# Nonfragile Synchronization of Semi-Markovian Jumping Neural Networks with Time Delays via Sampled-Data Control and Application to Chaotic Systems

Authors: K. Sivaranjani; M. Sivakumar; S. Dharani; K. Loganathan; Ngawang Ngmgyel
Journal: Journal of Mathematics
 (2021)

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## Abstract

This study discusses the synchronization problem for delayed neural networks with semi-Markovian jumping parameters. With the support of Jenson’s inequality and Writinger-based integral inequality, a suitable Lyapunov–Krasovskii functional was constructed, and a synchronization criterion for the considered system was derived in the form of LMIs. In order to cope with system uncertainties, the nonfragile controller was taken into account. Also, sampled-data controller is used to improve the effectiveness of the bandwidth usage. In order to achieve the benefits of both control techniques, the nonfragile sampled-data controller was considered for synchronization of semi-Markovian jumping neural networks and to assure that the error system is asymptotically stable. At last, numerical simulations are exhibited to validate the proposed technique.


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## Body

## 1. Introduction

Due to their effective implementation in cryptography, image analysis, associative memory, model identification, and so on, more attention has been given to different neural network (NN) models over the past centuries [1–4]. In latest years, the stability assessment of the constructed networks has appeared as a significant study subject due to the dynamic nature of these applications. In many of the engineering and neural systems, the communication transmission in a network is frequently interrupted by some exterior factors, which may direct to adverse dynamic behaviors such as oscillation and instability, and there arises time-delay. In addition to the fact that time delays in NNs are unavoidable, the literature has explored the stability assessment of delayed NNs well [5–8]. For example, synchronization problem for coupled inertial neural networks with reaction diffusion terms and time-varying delays via pinning sampled-data control has been considered in [5]. Finite time synchronization of drive response networks with discontinuous nodes and noise distribution has been analyzed in [9]. H∞ filtering problem for fuzzy stochastic NNs with mixed time-delays has been investigated in [10].In the actual application, since the system’s state equation tends to have some randomness, a linear time-invariant system cannot generally describe such systems. The Markov jump system, however, can describe such dynamic systems accurately, which has led to extensive research by scholars [11, 12]. Since the jump time of a Markov chain is exponentially distributed, Markovian jumping parameters have severe limitations in applications. A semi-Markovian process is a continuous stochastic process whose sojourn time fits a variety of probability distributions, such as the Weibull and Gaussian distributions. As a result of the relaxed conditions on the probability distributions, semi-Markovian jumping parameters are more general than Markovian jumping parameters in modelling realistic systems. Semi-MJSs have a fixed matrix of transition probabilities and a matrix of sojourn time probability density functions, while Markovian jumping systems (MJSs) have constant transition rates. This means that the transition rates in S-MJSs are time-varying. Due to these advantages, researchers paid their attention towards semi-Markovian jumping systems [13–16]. Recently, in [17], stochastic synchronization problem for semi-Markovian jumping Lure’s system with packet dropouts subject to multiple sampling periods has been discussed. In [18], network-based nonlinear semi-Markovian jump systems with randomly occurring parameter uncertainties and transmission delay have been taken into account. Event-triggered synchronization problem for semi-Markovian jumping complex dynamical networks with a reliable control technique has been investigated in [19].To stabilize the NNs, there are many control techniques such as impulsive control [20, 21], feedback control [22], sampled-data control [23–25], intermittent control [26, 27], and so on. It should be noted that the sampled-data control will improve the efficacy of the use of bandwidth by radically deducing the amount of information transmitted. Analog signal processing techniques are frequently replaced by digital signal processing techniques to achieve improved efficiency, reliability, and precision, in addition to the hasty developments in discrete measurement and intelligent instruments. Sampled-data control systems are continuous-time systems operated by a digital controller, and they are usually made up of continuous-time plants to be controlled, a discrete-time controller to control them, and an ideal sampler and ZOH to transform continuous-time signals into discrete-time signals and vice versa. The sampled-data systems are hybrid in nature because they operate in the continuous-time domain with both continuous-time and discrete-time signals. One of the discrete controllers, the sampled-data control system, allows control signals to adjust only at discrete sampling, resulting in a significant reduction in communication traffic and energy savings. As a result, sampled-data controllers have received a lot of attention in recent decades, with related findings published in the literature [28, 29].Practically, the designed controller should be able to tolerate some uncertainties in its coefficients because uncertainty cannot be avoided for many reasons, such as the inherent imprecision in analog systems and additional parameter tuning in the final implementation of the controller. Due to this fact, the nonfragile controller has been studied by many researchers [30–32]. Nonfragile synchronization for chaotic time-delay neural networks with semi-Markovian jump parameters has been discussed in [32]. Synchronization stability criteria for delay-coupled fractional-order complex Cohen–Grossberg neural networks under parameter uncertainties are discussed in [33].By the impact of the preceding facts, this manuscript discusses the synchronization problem for SMJNN with hybrid control strategy. At first, synchronization analysis has been performed to consider SMJNNs with recently introduced integral inequality techniques and proposed control strategy. Later, the synchronization criteria for SMJNNs have been explored with the nonfragile control technique. Finally, in numerical simulations, chaotic NNs are considered to verify the designed control technique.The main contributions and features of this study are presented as follows:(i)
Distinguished from the previous works, this article aims to study the synchronization issue for SMJNNs with time delays by using nonfragile sampled-data control(ii)
Moreover, in an aim to enjoy the benefits of the nonfragile control technique and sampled-data control technique, the nonfragile sampled-data control which has the features of both control techniques has been adopted for achieving synchronization(iii)
By utilizing novel integral inequalities and designed controller, synchronization criteria have been given in the form of LMIs. The resulting LMIs are solved with the help of Matlab LMI toolbox.(iv)
Finally, numerical simulations are presented to validate the correctness of the proposed control technique


