# Distributed Event-Triggered Output Synchronization of Complex-Valued Memristive Reaction-Diffusion Complex Networks with Spatial Sampled-Data

Authors: Tiane Chen; Zaihe Cheng
Journal: Complexity
 (2021)

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## Abstract

This study addresses the problem of output quasisynchronization for coupled complex-valued memristive reaction-diffusion complex networks via the distributed event-triggered control scheme. First, by using the separate method, set value mapping, and intermediate value theorem, the complex-valued memristive reaction-diffusion complex networks can be transferred into two semi-uncertain real-valued reaction-diffusion complex networks. Second, a distributed output piecewise event-triggered control (OPETC) scheme with spatial sampled-data is first proposed including a spatial sampling event-triggered generator and spatiotemporal sampling state feedback controller. Furthermore, this scheme can effectively save the measurement resources and lower the update rate of controllers in spatial and time domain. Third, the synchronization analysis is considered by utilizing an appropriate Lyapunov function, the Halanay inequality, and the improved Wirtinger inequality. Subsequently, several output event-triggered quasisynchronization criteria are derived. The relations among event trigger conditions, spatial sampling interval, convergence rate, and control gain are given by rigorous mathematical derivation. Finally, multiple simulations are compared to substantiate the validation of the OPETC scheme.


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## Body

## 1. Introduction

Complex networks have excited continuous interest for their nonlinear learning characteristics and have been extensively implemented in various domains such as deep learning [1–3], natural language analyzation, optimization computing, and image segments. When two or more networks interact, coupled complex networks, such as coupled neural networks (CNNs), can be formed and have attracted much considerable attentions for their significant collective dynamical behaviour. As an important collective dynamical behaviour, synchronization means that the states of complex networks are always consistent over time. Many crucial results about the synchronization of CNNs have been achieved in literatures [4–7]. For instance, Bai and Xu [4] studied the synchronization of CNNs with hybrid coupling, Luo and Yao [5] investigated the finite-time synchronization of uncertain complex dynamic networks, Wang et al. [6] researched the synchronization of coupled CNNs via the pinning method, and Ding et al. [7] considered the impulsive synchronization of complex networks.It should be highlighted that most research studies suppose that the electromagnetic fields of complex networks are uniform, and the networks states vary only with time. In fact, inhomogeneous electromagnetic fields inevitably exist in cellular networks [8], genetic regulatory networks, and traffic networks. In nonuniform electromagnetic fields, the networks states in different spatial positions are distinct and change simultaneously. This phenomenon can be expressed as the reaction-diffusion (RD) term in mathematical models. Then, the CNNs model is transformed into coupled RD neural networks (CRDNNs) with infinite dimensional states. Up to now, plenty of impressive studies on the synchronization of CRDNNs have been achieved. The authors investigated the impulsive synchronization [9], the adaptive synchronization [10], and the passivity [11] of CRDNNs. The abovementioned works assumed that the global spatial state information was known, and all these research studies aimed to synchronize all states. In some applications, it is costly and not necessary to measure the whole infinite dimensional state and to synchronize all states. Therefore, it is valuable to research partial state synchronization using incomplete measurements. Partial state synchronization named output synchronization (OS) has been investigated in literatures [12–15]. The authors [16] studied OS of CRDNNs with input constraints via continuous time control, and the authors [17] investigated OS of CRDNNs with spatial sampled-data via continuous time control. To our knowledge, few results about OS of CRDNNs have been investigated via noncontinuous time control. This is our first motivation for this study.Abovementioned investigations of complex networks have intensively emerged with state-independent weights realized by resistors. As a special resistor, memristor has been proposed by Chua [18] and has been realized first by HP Labs. Memristor can retain its history state owing to its switch characteristic which makes the memristor an attractive candidate to emulate the biological brain to improve machine intelligence [19, 20]. Consequently, memristive reaction-diffusion neural networks (MRDNNs) can be constructed with richer properties than common CRDNNs. Accordingly, it is significant to study MRDNNs with state-dependent weight. Until now, many worthwhile results about MRDNNs have been achieved. Global synchronization [21, 22], exponential Lag synchronization [23], and fixed-time synchronization [24] of MRDNNs have been studied. Nonetheless, the mismatch of the master-slave system due to state-dependent parameters inevitably causes the synchronization error of the collective behaviour to change in a certain range. That is quasisynchronization [25]. Actually, the research results on quasisynchronization of MRDNNs have not yet been published. This is our second motivation for this study.Different from real-valued neural networks, complex-valued neural networks own more complicated properties. Some scientific questions, such as the phase-sensitive detection [26] and exclusive or problem [27] in optical signal processing, can only be addressed by complex-valued neural networks. Complex-valued MRDNNs (CMRDNNs) have complex-valued states, state-dependent parameter, and complex-valued activation function which are more general than real-valued MRDNNs. Hence, it is necessary to investigate CMRDNNs. Besides, the existing difficulty is that complex-valued functions may not be bounded or analytic [28]. Many assumptions about real-valued functions cannot be utilized in the complexed-valued domain without any preprocessing. Thus, the general approach is to convert complex-valued functions into two parts: real and imaginary parts. Using this conversion method, intermittent control synchronization [28], exponential sampled-data synchronization, [29] and pinning control synchronization [30] of complex-valued memristive networks were investigated without considering RD term. Nevertheless, the synchronization problem of CMRDNNs has rarely been addressed. Song et al. [31] studied the finite-time synchronization of CMRDNNs with known full state information and activation function bounds. Therefore, it is meaningful to further research the synchronization problem of CMRDNNs with incomplete measurements.To achieve the synchronization of complex networks, many control strategies have been studied, such as continuous time control [16], sampled-data estimation and control [29, 32], and intermittent control [28]. Among these control strategies, sampled-data control has received wider attention because this method can reduce communication bandwidth. Sampled-data control includes time-triggered control and event-triggered control [33–35]. Compared with the time-triggered control mechanism, the event-triggered control scheme updates information only when the measurement error satisfies the triggering threshold condition, which leads to a lower information exchange rate. Therefore, the event-triggered sampling control strategy performs better than the time-triggered method [36]. Event trigger’s form can be considered as ex,t2>γzx,t2+ηt. The event trigger is considered static one when the function ηt is a constant, and the event trigger is considered dynamic one when the derivative of the function ηt is not zero [37]. Thus, the event-triggered control strategy can be classified into dynamical event triggers and static event triggers according to the function ηt. Meanwhile, the event-triggered control can be classified into centralized control and distributed control according to the scope of the controller. In large-scale coupled networks, it is difficult to realize complete centralized control. Thus, further research on distributed event-triggered control is crucial, and one of the primary problems to be prevented by event-triggered control scheme is Zeno phenomenon. As far as we know, the distributed event-triggered control strategy for synchronization of CMRDNNs with incomplete spatial measurement has not been studied so far. This is our third motivation for this study.Motivated by the abovementioned analysis, the main aim of this work is to establish the output quasisynchronization conditions of CMRDNNs via distributed event-triggered partial spatial control strategy. More specially, some questions should be solved: How to obtain the error model of the master-slave networks systems with state-dependent parameters? How to design an efficient distributed piecewise event-triggered control strategy using incomplete measurement to achieve output synchronization? What are the effects of dynamic event trigger and static event trigger on quasisynchronization? To achieve the control goal and answer the above three questions, there are three innovations in this study:(1)
A general CMRDNNs mathematical model with the output matrix is proposed. By using set-valued mappings, differential inclusions, and nonfragile techniques, the real-valued synchronization error model with uncertainty parameters is derived, and the assumption conditions about activation functions are more relaxed.(2)
We establish a distributed output piecewise event-triggered control (OPETC) strategy with spatial sampled-data to ensure the convergent speed and to reduce the communication cost. By using some inequalities and an appropriate Lyapunov functional, several universal criteria guaranteeing the output quasisynchronization of the discussed networks are established, where the relations among diffusion coefficients, control gains, memristive complex-valued range, and time delays are clear.(3)
By comparing the simulation results of time trigger, dynamical event trigger, static event trigger, and piecewise event trigger, the high efficiency of the OPETC strategy is verified.The structure of this study is as follows: the mathematical model is established and some assumptions are given. Then, the real-valued synchronization error models with uncertainty parameters are deduced in Section2. A distributed piecewise event-triggered control strategy with spatial sampled-data is designed, and several universal criteria guaranteeing the output quasisynchronization of the discussed networks are deduced in Section 3. Simulation examples are given to compare the effect of different triggers and to corroborate the quasisynchronization conditions in Section 4. Section 5 draws the conclusion of this study and clarifies our future research.


## 2. Preliminaries

In this section, we provide some necessary lemmas and show the process of deriving the mathematical model of the master-slave networks system to two real-valued error models.


### 2.1. Notations

LetN=0,1,2,… and N+=1,2,…. For each n∈N+,Nn=0,1,2,…,n and Nn+=1,2,…,n. ℛ and C denote the set of real-valued space and complex-valued space, respectively. Similarly, ℛn and Cn represent the sets of n-dimensional real-valued and n-dimensional complex-valued vectors. ℛn×N and Cn×N describe the sets of n×N real-valued matrices and complex-valued matrices. The Hilbert space of square integrable functions over Ω is denoted by ℒn2Ω with norm zx,⋅2=∫ΩzTx,⋅zx,⋅dx, Ω=0,l, and n∈N+. ℐn is a n-dimensional identity matrix. ℋn1Ω=z∈ℒn2Ω|/dzdx∈ℒn2Ω is a Sobolev space which is also a subspace of ℒn2Ω. T denotes the matrix transposition, and ⊗ represents the Kronecker product. l denotes the imaginary part with ι=−1. zx,⋅=zRx,⋅+ιzIx,⋅, zRx,⋅∈ℛ, and zIx,⋅∈ℛ.


