# Synchronization in Finite Time of Fuzzy Neural Networks with Hybrid Delays and Uncertain Nonlinear Perturbations

Authors: Shuyue Zhao; Kelin Li; Weiyi Hu
Journal: Advances in Fuzzy Systems
 (2022)

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## Abstract

This paper deals with the finite-time synchronization problem of a class of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations. By applying the famous finite-time stability theory, combining differential inequality techniques, and the analysis approach, several new algebraic sufficient criteria are obtained to realize finite-time synchronization between the drive system and the response system by designing a state feedback controller and an adaptive controller. Taking discrete delays, distributed delays, and uncertain nonlinear perturbations into account in fuzzy cellular neural networks makes the neural system more general than most existing cellular neural networks. Two different novel types of controllers designed to achieve finite-time synchronization can not only effectively overcome the influence of time delays and perturbations but also change their form according to the change of system state or perturbation to achieve a better control effect. Meanwhile, some algebraic sufficient criteria obtained in this paper can be proved by the parameters of the system itself, and the complex calculation of matrix inequality is avoided. Finally, the validity of our proposed results is confirmed by several examples and simulations. Furthermore, a secure communication problem is presented to further illustrate the fact of the obtained results.


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## Body

## 1. Introduction

Since Yang and Yang [1, 2] proposed fuzzy cellular neural networks (FCNNs) on the basis of traditional cellular neural networks (CNNs) in 1996, many researchers have performed extensive work on this topic due to their application in image processing and pattern recognition, see [3–6]. However, in the realization of neural networks, the emergence of time delays is unavoidable due to the limitation of velocity information. For one thing, discrete time delays often occur due to the limited switching speed of neurons and amplifiers [7–10]. For another, neural networks usually have spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, the propagation velocity distribution of these paths usually results in the propagation distribution delays [11–13]. The presence of time delays can lead to instability, chaos, oscillation, and other performance degradations of the nervous system. In this case, considering the combination of discrete time delays and distributed time delays is of great significance to the study of neural networks.Synchronization plays an essential role in practical applications such as biological systems, secure communications, and image protection. Therefore, synchronization has become a hot spot studying the dynamic behavior of neural networks. So far, many types of synchronization for fuzzy neural networks have been proposed, for instance, exponential synchronization, antisynchronization, projective synchronization, and adaptive synchronization, for example, see [14–19]. However, in practical applications, we are not only interested in the synchronization performance of the system but also more concerned with the convergence time of the system. Finite-time synchronization has the best convergence time in neural networks compared with infinite-time synchronization. That is because finite-time synchronization has better robustness and anti-interference ability. Hence, many scholars are interested in the finite-time synchronization of fuzzy neural networks and have studied it extensively [13, 20–26]. Abdurahman et al. [20] and Wang [21] investigated the finite-time synchronization problem of FCNNs with time-varying delays or time-varying coefficients and proportional delays based on the finite-time stability theory, and inequality techniques, and some criteria of finite-time synchronization for the addressed network are derived. In [22], Duan et al. studied the finite-time synchronization of delayed FCNNs with discontinuous activations. Under the framework of differential inclusions, by utilizing the discontinuous state feedback control method and constructing Lyapunov functionals, new finite-time synchronization criteria for the considered networks are established. In the same year, Tang et al. [13] further considered the finite-time cluster synchronization issue for coupled FCNNs with Markovian switching topology, discontinuous activation functions, proportional leakage, and time-varying unbounded delays, and novel quantization controllers without the sign function are designed to avoid the chattering and save communication resources. Several sufficient conditions are derived to guarantee the finite-time cluster synchronization by constructing new Lyapunov–Krasovskii functionals and utilizing M-matrix methods. What’s more, Jian and Duan [23] researched the synchronization in finite time of fuzzy neutral-type inertial neural networks with time-varying coefficients and proportional delays. To end this, by choosing proper variable transformation, the original system can be rewritten as the first-order differential system. Based on finite-time stability theory and some inequality techniques, several criteria are established to ensure that the drive-response systems can achieve finite-time synchronization. Duan et al. [24] focused on the finite-time synchronization for inertial fuzzy neural networks with time delays. The Lyapunov function ⋅pp≥1 constructed is more general than the used Lyapunov functions ⋅1 or ⋅2 in [23] to realize the finite-time synchronization objective.On the other hand, uncertain perturbations are inevitable in practical situations because the actual neural networks are always in a constantly changing environment, and some unknown factors in the environment changes will always interfere with them. It is of great practical significance to consider the uncertainty perturbations in the neural network models, which makes the models more realistic. Recently, many studies have concentrated on the synchronization of neural networks with external perturbations [27–32]. According to the previous literature review, although many synchronization results have been established, these results only consider the influence of hybrid delays and uncertain perturbations. However, when modeling real-world problems, we encounter other problems such as complexity, uncertainty or ambiguity, and time delays and other external perturbations. It has been proved that fuzzy logic theory has been developed into an effective method to model and deal with complex nonlinear systems [1, 3–6, 11, 13–26, 32, 33]. Therefore, considering fuzzy logic into the model will not only make the model more complete but also have more practical significance. It is worth noting that there are few papers published on the synchronization in finite time of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations.Motivated by the above discussions, we study the finite-time synchronization issue of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations. The main contributions of this manuscript are as follows:(a)
A new general neural network model is formulated, and the model assembles fuzzy cellular neural networks, discrete time delays, distributed time delays, and uncertain nonlinear perturbations. Compared with the existing literature on neural networks [20, 21, 31, 33, 34], this model is more general and extensive.(b)
Some algebraic sufficient criteria are established to ensure that the drive-response system can be synchronized in finite time. Compared with linear matrix inequality, the finite-time synchronization criterion in this paper is easy to verify with the parameters of the system and easy to implement in practice.(c)
The designed state feedback controller and the adaptive controller can not only eliminate the influence of time delays and uncertain nonlinear perturbations but also change their form according to the change of system state or perturbation to achieve a better control effect.