## 2. Problem Formulation

Letβt,t≥0 be a discrete-state continuous-time semi-Markov process and assume the values in the finite set =1,2,…,N are given by(1)Prβt+l=j|βt=i=αijll+oli≠j1+αiill+oli=j,where Δ=αijl denotes the transition probability matrix, liml⟶0ol/l=0, and αijl≥0, for i≠j, is the transition rate from mode i at time t to mode j at time t+l, and αiil=∑j∈S,j≠iαijl.Consider the SMJ-delayed neural networks:(2)ζ˙t=−Dβtζt+Aβtfζt+Bβtfζt−ϱt+Jt,where ζt=ζ1t,ζ2t,…,ζnt represents the state vector. fxt=f1xt,…,fnxt stands for neuron activation function and τt is the time-varying delay with 0≤ϱt≤ϱ. D=diagd1,d2,…,dn with di>0. A and B are connection and delayed connection weight matrices. The external input is denoted by Jt. The slave system is described as(3)ξ˙t=−Dβtξt+Aβtfξt+Bβtfξt−ϱt+Jt+ut,where ut∈Rn is the control input. By letting the error as ψt=ζt−ξt, the error system is(4)ψ˙t=−Dβtψt+Aβtfψt+Bβtfψt−ϱt+ut.Consider the nonfragile sampled-data control as(5)uit=K+ΔKtψtk=K¯ψtk,where K is the gain matrix to be designed, and tk denotes the sampling instant and satisfies 0=t0<t1<⋯<tk<⋯<limk⟶∞tk. ΔKt stands for the controller gain fluctuations. It takes the form(6)ΔKt=MFtN,where FTtFt≤I. Using the input-delay approach t−tk=σt,tk≤t<tk+1, and 0<σt<σ, then the controller becomes(7)uit=K¯ψt−σt.Then, (4) can be written as(8)ψ˙t=−Dβtψt+Aβtfψt+Bβtfψt−ϱt+K¯ψt−σt.Lemma 1 (see [34]).
For any two scalarsυ2≥υ1>0, constant matrix H∈ℝn×n,H=HT>0, such that the integrations concerned are well defined:(9)−υ2−υ1∫t−υ2t−υ1ξTsHξsds≤−∫t−υ2tξsdsTH∫tt−υ1ξsds.Lemma 2 (see [35]).
For any matrixE∈ℝn×m,E=ET>0, differentiable function θ from a,b⟶ℝn, the succeeding inequality holds:(10)∫abz˙TsEz˙sds≥ϑTΥ1TEΥ1+π2Υ2TEΥ2ϑb−a,where ϑ=zTbzTa∫abzTs/b−adsT,Υ1=I−I0, and Υ2=I/2I/2−I.Lemma 3 (see [36]).
LetS=ST,U and V be the real constant matrices of appropriate dimensions S+UFtV+VTFTtUT<0 for F, satisfying FTtFt=I if and only if there exists a scalar ε>0, such that S+ε−1UUT+εVTV<0.Assumption 1.
Each activation functionfi⋅ is continuous and bounded and there exist constants Fi− and Fi+ such that(11)Fi−≤fik1−fik2k1−k2≤Fi+,i=1,2,…,n,where k1,k2∈R and k1≠k2.