### 2.2. Model Transform

In a master-slave system scheme, there are two coupled networks with the same structure and diverse initial values. The master-coupled complex-valued memristive reaction-diffusion networks can be expressed as(1)∂z˜ix,t∂t=−Az˜ix,t+B∂2z˜ix,t∂x2+cz˜ix,tfiz˜ix,t+∑j=1NdijΓgjz˜jx,t−τjt,y˜ix,t=βz˜ix,t,where z˜ix,t=z˜i1x,t,⋯,z˜inx,tT∈ℋn1Ω denotes the state information of the ith networks with i∈NN+, t∈t0,+∞ and t0≥0; x∈Ω corresponds to the space variable; A∈ℛn×n represents the self-inhibition coefficient; B∈ℛn×n denotes the diffusion coefficient; τjt represents the nonnegative time-varying transmission delay from the jth networks to the ith networks; τ¯=maxτjt; cz˜ix,t is the connection weights with memristive characteristic; fi⋅ corresponds to the neural activation vector function of the ith networks; Γ∈ℛn×n is the inner coupling matrix; D=dijN×N∈CN×N denotes the coupling strength matrix where dii=−∑j=1,j≠iNdij​; gj⋅ denotes the complex-valued state function; y˜ix,t∈Cr×1 is the output vector; β∈ℛr×n represents the output matrix; and βTβ is the idempotent matrix.The initial values and the Dirichlet boundary values of the master networks are given by(2)z^ix,s=ν^ix,s,x,t∈Ω×t0−τ¯,t0,z^i0,t=z^il,t=0,x,t∈∂Ω×t0−τ¯,+∞,,where ν˜ix,s is a complex-valued function.Similarly, the slave networks can be designed as(3)∂z^ix,t∂t=−Az^ix,t+B∂2z^ix,t∂x2+ciz^ix,tfiz^ix,t+∑j=1NdijΓgjz^jx,t−τjt​+u^ix,t,y^ix,t=βz^ix,t,where u^ix,t∈ℋn1Ω is the control input of the ith networks to achieve output synchronization; the initial condition and the Dirichlet boundary condition of the slave networks are given by(4)z^ix,s=ν^ix,s,x,t∈Ω×t0−τ¯,t0,z^i0,t=z^il,t=0,x,t∈∂Ω×t0−τ¯,+∞,where ν^ix,s is a complex-valued function.Suppose thatci0 and ci1 are two complex-valued constants and υix,t∈C. According to the switch property of memristor, the connection weight ciυix,t can be defined as(5)ciυix,t=ci0,υi⋅,t↑,ci1,υi⋅,t↓,lims⟶t−ciυi⋅,t,υi⋅,tunchange,where ci0=ci0R+ιci0I, ci1=ci1R+ιci1I, c¯iR=maxci0R,ci1R, c¯iR=minci0R,ci1R, c¯iI=maxci0I,ci1I, and c¯iI=minci0I,ci1I.We can formulate equations (1) and (3) in a compact matrix form as(6)∂z˜x,t∂t=−IN⊗Az˜x,t+IN⊗B∂2z˜x,t∂x2+Cz˜x,tfz˜x,t+D⊗Γgz˜x,t−τt,y˜x,t=IN⊗βz˜x,t,∂z^x,t∂t=−IN⊗Az^x,t+IN⊗B∂2z^x,t∂x2+Cz^x,tfz^x,t+D⊗Γgz^x,t−τt​+ux,t,y^x,t=IN⊗βz^x,t,where z˜x,t=z˜1Tx,t,…,z˜NTx,tT, z^x,t=z^1Tx,t,…,z^NTx,tT, ux,t=u1Tx,t,u2Tx,t,…,uNTx,tT, Cz˜x,t=c1z˜1Tx,t,…,cNz˜NTx,tT, Cz^x,t=c1z^1Tx,t,…,cNz^NTx,tT, f⋅=f1T⋅,…,fNT⋅T, g⋅=g1T⋅,…,gNT⋅T, y˜x,t=y˜1Tx,t,…,y˜NTx,tT, and y^x,t=y^1Tx,t…,y^NTx,tT.Assume that the master-slave networks (equations (1) and (3)) with Dirichlet boundary conditions have unique continuous solutions z˜ix,t and z^x,t, respectively.Remark 1.
The authors [17] considered the reaction-diffusion term and output matrix. The mathematical model in [38] included memristive parameters and reaction-diffusion terms without complex-valued state and output matrix. The authors [31] investigated complex-valued states, memristive parameters, and reaction-diffusion term without considering output matrix. Different from literatures [17, 31, 38], this study introduces complex value states, memristive parameters, reaction-diffusion terms, and output matrices into the mathematical model, which are more general and practical in applications.
Our goal is to synchronize the master system (equation (1)) and the slave system (equation (3)). Then, we naturally need to derivate the error model of the master-slave system. However, many assumptions about real-valued functions cannot be utilized in the complexed-valued domain. Meanwhile, the existence of memristive parameters leads to parameter mismatch of the master networks system and the slave one. Obviously, there are some difficulties in deriving the error model directly. Thus, the model should be converted. First, the complex-valued model is converted into two real-valued models by using some assumptions. Second, the real-valued synchronization error model can be obtained by using different techniques.Assumption 1.
zx,t, Czx,t, and fzx,t can be divided into real and imaginary parts, respectively.(7)zx,t=zRx,t+ιzIx,t,Czx,t=CRzx,t+ιCIzx,t,fzx,t=fRzx,t+ιfIzx,t,where zRx,t,CRzx,t, and fRzx,t are the real parts; zIx,t,CIzx,t, and fIzx,t are the imaginary parts.Lemma 1 see ([39]).
For any functionυξ|ξ∈a,b,υξ∈ℋn1a,b. If xa=0 or xb=0, then(8)∫abυTξυξdξ≤4b−a2π2∫ab∂υξ∂ξT∂υξ∂ξdξ.Corollary 1 see ([40, 41]).
For any functionxξ|ξ∈a,b,xξ∈ℋn1a,b, c=aorb. If r0,r1,r2,r3>0, then the following integral inequalities hold,(9)−r0∫ab∂xξ∂ξ2dξ−r1∫abx2cdξ≤−r2∫abx2ξdξ,where r2=π2r0r1/π2r0+4b−a2r1.
Similarly, we can get(10)−r0∫ab∂xξ∂ξ2dξ+r1x2c≤r3∫abx2ξdξ,where r3=π2r0r1/π2r0−4b−a2r1.Lemma 2 see ([25, 42]).
Assume that the functionWt≥0 for all t∈−∞,+∞,(11)dWtdt≤αtWt+βtsupt−τt≤s≤tWs+γt,t≥t0,Wt=ψt,t≤t0,where ψt is bounded and continuous and three functions −αt, βt, and γt are continuous and positive. If there exists σ>0 and αt+βt≤−σ for t≥t0, then we have Wt≤γ¯/σ+supt−τt≤s≤tWsexp−η¯t−t0, where γ¯=supt0≤tγt and η¯=inft0≤tηt:ηt+αt+βtexpηtτt=0.
Under Assumption1, equation (6) can be divided into two real-valued mathematical models as(12)∂z˜Rx,t∂t=−IN⊗Az˜Rx,t+IN⊗B∂2z˜Rx,t∂x2+CRz˜x,tfRz˜x,t,−CIz˜x,tfIz˜x,t+D⊗ΓgRz˜x,t−τt,y˜Rx,t=IN⊗βz˜Rx,t,∂z˜Ix,t∂t=−IN⊗Az˜Ix,t+IN⊗B∂2z˜Ix,t∂x2+CRz˜x,tfIz˜x,t,+CIz˜x,tfRz˜x,t+D⊗ΓgIz˜x,t−τt,y˜Ix,t=IN⊗βz˜Ix,t,(13)∂z^Rx,t∂t=−IN⊗Az^Rx,t+IN⊗B∂2z^Rx,t∂x2+CRz^x,tfRz^x,t,−CIz^x,tfIz^x,t+D⊗ΓgRz^x,t−τt​+uRx,t,y^Rx,t=IN⊗βz^Rx,t,∂z^Ix,t∂t=−IN⊗Az^Ix,t+IN⊗B∂2z^Ix,t∂x2+CRz^x,tfIz^x,t,+CIz^x,tfRz^x,t+D⊗ΓgIz^x,t−τt​+uIx,t,y^Ix,t=IN⊗βz^Ix,t.
LetC¯i=diagc¯1,…,c¯n, C¯i=diagc¯1,…,c¯n, C¯=diagC¯iN×N∈CnN×nN, C¯=diagC¯iN×N∈CnN×nN, and Cυx,t∈coC¯,C¯. Using some properties of set-valued mapping [43], equations (12) and (13) can be expressed as the differential inclusions:(14)∂z˜Rx,t∂t∈−IN⊗Az˜Rx,t+IN⊗B∂2z˜Rx,t∂x2+coC¯R,C¯RfRz˜x,t−coC¯I,C¯IfIz˜x,t+D⊗ΓgRz˜x,t−τt,∂z˜Ix,t∂t∈−IN⊗Az˜Ix,t+IN⊗B∂2z˜Ix,t∂x2+coC¯R,C¯RfIz˜x,t+coC¯I,C¯IfRz˜x,t+D⊗ΓgIz˜x,t−τt,∂z^Rx,t∂t∈−IN⊗Az^Rx,t+IN⊗B∂2z^Rx,t∂x2+coC¯R,C¯RfRz^x,t−coC¯I,C¯IfIz^x,t+D⊗ΓgRz^x,t−τt​+uRx,t,∂z^Ix,t∂t∈−IN⊗Az^Ix,t+IN⊗B∂2z^Ix,t∂x2+coC¯R,C¯RfIz^x,t+coC¯I,C¯IfRz^x,t+D⊗ΓgIz^x,t−τt​+uIx,t,where C¯iR=diagc¯1R,…,c¯nR,C¯iR=diagc¯1R,…,c¯nR, C¯iI=diagc¯1I,…,c¯nI,C¯iI=diagc¯1I,…,c¯nI, C¯R=diagC¯iRN×N, C¯R=diagC¯iRN×N, C¯I=diagC¯iIN×N, C¯I=diagC¯iIN×N, C¯R,C¯R,C¯I,C¯I∈CnN×nN, CRz˜x,t∈coC¯R,C¯R,andCIz˜x,t∈coC¯I,C¯I.Assumption 2.
fR⋅,fI⋅,gR⋅, and gI⋅ are the bounded activation functions and are Lipschitz continuous. There exist positive values hfR,, hfI,hgR,hgI as(15)fRξ1−fRξ22≤hfRξ1−ξ22,fIξ1−fIz˜x,t2≤hfIξ1−ξ22,gRξ1−gRξ22≤hgRξ1−ξ22,gIξ1−gIz˜x,t2≤hgIξ1−ξ22,that hold for any ξ1,ξ2∈ℋn1a,b.Remark 2.
Assumption2 is more relaxed than the one in the literatures [36, 38] and does not need to know the bounds of the state.Assumption 3.
The activation function term in equation (14) satisfies the following conditions:(16)coC¯R,C¯RfRz˜x,t−coC¯R,C¯RfRz^x,t⊆coC¯R,C¯RfRz˜x,t−fRz^x,t,coC¯I,C¯IfIz˜x,t−coC¯I,C¯IfIz^x,t⊆coC¯I,C¯IfIz˜x,t−fIz^x,t,coC¯R,C¯RfIz˜x,t−coC¯R,C¯RfIz^x,t⊆coC¯R,C¯RfIz˜x,t−fIz^x,t,coC¯I,C¯IfRz˜x,t−coC¯I,C¯IfRz^x,t⊆coC¯I,C¯IfRz˜x,t−fRz^x,t,where z˜x,t,z^x,t are the solutions of equation (6) with the Dirichlet boundary condition.
Denote the error state, the error output, and the error activation function aszRx,t=z^Rx,t−z˜Rx,t,zIx,t=z^Ix,t−z˜Ix,t,yRx,t=y^Rx,t−y˜Rx,t,yIx,t=y^Ix,t−y˜Ix,t,HfRz^x,t−z˜x,t=fRz^x,t−fRz˜x,t,HfIz^x,t−z˜x,t=fIz^x,t−fIz˜x,t, and HgRz^x,t−z˜x,t=gRz^x,t−gRz˜x,t,HgIz^x,t−z˜x,t=gIz^x,t−gIz˜x,t.
Under Assumption3, the error system of equation (14) can be rephrased as(17)∂zRx,t∂t∈−IN⊗AzRx,t+IN⊗B∂2zRx,t∂x2+coC¯R,C¯RHfRzx,t−coC¯I,C¯IHfIzx,t+D⊗ΓHgRzx,t−τt+uRx,t,yRx,t=IN⊗βzRx,t,∂zIx,t∂t∈−IN⊗AzIx,t+IN⊗B∂2zIx,t∂x2+coC¯R,C¯RHfIzx,t+coC¯I,C¯IHfRzx,t+D⊗ΓHgRzx,t−τt+uIx,t,yIx,t=IN⊗βzIx,t.
There exist measurable matricesC˜Rx,t∈coC¯R,C¯R,C˜Ix,t∈coC¯I,C¯I, and one has(18)∂zRx,t∂t=−IN⊗AzRx,t+IN⊗B∂2zRx,t∂x2+C˜Rx,tHfRzx,t−C˜Ix,tHfIzx,t+D⊗ΓHgRzx,t−τt+uRx,t,yRx,t=IN⊗βzRx,t,∂zIx,t∂t=−IN⊗AzIx,t+IN⊗B∂2zIx,t∂x2+C˜Rx,tHfIzx,t+C˜Ix,tHfRzx,t+D⊗ΓHgRzx,t−τt+uIx,t,yIx,t=IN⊗βzIx,t,with the initial values and the boundary values of networks as(19)z0,t=zl,t=0,t∈t0−τ¯,+∞,zx,t=νx,t,νx,t=ν^x,t−ν˜x,t,x,t∈Ω×t0−τ¯,t0.
Because the time-varying parameter is known, upper and lower bounds can be converted into a general semi-uncertain parameter by using the intermediate value theorem. There exists anyςix,t satisfied 0≤ςix,t≤1, and c˜iRx,t=c¯iR+c¯iR−c¯iRςiRx,t,c˜iIx,t=c¯iI+c¯iI−c¯iIςiIx,t. Then, C˜x,t varies over time and can be expressed as semi-uncertain parameters:(20)C˜Rx,t=C¯R+C¯R−C¯RFRx,t,C˜Ix,t=C¯I+C¯I−C¯IFIx,t,where FTx,tFx,t≤InN, FRx,t=diagF1Rx,t,…,FNRx,t,FiRx,t=diagς1Rx,t,…,ςnRx,t, FIx,t=diagF1Ix,t,…,FNIx,t,FiIx,t=diagς1Ix,t,…,ςnIx,t, and OR=C¯R−C¯R,OI=C¯I−C¯I.Remark 3.
Equations (18) and (20) can be seen as real-valued error models with semi-uncertainty parameters.