## 2. Preliminaries

In this paper, we concern with the fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations described by(1)x˙kt=−ckxkt+∑i=1nakifixit+∑i=1nbkivi+∧i=1nSkivi+∨i=1nHkivi+∧i=1nαkifixit−τit+∨i=1nα˜kifixit−τit+∧i=1nβki∫t−σittfixisds+∨i=1nβ˜ki∫t−σittfixisds+δkxt,where k∈ℐ≜1,2,…,n,n≥2; n corresponds to the number of neurons in the delayed network system; xkt denotes the state of the kth unit at moment t; ck represents the passive decay rates to the state of kth unit; aki and bki are the connection weight of the feedback template and feed-forward template; αki,α˜ki,βki, and β˜ki denote the connection weights of the elements of the discrete fuzzy feedback MIN template, the discrete fuzzy feedback MAX template, the distributed fuzzy feedback MIN template, and the distributed fuzzy feedback MAX template, respectively; Ski and Hki are the elements of the fuzzy feed-forward MIN template and the fuzzy feed-forward MAX template; ∧ and ∨ denote the fuzzy AND and fuzzy OR operations, respectively; vi presents input and bias of the ith neuron; fi⋅ denotes the signal activation function of the ith neuron at moment t; τit and σit correspond to the discrete time-varying delay and distributed time-varying delay and satisfy 0≤τit≤τi≤τ,0≤σit≤σi≤σ, where τ=maxi∈ℐτi and σ=maxi∈ℐσi are nonnegative constants; δkxt≜δkt,xt,xt−τt,∫t−σttxsds represents the nonlinear perturbations, where xt=x1t,x2t,…,xntT,xt−τt=x1t−τ1t,x2t−τ2t,…,xnt−τntT and ∫t−σttxsds=∫t−σ1ttx1sds,∫t−σ2ttx2sds,…,∫t−σnttxnsdsT. Moreover, let ζ=maxτ,σ.According to the concept of drive-response synchronization first proposed by Pecora and Carrol in [35], we take (1) as the drive system; then, the corresponding response system is described as follows:(2)y˙kt=−ckykt+∑i=1nakifiyit+∑i=1nbkivi+∧i=1nSkivi+∨i=1nHkivi+∧i=1nαkifiyit−τit+∨i=1nα˜kifiyit−τit+∧i=1nβki∫t−σittfiyisds+∨i=1nβ˜ki∫t−σittfyisids+δkyt+ukt,where δkyt≜δkt,yt,yt−τt,∫t−σttysds represents the nonlinear perturbations to system (2), and ut=u1t,u2t,…,untT is the suitable controller to be designed for realizing synchronization of the drive-response system. The other parameters are the same as those defined in system (1).We defineekt=ykt−xkt as the synchronization error of system (1) and (2). Then, subtracting (1) from (2) yields the following error system:(3)e˙kt=−ckekt+∑i=1nakifieit+∧i=1nαkifieit−τit+∨i=1nα˜kifieit−τit+∧i=1nβki∫t−σittfeisids+∨i=1nβ˜ki∫t−σittfieisds+δkyt−δkxt+ukt,where fiei⋅=fiyi⋅−fixi⋅; ∧i=1nαkifiei⋅=∧i=1nαkifiyi⋅−∧i=1nαkifixi⋅; ∨i=1nα˜kifiei⋅=∨i=1nα˜kifiyi⋅−∨i=1nα˜kifixi⋅.LetPC=C−ζ,0,ℝn, which is the Banach space of all continuous functions mapping −ζ,0⟶ℝn with the norm φ=sup−ζ≤s≤0∑i=1nφks. The initial values associated with systems (1) and (2) are assumed to be xks=φks,yks=ϕks,k∈ℐ, respectively, s∈−ζ,0, where φs,ϕs∈PC. Then, the initial value of the error system is described as ψks=ϕks−φks,s∈−ζ,0, and ψs∈PC.The other parameters and symbols can be seen in.Table 1
The meanings of sets, parameters, and vectors.
MeaningSetℝThe set of all the real numbersℝ+The set of all positive real numbersℝnThe set of alln-dimensional real vectorsVectorxt=x1t,x2t,…,xntTThe state vector of neurons in the drive system at timetyt=y1t,y2t,…,yntTThe state vector of neurons in the response system at timetet=e1t,e2t,…,entTThe state vector of neurons in the error system at timetut=u1t,u2t,…,untTControl inputδktThe nonlinear vector contains parametric perturbations and other external disturbancesParametersnThe number of neurons in networksckThe positive constantakiConnection weight of the feedback templatebkiConnection weight of the feed-forward templateSki/HkiThe elements of the fuzzy feed-forward MIN/MAX templateαki/α˜kiThe elements of the discrete fuzzy feedback MIN/MAX templateβki/β˜kiThe elements of the distributed fuzzy feedback MIN/MAX template∧/∨Fuzzy AND/OR operationviInput and bias of thei th neuronτitThe discrete time-varying delaysσitThe distributed time-varying delaysfiThe activation functions of theith neuronFor research purposes, we make some basic assumptions and introduce the definition and the lemmas that we need to use.(i)
A1 For each k∈ℐ, there exist a nonnegative constant Fk such that(4)fkx−fky≤Fkx−y,for allx,y∈ℝ.(ii)
A2 For each k∈ℐ, there exist nonnegative constants Ak,Bk,Ck, and Dk such that(5)δkyt−δkxt≤Ak∑i=1neit+Bk∑i=1neit−τit+Ck∑i=1n∫t−σitteisds+Dk.Definition 1.
(see [36]). The neural network (2) is said to be synchronized with (1) in finite time if, for a suitable designed controller, there exists a constant T>0 (T>0 depends on the initial state vector error value ψs), such that eT=0 and et=yt−xt≡0 for t>T, where the T is called the setting time.Lemma 1 (see [37]).
We assume that there exist a continuous, positive-definite functionVt:D⟶ℝ+, constants α,β>0, 0<η<1, and an open neighborhood V⊂D of the origin such that(6)V˙t≤−αVηt−βVt,∀t∈V0.Then, the origin of system is finite-time stable. The settling timeT satisfies(7)T≤t0+ln1+β/αV1−ηt0β1−η.Moreover, ifV=ℝ+, Vt is proper and radially unbounded, then the origin is globally finite-time stable.Lemma 2 (see [38])..
We assume that a continuous, positive-definite functionVt and constants α>0,0<η<1 satisfy the following differential inequality(8)V˙t≤−αVηt,∀t≥t0,Vt0≥0.Then, for any givent0,Vt satisfies the following inequality(9)V1−ηt≤V1−ηt0−α1−ηt−t0,t0≤t≤T,and Vt≡0 for all t≥T with the settling T given by(10)T=t0+V1−ηt0α1−η.Lemma 3 (see [39]).
Letet be a solution of the error system, which is defined on 0,T, T∈0,+∞. Then, function et is absolutely continuous and(11)ddtet=υtTe˙t=∑k=1nυkte˙ktfora.a.t∈0,T,where υkt=signekt, if ekt≠0, while υkt can be arbitrarily chosen in −1,1, if ekt=0.Lemma 4 (see [1]).
If supposex and x˜ are two states of neural networks (1), then we have(12)∧i=1nαkifixi−∧i=1nαkifix˜i≤∑i=1nαki‖fixi−fix˜i,∨i=1nα˜kifixi−∨i=1nα˜kifix˜i≤∑i=1nα˜ki‖fixi−fix˜i.Lemma 5 (see [20])..
If supposen is a positive integer and a1,a2,…,an and 0<q<p are real numbers, then the following inequality holds(13)a1q+a2q+⋯+anq≥a1p+a2p+⋯+anpq/p.Lemma 6 (see [40]).
Leta≥0,bk≥0k=1,2,…,l, then(14)a∏k=1lbkqk≤1rar+∑k=1lqkbkr,where qk≥0k=1,2,…,l are some constants, ∑k=1lqk=r−1, and r≥1.


## 3. Main Results

In this section, two different types of controllers should be designed for the finite-time stability of the zero solution of the error system (3), which is equivalent to the finite-time synchronization between system (1) and system (2). Specifically, a state feedback controller is first designed for the finite-time synchronization problem. Then, an adaptive controller is considered based on the state feedback controller so that the control strength can be adjusted automatically. At the same time, several sufficient conditions that the considered system can achieve finite-time synchronization are obtained through rigorous mathematical proofs.