## 3. Nonfragile Sampled-Data Synchronization

This section derives some sufficient conditions for the synchronization of considered system (8), which can be seen through the subsequent theorem.Theorem 1.
The system (8) is asymptotically synchronized, if there exist matrices P>0,Q1>0,Q2>0,ℜ1>0,ℜ2>0,ℨ1>0,ℨ2>0 and matrices S,L,G, for a given scalar γ, such that the following LMI holds:(12)Υδ<0,where(13)Υ1,1=2P+Q1+Q3+τ12ℜ1+ϱℜ2+ϱ24T1+ℨ1−2GDi−ℨ2+π24ℨ2−ℜ1+π24ℜ1−F1Λ1i+∑j=1NαijδPj,Υ1,2=GAi+F2Λ1i,Υ1,3=GBi,Υ1,4=GK¯+ℨ2−π24ℨ2,Υ1,5=ℜ1−π24ℜ1,Υ1,7=−G+Pi,Υ1,9=π241ϱℜ1,Υ1,12=−π22ℨ2,Υ2,2=−Λ1i+Q2,Υ2,5=F2Λ˜2i,Υ2,7=GAi,Υ3,3=1−μQ2,Υ3,7=γGBi,Υ4,4=2−ℨ2−π24ℨ2,Υ4,7=GK¯,Υ5,5=−Q1−2ℜ1+π24ℜ1Υ5,6=ℜ1−π24ℜ1,Υ5,7=γGBi,Υ5,8=π24ℜ1,Υ5,9=π22ℜ1,Υ6,6=−Q3−ℨ2+π24ℨ2,Υ7,7=τ2ℜ1+η2ℨ2−2γG,Υ8,8=−π2ℜ1,Υ9,9=−π2ℜ1,Υ10,10=−ℨ1−ℨ2−π24ℨ2,Υ10,11=π22ℨ2,Υ11,11=−π2ℨ2,Υ12,12=−π2ℨ2,Υ13,13=−T1.Proof.
Consider the following Lyapunov–Krasovskii functional:(14)Vt=∑i=15Vit,where(15)V1t=ψTtPiψt,V2t=∫t−ϱttψTsQ1ψsds+∫t−ϱttgTψsQ2gψsds+∫t−ϱtψTsQ3ψsds,V3t=ϱ∫−ϱ0∫t+θtψ˙Tsℜ1ψ˙sdsdθ+∫−ϱ0∫t+θtψTsℜ2ψsdsdθ,V4t=ϱ2∫−ϱ0∫θ0∫t+λtψTsT1ψsdsdλdθ,V5t=∫t−σmtψTsℨ1ψsds+σm∫−σm0∫t+θtψ˙Tsℨ2ψ˙sdsdθ.