## 2.1. Notations

LetN=0,1,2,… and N+=1,2,…. For each n∈N+,Nn=0,1,2,…,n and Nn+=1,2,…,n. ℛ and C denote the set of real-valued space and complex-valued space, respectively. Similarly, ℛn and Cn represent the sets of n-dimensional real-valued and n-dimensional complex-valued vectors. ℛn×N and Cn×N describe the sets of n×N real-valued matrices and complex-valued matrices. The Hilbert space of square integrable functions over Ω is denoted by ℒn2Ω with norm zx,⋅2=∫ΩzTx,⋅zx,⋅dx, Ω=0,l, and n∈N+. ℐn is a n-dimensional identity matrix. ℋn1Ω=z∈ℒn2Ω|/dzdx∈ℒn2Ω is a Sobolev space which is also a subspace of ℒn2Ω. T denotes the matrix transposition, and ⊗ represents the Kronecker product. l denotes the imaginary part with ι=−1. zx,⋅=zRx,⋅+ιzIx,⋅, zRx,⋅∈ℛ, and zIx,⋅∈ℛ.


## 2.2. Model Transform

In a master-slave system scheme, there are two coupled networks with the same structure and diverse initial values. The master-coupled complex-valued memristive reaction-diffusion networks can be expressed as(1)∂z˜ix,t∂t=−Az˜ix,t+B∂2z˜ix,t∂x2+cz˜ix,tfiz˜ix,t+∑j=1NdijΓgjz˜jx,t−τjt,y˜ix,t=βz˜ix,t,where z˜ix,t=z˜i1x,t,⋯,z˜inx,tT∈ℋn1Ω denotes the state information of the ith networks with i∈NN+, t∈t0,+∞ and t0≥0; x∈Ω corresponds to the space variable; A∈ℛn×n represents the self-inhibition coefficient; B∈ℛn×n denotes the diffusion coefficient; τjt represents the nonnegative time-varying transmission delay from the jth networks to the ith networks; τ¯=maxτjt; cz˜ix,t is the connection weights with memristive characteristic; fi⋅ corresponds to the neural activation vector function of the ith networks; Γ∈ℛn×n is the inner coupling matrix; D=dijN×N∈CN×N denotes the coupling strength matrix where dii=−∑j=1,j≠iNdij​; gj⋅ denotes the complex-valued state function; y˜ix,t∈Cr×1 is the output vector; β∈ℛr×n represents the output matrix; and βTβ is the idempotent matrix.The initial values and the Dirichlet boundary values of the master networks are given by(2)z^ix,s=ν^ix,s,x,t∈Ω×t0−τ¯,t0,z^i0,t=z^il,t=0,x,t∈∂Ω×t0−τ¯,+∞,,where ν˜ix,s is a complex-valued function.Similarly, the slave networks can be designed as(3)∂z^ix,t∂t=−Az^ix,t+B∂2z^ix,t∂x2+ciz^ix,tfiz^ix,t+∑j=1NdijΓgjz^jx,t−τjt​+u^ix,t,y^ix,t=βz^ix,t,where u^ix,t∈ℋn1Ω is the control input of the ith networks to achieve output synchronization; the initial condition and the Dirichlet boundary condition of the slave networks are given by(4)z^ix,s=ν^ix,s,x,t∈Ω×t0−τ¯,t0,z^i0,t=z^il,t=0,x,t∈∂Ω×t0−τ¯,+∞,where ν^ix,s is a complex-valued function.Suppose thatci0 and ci1 are two complex-valued constants and υix,t∈C. According to the switch property of memristor, the connection weight ciυix,t can be defined as(5)ciυix,t=ci0,υi⋅,t↑,ci1,υi⋅,t↓,lims⟶t−ciυi⋅,t,υi⋅,tunchange,where ci0=ci0R+ιci0I, ci1=ci1R+ιci1I, c¯iR=maxci0R,ci1R, c¯iR=minci0R,ci1R, c¯iI=maxci0I,ci1I, and c¯iI=minci0I,ci1I.We can formulate equations (1) and (3) in a compact matrix form as(6)∂z˜x,t∂t=−IN⊗Az˜x,t+IN⊗B∂2z˜x,t∂x2+Cz˜x,tfz˜x,t+D⊗Γgz˜x,t−τt,y˜x,t=IN⊗βz˜x,t,∂z^x,t∂t=−IN⊗Az^x,t+IN⊗B∂2z^x,t∂x2+Cz^x,tfz^x,t+D⊗Γgz^x,t−τt​+ux,t,y^x,t=IN⊗βz^x,t,where z˜x,t=z˜1Tx,t,…,z˜NTx,tT, z^x,t=z^1Tx,t,…,z^NTx,tT, ux,t=u1Tx,t,u2Tx,t,…,uNTx,tT, Cz˜x,t=c1z˜1Tx,t,…,cNz˜NTx,tT, Cz^x,t=c1z^1Tx,t,…,cNz^NTx,tT, f⋅=f1T⋅,…,fNT⋅T, g⋅=g1T⋅,…,gNT⋅T, y˜x,t=y˜1Tx,t,…,y˜NTx,tT, and y^x,t=y^1Tx,t…,y^NTx,tT.Assume that the master-slave networks (equations (1) and (3)) with Dirichlet boundary conditions have unique continuous solutions z˜ix,t and z^x,t, respectively.Remark 1.
The authors [17] considered the reaction-diffusion term and output matrix. The mathematical model in [38] included memristive parameters and reaction-diffusion terms without complex-valued state and output matrix. The authors [31] investigated complex-valued states, memristive parameters, and reaction-diffusion term without considering output matrix. Different from literatures [17, 31, 38], this study introduces complex value states, memristive parameters, reaction-diffusion terms, and output matrices into the mathematical model, which are more general and practical in applications.
Our goal is to synchronize the master system (equation (1)) and the slave system (equation (3)). Then, we naturally need to derivate the error model of the master-slave system. However, many assumptions about real-valued functions cannot be utilized in the complexed-valued domain. Meanwhile, the existence of memristive parameters leads to parameter mismatch of the master networks system and the slave one. Obviously, there are some difficulties in deriving the error model directly. Thus, the model should be converted. First, the complex-valued model is converted into two real-valued models by using some assumptions. Second, the real-valued synchronization error model can be obtained by using different techniques.Assumption 1.