### 3.1. State Feedback Control

State feedback is to multiply each state variable of the system by the corresponding feedback coefficient and feed back to the input end, which is added with the reference input as the control input of the controlled system. The basic structure of state feedback control system is shown in Figure1. In order to realize the finite-time synchronization of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations, we design a new state feedback controller uk for the response system.Figure 1
Structure diagram of state feedback control system.The aforementioned controlleruk is designed in the following form:(15)ukt=−ηkekt−signektξk+λektμ−∑i=1nγkisignekteit−τit−∑i=1nωkisignekt∫t−σitteisds,where k,i∈ℐ, the positive constants ηk,ξk,γki,ωki are the gain coefficients to be determined, λ is a tunable positive constant, and the real number μ satisfies 0<μ<1.Theorem 1.
suppose that the assumptionsA1 and A2 are satisfied, then the neural network (1) and (2) can be synchronized in a finite time under controller (15) if for any k∈ℐ, and the following conditions hold:(16)d=mink∈ℐck+ηk−akkFk−Ak−Ak−ℬk>0,ξk≥Dk,(17)γki≥αkiFi+α˜kiFi+Bk,ωki≥βkiFi+β˜kiFi+Ck,where Ak=1/p∑i=1,i≠knaikpιikL1+1FkpκikL1+1+1/p∑i=1,i≠kn∑l=1L1qlakipιkil/qlFipκkil/ql, ℬk=1/p∑i=1,i≠knAipςikL2+1+1/p∑i=1,i≠kn∑l=1L2qlAkpςkil/ql, and ∑l=1L1ql+1=∑l=1L2ql+1=p,∑l=1L1+1ιkil=∑l=1L1+1κkil=∑l=1L2+1ςkil=1. Moreover, the settling time can be estimated as(18)T=ln1+d/λV1−μ/p0d1−μ,in which V0=∑i=1nek0p, and p≥1 is a positive integer.Proof.
define the following Lyapunov function:(19)Vt=∑k=1nektp.
Taking the derivative ofVt along the trajectories of system (3) and using Lemma 3, we obtain that(20)V˙t=∑k=1npektp−1υkte˙kt=∑k=1npektp−1υkt−ckekt+∑i=1nakifieit+∧i=1nαkifieit−τit+∨i=1nα˜kifieit−τit+∧i=1nβki∫t−σittfieisds+δkyt−δkxt+∨i=1nβ˜ki∫t−σittfieisds+ukt=∑k=1npektp−1υkt−ckekt+∑i=1nakifieit+∧i=1nαkifieit−τit+∨i=1nα˜kifieit−τit+∧i=1nβki∫t−σittfieisds+δkyt−δkxt+∨i=1nβ˜ki∫t−σittfieisds−ηkekt−signektξk+λektμ−∑i=1nγkisignekteit−τit−∑i=1nωkisignekt∫t−σitteisds≤−p∑k=1nckektp+p∑k=1n∑i=1nektp−1aki‖fieit+p∑k=1nektp−1∧i=1nαkifieit−τit+p∑k=1nektp−1∨i=1nαkifieit−τit+p∑k=1nektp−1∧i=1nβki∫t−σittfieisds+p∑k=1nektp−1∨i=1nβ˜ki∫t−σittfieisds+p∑k=1nektp−1υktδkyt−δkxt−p∑k=1nηkektp−p∑k=1nξkektp−1−pλ∑k=1nektp−1+μ−p∑k=1n∑i=1nγkiektp−1eit−τit−p∑k=1n∑i=1nωkiektp−1∫t−σitteisds.
According toA1 and Lemma 4, we have(21)fieit≤Fieit,(22)∧i=1nαkifieit−τit≤∑i=1nαkiFieit−τit,(23)∨i=1nα˜kifieit−τit≤∑i=1nα˜kiFieit−τit,(24)∧i=1nβki∫t−σittfieisds≤∑i=1nβkiFi∫t−σitteisds,(25)∨i=1nβ˜ki∫t−σittfieisds≤∑i=1nβ˜kiFi∫t−σitteisds.
Substituting inequalities (21–25) into (20), it produces that(26)V˙t≤−p∑k=1nck+ηkektp+p∑k=1n∑i=1nakiFiektp−1eit+p∑k=1n∑i=1nαkiFi+α˜kiFi−γkiektp−1eit−τit+p∑k=1n∑i=1nβkiFi+β˜kiFi−ωkiektp−1∫t−σitteisds+p∑k=1nektp−1υktδkyt−δkxt−p∑k=1nξkektp−1−pλ∑k=1nektp−1+μ.
Based on Lemma6, we can infer(27)p∑k=1n∑i=1nakiFi|ekt|p−1|eit|=p∑k=1nakkFk|ekt|p+p∑k=1n∑i=1,i≠knakiFi|ekt|p−1|eit|=p∑k=1nakkFk|ekt|p+p∑k=1n∑i=1,i≠knaki|ιkiL1+1FiκkiL1+1|eit∏l=1L1aki|ιkil/qlFiκkil/ql|ekt|)ql≤p∑k=1nakkFk|ekt|p+∑k=1n∑i=1,i≠kn|aki|pιkiL1+1FipκkiL1+1|eit|p+∑l=1L1ql|aki|pιkil/qlFipκkil/ql|ekt|p=p∑k=1nakkFk|ekt|p+∑k=1n∑i=1,i≠kn|aik|pιikL1+1FkpκikL1+1+∑i=1,i≠kn∑l=1L1ql|aki|pιkil/qlFipκkil/ql|ekt|p=p∑k=1nakk|Fk+Ak|ekt|p.
Similarly, it follows fromA2 and Lemma 6 that(28)p∑k=1nektp−1υktδkyt−δkxt≤p∑k=1n∑i=1nAkektp−1eit+p∑k=1n∑i=1nBkektp−1eit−τit+p∑k=1n∑i=1nCkektp−1∫t−σitteisds+p∑k=1nDkektp−1≤p∑k=1nAkektp+∑k=1n∑i=1,i≠knAipςikL2+1+∑i=1,i≠kn∑l=1L2wlAkpςkil/wlektp+∑k=1n∑i=1nBkektp−1eit−τit+p∑k=1nDkektp−1+p∑k=1n∑i=1nCkektp−1∫t−σitteisds≤p∑k=1nAk+ℬkektp+∑k=1n∑i=1nBkektp−1eit−τit+p∑k=1nDkektp−1+p∑k=1n∑i=1nCkektp−1∫t−σitteisds.
Substituting (27) and (28) into (26) yields(29)V˙t≤−p∑k=1nck+ηk−akkFk−Ak−Ak−ℬkektp+p∑k=1nDk−ξkektp−1+p∑k=1n∑i=1nαki+α˜ki+Bk−γkiektp−1eit−τit+p∑k=1n∑i=1nβki+β˜ki+Ck−ωkiektp−1∫t−σitteisds−pλ∑k=1nektp−1+μ.
By Lemma5 and 0<μ<1, one can derive that(30)∑k=1nektp−1+μ≥∑k=1nektpp−1+μ/p.
In view of(31)d=mink∈ℐck+ηk−akkFk−Ak−Ak−ℬk>0,ξk≥Dk,γki≥αkiFi+α˜kiFi+Bk,ωki≥βkiFi+β˜kiFi+Ck,and combining (29) and (30), we can conclude that(32)V˙t≤−pdVt−pλ∑k=1nektp−1+μ≤−pdVt−pλ∑k=1nektpp−1+μ/p=−pdVt−pλVp−1+μ/pt.
According to Lemma1 and (32), the error system (3) will converge to zero within T=ln1+d/λV1−μ/p0/d1−μ. Therefore, the response system (2) is synchronized with the drive system (1) in finite time T under the controller (15). The proof of Theorem 1 is completed.
If there is no uncertain nonlinear perturbation in the drive system (1) and the response system (2), that is, δkxt=δkyt=0, then the Assumption 2 is no longer needed and the control parameter ξk in the state feedback controller (15) designed can also be removed. At the same time, Theorem 1 can be simplified. Therefore, we have the following result.Corollary 1.
suppose that the assumptionA1 and δkxt=δkyt=0 are satisfied, then under the following controller:(33)ukt=−ηkekt−λsignektektμ−∑i=1nγkisignekt|eit−τit−∑i=1nωkisignekt∫t−σitteisds,the response system (2) can be synchronized with drive system (1) in finite time T if for any k∈ℐ, and the following conditions hold:(34)d=mink∈ℐck+ηk−akkFp−Ak>0,γki≥αkiFi+α˜kiFi,ωki≥βkiFi+β˜kiFi.When the drive-response system is considered without discrete time delay (or without distributed time delay), that is,αki=α˜ki=0 (or βki=β˜ki=0), k,i∈ℐ in (1) and (2), the influence of discrete time delay (or distributed time delay) cannot be considered in the uncertain nonlinear perturbation. Therefore, through the similar discussion to Theorem 1, the following two corollaries can be drawn.Corollary 2.
suppose that the assumptionsA1 and A2 are satisfied, when the drive-response system without discrete time delays is concerned, that is, αki=α˜ki=0, k,i∈ℐ in (1) and (2); under the following controller:(35)ukt=−ηkekt−signektξk+λektμ−∑i=1nωkisignekt∫t−σitteisds,the response system (2) is synchronized with the drive system (1) in finite time T if (16) and the following condition hold:(36)ωki≥βkiFi+β˜kiFi+Ck.Corollary 3.
suppose that the assumptionsA1 and A2 are satisfied, when the drive-response system without distributed time delays is concerned, that is, βki=β˜ki=0, k,i∈ℐ in (1) and (2); under the following controller:(37)ukt=−ηkekt−signektξk+λektμ−∑i=1nγkisignekteit−τit,the response system (2) is synchronized with the drive system (1) in finite time T if (16) and the following condition hold:(38)γki≥αkiFi+α˜kiFi+Bk.


