rt⩽1.7,−0.32,aδ1Rt>1.7,p12Raδ1Rt=−0.32,aδ1Rt⩽1.7,−0.36,aδ1Rt>1.7,p13Raδ1Rt=−0.28,aδ1Rt⩽1.7,−0.40,aδ1Rt>1.7,p21Raδ2Rt=0.33,aδ2Rt⩽1.7,−0.28,aδ2Rt>1.7,p22Raδ2Rt=−0.29,aδ2Rt⩽1.7,0.36,aδ2Rt>1.7,p23Raδ2Rt=0.20,aδ2Rt⩽1.7,−0.19,aδ2Rt>1.7,p31Raδ3Rt=0.24,aδ3Rt⩽1.7,−0.13,aδ3Rt>1.7,p32Raδ3Rt=0.39,aδ3Rt⩽1.7,0.26,aδ3Rt>1.7,p33Raδ3Rt=−0.34,aδ3Rt⩽1.7,−0.27,aδ3Rt>1.7,p11Iaδ1It=−0.34,aδ1It⩽1.7,0.25,aδ1It>1.7,p12Iaδ1It=0.24,aδ1It⩽1.7,0.17,aδ1It>1.7,p13Iaδ1It=−0.42,aδ1It⩽1.7,0.36,aδ1It>1.7,p21Iaδ2It=0.27,aδ2It⩽1.7,−0.14,aδ2It>1.7,p22Iaδ2It=0.26,aδ2It⩽1.7,0.17,aδ2It>1.7,p23Iaδ2It=−0.19,aδ2It⩽1.7,−0.21,aδ2It>1.7,p31Iaδ3It=−0.25,aδ3It⩽1.7,0.31,aδ3It>1.7,p32Iaδ3It=0.27,aδ3It⩽1.7,0.26,aδ3It>1.7,p33Iaδ3It=−0.38,aδ3It⩽1.7,0.40,aδ3It>1.7,K1=−0.60.20.10.100.20.3−0.70.10.10.10.10.10.2−0.80.10.30.10.20.10.3−1.10.40.10.10.10.20.2−0.90.30.20.5000.3−1.0,K2=−0.70.10.20.30.100.2−0.900.30.30.10.20.2−0.900.30.200.40.2−0.80.10.10.30.10.40.1−1.10.20.10.10.20.20−0.6,K3=−0.80.300.20.10.20.4−0.80.10.200.100−1.00.50.20.30.30.10.4−1.20.20.20.20.100.3−0.90.30.20.400.20.3−1.1.
It is readily seen that Assumption1 is satisfied with YδR=YδI=0.4 and jδR=jδI=0.4. Choose the parameters in the controller (15) as follows: ΘR=diagΘ1R,Θ2R,…,Θ6R=diag1.01,0.98,0.76,0.69,1.10,1.05,0.86,0.79,0.91,1.23,1.54,1.71,1.82,1.79,1.79,1.06,0.86,0.78, ΘI=diagΘ1I,Θ2I,…,Θ6I=diag0.92,0.76,0.87,0.92,0.71,0.78,0.86,0.99,0.94,0.85,0.69,0.76,0.59,0.71,0.77,0.89,0.83,0.90.
By making use of the YALMIP Toolbox in MATLAB, we can obtain(74)λΦ1R=−6.2147,−5.8330,−4.5881,−4.2615,−3.9606,−3.8739,−3.6252,−3.5277,−2.7759,−2.5151,−2.3761,−2.1608,−2.0750,−1.9064,−1.8040,−1.6740,−1.5948,−1.1972,λΦ1I=−4.7743,−4.3922,−4.0962,−4.0088,−3.5890,−2.3394,−2.1504,−2.0480,−1.9877,−1.9329,−1.8667,−1.7628,−1.7475,−1.7018,−1.5975,−1.5615,−1.4166,−1.0108,which meet condition (17). By Theorem 1, it implies that the network (72) has the capability of realizing antisynchronization under the pinning adaptive controller (15). Figure 1 depicts the antisynchronization simulation result of network (72). In order to display the effect of multiple coupling matrices, Figure 2 depicts the single dynamical change process of uncoupled NN in (72).Figure 1
Time evolution ofzιt, ι=1,2,…,6.Figure 2
The single dynamical change process of uncoupled NN in (72).Example 2.