zx,t, Czx,t, and fzx,t can be divided into real and imaginary parts, respectively.(7)zx,t=zRx,t+ιzIx,t,Czx,t=CRzx,t+ιCIzx,t,fzx,t=fRzx,t+ιfIzx,t,where zRx,t,CRzx,t, and fRzx,t are the real parts; zIx,t,CIzx,t, and fIzx,t are the imaginary parts.Lemma 1 see ([39]).
For any functionυξ|ξ∈a,b,υξ∈ℋn1a,b. If xa=0 or xb=0, then(8)∫abυTξυξdξ≤4b−a2π2∫ab∂υξ∂ξT∂υξ∂ξdξ.Corollary 1 see ([40, 41]).
For any functionxξ|ξ∈a,b,xξ∈ℋn1a,b, c=aorb. If r0,r1,r2,r3>0, then the following integral inequalities hold,(9)−r0∫ab∂xξ∂ξ2dξ−r1∫abx2cdξ≤−r2∫abx2ξdξ,where r2=π2r0r1/π2r0+4b−a2r1.
Similarly, we can get(10)−r0∫ab∂xξ∂ξ2dξ+r1x2c≤r3∫abx2ξdξ,where r3=π2r0r1/π2r0−4b−a2r1.Lemma 2 see ([25, 42]).
Assume that the functionWt≥0 for all t∈−∞,+∞,(11)dWtdt≤αtWt+βtsupt−τt≤s≤tWs+γt,t≥t0,Wt=ψt,t≤t0,where ψt is bounded and continuous and three functions −αt, βt, and γt are continuous and positive. If there exists σ>0 and αt+βt≤−σ for t≥t0, then we have Wt≤γ¯/σ+supt−τt≤s≤tWsexp−η¯t−t0, where γ¯=supt0≤tγt and η¯=inft0≤tηt:ηt+αt+βtexpηtτt=0.
Under Assumption1, equation (6) can be divided into two real-valued mathematical models as(12)∂z˜Rx,t∂t=−IN⊗Az˜Rx,t+IN⊗B∂2z˜Rx,t∂x2+CRz˜x,tfRz˜x,t,−CIz˜x,tfIz˜x,t+D⊗ΓgRz˜x,t−τt,y˜Rx,t=IN⊗βz˜Rx,t,∂z˜Ix,t∂t=−IN⊗Az˜Ix,t+IN⊗B∂2z˜Ix,t∂x2+CRz˜x,tfIz˜x,t,+CIz˜x,tfRz˜x,t+D⊗ΓgIz˜x,t−τt,y˜Ix,t=IN⊗βz˜Ix,t,(13)∂z^Rx,t∂t=−IN⊗Az^Rx,t+IN⊗B∂2z^Rx,t∂x2+CRz^x,tfRz^x,t,−CIz^x,tfIz^x,t+D⊗ΓgRz^x,t−τt​+uRx,t,y^Rx,t=IN⊗βz^Rx,t,∂z^Ix,t∂t=−IN⊗Az^Ix,t+IN⊗B∂2z^Ix,t∂x2+CRz^x,tfIz^x,t,+CIz^x,tfRz^x,t+D⊗ΓgIz^x,t−τt​+uIx,t,y^Ix,t=IN⊗βz^Ix,t.
LetC¯i=diagc¯1,…,c¯n, C¯i=diagc¯1,…,c¯n, C¯=diagC¯iN×N∈CnN×nN, C¯=diagC¯iN×N∈CnN×nN, and Cυx,t∈coC¯,C¯. Using some properties of set-valued mapping [43], equations (12) and (13) can be expressed as the differential inclusions:(14)∂z˜Rx,t∂t∈−IN⊗Az˜Rx,t+IN⊗B∂2z˜Rx,t∂x2+coC¯R,C¯RfRz˜x,t−coC¯I,C¯IfIz˜x,t+D⊗ΓgRz˜x,t−τt,∂z˜Ix,t∂t∈−IN⊗Az˜Ix,t+IN⊗B∂2z˜Ix,t∂x2+coC¯R,C¯RfIz˜x,t+coC¯I,C¯IfRz˜x,t+D⊗ΓgIz˜x,t−τt,∂z^Rx,t∂t∈−IN⊗Az^Rx,t+IN⊗B∂2z^Rx,t∂x2+coC¯R,C¯RfRz^x,t−coC¯I,C¯IfIz^x,t+D⊗ΓgRz^x,t−τt​+uRx,t,∂z^Ix,t∂t∈−IN⊗Az^Ix,t+IN⊗B∂2z^Ix,t∂x2+coC¯R,C¯RfIz^x,t+coC¯I,C¯IfRz^x,t+D⊗ΓgIz^x,t−τt​+uIx,t,where C¯iR=diagc¯1R,…,c¯nR,C¯iR=diagc¯1R,…,c¯nR, C¯iI=diagc¯1I,…,c¯nI,C¯iI=diagc¯1I,…,c¯nI, C¯R=diagC¯iRN×N, C¯R=diagC¯iRN×N, C¯I=diagC¯iIN×N, C¯I=diagC¯iIN×N, C¯R,C¯R,C¯I,C¯I∈CnN×nN, CRz˜x,t∈coC¯R,C¯R,andCIz˜x,t∈coC¯I,C¯I.Assumption 2.
fR⋅,fI⋅,gR⋅, and gI⋅ are the bounded activation functions and are Lipschitz continuous. There exist positive values hfR,, hfI,hgR,hgI as(15)fRξ1−fRξ22≤hfRξ1−ξ22,fIξ1−fIz˜x,t2≤hfIξ1−ξ22,gRξ1−gRξ22≤hgRξ1−ξ22,gIξ1−gIz˜x,t2≤hgIξ1−ξ22,that hold for any ξ1,ξ2∈ℋn1a,b.Remark 2.
Assumption2 is more relaxed than the one in the literatures [36, 38] and does not need to know the bounds of the state.Assumption 3.
The activation function term in equation (14) satisfies the following conditions:(16)coC¯R,C¯RfRz˜x,t−coC¯R,C¯RfRz^x,t⊆coC¯R,C¯RfRz˜x,t−fRz^x,t,coC¯I,C¯IfIz˜x,t−coC¯I,C¯IfIz^x,t⊆coC¯I,C¯IfIz˜x,t−fIz^x,t,coC¯R,C¯RfIz˜x,t−coC¯R,C¯RfIz^x,t⊆coC¯R,C¯RfIz˜x,t−fIz^x,t,coC¯I,C¯IfRz˜x,t−coC¯I,C¯IfRz^x,t⊆coC¯I,C¯IfRz˜x,t−fRz^x,t,where z˜x,t,z^x,t are the solutions of equation (6) with the Dirichlet boundary condition.
Denote the error state, the error output, and the error activation function aszRx,t=z^Rx,t−z˜Rx,t,zIx,t=z^Ix,t−z˜Ix,t,yRx,t=y^Rx,t−y˜Rx,t,yIx,t=y^Ix,t−y˜Ix,t,HfRz^x,t−z˜x,t=fRz^x,t−fRz˜x,t,HfIz^x,t−z˜x,t=fIz^x,t−fIz˜x,t, and HgRz^x,t−z˜x,t=gRz^x,t−gRz˜x,t,HgIz^x,t−z˜x,t=gIz^x,t−gIz˜x,t.
Under Assumption3, the error system of equation (14) can be rephrased as(17)∂zRx,t∂t∈−IN⊗AzRx,t+IN⊗B∂2zRx,t∂x2+coC¯R,C¯RHfRzx,t−coC¯I,C¯IHfIzx,t+D⊗ΓHgRzx,t−τt+uRx,t,yRx,t=IN⊗βzRx,t,∂zIx,t∂t∈−IN⊗AzIx,t+IN⊗B∂2zIx,t∂x2+coC¯R,C¯RHfIzx,t+coC¯I,C¯IHfRzx,t+D⊗ΓHgRzx,t−τt+uIx,t,yIx,t=IN⊗βzIx,t.
There exist measurable matricesC˜Rx,t∈coC¯R,C¯R,C˜Ix,t∈coC¯I,C¯I, and one has(18)∂zRx,t∂t=−IN⊗AzRx,t+IN⊗B∂2zRx,t∂x2+C˜Rx,tHfRzx,t−C˜Ix,tHfIzx,t+D⊗ΓHgRzx,t−τt+uRx,t,yRx,t=IN⊗βzRx,t,∂zIx,t∂t=−IN⊗AzIx,t+IN⊗B∂2zIx,t∂x2+C˜Rx,tHfIzx,t+C˜Ix,tHfRzx,t+D⊗ΓHgRzx,t−τt+uIx,t,yIx,t=IN⊗βzIx,t,with the initial values and the boundary values of networks as(19)z0,t=zl,t=0,t∈t0−τ¯,+∞,zx,t=νx,t,νx,t=ν^x,t−ν˜x,t,x,t∈Ω×t0−τ¯,t0.
Because the time-varying parameter is known, upper and lower bounds can be converted into a general semi-uncertain parameter by using the intermediate value theorem. There exists anyςix,t satisfied 0≤ςix,t≤1, and c˜iRx,t=c¯iR+c¯iR−c¯iRςiRx,t,c˜iIx,t=c¯iI+c¯iI−c¯iIςiIx,t. Then, C˜x,t varies over time and can be expressed as semi-uncertain parameters:(20)C˜Rx,t=C¯R+C¯R−C¯RFRx,t,C˜Ix,t=C¯I+C¯I−C¯IFIx,t,where FTx,tFx,t≤InN, FRx,t=diagF1Rx,t,…,FNRx,t,FiRx,t=diagς1Rx,t,…,ςnRx,t, FIx,t=diagF1Ix,t,…,FNIx,t,FiIx,t=diagς1Ix,t,…,ςnIx,t, and OR=C¯R−C¯R,OI=C¯I−C¯I.Remark 3.
Equations (18) and (20) can be seen as real-valued error models with semi-uncertainty parameters.