### 3.2. Adaptive Control

It is well known that adaptive controller can automatically change the control intensity and obtain satisfactory results in practical application. Therefore, by applying adaptive technology to state feedback controller, we provide an adaptive controlleruk such that the drive-response system can be synchronized in a finite time.The adaptive controller and the update rules be expressed by:(39)ukt=−ηktekt−ξktsignekt−∑i=1nγkisignekteit−τit−∑i=1nωkisignekt∫t−σitteisdsη˙kt=ρkekt2−signηkt−ηkξ˙kt=ϱkekt−signξkt−ξk,where k,i∈ℐ, ρk>0, and ϱk>0 are constants, ηk>0 and ξk>0 are constants to be determined.Theorem 2.
suppose that the assumptionsA1 and A2 are satisfied; then, the neural network (1) and (2) can be synchronization in a finite time if for any k∈ℐ such that (17) and the following conditions hold:(40)ηk≥akkFk+Ak+Ak∗+ℬk∗−ck,ξk>Dk,where Ak∗=1/2∑i=1,i≠knaik2ιikL1+1Fk2κikL1+1+1/2∑i=1,i≠kn∑l=1L1qlaki2ιkil/qlFi2κkil/ql, ℬk∗=1/2∑i=1,i≠knAi2ςikL2+1+1/2∑i=1,i≠kn∑l=1L2wlAk2ςkil/wl, and ∑l=1L1+1ιkil=∑l=1L1+1κkil=∑l=1L2+1ςkil=∑l=1L1ql=∑l=1L2wl=1. Moreover, the settling time can be estimated as(41)T˜=2V˜1/20F,in which V˜0=1/2∑k=1nek02+1/2∑k=1nηk0−ηk2/ρk+1/2∑k=1nξk0−ξk2/ϱk,F=mink∈ℐϑk,1/ρk,1/ϱk,ϑk=ξk−Dk.Proof.
define the following Lyapunov function:(42)V˜t=12∑k=1nekt2+12∑k=1nηkt−ηk2ρk+12∑k=1nξkt−ξk2〉k.
Taking the derivative ofV˜t along the trajectories of system (3), applying Lemma 3 and inequalities (21–25), one obtains(43)V˜˙t=∑k=1nektυkte˙kt+∑k=1nηkt−ηkρkη˙t+∑k=1nξkt−ξkϱkξ˙t=∑k=1nektυkt−ckekt+∑i=1nakifieit+∧i=1nαkifieit−τit+∨i=1nα˜kifieit−τit+∧i=1nβki∫t−σittfieisds+δkyt−δkxt+∨i=1nβ˜ki∫t−σittfieisds−ηktekt−ξktsignekt−∑i=1nγkisignekteit−τit−∑i=1nωkisignekt∫t−σitteisds+∑k=1nηkt−ηkρkρkekt2−signηkt−ηk+∑k=1nξkt−ξkϱkϱkekt−signξkt−ξk≤∑k=1n−ck+ηkekt2+∑k=1n∑i=1nakiFiekt‖eit+∑k=1n∑i=1nαkiFi+α˜kiFi−γkiekt‖eit−τit+∑k=1n∑i=1nβkiFi+β˜kiFi−ωkiekt∫t−σitteisds+∑k=1nektυktδkyt−δkxt−∑k=1nξk−Dkekt−∑k=1nηkt−ηkρk−∑k=1nξkt−ξkϱk.
It can be derived from Lemma6 and (27) that(44)∑k=1n∑i=1nakiFiekt‖eit≤∑k=1nakkFkekt2+12∑k=1n∑i=1,i≠knaik2ιikL1+1Fk2κikL1+1+∑i=1,i≠kn∑l=1L1qlaki2ιkil/qlFi2κkil/qlekt2=∑k=1nakkFk+Ak∗ekt2.
Similarly, according toA2 and (28), we get(45)∑k=1nektυktδkyt−δkxt≤∑k=1nAkekt2+12∑k=1n∑i=1,i≠knAi2ςikL2+1+∑i=1,i≠kn∑l=1L2wlAk2ςkil/wlekt2+∑k=1n∑i=1nBkekteit−τit+∑k=1n∑i=1nCkekt∫t−σitteisds+∑k=1nDkekt=∑k=1nAk+ℬk∗ekt2+∑k=1n∑i=1nBkekt‖eit−τit+∑k=1n∑i=1nCkekt∫t−σitteisds+∑k=1nDkekt.
Substituting (44) and (45) into (43), we have(46)V˜˙t≤∑k=1n−ck+ηk−akkFk−Ak−Ak∗−ℬk∗ekt2+∑k=1n∑i=1nαkiFi+α˜kiFi+Bk−γkiekt‖eit−τit+∑k=1n∑i=1nβkiFi+β˜kiFi+Ck−ωkiekt∫t−σitteisds−∑k=1nξk−Dkekt−∑k=1nηkt−ηkρk−∑k=1nξkt−ξkϱk.
By virtue of(47)ηk≥akkFk+Ak+Ak∗+ℬk∗−ck,ξk>Dk,γki≥αkiFi+α˜kiFi+Bk,ωki≥βkiFi+β˜kiFi+Ck.
Letϑk=ξk−Dk and choose ϑk>0, it is derived from Lemma 5 that(48)V˜˙t≤−∑k=1nϑkekt−∑k=1nηkt−ηkρk−∑k=1nξkt−ξk〉k≤−F∑k=1nekt+∑k=1nηkt−ηkρk+∑k=1nξkt−ξk〉k≤−F∑k=1nekt2+∑k=1nηkt−ηk2ρk+∑k=1nξkt−ξk〉k21/2=−2FV˜1/2t.
According to Lemma2 and (48), the error system (3) will converge to zero within T˜=2V˜1/20/F. Therefore, the response system (2) is synchronized with the drive system (1) in finite time T˜ under the controller (39). The proof of Theorem 2 is completed.Remark 1.
Compared with traditional state feedback controllerukt=−ηkekt, the state feedback controller (15) includes many compensation terms. Such a novel controller can effectively deal with the finite-time synchronization problems. From Theorem 1, it can be seen that control parameters ηk,ξk,γki, and ωki play an important role in the finite-time synchronization of drive system (1) and response system (2). In addition, the settling time T is inversely proportional to the tunable constant λ, which means that a great λ results in short time T.Remark 2.
It is obvious from Theorem2 that the settling time of finite-time synchronization can be effectively adjusted by control parameter ξk,ρk, and ϱk of the adaptive controller (35). Besides, adding ∑i=1nγkisignekteit−τit and ∑i=1nωkisignekt∫t−σitteisds into the adaptive controller (35) can further eliminate the influence of time delays and perturbations upon the states of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations.Remark 3.
If we make the perturbation termδkt=Ik in the system, then many previous neural network models are the special cases of system (1). It is also clear from Table 2 that the model of this paper is more general. Where “√” means that the model contains this component, “×” means that the model does not contain this component. Moreover, since there is no finite-time synchronization of the fuzzy neural network as in form (1), the two different forms of controllers designed in this paper can successfully achieve the finite-time synchronization of the considered drive-response system. This shows that the finite-time synchronization results obtained in this paper are new.Table 2
Model comparisons.
Models[20][31][41][32][34]This paperFuzzy terms√××√×√Time-varying delays√√√√√√Distributed time-varying delays×√√××√Perturbation terms×√××√√Remark 4.
Since controllers (15), (33), (35), (37) and (39) contain discrete symbolic functions, some undesirable chattering may occur when synchronization is implemented. To prevent this from happening, the methods in [20] are used, and the continuous tanh⋅ function can be used instead of the discontinuous symbolic function. For example, the control law (15) can be redesigned as(49)ukt=−ηkekt−tanhϖkektξk+λektμ−∑i=1nγkitanhϖkekteit−τit−∑i=1nωkitanhϖkekt∫t−σitteisds,where ϖk>0 and k∈ℐ.