Take into account the following MWCCVDMNNs with coupling delays:(75)A˙δt=−DAδt+PAδtyAδt¯+cδt+l1∑κ=1NKικ1H1Aκt^+l2∑κ=1NKικ2H2Aκt^+l3∑κ=1NKικ3H3Aκt^,where δ=1,2,…,6, yiRμ=yiIμ=μ+1−μ−1/4i=1,2,3, D=diag0.8,1.0,0.7, l1=0.4, l2=0.3, l3=0.1, uht=1−1/2+hz−t, u=1, ϱh=1/2+h, wht=1/20−1/10h+4z−t, ϱ¯h=1/10h+4, h=1,2,3, and the matricesHr, Kr=Kδκr6×6r=1,2,3 and the elements in the matrices PAδtδ=1,2,…,6 are selected as follows:(76)H1=0.18000.040.190.06000.18,H2=0.2700.0900.0700.0400.12,H3=0.13000.040.150.06000.15,p11Raδ1Rt=0.53,aδ1Rt⩽1.5,0.47,aδ1Rt>1.5,p12Raδ1Rt=0.23,aδ1Rt⩽1.5,−0.53,aδ1Rt>1.5,p13Raδ1Rt=0.32,aδ1Rt⩽1.5,0.42,aδ1Rt>1.5,p21Raδ2Rt=−0.47,aδ2Rt⩽1.5,−0.53,aδ2Rt>1.5,p22Raδ2Rt=0.31,aδ2Rt⩽1.5,0.24,aδ2Rt>1.5,p23Raδ2Rt=−0.41,aδ2Rt⩽1.5,0.51,aδ2Rt>1.5,p31Raδ3Rt=−0.27,aδ3Rt⩽1.5,−0.43,aδ3Rt>1.5,p32Raδ3Rt=0.37,aδ3Rt⩽1.5,0.25,aδ3Rt>1.5,p33Raδ3Rt=−0.38,aδ3Rt⩽1.5,0.29,aδ3Rt>1.5,p11Iaδ1It=−0.43,aδ1Rt⩽1.5,0.32,aδ1Rt>1.5,p12Iaδ1It=0.35,aδ1Rt⩽1.5,−0.27,aδ1Rt>1.5,p13Iaδ1It=0.41,aδ1Rt⩽1.5,0.40,aδ1Rt>1.5,p21Iaδ2It=0.32,aδ2Rt⩽1.5,−0.38,aδ2Rt>1.5,p22Iaδ2It=0.42,aδ2Rt⩽1.5,0.45,aδ2Rt>1.5,p23Iaδ2It=−0.52,aδ2Rt⩽1.5,−0.71,aδ2Rt>1.5,p31Iaδ3It=−0.32,aδ3Rt⩽1.5,0.41,aδ3Rt>1.5,p32Iaδ3It=0.71,aδ3Rt⩽1.5,0.54,aδ3Rt>1.5,p33Iaδ3It=0.60,aδ3Rt⩽1.5,0.57,aδ3Rt>1.5,K1=−0.70.10.300.10.20.3−0.80.10.20.10.10.10.2−1.10.50.10.200.20.4−1.10.20.30.20.100.2−1.00.50.10.20.40.10−0.8,K2=−0.80.20.20.30.100.1−1.00.20.20.30.20.30−0.900.30.30.20.10.1−0.80.30.100.30.40.2−1.10.20.20.10.30.10.2−0.9,K3=−0.90.30.10.10.20.20.2−0.800.40.10.100−1.00.50.20.30.20.10.2−0.700.20.10.10.20.2−0.90.30.300.10.20.4−1.0.
It is readily seen that Assumption1 is satisfied with YδR=YδI=0.5 and jδR=jδI=0.5. Select the parameters in the controller (47) as follows: ΘR=diagΘ1R,Θ2R,…,Θ6R=diag0.89,0.78,0.81,0.64,0.71,0.90,0.89,0.65,0.71,0.67,0.74,0.57,0.73,0.77,0.89,0.75,0.88,0.73andΘI=diagΘ1I,Θ2I,…,Θ6I=diag0.74,0.65,0.54,0.71,0.93,0.73,0.89,0.65,0.52,0.67,0.51,0.49,0.81,0.60,0.83,0.77,0.81,0.74.