## 3. Output Event-Triggered Synchronization of CMRDNNs

### 3.1. Piecewise Event-Triggered Control Strategy

The state feedback input controller of the slave networks system can be designed as follows.(21)uix,t=−∑q=1Quixq,t=−∑q=1QαxqLiβTβzixq,tki,where Li=diagLi1,Li2,…,Lin, i∈Nn; αxq denotes a spatial distributed operator with ∫Ωzx,tαxqdx=zxq,t, and Q represents the number of the spatial sampled-data. For simplicity, αxq can be replaced by α.Remark 4.
uixq,t only uses the output state in the partial spatial domain xq and at some discrete time instantstk. Thus, uixq,t can be seen as a distributed spatiotemporal sampled-data controller. Different from the spatial sampled-data controller designed in [17], this controller designed in equation (21) is a spatial and temporal sampling control.
Letux,t=u1Tx,t,u2Tx,t,…,uNTx,t​T, L=diagL1,L2,…,LN, and ux,t can be formulated into the real-valued compact matrix form:(22)uRx,t=−∑q=1QαLRIN⊗βTβzRxq,tk,uIx,t=−∑q=1QαLIIN⊗βTβzIxq,tk,where zRxq,tk=z1RTxq,tk1,…,zNRTxq,tkNT, and zIxq,tk=z1ITxq,tk1,…,zNITxq,tkNT.
The transmission time sequence of the input controller can be determined by event-triggered conditions.
The information flow diagram of the distributed event-triggered control mechanism corresponding to theith slave system is shown in Figure 1, where the event trigger is used to determine whether the controller updates with the current spatial sampling signal. That is, an update event is determined by the event-triggered condition [44]:
The distributed output event-triggered condition is designed as(23)tk+1i=inft:t≥tki,∑q=1Qβeixq,tTβeixq,t−∑q=1Qγiβzixq,tTβzixq,t−η¯it>0,where eixq,t=zixq,tki−zixq,t;tk+1i denotes the next event-triggered time instants of the ith networks; and γi,η¯it>0 represent parts of the event trigger which can be adaptive derived later.
The condition of equation (23) can generate a triggering time sequence t0i,⋯,tki,tk+1i,⋯ for the ith networks control.
Letex,t=e1Tx,t,…,eNTx,tT,ϒ=diagγ1,…γ,N, and the real-valued compact form of event-triggered condition can be expressed as(24)∑q=1QeRTxq,tIN⊗βTβeRxq,t>∑q=1QzRTxq,tϒ⊗βTβzRxq,t+η¯Rt,∑q=1QeITxq,tIN⊗βeIxq,t>∑q=1QzITxq,tϒ⊗βzIxq,t+η¯It,where η¯t=η¯Rt+ιη¯It, η¯Rt=∑i=1Nη¯iRt,andη¯It=∑i=1Nη¯iIt.
Assumeη¯it is a piecewise function as(25)η¯it=η¯i0R+ιη¯i0Ifft,fft=exp−κ1t−t0,t0<t<tf,exp−κ1tf−t0,t≥tf,where four parameters η¯i0R,η¯i0I,η¯i0R, and η¯i0I are the positives, η¯0R=∑i=1Nη¯iRt0,andη¯0I=∑i=1Nη¯iIt0.
From equation (25), we find that tk+1i>tki is always satisfied. Therefore, the Zeno behaviour can be naturally excluded.Figure 1
Theith event-triggered controller structure of the ith master-slave system.Remark 5.
The distributed event-triggered method in [38] defined an event trigger for each neural network, and each event trigger required the entire state information of the corresponding neural networks. Different from the definition in [38], the distributed event trigger defined in equation (14) uses spatial sampled-data, which reduces the communication load and decreases the measurement cost.Remark 6.
Because the semi-uncertain error system always has a certain range of error [25], quasisynchronization is a more realistic goal. Meanwhile, to better solving the Zeno problem, the function η¯it in the event-triggered condition must be positive. Therefore, the piecewise function η¯it in equation (25) is a good selection which is different from the exponential function in literatures [45, 46].
Using the above distributed output piecewise event-triggered control strategy (equations (24) and (25)), the quasisynchronization conditions are obtained, and a theorem is constructed for the master-slave networks system in Section 3.2.