## 3.1. State Feedback Control

State feedback is to multiply each state variable of the system by the corresponding feedback coefficient and feed back to the input end, which is added with the reference input as the control input of the controlled system. The basic structure of state feedback control system is shown in Figure1. In order to realize the finite-time synchronization of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations, we design a new state feedback controller uk for the response system.Figure 1
Structure diagram of state feedback control system.The aforementioned controlleruk is designed in the following form:(15)ukt=−ηkekt−signektξk+λektμ−∑i=1nγkisignekteit−τit−∑i=1nωkisignekt∫t−σitteisds,where k,i∈ℐ, the positive constants ηk,ξk,γki,ωki are the gain coefficients to be determined, λ is a tunable positive constant, and the real number μ satisfies 0<μ<1.Theorem 1.
suppose that the assumptionsA1 and A2 are satisfied, then the neural network (1) and (2) can be synchronized in a finite time under controller (15) if for any k∈ℐ, and the following conditions hold:(16)d=mink∈ℐck+ηk−akkFk−Ak−Ak−ℬk>0,ξk≥Dk,(17)γki≥αkiFi+α˜kiFi+Bk,ωki≥βkiFi+β˜kiFi+Ck,where Ak=1/p∑i=1,i≠knaikpιikL1+1FkpκikL1+1+1/p∑i=1,i≠kn∑l=1L1qlakipιkil/qlFipκkil/ql, ℬk=1/p∑i=1,i≠knAipςikL2+1+1/p∑i=1,i≠kn∑l=1L2qlAkpςkil/ql, and ∑l=1L1ql+1=∑l=1L2ql+1=p,∑l=1L1+1ιkil=∑l=1L1+1κkil=∑l=1L2+1ςkil=1. Moreover, the settling time can be estimated as(18)T=ln1+d/λV1−μ/p0d1−μ,in which V0=∑i=1nek0p, and p≥1 is a positive integer.Proof.
define the following Lyapunov function:(19)Vt=∑k=1nektp.
Taking the derivative ofVt along the trajectories of system (3) and using Lemma 3, we obtain that(20)V˙t=∑k=1npektp−1υkte˙kt=∑k=1npektp−1υkt−ckekt+∑i=1nakifieit+∧i=1nαkifieit−τit+∨i=1nα˜kifieit−τit+∧i=1nβki∫t−σittfieisds+δkyt−δkxt+∨i=1nβ˜ki∫t−σittfieisds+ukt=∑k=1npektp−1υkt−ckekt+∑i=1nakifieit+∧i=1nαkifieit−τit+∨i=1nα˜kifieit−τit+∧i=1nβki∫t−σittfieisds+δkyt−δkxt+∨i=1nβ˜ki∫t−σittfieisds−ηkekt−signektξk+λektμ−∑i=1nγkisignekteit−τit−∑i=1nωkisignekt∫t−σitteisds≤−p∑k=1nckektp+p∑k=1n∑i=1nektp−1aki‖fieit+p∑k=1nektp−1∧i=1nαkifieit−τit+p∑k=1nektp−1∨i=1nαkifieit−τit+p∑k=1nektp−1∧i=1nβki∫t−σittfieisds+p∑k=1nektp−1∨i=1nβ˜ki∫t−σittfieisds+p∑k=1nektp−1υktδkyt−δkxt−p∑k=1nηkektp−p∑k=1nξkektp−1−pλ∑k=1nektp−1+μ−p∑k=1n∑i=1nγkiektp−1eit−τit−p∑k=1n∑i=1nωkiektp−1∫t−σitteisds.
According toA1 and Lemma 4, we have(21)fieit≤Fieit,(22)∧i=1nαkifieit−τit≤∑i=1nαkiFieit−τit,(23)∨i=1nα˜kifieit−τit≤∑i=1nα˜kiFieit−τit,(24)∧i=1nβki∫t−σittfieisds≤∑i=1nβkiFi∫t−σitteisds,(25)∨i=1nβ˜ki∫t−σittfieisds≤∑i=1nβ˜kiFi∫t−σitteisds.
Substituting inequalities (21–25) into (20), it produces that(26)V˙t≤−p∑k=1nck+ηkektp+p∑k=1n∑i=1nakiFiektp−1eit+p∑k=1n∑i=1nαkiFi+α˜kiFi−γkiektp−1eit−τit+p∑k=1n∑i=1nβkiFi+β˜kiFi−ωkiektp−1∫t−σitteisds+p∑k=1nektp−1υktδkyt−δkxt−p∑k=1nξkektp−1−pλ∑k=1nektp−1+μ.
Based on Lemma6, we can infer(27)p∑k=1n∑i=1nakiFi|ekt|p−1|eit|=p∑k=1nakkFk|ekt|p+p∑k=1n∑i=1,i≠knakiFi|ekt|p−1|eit|=p∑k=1nakkFk|ekt|p+p∑k=1n∑i=1,i≠knaki|ιkiL1+1FiκkiL1+1|eit∏l=1L1aki|ιkil/qlFiκkil/ql|ekt|)ql≤p∑k=1nakkFk|ekt|p+∑k=1n∑i=1,i≠kn|aki|pιkiL1+1FipκkiL1+1|eit|p+∑l=1L1ql|aki|pιkil/qlFipκkil/ql|ekt|p=p∑k=1nakkFk|ekt|p+∑k=1n∑i=1,i≠kn|aik|pιikL1+1FkpκikL1+1+∑i=1,i≠kn∑l=1L1ql|aki|pιkil/qlFipκkil/ql|ekt|p=p∑k=1nakk|Fk+Ak|ekt|p.
Similarly, it follows fromA2 and Lemma 6 that(28)p∑k=1nektp−1υktδkyt−δkxt≤p∑k=1n∑i=1nAkektp−1eit+p∑k=1n∑i=1nBkektp−1eit−τit+p∑k=1n∑i=1nCkektp−1∫t−σitteisds+p∑k=1nDkektp−1≤p∑k=1nAkektp+∑k=1n∑i=1,i≠knAipςikL2+1+∑i=1,i≠kn∑l=1L2wlAkpςkil/wlektp+∑k=1n∑i=1nBkektp−1eit−τit+p∑k=1nDkektp−1+p∑k=1n∑i=1nCkektp−1∫t−σitteisds≤p∑k=1nAk+ℬkektp+∑k=1n∑i=1nBkektp−1eit−τit+p∑k=1nDkektp−1+p∑k=1n∑i=1nCkektp−1∫t−σitteisds.
Substituting (27) and (28) into (26) yields(29)V˙t≤−p∑k=1nck+ηk−akkFk−Ak−Ak−ℬkektp+p∑k=1nDk−ξkektp−1+p∑k=1n∑i=1nαki+α˜ki+Bk−γkiektp−1eit−τit+p∑k=1n∑i=1nβki+β˜ki+Ck−ωkiektp−1∫t−σitteisds−pλ∑k=1nektp−1+μ.
By Lemma5 and 0<μ<1, one can derive that(30)∑k=1nektp−1+μ≥∑k=1nektpp−1+μ/p.
In view of(31)d=mink∈ℐck+ηk−akkFk−Ak−Ak−ℬk>0,ξk≥Dk,γki≥αkiFi+α˜kiFi+Bk,ωki≥βkiFi+β˜kiFi+Ck,and combining (29) and (30), we can conclude that(32)V˙t≤−pdVt−pλ∑k=1nektp−1+μ≤−pdVt−pλ∑k=1nektpp−1+μ/p=−pdVt−pλVp−1+μ/pt.
According to Lemma1 and (32), the error system (3) will converge to zero within T=ln1+d/λV1−μ/p0/d1−μ. Therefore, the response system (2) is synchronized with the drive system (1) in finite time T under the controller (15). The proof of Theorem 1 is completed.
If there is no uncertain nonlinear perturbation in the drive system (1) and the response system (2), that is, δkxt=δkyt=0, then the Assumption 2 is no longer needed and the control parameter ξk in the state feedback controller (15) designed can also be removed. At the same time, Theorem 1 can be simplified. Therefore, we have the following result.Corollary 1.
suppose that the assumptionA1 and δkxt=δkyt=0 are satisfied, then under the following controller:(33)ukt=−ηkekt−λsignektektμ−∑i=1nγkisignekt|eit−τit−∑i=1nωkisignekt∫t−σitteisds,the response system (2) can be synchronized with drive system (1) in finite time T if for any k∈ℐ, and the following conditions hold:(34)d=mink∈ℐck+ηk−akkFp−Ak>0,γki≥αkiFi+α˜kiFi,ωki≥βkiFi+β˜kiFi.When the drive-response system is considered without discrete time delay (or without distributed time delay), that is,αki=α˜ki=0 (or βki=β˜ki=0), k,i∈ℐ in (1) and (2), the influence of discrete time delay (or distributed time delay) cannot be considered in the uncertain nonlinear perturbation. Therefore, through the similar discussion to Theorem 1, the following two corollaries can be drawn.Corollary 2.
suppose that the assumptionsA1 and A2 are satisfied, when the drive-response system without discrete time delays is concerned, that is, αki=α˜ki=0, k,i∈ℐ in (1) and (2); under the following controller:(35)ukt=−ηkekt−signektξk+λektμ−∑i=1nωkisignekt∫t−σitteisds,the response system (2) is synchronized with the drive system (1) in finite time T if (16) and the following condition hold:(36)ωki≥βkiFi+β˜kiFi+Ck.Corollary 3.
suppose that the assumptionsA1 and A2 are satisfied, when the drive-response system without distributed time delays is concerned, that is, βki=β˜ki=0, k,i∈ℐ in (1) and (2); under the following controller:(37)ukt=−ηkekt−signektξk+λektμ−∑i=1nγkisignekteit−τit,the response system (2) is synchronized with the drive system (1) in finite time T if (16) and the following condition hold:(38)γki≥αkiFi+α˜kiFi+Bk.