We select the first 5 nodes as pinned nodes. Chooseρ^1R=diagρ^11R,ρ^21R,…,ρ^51R,0=diag0.1,0.2,0.3,0.4,0.5,0,ρ^2R=diagρ^12R,ρ^22R,…,ρ^52R,0=diag0.3,0.6,0.9,1.2,1.5,0, ρ^3R=diagρ^13R,ρ^23R,…,ρ^53R,0=diag0.2,0.4,0.6,0.8,1.0,0, ρ^1I=diagρ^11I,ρ^21I,…,ρ^51I,0=diag0.2,0.4,0.6,0.8,1.0,0, ρ^2I=diagρ^12I,ρ^22I,⋯,ρ^52I,0=diag0.4,0.8,1.2,1.6,2.0,0, and ρ^3I=diagρ^13I,ρ^23I,…,ρ^53I,0=diag0.3,0.6,0.9,1.2,1.5,0.
By making use of the YALMIP Toolbox of MATLAB, the followingΨϵϵ=1,2,3 satisfying (68) can be computed: Ψ1=diag0.5033,0.5033,0.4082,0.4840,0.5203,0.4739,0.5319,0.5095,0.4281,0.5077,0.5072,0.4086,0.5230,0.5148,0.5041,0.4876,0.5213,0.4381, Ψ2=diag0.3954,0.3820,0.3388,0.3965,0.3908,0.3726,0.4044,0.3796,0.3394,0.3947,0.3783,0.3282,0.4068,0.3867,0.3864,0.3933,0.3897,0.3518, and Ψ3=diag0.1188,0.1191,0.1194,0.1181,0.1184,0.1190,0.1186,0.1195,0.1200,0.1186,0.1190,0.1190,0.1187,0.1191,0.1201,0.1192,0.1194,0.1209.
Based on Theorem4, we conclude that the pinning antisynchronization of network (75) is realized with the controller (47). Figures 3 and 4 show the simulation results. For comparison, Figure 5 shows the variation trajectory of error variables xιt without pinned nodes.Figure 3
Time evolution ofzιt under pinning controller (47), where ι=1,2,…,6.Figure 4
ριrt when ζιr=0.5, ι=1,2,…,5, r=1,2,3.Figure 5
Time evolution ofzιt without pinned nodes, where ι=1,2,…,6.


## 6. Conclusions

This paper has investigated the antisynchronization of MWCCVDMNNs without and with coupling delays. On the one hand, we have presented some sufficient conditions for reaching antisynchronization of the proposed network models. On the other hand, some generalized pinning antisynchronization criteria on the basis of the designed pinning control strategy have been established to ensure that the considered MWCCVDMNNs with and without coupling delays realize generalized pinning antisynchronization, respectively. Furthermore, two numerical examples have been shown to verify the correctness of the derived results. This paper is an extended version of our previous work published in [37]. Based on the derived antisynchronization results of MWCCVDMNNs with and without coupling delays in [37], we have further investigated the generalized pinning antisynchronization of the considered networks by designing a novel generalized pinning adaptive controller in this paper. More specifically, the pinning controller we designed in this paper consists of two parts, one part controls all the nodes, and the other part controls the first m nodes of the considered network, which is different from the classical pinning control. To the best of the authors’ knowledge, this is the first paper toward to researching the pinning antisynchronization of the MWCCVDMNNs. However, there are several interesting problems for further study. On the one hand, some scholars have investigated the dynamical behavior of a new type of coupled complex-valued networks with intermittent coupling recently [38]. In contrast to the common continuous coupling in this paper, intermittent coupling is a discontinuous form of communication which has greater flexibility for nodes because they are not constrained by communication requirements in decoupling time, which unavoidably result in the difference of dynamics of nodes between the coupling time and decoupling period. Therefore, it would be very interesting to take the intermittent coupling into consideration when studying the pinning antisynchronization of MWCCVDMNNs in our future work. On the other hand, it is known to all that fractional calculus is a theory that generalizes the concept of calculus from the integer order to arbitrary order. Up to now, fractional-order systems have been applied in some new mechanical models due to their non-Markovian and non-Gaussian properties during the studying of dynamical systems. In [39], a novel criterion for achieving synchronization of fractional-order chaotic and hyperchaotic systems was proposed. Motivated by this work, it would be also a very interesting problem of research to insert fractional operators into the proposed MWCCVDMNNs in this paper and study the antisynchronization of this kind of the fractional-order network model.



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