### 3.2. Synchronization Analysis

Theorem 1.
Under Assumptions1, 2, and 3, using the distributed piecewise event-triggered mechanism (equation (23)) and the sampling controller (equation (21)), the master-slave networks systems (equations (1) and (3)) can be output quasisynchronized within the error Θ, and the error system (equation (18)) converges exponentially to the error level Θ with the convergence index κ¯2>0, if the criteria are satisfied:(26)1ε22ε−1>ϒ,λ1>λ2expκ2tτt>0,where(27)Θ=ελ1−λ2λmaxLRη¯0R+λmaxLIη¯0Isupt≥t0fft,fft=exp−κ1t−t0,t0<t<tf,exp−κ1tf−t0,t≥tf,κ¯2=inft0≤tκ2t:κ2t−λ1+λ2expκ2tτt=0,λ1=λmin2IN⊗A+ℛ1R−3εINn−εhfRP1R+P1I,2IN⊗A+ℛ1I−3εINn−εhfIP1R+P1I,λ2=λmaxIN⊗βTβεhgRP2,IN⊗βTβεhgIP2ℛ1R=M1M2LRM1+2ξ1M2LR−1,ℛ1I=M1M2LIM1+2ξ1M2LI−1,P1R=CRTCR+ORTOR,P1I=CITCI+OITOI,P2=DTD⊗ΓTΓM1=2IN⊗B,M2=2−1εIN−εϒ⊗In,ξ1=1π2max2≤q≤Qxq−xq−1,x1,l−xQ2,where λmin and λmax mean the minimum and maximum eigenvalues, respectively.Proof.
Construct the following Lyapunov functional for the system (equation (18)) as(28)Vt=∫ΩzRTx,tIN⊗βTIN⊗βzRx,tdx+∫ΩzITx,tIN⊗βTIN⊗βzIx,tdx.
By using integral formula of division and equation (18), we can derive(29)∫ΩzRTx,t2IN⊗βTβB∂2zRx,t∂x2dx≤−∫Ω∂zRTx,t∂x2IN⊗βTβB∂zRx,t∂xdx,∫ΩzITx,t2IN⊗βTβB∂2zIx,t∂x2dx≤−∫Ω∂zITx,t∂x2IN⊗βTβB∂zIx,t∂xdx.
According to equation (24) and by using Cauchy inequality yields(30)−2∫ΩzRTx,tIN⊗βTβ∑q=1QαIN⊗βTβLRzRxq,tkdx=−2∑q=1QzRTxq,tIN⊗βTβLReRxq,t+zRxq,t≤∑q=1Q1ε−2zRTxq,tIN⊗βTβLRzRxq,t+εeRTxq,tIN⊗βTβLReRxq,t=−∑q=1QzRTxq,t2−1εIN−εϒ⊗βTβLRzRxq,t+ηRt.
Similarly,(31)−2∑q=1QzITxq,tIN⊗βTβLIzIxq,tk≤−∑q=1QzITxq,t2−1εIN−εϒ⊗βTβLIzIxq,t+ηIt.where ηRt=ελmLRη¯Rt and ηIt=ελmLIη¯It.
According to Lemma1, Corollary 1, and the boundary condition of equation (18),(32)−∫Ω∂zRTx,t∂x2IN⊗βTβB∂zRx,t∂xdx−∑q=1QzRTxq,t2−1εIN−εϒ⊗βTβLRzRxq,t≤−∫ΩzRTx,tIN⊗βTβℛ1RzRx,tdx.
Similarly,(33)−∫Ω∂zITx,t∂x2IN⊗βTβB∂zIx,t∂xdx−∑q=1QzITxq,t2−1εIN−εϒ⊗βTβLIzIxq,t≤−∫ΩzITx,tIN⊗βTβℛ1IzIx,tdx,where ℛ1R=M1M2LRM1+2ξ1M2LR−1, ℛ1I=M1M2LIM1+2ξ1M2LI−1, M1=2IN⊗B, ϒ<1/ε22ε−1IN, M2=2−1/εIN−εϒ⊗In, andξ1=1/π2max2≤q≤Qxq−xq−1,x1,l−xQ2.
According to Assumption2 and Cauchy inequality, the real part and imaginary part of the activation function can be derived:(34)2∫ΩzRTx,tIN⊗βTβC˜Rx,tΗfRzx,tdx−2∫ΩzRTx,tIN⊗βTβC˜Ix,tΗfIzx,tdx≤∫ΩzRTx,tIN⊗βTβ2εINn+εhfRP1RzRx,tdx+∫ΩzITx,tIN⊗βTβεhfIP1IzIx,tdx,2∫ΩzITx,tIN⊗βTβC˜Rx,tΗfIzx,t+C˜Ix,tΗfRzx,tdx≤∫ΩzITx,tIN⊗βTβ2εINn+εhfIP1RzIx,tdx+∫ΩzRTx,tIN⊗βTβεhfRP1IzRx,tdx,whereP1R=CRTCR+ORTOR and P1I=CITCI+OITOI.
Real part and imaginary part of the state-coupling part can be processed as follows:(35)2∫ΩzRTx,tIN⊗βTβD⊗ΓΗgRzx,t−τtdx≤∫ΩzRTx,t1εIN⊗βTβzRx,tdx+∫ΩzRTx,t−τtIN⊗βTβεhgRP2zRx,t−τtdx,2∫ΩzITx,tIN⊗βTβD⊗ΓΗgIzx,t−τtdx≤∫ΩzITx,t1εIN⊗βTβzIx,tdx+∫ΩzITx,t−τtIN⊗βTβεhgIP2zIx,t−τtdx.where P2=DTD⊗ΓTΓ.
According to equation (28) to equation (35),(36)V˙t=2∫ΩzRTx,tIN⊗βTβ∂zRx,t∂tdx+2∫ΩzITx,tIN⊗βTβ∂zIx,t∂tdx≤∫ΩzRTx,tIN⊗βTβ−2IN⊗A−ℛ1R+3εINn+εhfRP1R+P1IzRx,t+∫ΩzITx,tIN⊗βTβ−2IN⊗A−ℛ1I+3εINn+εhfIP1R+P1IzIx,tdx+∫ΩzRTx,t−τtIN⊗βTβεhgRP2zRx,t−τtdx+∫ΩzITx,t−τtIN⊗βTβεhgIP2zIx,t−τtdx+ελmLRη¯Rt+λmLIη¯It≤−λ1Vt+λ2Vt−τt+ελmaxLRη¯Rt+λmaxLIη¯It,with λ0,λ1,λ2 defined in Theorem 1.
From Lemma2, we can obtain(37)Vt≤Θ+supt−τt≤s≤tVsexp−κ¯2t−t0,where Θ=ε/λ1−λ2λmaxLRη¯0R+λmaxLIη¯0Isupt≥t0fft,κ¯2=inft0≤tκ2t:κ2t−λ1+λ2expκ2tτt=0, and λ1>λ2expκ2tτt.
Iffft is set as equation (25) and κ1≥κ¯2, then the error system in equations (18) and (20) converges exponentially to Θ with the exponential index κ¯2. That is, the master-slave system (equations (1) and (3)) achieves quasisynchronization within the error levelΘ.


## 3.1. Piecewise Event-Triggered Control Strategy

The state feedback input controller of the slave networks system can be designed as follows.(21)uix,t=−∑q=1Quixq,t=−∑q=1QαxqLiβTβzixq,tki,where Li=diagLi1,Li2,…,Lin, i∈Nn; αxq denotes a spatial distributed operator with ∫Ωzx,tαxqdx=zxq,t, and Q represents the number of the spatial sampled-data. For simplicity, αxq can be replaced by α.Remark 4.
uixq,t only uses the output state in the partial spatial domain xq and at some discrete time instantstk. Thus, uixq,t can be seen as a distributed spatiotemporal sampled-data controller. Different from the spatial sampled-data controller designed in [17], this controller designed in equation (21) is a spatial and temporal sampling control.
Letux,t=u1Tx,t,u2Tx,t,…,uNTx,t​T, L=diagL1,L2,…,LN, and ux,t can be formulated into the real-valued compact matrix form:(22)uRx,t=−∑q=1QαLRIN⊗βTβzRxq,tk,uIx,t=−∑q=1QαLIIN⊗βTβzIxq,tk,where zRxq,tk=z1RTxq,tk1,…,zNRTxq,tkNT, and zIxq,tk=z1ITxq,tk1,…,zNITxq,tkNT.
The transmission time sequence of the input controller can be determined by event-triggered conditions.
The information flow diagram of the distributed event-triggered control mechanism corresponding to theith slave system is shown in Figure 1, where the event trigger is used to determine whether the controller updates with the current spatial sampling signal. That is, an update event is determined by the event-triggered condition [44]:
The distributed output event-triggered condition is designed as(23)tk+1i=inft:t≥tki,∑q=1Qβeixq,tTβeixq,t−∑q=1Qγiβzixq,tTβzixq,t−η¯it>0,where eixq,t=zixq,tki−zixq,t;tk+1i denotes the next event-triggered time instants of the ith networks; and γi,η¯it>0 represent parts of the event trigger which can be adaptive derived later.
The condition of equation (23) can generate a triggering time sequence t0i,⋯,tki,tk+1i,⋯ for the ith networks control.
Letex,t=e1Tx,t,…,eNTx,tT,ϒ=diagγ1,…γ,N, and the real-valued compact form of event-triggered condition can be expressed as(24)∑q=1QeRTxq,tIN⊗βTβeRxq,t>∑q=1QzRTxq,tϒ⊗βTβzRxq,t+η¯Rt,∑q=1QeITxq,tIN⊗βeIxq,t>∑q=1QzITxq,tϒ⊗βzIxq,t+η¯It,where η¯t=η¯Rt+ιη¯It, η¯Rt=∑i=1Nη¯iRt,andη¯It=∑i=1Nη¯iIt.
Assumeη¯it is a piecewise function as(25)η¯it=η¯i0R+ιη¯i0Ifft,fft=exp−κ1t−t0,t0<t<tf,exp−κ1tf−t0,t≥tf,where four parameters η¯i0R,η¯i0I,η¯i0R, and η¯i0I are the positives, η¯0R=∑i=1Nη¯iRt0,andη¯0I=∑i=1Nη¯iIt0.
From equation (25), we find that tk+1i>tki is always satisfied. Therefore, the Zeno behaviour can be naturally excluded.Figure 1
Theith event-triggered controller structure of the ith master-slave system.Remark 5.
The distributed event-triggered method in [38] defined an event trigger for each neural network, and each event trigger required the entire state information of the corresponding neural networks. Different from the definition in [38], the distributed event trigger defined in equation (14) uses spatial sampled-data, which reduces the communication load and decreases the measurement cost.Remark 6.
Because the semi-uncertain error system always has a certain range of error [25], quasisynchronization is a more realistic goal. Meanwhile, to better solving the Zeno problem, the function η¯it in the event-triggered condition must be positive. Therefore, the piecewise function η¯it in equation (25) is a good selection which is different from the exponential function in literatures [45, 46].
Using the above distributed output piecewise event-triggered control strategy (equations (24) and (25)), the quasisynchronization conditions are obtained, and a theorem is constructed for the master-slave networks system in Section 3.2.