## 3.2. Adaptive Control

It is well known that adaptive controller can automatically change the control intensity and obtain satisfactory results in practical application. Therefore, by applying adaptive technology to state feedback controller, we provide an adaptive controlleruk such that the drive-response system can be synchronized in a finite time.The adaptive controller and the update rules be expressed by:(39)ukt=−ηktekt−ξktsignekt−∑i=1nγkisignekteit−τit−∑i=1nωkisignekt∫t−σitteisdsη˙kt=ρkekt2−signηkt−ηkξ˙kt=ϱkekt−signξkt−ξk,where k,i∈ℐ, ρk>0, and ϱk>0 are constants, ηk>0 and ξk>0 are constants to be determined.Theorem 2.
suppose that the assumptionsA1 and A2 are satisfied; then, the neural network (1) and (2) can be synchronization in a finite time if for any k∈ℐ such that (17) and the following conditions hold:(40)ηk≥akkFk+Ak+Ak∗+ℬk∗−ck,ξk>Dk,where Ak∗=1/2∑i=1,i≠knaik2ιikL1+1Fk2κikL1+1+1/2∑i=1,i≠kn∑l=1L1qlaki2ιkil/qlFi2κkil/ql, ℬk∗=1/2∑i=1,i≠knAi2ςikL2+1+1/2∑i=1,i≠kn∑l=1L2wlAk2ςkil/wl, and ∑l=1L1+1ιkil=∑l=1L1+1κkil=∑l=1L2+1ςkil=∑l=1L1ql=∑l=1L2wl=1. Moreover, the settling time can be estimated as(41)T˜=2V˜1/20F,in which V˜0=1/2∑k=1nek02+1/2∑k=1nηk0−ηk2/ρk+1/2∑k=1nξk0−ξk2/ϱk,F=mink∈ℐϑk,1/ρk,1/ϱk,ϑk=ξk−Dk.Proof.
define the following Lyapunov function:(42)V˜t=12∑k=1nekt2+12∑k=1nηkt−ηk2ρk+12∑k=1nξkt−ξk2〉k.
Taking the derivative ofV˜t along the trajectories of system (3), applying Lemma 3 and inequalities (21–25), one obtains(43)V˜˙t=∑k=1nektυkte˙kt+∑k=1nηkt−ηkρkη˙t+∑k=1nξkt−ξkϱkξ˙t=∑k=1nektυkt−ckekt+∑i=1nakifieit+∧i=1nαkifieit−τit+∨i=1nα˜kifieit−τit+∧i=1nβki∫t−σittfieisds+δkyt−δkxt+∨i=1nβ˜ki∫t−σittfieisds−ηktekt−ξktsignekt−∑i=1nγkisignekteit−τit−∑i=1nωkisignekt∫t−σitteisds+∑k=1nηkt−ηkρkρkekt2−signηkt−ηk+∑k=1nξkt−ξkϱkϱkekt−signξkt−ξk≤∑k=1n−ck+ηkekt2+∑k=1n∑i=1nakiFiekt‖eit+∑k=1n∑i=1nαkiFi+α˜kiFi−γkiekt‖eit−τit+∑k=1n∑i=1nβkiFi+β˜kiFi−ωkiekt∫t−σitteisds+∑k=1nektυktδkyt−δkxt−∑k=1nξk−Dkekt−∑k=1nηkt−ηkρk−∑k=1nξkt−ξkϱk.
It can be derived from Lemma6 and (27) that(44)∑k=1n∑i=1nakiFiekt‖eit≤∑k=1nakkFkekt2+12∑k=1n∑i=1,i≠knaik2ιikL1+1Fk2κikL1+1+∑i=1,i≠kn∑l=1L1qlaki2ιkil/qlFi2κkil/qlekt2=∑k=1nakkFk+Ak∗ekt2.
Similarly, according toA2 and (28), we get(45)∑k=1nektυktδkyt−δkxt≤∑k=1nAkekt2+12∑k=1n∑i=1,i≠knAi2ςikL2+1+∑i=1,i≠kn∑l=1L2wlAk2ςkil/wlekt2+∑k=1n∑i=1nBkekteit−τit+∑k=1n∑i=1nCkekt∫t−σitteisds+∑k=1nDkekt=∑k=1nAk+ℬk∗ekt2+∑k=1n∑i=1nBkekt‖eit−τit+∑k=1n∑i=1nCkekt∫t−σitteisds+∑k=1nDkekt.
Substituting (44) and (45) into (43), we have(46)V˜˙t≤∑k=1n−ck+ηk−akkFk−Ak−Ak∗−ℬk∗ekt2+∑k=1n∑i=1nαkiFi+α˜kiFi+Bk−γkiekt‖eit−τit+∑k=1n∑i=1nβkiFi+β˜kiFi+Ck−ωkiekt∫t−σitteisds−∑k=1nξk−Dkekt−∑k=1nηkt−ηkρk−∑k=1nξkt−ξkϱk.
By virtue of(47)ηk≥akkFk+Ak+Ak∗+ℬk∗−ck,ξk>Dk,γki≥αkiFi+α˜kiFi+Bk,ωki≥βkiFi+β˜kiFi+Ck.
Letϑk=ξk−Dk and choose ϑk>0, it is derived from Lemma 5 that(48)V˜˙t≤−∑k=1nϑkekt−∑k=1nηkt−ηkρk−∑k=1nξkt−ξk〉k≤−F∑k=1nekt+∑k=1nηkt−ηkρk+∑k=1nξkt−ξk〉k≤−F∑k=1nekt2+∑k=1nηkt−ηk2ρk+∑k=1nξkt−ξk〉k21/2=−2FV˜1/2t.
According to Lemma2 and (48), the error system (3) will converge to zero within T˜=2V˜1/20/F. Therefore, the response system (2) is synchronized with the drive system (1) in finite time T˜ under the controller (39). The proof of Theorem 2 is completed.Remark 1.
Compared with traditional state feedback controllerukt=−ηkekt, the state feedback controller (15) includes many compensation terms. Such a novel controller can effectively deal with the finite-time synchronization problems. From Theorem 1, it can be seen that control parameters ηk,ξk,γki, and ωki play an important role in the finite-time synchronization of drive system (1) and response system (2). In addition, the settling time T is inversely proportional to the tunable constant λ, which means that a great λ results in short time T.Remark 2.
It is obvious from Theorem2 that the settling time of finite-time synchronization can be effectively adjusted by control parameter ξk,ρk, and ϱk of the adaptive controller (35). Besides, adding ∑i=1nγkisignekteit−τit and ∑i=1nωkisignekt∫t−σitteisds into the adaptive controller (35) can further eliminate the influence of time delays and perturbations upon the states of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations.Remark 3.
If we make the perturbation termδkt=Ik in the system, then many previous neural network models are the special cases of system (1). It is also clear from Table 2 that the model of this paper is more general. Where “√” means that the model contains this component, “×” means that the model does not contain this component. Moreover, since there is no finite-time synchronization of the fuzzy neural network as in form (1), the two different forms of controllers designed in this paper can successfully achieve the finite-time synchronization of the considered drive-response system. This shows that the finite-time synchronization results obtained in this paper are new.Table 2
Model comparisons.
Models[20][31][41][32][34]This paperFuzzy terms√××√×√Time-varying delays√√√√√√Distributed time-varying delays×√√××√Perturbation terms×√××√√Remark 4.
Since controllers (15), (33), (35), (37) and (39) contain discrete symbolic functions, some undesirable chattering may occur when synchronization is implemented. To prevent this from happening, the methods in [20] are used, and the continuous tanh⋅ function can be used instead of the discontinuous symbolic function. For example, the control law (15) can be redesigned as(49)ukt=−ηkekt−tanhϖkektξk+λektμ−∑i=1nγkitanhϖkekteit−τit−∑i=1nωkitanhϖkekt∫t−σitteisds,where ϖk>0 and k∈ℐ.