## 3.2. Synchronization Analysis

Theorem 1.
Under Assumptions1, 2, and 3, using the distributed piecewise event-triggered mechanism (equation (23)) and the sampling controller (equation (21)), the master-slave networks systems (equations (1) and (3)) can be output quasisynchronized within the error Θ, and the error system (equation (18)) converges exponentially to the error level Θ with the convergence index κ¯2>0, if the criteria are satisfied:(26)1ε22ε−1>ϒ,λ1>λ2expκ2tτt>0,where(27)Θ=ελ1−λ2λmaxLRη¯0R+λmaxLIη¯0Isupt≥t0fft,fft=exp−κ1t−t0,t0<t<tf,exp−κ1tf−t0,t≥tf,κ¯2=inft0≤tκ2t:κ2t−λ1+λ2expκ2tτt=0,λ1=λmin2IN⊗A+ℛ1R−3εINn−εhfRP1R+P1I,2IN⊗A+ℛ1I−3εINn−εhfIP1R+P1I,λ2=λmaxIN⊗βTβεhgRP2,IN⊗βTβεhgIP2ℛ1R=M1M2LRM1+2ξ1M2LR−1,ℛ1I=M1M2LIM1+2ξ1M2LI−1,P1R=CRTCR+ORTOR,P1I=CITCI+OITOI,P2=DTD⊗ΓTΓM1=2IN⊗B,M2=2−1εIN−εϒ⊗In,ξ1=1π2max2≤q≤Qxq−xq−1,x1,l−xQ2,where λmin and λmax mean the minimum and maximum eigenvalues, respectively.Proof.
Construct the following Lyapunov functional for the system (equation (18)) as(28)Vt=∫ΩzRTx,tIN⊗βTIN⊗βzRx,tdx+∫ΩzITx,tIN⊗βTIN⊗βzIx,tdx.
By using integral formula of division and equation (18), we can derive(29)∫ΩzRTx,t2IN⊗βTβB∂2zRx,t∂x2dx≤−∫Ω∂zRTx,t∂x2IN⊗βTβB∂zRx,t∂xdx,∫ΩzITx,t2IN⊗βTβB∂2zIx,t∂x2dx≤−∫Ω∂zITx,t∂x2IN⊗βTβB∂zIx,t∂xdx.
According to equation (24) and by using Cauchy inequality yields(30)−2∫ΩzRTx,tIN⊗βTβ∑q=1QαIN⊗βTβLRzRxq,tkdx=−2∑q=1QzRTxq,tIN⊗βTβLReRxq,t+zRxq,t≤∑q=1Q1ε−2zRTxq,tIN⊗βTβLRzRxq,t+εeRTxq,tIN⊗βTβLReRxq,t=−∑q=1QzRTxq,t2−1εIN−εϒ⊗βTβLRzRxq,t+ηRt.
Similarly,(31)−2∑q=1QzITxq,tIN⊗βTβLIzIxq,tk≤−∑q=1QzITxq,t2−1εIN−εϒ⊗βTβLIzIxq,t+ηIt.where ηRt=ελmLRη¯Rt and ηIt=ελmLIη¯It.
According to Lemma1, Corollary 1, and the boundary condition of equation (18),(32)−∫Ω∂zRTx,t∂x2IN⊗βTβB∂zRx,t∂xdx−∑q=1QzRTxq,t2−1εIN−εϒ⊗βTβLRzRxq,t≤−∫ΩzRTx,tIN⊗βTβℛ1RzRx,tdx.
Similarly,(33)−∫Ω∂zITx,t∂x2IN⊗βTβB∂zIx,t∂xdx−∑q=1QzITxq,t2−1εIN−εϒ⊗βTβLIzIxq,t≤−∫ΩzITx,tIN⊗βTβℛ1IzIx,tdx,where ℛ1R=M1M2LRM1+2ξ1M2LR−1, ℛ1I=M1M2LIM1+2ξ1M2LI−1, M1=2IN⊗B, ϒ<1/ε22ε−1IN, M2=2−1/εIN−εϒ⊗In, andξ1=1/π2max2≤q≤Qxq−xq−1,x1,l−xQ2.
According to Assumption2 and Cauchy inequality, the real part and imaginary part of the activation function can be derived:(34)2∫ΩzRTx,tIN⊗βTβC˜Rx,tΗfRzx,tdx−2∫ΩzRTx,tIN⊗βTβC˜Ix,tΗfIzx,tdx≤∫ΩzRTx,tIN⊗βTβ2εINn+εhfRP1RzRx,tdx+∫ΩzITx,tIN⊗βTβεhfIP1IzIx,tdx,2∫ΩzITx,tIN⊗βTβC˜Rx,tΗfIzx,t+C˜Ix,tΗfRzx,tdx≤∫ΩzITx,tIN⊗βTβ2εINn+εhfIP1RzIx,tdx+∫ΩzRTx,tIN⊗βTβεhfRP1IzRx,tdx,whereP1R=CRTCR+ORTOR and P1I=CITCI+OITOI.
Real part and imaginary part of the state-coupling part can be processed as follows:(35)2∫ΩzRTx,tIN⊗βTβD⊗ΓΗgRzx,t−τtdx≤∫ΩzRTx,t1εIN⊗βTβzRx,tdx+∫ΩzRTx,t−τtIN⊗βTβεhgRP2zRx,t−τtdx,2∫ΩzITx,tIN⊗βTβD⊗ΓΗgIzx,t−τtdx≤∫ΩzITx,t1εIN⊗βTβzIx,tdx+∫ΩzITx,t−τtIN⊗βTβεhgIP2zIx,t−τtdx.where P2=DTD⊗ΓTΓ.
According to equation (28) to equation (35),(36)V˙t=2∫ΩzRTx,tIN⊗βTβ∂zRx,t∂tdx+2∫ΩzITx,tIN⊗βTβ∂zIx,t∂tdx≤∫ΩzRTx,tIN⊗βTβ−2IN⊗A−ℛ1R+3εINn+εhfRP1R+P1IzRx,t+∫ΩzITx,tIN⊗βTβ−2IN⊗A−ℛ1I+3εINn+εhfIP1R+P1IzIx,tdx+∫ΩzRTx,t−τtIN⊗βTβεhgRP2zRx,t−τtdx+∫ΩzITx,t−τtIN⊗βTβεhgIP2zIx,t−τtdx+ελmLRη¯Rt+λmLIη¯It≤−λ1Vt+λ2Vt−τt+ελmaxLRη¯Rt+λmaxLIη¯It,with λ0,λ1,λ2 defined in Theorem 1.
From Lemma2, we can obtain(37)Vt≤Θ+supt−τt≤s≤tVsexp−κ¯2t−t0,where Θ=ε/λ1−λ2λmaxLRη¯0R+λmaxLIη¯0Isupt≥t0fft,κ¯2=inft0≤tκ2t:κ2t−λ1+λ2expκ2tτt=0, and λ1>λ2expκ2tτt.
Iffft is set as equation (25) and κ1≥κ¯2, then the error system in equations (18) and (20) converges exponentially to Θ with the exponential index κ¯2. That is, the master-slave system (equations (1) and (3)) achieves quasisynchronization within the error levelΘ.


## 4. Numerical Examples

This section mainly completes multiple numerical simulations to testify the theoretical results.Consider the master-slave error networks system as(38)∂zix,t∂t=−Azix,t+B∂2zix,t∂x2+cizix,tfizix,t+∑j=1NdijΓgjzjx,t−τjt​−∑q=1QαxqLiβTβzixq,tk,where i=1,2,N=2,n=3,gi⋅=fi⋅=tanh⋅;(39)c1v⋅,s=−0.5−1.2ι,v⋅,s↓,−0.4−1.1ιv⋅,s↑,lims⟶t−v⋅,sv⋅,sunchange,c2v⋅,s=−1−1ι,v⋅,s↓,−0.9−0.9ι,v⋅,s↑,lims⟶t−v⋅,s,v⋅,sunchange,A=0.10000.10000.1,B=0.50000.50000.5,D=−0.30.30.3−0.3,Γ=0.50000.50000.5.The positions of two spatial point measurements arex1=0.25l,x2=0.75l. The initial conditions of the master system are(40)z˜1x,0=2−2cos3πx+i1−cos4πx1−3sin5πx+i2−3sin4πx−1.5+0.5sin4πx+i−0.5+sin3πx,z˜2x,0=cos5πx+i1+2cos6πx3−3sin3πx+i4−sin2πx1.5+2sin2πx+i3+3sin2πx.The initial conditions of the slave system are(41)z^1x,0=3cos2πx−2+4cosπx−3ι,3+1.5ι,1+1ιT,z^2x,0=5cos7πx−3+5cos7πx−3ι,1+1ι,1+1ιT.


### 4.1. Output Static Event-Triggered Control (OSETC) withβ=100;010

Event trigger’s parameters satisfied with equation (26) are set as γi=0.02,, η¯it=0.01, and Lij=10. And the simulation results are shown in Figures 2–4. Figure 2 represents the jth dimensional state norm evolution of the ith error networks over time. The real-valued state norm evolution and the imaginary-valued one can be formulated as(42)zijRx,t2=∫ΩzijRTx,t,zijRx,tdx,zijIx,t2=∫ΩzijITx,t,zijIx,tdx,i=1,2,j=1,2,3.Figure 2
Real part and imaginary part of state error norm evolution under the output static event-triggered control. net11, net12, and net13 represent three-dimensional state norm evolution of the 1st error network. net21, net22, and net23 represent three-dimensional state norm evolution of the 2nd error network. (a) Real part of state error evolution. (b) Imaginary part of state error evolution.
(a)(b)Figure 3
Real part of state error evolution under the output static event-triggered control. net11, net12, and net13 represent 3-dimensional real-valued state evolution of the 1sterror network. net21, net22, and net23 represent 3-dimensional real-valued state evolution of the 2nd error network. “real” denotes real part. (a) net11-real. (b) net12-real. (c) net13-real. (d) net21-real. (e) net22-real. (f) net23-real.
(a)(b)(c)(d)(e)(f)Figure 4
Imaginary part of state error evolution under the output static event-triggered control. net11, net12, and net13 represent 3-dimensional state evolution of the 1st error network. net21, net22, and net23 represent 3-dimensional state evolution of the 2nd error network. “im” denotes imaginary part. (a) net11-im. (b) net12-im. (c) net13-im. (d) net21-im. (e) net22-im. (f) net23-im.
(a)(b)(c)(d)(e)(f)Figures2–4 show that the first dimension and the second dimension of the error state converge over time, while the third dimension does not converge. That is, the output synchronization has been realized which matches the output matrix. Figure 5 shows that the four triggered time series are different which are consistent with their distribution characteristics of the four event triggers.Figure 5
The event-triggered time instantstki in the error networks (17) under the output static event-triggered control. net1r and net2r mean the real-valued parts of the slave networks 1 and 2. net1i and net2i represent the imaginary parts of the slave networks 1 and 2, respectively.


### 4.2. Output Piecewise Event-Triggered Control (OPETC) withβ=100;010;001

The output matrixβ is a unit matrix, and the output states are common full-dimensional states. Event trigger’s parameters are set as γi=0.02, η¯it=0.01exp−0.5t for t≤7, η¯it=0.01exp−3.5 for t>7, and the input control gain is Lij=10. These parameters are satisfied with equation (26), and the simulation results are shown in Figures 6 and 7. Figure 6 indicates that the real part (Figure 6(a)) and the imaginary part (Figure 6(b)) of the error state converge over time. While, Figure 7 denotes that the four triggered time series that are independent of each other.Figure 6
Real part and imaginary part of state error evolution under the piecewise event-triggered control. (a) Real part of state error evolution. (b) Imaginary part of state error evolution.
(a)(b)Figure 7
The event-triggered time instantstkiin the error networks (17) under the piecewise event-triggered control.