## 4. Illustrative Examples

Here, three numerical examples are provided to demonstrate that our obtained theoretical results are effectiveness.Example 1.
We consider the following 2-D neural network model as the drive system:(50)x˙kt=−ckxkt+∑i=12akifixit+∑i=12bkivi+∧i=12Skivi+∨i=12Hkivi+δkxt+∧i=12αkifixit−τit+∨i=12α˜kifixit−τit+∧i=12βki∫t−σittfixisds+∨i=12β˜ki∫t−σittfxisids,for k∈1,2, where xt=x1t,x2tT. Other parameters of system (50) are as follows:(51)c1=1.2,c2=1fix=tanhx,Fi=1,i=1,2aki2×2=1.2−1.5−0.12.5,bki2×2=Ski2×2=Hki2×2=0.1000.2αki2×2=−0.1−0.4−1.5−1.8,α˜ki2×2=−0.21.2−1.6−2.5βki2×2=−1.20.12.8−1,β˜ki2×2=−1.80.22.5−1.5v1=v2=1,τit=σit=1δ1xt=0.1∑i=12xit+0.1∑i=12xit−1+0.1∑i=12∫t−1txsdsi+0.1δ2xt=0.2∑i=12xit+0.2∑i=12xit−1+0.2∑i=12∫t−1txisds+0.2ζ=1,A1=B1=C1=D1=0.1,A2=B2=C2=D2=0.2.
The corresponding response system is depicted as follows:(52)y˙kt=−ckykt+∑i=12akifiyit+∑i=12bkivi+∧i=12Skivi+∨i=12Hkivi+δkyt+∧i=12αkifiyit−τit+∨i=12α˜kifiyit−τit+∧i=12βki∫t−σittfiyisds+∨i=12β˜ki∫t−σittfiyisds+ukt,for k∈1,2, in which the parameters in system (52) are the same as the drive system (50). If u1t=u2t≡0, in response system (52), then state trajectories and synchronization error trajectories of (50)–(52) with the initial conditions φs=1.8,−2.8T, ϕs=−2.6,1.5T, and s∈−1,0 are presented in Figures 2 and 3. It can be clearly seen that system (50) and system (52) are not synchronized. Therefore, to enable them to synchronize in a finite time, the controller is given by(53)ukt=−ηkekt−tanhektξk+2ekt0.8−∑i=12γkitanhekteit−τit−∑i=2nωkitanhekt∫t−σitteisds,where ekt=ykt−xkt for k=1,2. By simple calculations, we can select the following control gains(54)η1=2,η2=8.5,ξ1=0.12,ξ2=0.25γki2×2=0.51.83.44.6,ωki2×2=3.20.54.62.8p=2,ιkil=κkil=ςkil=0.5,l=1,2,q1=w1=1.
It is obvious that all the conditions for Theorem1 are completely satisfied. Therefore, system (52) is synchronized with system (50) in finite time T=2.74. The state trajectories of the drive-response system under the controller (53) are shown in Figure 4. The synchronization error trajectories between the drive system (50) and the corresponding response system (52) under the controller (53) are presented in Figure 5. The validity of our obtained Theorem 1 can be seen intuitively in Figures 4 and 5.Figure 2
(a) Trajectories ofx1t and y1t for drive-response system without controller; (b) trajectories of x2t and y2t for drive-response system without controller.
(a)(b)Figure 3
Trajectories ofe1t and e2t for drive-response system without controller.Figure 4
(a) Trajectories ofx1t and y1t for drive-response system under controller (53); (b) trajectories of x2t and y2t for drive-response system under controller (53).
(a)(b)Figure 5
Trajectories ofe1t and e2t for drive-response system under controller (53).Example 2.
We consider the following 3-D neural network model as the drive system:(55)x˙kt=−ckxkt+∑i=13akifixit+∑i=13bkivi+∧i=13Skivi+∨i=13Hkivi+δkxt+∧i=13αkifixit−τit+∨i=13α˜kifixit−τit+∧i=13βki∫t−σittfixisds+∨i=13β˜ki∫t−σittfixisds,for k∈1,2,3, where xt=x1t,x2t,x3tT and the other coefficients and functions are taken as(56)c1=1.8,c2=1.5,c3=1.2,fix=12x+1−x−1,Fi=1,i=1,2,3,aki3×3=0.8−2−1.2−1.60.62.8−2.5−2.20.4,bki3×3=Ski3×3=Hki3×3=0.10000.20000.3,αki3×3=−1.8−1.5−0.7−2.5−1.6−2.8−0.3−1.4−1.2,α˜ki3×3=−2−1.2−1.5−1.4−2.2−2.6−0.4−1.6−1,βki3×3=−2.12.52.60.2−1.2−2.81.81.6−1.4,β˜ki3×3=−1.22.61.41−1.8−2.61.41.8−1.6,v1=v2=v3=1,τit=sint,σit=cost,δ1xt=δ2xt=δ3xt=0.3∑i=13xit+0.3∑i=13xit−sint+0.3∑i=13∫t−costtxisds+0.3,ζ=1,Ak=Bk=Ck=Dk=0.3.
The corresponding response system is depicted as follows:(57)y˙kt=−ckykt+∑i=13akifiyit+∑i=13bkivi+∧i=13Skivi+∨i=13Hkivi+δkyt+∧i=13αkifiyit−τit+∨i=13α˜kifiyit−τit+∧i=13βki∫t−σittfiyisds+∨i=13β˜ki∫t−σittfiyisds+ukt,for k∈1,2,3, in which the parameters in system (57) are the same as the drive system (55). If u1t=u2t=u3t≡0 in response system (57), then state trajectories and error trajectories of (55)–(57) with the initial conditions φs=0.6,0.4,−0.2T, ϕs=−0.5,−0.1,0.3T, and s∈−1,0 are presented in Figure 6. It can be clearly seen that system (55) and system (57) are not synchronized. Therefore, to enable them to synchronize in a finite time, the controller is given by(58)ukt=−ηktekt−ξktsignekt−∑i=13γkisignekteit−τit−∑i=13ωkisignekt∫t−σitteisdsη˙kt=ekt2−signηkt−ηkξ˙kt=ekt−signξkt−ξk,where ekt=ykt−xkt for k=1,2,3. By simple calculations, we can select(59)η1=3.5,η2=4.2,η3=4.5,ξ1=ξ2=ξ3=1.3γki3×3=4.23.22.64.44.35.80.83.42.4,ωki3×3=3.85.64.51.63.45.83.63.83.4ρk=〉k=1p=2,ιkil=κkil=ςkil=0.5,l=1,2,q1=w1=1.
It is easy to verify that the above parameters satisfy all the conditions of Theorem2. Hence, the neural networks (55)–(57) can be synchronized with finite time T˜=3.46. The trajectories of the control gains of controller (58) with the initial conditions ηs=1.5,2.4,3.2 and ξs=0.3,0.8,1, s∈−1,0 are shown in Figure 7. And the state trajectories and error trajectories of the drive-response system are presented in Figure 8 under the adaptive controller (58). The effectiveness of our obtained Theorem 2 can be seen intuitively in Figure 8.Figure 6
(a) Trajectories ofx1t and y1t for drive-response system without controller; (b) trajectories of x2t and y2t for drive-response system without controller; (c) trajectories of x3t and y3t for drive-response system without controller; (d) trajectories of e1t,e2t and e3t for drive-response system without controller.
(a)(b)(c)(d)Figure 7
(a) Trajectories of control parametersη1t,η2t, and η3t with the adaptive controller (58); (b) trajectories of control parameters ξ1t,ξ2t, and ξ3t with the adaptive controller (58).
(a)(b)Figure 8
(a) Trajectories ofx1t and y1t for drive-response system under the controller (58); (b) trajectories of x2t and y2t for drive-response system under the controller (58); (c) trajectories of x3t and y3t for drive-response system under the controller (58); (d) trajectories of e1t,e2t, and e3t for drive-response system under the controller (58).
(a)(b)(c)(d)Example 3.
Ifαki=α˜ki=0, βki=β˜ki=0, and δkt=Ik, then system (1) is transformed into the following neural networks model:(60)x˙kt=−ckxkt+∑i=12akifixit+Ik,for k∈1,2, where xt=x1t,x2tT. The parameters ck, aki, and fi of system (60) are the same as those of system (50) and Ik=0. Taking system (60) as the drive system, the response system has the following form:(61)y˙kt=−ckykt+∑i=12akifiyit+Ik+ukt,for k∈1,2, in which the parameters in system (61) are the same as the drive system (60). Meanwhile, the initial conditions for the drive system (60) and response system (61) are the same as in Example 1. If u1t=u2t≡0 in response system (61), then state trajectories and synchronization error trajectories of (60) and (61) are shown in Figures 9 and 10. It is obvious that the systems are not synchronized. Therefore, to enable them to synchronize in a finite time, we change the controller (15) to the following form:(62)ukt=−ηkekt−λtanhektektμ,where ek=yk−xk for k=1,2. The values of η,λ,μ,p,q,l,ι,κ, and F are the same as in Example 1. In the case, the condition in Theorem 1 is satisfied. Therefore, system (61) is synchronized with system (60) in finite time T=2.60. The state trajectories of the drive-response system under the controller (62) are described in Figure 11. The synchronization error trajectories between the drive system (60) and the corresponding response system (61) under the controller (62) are presented in Figure 12.
The validity of Theorem1 and Theorem 2 we have obtained are evident from Example 1 and Example 2. It can be seen from Example 3 that the controller designed in this paper can change its form according to the change of the state or disturbance of the system so that the considered system can achieve finite-time synchronization. In addition, the convergence time is shortened when the complexity of the system is reduced. Therefore, the controller designed by this paper is more general and practical.Figure 9
(a) Trajectories ofx1t and y1t for drive-response system without controller; (b) trajectories of x2t and y2t for drive-response system without controller.
(a)(b)Figure 10
Trajectories ofe1t and e2t for drive-response system without controller.Figure 11
(a) Trajectories ofx1t and y1t for drive-response system under controller (62); (b) trajectories of x2t and y2t for drive-response system under controller (62).
(a)(b)Figure 12
Trajectories ofe1t and e2t for drive-response system under controller (62).


## 5. Discussion

Based on the numerical simulations given in Section4, it can be intuitively seen that both controllers designed in this paper can achieve finite-time synchronization of the considered drive-response system. Among them, since the internal characteristics of the system can be fully reflected by the state variables, the state feedback is beneficial to improving the control performance of the system. However, in the actual system, the introduced unknown feedback gain needs to be solved, but most of the obtained feedback gains are large and difficult to achieve. The adaptive controller can avoid large control gains and is also suitable when the system parameters are unknown or the model is uncertain, but its cost is very high. Therefore, in practical applications, the designer should select the corresponding controller according to their needs to achieve better control effects.


### 5.1. Comparative Analysis

In Section3, finite-time synchronization of system (1) is realized by designing state feedback controller and adaptive controller. In the existing research, many scholars also use feedback control to realize finite-time synchronization of neural networks. Abdurahman et al. [20] realized the finite-time synchronization of the following neural networks based on the theory of finite-time convergence via controller ukt=−ηkekt−ξksignekt−λsignektektμ.(63)x˙kt=−ckxkt+∑i=1nakifixit+∑i=1nbkifixit−τit+∑i=1nbkivi+∧i=1nSkivi+∨i=1nHkivi+∧i=1nαkifixit−τit+∨i=1nα˜kifixit−τit+Ik.If we makeβki=β˜ki=0 and replace the perturbation term δkxt with Ik in system (1), and make bki=0 in the above system, then the two systems are identical. At the same time, finite-time synchronization of the transformed system can be achieved by changing the form of the controller (15). It is worth noting that the FCNNs considered in [20] have no distributed delay, and its results cannot be adopted in the fuzzy neural networks with distributed delays. Therefore, the results obtained in this paper are more general. Besides, in [42–44], the authors put forward some criteria to ensure the finite-time synchronization of complex networks and coupled neural networks according to matrix inequalities. Compared with the sufficient criterion of the matrix inequalities obtained by them, several criteria in this paper are easier to verify with parameters and avoid the complicated calculation of the matrix inequalities. Compared with [20, 21, 27, 31, 36, 45, 46], the delay-independent feedback controllers they designed are not suitable for model (1), while the feedback controller (15) can change its form according to the state of the system or the change of perturbation to achieve finite-time synchronization of the models studied by them. In addition, for some uncertain systems and systems with unknown parameters, the adaptive controller (35) can more effectively achieve finite-time synchronization of these systems. Overall, the controller designed in this paper is more flexible and suitable for more complex systems.


### 5.2. Application in Secure Communication

Secure communication using synchronization between chaotic systems (chaotic secure communication, for short) is a new concept of secure communication [47]. In secure communication, message encryption is performed when two people want to communicate with each other over an insecure communication channel, but a third person cannot identify the message by listening. Therefore, one of the main objectives of chaotic secure communication is to protect information from eavesdropping and interception. It is worth noting that we can regard the communication network as a chaotic neural network, in which the transmitter and receiver correspond to the drive system and the response system, respectively. We apply the feedback controller synchronization designed in Theorem 1 to make the receiver track the sender accurately in finite time. An information signal ht containing the message to be transmitted can be masked by a chaotic signal xt and recovered using finite-time synchronization. Different strategies can be used to make the actual transmitted signal pt as broadband as possible. In general, three strategies are involved in secure communications with chaos [48]:(1)
Signal masking, wherept=xt+πht;(2)
Modulation, wherept=xtht;(3)
Masking and modulation, wherept=xt1+πht.For the sake of illustration, we only use signal masking here, which ispt=xt+πht. For another, we can extract messages from receivers in a communication network through finite-time synchronization. The proposed communication system consisting of a transmitter and a receiver is shown in Figure 13. Masking techniques are proposed as follows. The transmitter is described as(64)x˙1t=−c1x1t+∑i=12a1ifixit+∑i=12b1ivi+∧i=12S1ivi+∨i=12H1ivi+δ1xt+∧i=12α1ifixit−τit+∨i=12α˜1ifixit−τit+∧i=12β1i∫t−σittfixisds+∨i=12β˜1i∫t−σittfixisds+h1t,x˙2t=−c2x2t+∑i=12a2ifixit+∑i=12b2ivi+∧i=12S2ivi+∨i=12H2ivi+δ2xt+∧i=12α2ifixit−τit+∨i=12α˜2ifixit−τit+∧i=12β2i∫t−σittfixisds+∨i=12β˜2i∫t−σittfixisds+h2t,where the parameters are the same as in Example 1. h1t=πht is the information message, and we set π=0.05 makes the energy of information signal much less than chaotic carrier signal. In order not to lose generality, we make h2t=0. Accordingly, the receiver is stated as(65)y˙1t=−c1y1t+∑i=12a1ifiyit+∑i=12b1ivi+∧i=12S1ivi+∨i=12H1ivi+δ1yt+∧i=12α1ifiyit−τit+∨i=12α˜1ifiyit−τit+∧i=12β1i∫t−σittfiyisds+∨i=12β˜1i∫t−σittfiyisds+u1t−y1t+P1t,y˙2t=−c2y2t+∑i=12a2ifiyit+∑i=12b2ivi+∧i=12S2ivi+∨i=12H2ivi+δ2yt+∧i=12α2ifiyit−τit+∨i=12α˜2ifiyit−τit+∧i=12β2i∫t−σittfiyisds+∨i=12β˜2i∫t−σittfiyisds+u2t−y2t+P2t,where P1t=pt=x1t+πht is the transmission signal of the system, P2t=x2t. The transmitted information signal can be recovered by transformation rt=π−1x1t+πht−y1t. We use the same functions and parameters as in Example 1, and the information signal is arbitrarily selected as ht=sint. The information signal ht, transmitted signal pt, recovered signal rt, and error between the information signal ht and the recovered signal rt are depicted in Figure 14. The simulation results show that the information signal can be recovered in finite time under the feedback controller (15).Figure 13
The schematic diagram of the proposed secure communication system.Figure 14
(a) The information signal; (b) the transmitted signal; (c) the recovered signal; (d) the error between the information signal and the recovered signal.
(a)(b)(c)(d)


## 5.1. Comparative Analysis

In Section3, finite-time synchronization of system (1) is realized by designing state feedback controller and adaptive controller. In the existing research, many scholars also use feedback control to realize finite-time synchronization of neural networks. Abdurahman et al. [20] realized the finite-time synchronization of the following neural networks based on the theory of finite-time convergence via controller ukt=−ηkekt−ξksignekt−λsignektektμ.(63)x˙kt=−ckxkt+∑i=1nakifixit+∑i=1nbkifixit−τit+∑i=1nbkivi+∧i=1nSkivi+∨i=1nHkivi+∧i=1nαkifixit−τit+∨i=1nα˜kifixit−τit+Ik.If we makeβki=β˜ki=0 and replace the perturbation term δkxt with Ik in system (1), and make bki=0 in the above system, then the two systems are identical. At the same time, finite-time synchronization of the transformed system can be achieved by changing the form of the controller (15). It is worth noting that the FCNNs considered in [20] have no distributed delay, and its results cannot be adopted in the fuzzy neural networks with distributed delays. Therefore, the results obtained in this paper are more general. Besides, in [42–44], the authors put forward some criteria to ensure the finite-time synchronization of complex networks and coupled neural networks according to matrix inequalities. Compared with the sufficient criterion of the matrix inequalities obtained by them, several criteria in this paper are easier to verify with parameters and avoid the complicated calculation of the matrix inequalities. Compared with [20, 21, 27, 31, 36, 45, 46], the delay-independent feedback controllers they designed are not suitable for model (1), while the feedback controller (15) can change its form according to the state of the system or the change of perturbation to achieve finite-time synchronization of the models studied by them. In addition, for some uncertain systems and systems with unknown parameters, the adaptive controller (35) can more effectively achieve finite-time synchronization of these systems. Overall, the controller designed in this paper is more flexible and suitable for more complex systems.


## 5.2. Application in Secure Communication

Secure communication using synchronization between chaotic systems (chaotic secure communication, for short) is a new concept of secure communication [47]. In secure communication, message encryption is performed when two people want to communicate with each other over an insecure communication channel, but a third person cannot identify the message by listening. Therefore, one of the main objectives of chaotic secure communication is to protect information from eavesdropping and interception. It is worth noting that we can regard the communication network as a chaotic neural network, in which the transmitter and receiver correspond to the drive system and the response system, respectively. We apply the feedback controller synchronization designed in Theorem 1 to make the receiver track the sender accurately in finite time. An information signal ht containing the message to be transmitted can be masked by a chaotic signal xt and recovered using finite-time synchronization. Different strategies can be used to make the actual transmitted signal pt as broadband as possible. In general, three strategies are involved in secure communications with chaos [48]:(1)
Signal masking, wherept=xt+πht;(2)
Modulation, wherept=xtht;(3)
Masking and modulation, wherept=xt1+πht.For the sake of illustration, we only use signal masking here, which ispt=xt+πht. For another, we can extract messages from receivers in a communication network through finite-time synchronization. The proposed communication system consisting of a transmitter and a receiver is shown in Figure 13. Masking techniques are proposed as follows. The transmitter is described as(64)x˙1t=−c1x1t+∑i=12a1ifixit+∑i=12b1ivi+∧i=12S1ivi+∨i=12H1ivi+δ1xt+∧i=12α1ifixit−τit+∨i=12α˜1ifixit−τit+∧i=12β1i∫t−σittfixisds+∨i=12β˜1i∫t−σittfixisds+h1t,x˙2t=−c2x2t+∑i=12a2ifixit+∑i=12b2ivi+∧i=12S2ivi+∨i=12H2ivi+δ2xt+∧i=12α2ifixit−τit+∨i=12α˜2ifixit−τit+∧i=12β2i∫t−σittfixisds+∨i=12β˜2i∫t−σittfixisds+h2t,where the parameters are the same as in Example 1. h1t=πht is the information message, and we set π=0.05 makes the energy of information signal much less than chaotic carrier signal. In order not to lose generality, we make h2t=0. Accordingly, the receiver is stated as(65)y˙1t=−c1y1t+∑i=12a1ifiyit+∑i=12b1ivi+∧i=12S1ivi+∨i=12H1ivi+δ1yt+∧i=12α1ifiyit−τit+∨i=12α˜1ifiyit−τit+∧i=12β1i∫t−σittfiyisds+∨i=12β˜1i∫t−σittfiyisds+u1t−y1t+P1t,y˙2t=−c2y2t+∑i=12a2ifiyit+∑i=12b2ivi+∧i=12S2ivi+∨i=12H2ivi+δ2yt+∧i=12α2ifiyit−τit+∨i=12α˜2ifiyit−τit+∧i=12β2i∫t−σittfiyisds+∨i=12β˜2i∫t−σittfiyisds+u2t−y2t+P2t,where P1t=pt=x1t+πht is the transmission signal of the system, P2t=x2t. The transmitted information signal can be recovered by transformation rt=π−1x1t+πht−y1t. We use the same functions and parameters as in Example 1, and the information signal is arbitrarily selected as ht=sint. The information signal ht, transmitted signal pt, recovered signal rt, and error between the information signal ht and the recovered signal rt are depicted in Figure 14. The simulation results show that the information signal can be recovered in finite time under the feedback controller (15).Figure 13
The schematic diagram of the proposed secure communication system.Figure 14
(a) The information signal; (b) the transmitted signal; (c) the recovered signal; (d) the error between the information signal and the recovered signal.
(a)(b)(c)(d)


## 6. Conclusion

In this paper, we discuss the finite-time synchronization of a class of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations. Through the application of famous finite-time stability theory, differential inequality techniques, and the analysis approach, useful state feedback controller and adaptive controller are involved, and several new algebraic sufficient criteria are derived to realize finite-time synchronization. In addition, the system studied in this paper can contain many previous neural networks, so the system in this paper is more general. Finally, the main results obtained are as follows:(a)
Some new algebraic sufficient criteria are proposed to ensure finite-time synchronization between the drive system and the response system, and the settling time is also estimated.(b)
The designed state feedback controller and adaptive controller not only realize the finite-time synchronization of the drive-response system but also can change their forms according to the change of the state or perturbation of the system to obtain better a control effect.(c)
The studied model and the designed controller can be effectively used in other practical applications such as secure communication.It is well known that the settling time for finite-time synchronization depends essentially on the initial conditions of the system. However, in many practical applications, it is difficult or impossible to obtain accurate values for the initial conditions of systems such as industrial systems, robots, and biological models, which greatly limits practical applications. While in [49], Polyakov proposed the concept of fixed-time stability and pointed out that the time of fixed-time synchronization is independent of the initial synchronization error, and the settling time can be estimated in advance. From this perspective, fixed-time synchronization has more advantages. In some circumstances, there are several abrupt changes at certain moments in physical systems since instantaneous disturbances, which are called impulsive effects [50]. The influence of impulsive phenomena may appear in some other fields, such as automatic control system and artificial intelligence. Therefore, for future research, we will consider fixed-time synchronization of fuzzy neural networks with time delays and impulsive effects.



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*Source: 1448819-2022-07-05.xml*