### 4.3. Multistrategy Comparison

The simulations of output static event-triggered control (OSETC), output dynamic event-triggered control (ODETC), and output time-triggered control (OTTC) have been realized where the scale parameter of event trigger isγi=0.02; the function parameter of the static event trigger is η¯it=0.01; the function parameter of the dynamic event trigger is η¯it=0.01exp−0.5t; the output matrix is β=100;010;001; and the time trigger interval is 88 ms. Some results have been achieved and given in Table 1. To compare the update numbers of the controller with different control strategies, the data in Table 1 can be analysed. The data in column 2 and column 3 of Table 1 show that the update rate of OTTC is greater than that of OSETC and ODETC with similar error decreased. The simulation results in columns 4 and 5 imply that the number of controller update increases over time even if the error norm is small enough. Thus, the simulation results demonstrate the first hint; output event-triggered control (OETC) is more effective than output time-triggered control (OTTC), and the similar conclusion can be referred from [36].Table 1
Six control strategies for synchronization of CMRDNNs.
Control strategyNumber of update, state norm≤0.1Time (s), state norm≤0.1Number of controller update,Tf = 8 sState norms∗103, Tf = 8 sFour values: real part of net1and net2 and imaginary part of net1 and net2OSETC25, 30, 26, 333.5, 6.4, 4.4, 6.425, 30, 28, 3356.4, 56.5, 6.9, 0.7.4ODETC18, 27, 20, 283.4, 6.4, 4.5, 4.923, 30, 25, 337.0, 28.8, 18.8, 57.4OTTC41, 45, 49, 633.6, 4.0, 4.3, 5.590, 90, 90, 900.1, 0.4, 0.2, 3.1The OSETC strategy and ODETC strategy can be further compared using the information in Table1. The number of controller update in OSETC shown in column 2 is greater than that of ODETC when state norm is below 0.1. Moreover, the growth rate of controller update in OSETC shown in columns 2 and 4 is lower than that of ODETC. Then, the second hint can be drawn; ODETC is more effective than OSETC at the first stage, and OSETC is better than ODETC at the following stage.Based the above analysis, the distributed output piecewise event-triggered control (OPETC) which combines OSETC and ODETC has been realized, and the results are given in Table2. Comparing the data in column 2 and column 3 of Table 2, we can find that OPETC has better performance than ODETC and OSETC. This simulation validates the similar theoretical analysis of Theorem 1.Table 2
Three control strategies for synchronization of CMRDNNs.
Control strategyTf = 8 s, controller update numbersTf = 15 s, controller update numbersOSETC29, 32, 29, 3131, 37, 34, 40ODETC26, 30, 24, 3136, 54, 34, 56OPETC26, 30, 24, 3128, 31, 29, 36


## 4.1. Output Static Event-Triggered Control (OSETC) withβ=100;010

Event trigger’s parameters satisfied with equation (26) are set as γi=0.02,, η¯it=0.01, and Lij=10. And the simulation results are shown in Figures 2–4. Figure 2 represents the jth dimensional state norm evolution of the ith error networks over time. The real-valued state norm evolution and the imaginary-valued one can be formulated as(42)zijRx,t2=∫ΩzijRTx,t,zijRx,tdx,zijIx,t2=∫ΩzijITx,t,zijIx,tdx,i=1,2,j=1,2,3.Figure 2
Real part and imaginary part of state error norm evolution under the output static event-triggered control. net11, net12, and net13 represent three-dimensional state norm evolution of the 1st error network. net21, net22, and net23 represent three-dimensional state norm evolution of the 2nd error network. (a) Real part of state error evolution. (b) Imaginary part of state error evolution.
(a)(b)Figure 3
Real part of state error evolution under the output static event-triggered control. net11, net12, and net13 represent 3-dimensional real-valued state evolution of the 1sterror network. net21, net22, and net23 represent 3-dimensional real-valued state evolution of the 2nd error network. “real” denotes real part. (a) net11-real. (b) net12-real. (c) net13-real. (d) net21-real. (e) net22-real. (f) net23-real.
(a)(b)(c)(d)(e)(f)Figure 4
Imaginary part of state error evolution under the output static event-triggered control. net11, net12, and net13 represent 3-dimensional state evolution of the 1st error network. net21, net22, and net23 represent 3-dimensional state evolution of the 2nd error network. “im” denotes imaginary part. (a) net11-im. (b) net12-im. (c) net13-im. (d) net21-im. (e) net22-im. (f) net23-im.
(a)(b)(c)(d)(e)(f)Figures2–4 show that the first dimension and the second dimension of the error state converge over time, while the third dimension does not converge. That is, the output synchronization has been realized which matches the output matrix. Figure 5 shows that the four triggered time series are different which are consistent with their distribution characteristics of the four event triggers.Figure 5
The event-triggered time instantstki in the error networks (17) under the output static event-triggered control. net1r and net2r mean the real-valued parts of the slave networks 1 and 2. net1i and net2i represent the imaginary parts of the slave networks 1 and 2, respectively.


## 4.2. Output Piecewise Event-Triggered Control (OPETC) withβ=100;010;001

The output matrixβ is a unit matrix, and the output states are common full-dimensional states. Event trigger’s parameters are set as γi=0.02, η¯it=0.01exp−0.5t for t≤7, η¯it=0.01exp−3.5 for t>7, and the input control gain is Lij=10. These parameters are satisfied with equation (26), and the simulation results are shown in Figures 6 and 7. Figure 6 indicates that the real part (Figure 6(a)) and the imaginary part (Figure 6(b)) of the error state converge over time. While, Figure 7 denotes that the four triggered time series that are independent of each other.Figure 6
Real part and imaginary part of state error evolution under the piecewise event-triggered control. (a) Real part of state error evolution. (b) Imaginary part of state error evolution.
(a)(b)Figure 7
The event-triggered time instantstkiin the error networks (17) under the piecewise event-triggered control.


## 4.3. Multistrategy Comparison

The simulations of output static event-triggered control (OSETC), output dynamic event-triggered control (ODETC), and output time-triggered control (OTTC) have been realized where the scale parameter of event trigger isγi=0.02; the function parameter of the static event trigger is η¯it=0.01; the function parameter of the dynamic event trigger is η¯it=0.01exp−0.5t; the output matrix is β=100;010;001; and the time trigger interval is 88 ms. Some results have been achieved and given in Table 1. To compare the update numbers of the controller with different control strategies, the data in Table 1 can be analysed. The data in column 2 and column 3 of Table 1 show that the update rate of OTTC is greater than that of OSETC and ODETC with similar error decreased. The simulation results in columns 4 and 5 imply that the number of controller update increases over time even if the error norm is small enough. Thus, the simulation results demonstrate the first hint; output event-triggered control (OETC) is more effective than output time-triggered control (OTTC), and the similar conclusion can be referred from [36].Table 1
Six control strategies for synchronization of CMRDNNs.
Control strategyNumber of update, state norm≤0.1Time (s), state norm≤0.1Number of controller update,Tf = 8 sState norms∗103, Tf = 8 sFour values: real part of net1and net2 and imaginary part of net1 and net2OSETC25, 30, 26, 333.5, 6.4, 4.4, 6.425, 30, 28, 3356.4, 56.5, 6.9, 0.7.4ODETC18, 27, 20, 283.4, 6.4, 4.5, 4.923, 30, 25, 337.0, 28.8, 18.8, 57.4OTTC41, 45, 49, 633.6, 4.0, 4.3, 5.590, 90, 90, 900.1, 0.4, 0.2, 3.1The OSETC strategy and ODETC strategy can be further compared using the information in Table1. The number of controller update in OSETC shown in column 2 is greater than that of ODETC when state norm is below 0.1. Moreover, the growth rate of controller update in OSETC shown in columns 2 and 4 is lower than that of ODETC. Then, the second hint can be drawn; ODETC is more effective than OSETC at the first stage, and OSETC is better than ODETC at the following stage.Based the above analysis, the distributed output piecewise event-triggered control (OPETC) which combines OSETC and ODETC has been realized, and the results are given in Table2. Comparing the data in column 2 and column 3 of Table 2, we can find that OPETC has better performance than ODETC and OSETC. This simulation validates the similar theoretical analysis of Theorem 1.Table 2
Three control strategies for synchronization of CMRDNNs.
Control strategyTf = 8 s, controller update numbersTf = 15 s, controller update numbersOSETC29, 32, 29, 3131, 37, 34, 40ODETC26, 30, 24, 3136, 54, 34, 56OPETC26, 30, 24, 3128, 31, 29, 36


## 5. Conclusions

The research model in this study is complex and general, including complex-valued states, memristive parameters, state coupling, reaction-diffusion term, and output matrix. First, by combining differential inclusion and nonfragile techniques, a general method is developed to overcome the difficulty of the master-slave system parameter mismatch caused by memristor. Second, an efficient distributed piecewise event-triggered control strategy using incomplete measurement has been designed to achieve output synchronization which not only reduces the update rate of the controller but also decreases the measurement cost. Finally, it is verified that the new distributed piecewise event trigger can achieve better quasisynchronous performance through theoretical analysis and experimental simulation. What is more, the distributed output piecewise event-triggered control strategy can be used for the synchronization control of general one-order reaction-diffusion neural networks without adding any other conditions. To verify wide effectiveness of this control strategy, the OPETC strategy will be extended to two-order or even higher-order neural networks, which is one of our future works.



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*Source: 5577675-2021-03-26.xml*