Calculating the time-derivative of (14) along (8) gives(16)LV1t=−2ψTtPiψt+ξTt∑j=1NπijδPjψt,LV2t=ψTtQ1ψt−1−μψTt−ϱtQ1ψt−ϱt+gTψtQ2gψt−1−μgTt−τtQ2gt−ϱt+ψTtQ3ψt−ψTt−ϱQ3ψt−ϱ,(17)LV3t=ϱ2ψ˙Ttℜ1ψ˙t−ϱ∫t−ϱtψ˙Tsℜ1ψ˙sds+ϱψTtℜ2ψt−∫t−ϱtψTsℜ2ψsds,(18)LV4t=ϱ44ψTtT1ψt−∫−ϱ0∫t+θtψTsdsdθT1∫−ϱ0∫t+θtψsdsdθ,(19)LV5t=ψTsℨ1ψs−ψTt−σmℨ1ψt−σm+σm2ψ˙Ttℨ2ψ˙t−σm∫t−σMtψ˙Tsℨ2ψ˙sds.
From Lemma1, it follows from equation (19) that(20)−∫t−ϱtψTsR2ψsds≤−1ϱ∫t−ϱt−ϱtψsds∫t−ϱttψsdsTℜ200ℜ2∫t−ϱt−ϱtψsds∫t−ϱttψsds.
From Lemma2, (19) becomes(21)−σm∫t−σmt−σtψ˙Tsℨ2ψ˙sds≤−ψt−σtψt−σ1σ∫t−σt−σtψsdsT×ℨ2+π24ℨ2−Z2+π24ℨ2−π24ℨ2−ℨ2+π24ℨ2Z2+π24ℨ2π22ℨ2−π24ℨ2−π24ℨ2π2ℨ2ψt−σtψt−σ1σ∫t−σt−σtψsdsand(22)−σm∫t−σttψ˙Tsℨ2ψ˙sds≤ψtψt−σt1σ∫t−σttψsdsT×ℨ2+π24ℨ2−ℨ2+π24ℨ2−π24ℨ2−ℨ2+π24ℨ2ℨ2+π24ℨ2π22ℨ2−π24ℨ2−π24ℨ2π2ℨ2ψtψt−σt1σ∫t−σttψsds.
From (17), we have(23)∫t−ϱt−ϱtψ˙Tsℜ1ψ˙sds≥1ϱψt−ϱtψt−ϱ1ϱ∫t−ϱtt−ϱtψsdsT×ℜ1+π24ℜ1−ℜ1+π24ℜ1−π24ℜ1−ℜ1+π24ℜ1ℜ1+π24ℜ1π22ℜ1−π24ℜ1−π24ℜ1π2ℜ1ψt−ϱtψt−ϱ1ϱ∫t−ϱtt−ϱtψsds,and(24)∫t−ϱttψ˙Tsℜ1ψ˙sds≥1ϱψtψt−ϱt1ϱ∫t−ϱtt−ϱtψsdsT×ℜ1+π24ℜ1−ℜ1+π24ℜ1−π24ℜ1−ℜ1+π24ℜ1ℜ1+π24ℜ1π22ℜ1−π24ℜ1−π24ℜ1π2ℜ1ψtψt−ϱt1ϱ∫t−ϱtt−ϱtψsds.
By Assumption1, for diagonal matrices Λ1i and Λ2i,(25)0≤−ψTtF1Λ1iψt+ψTtF2Λ1igψt+gTψtF2Λ˜1iψt−gTψtΛ1i˜gψt,(26)0≤−ψTt−τtF1Λ2iψt−τt+ψTt−τtF2Λ˜2igψt−ϱt+gψt−ϱtF2Λ˜2iψt−ϱt−gTψt−ϱtΛ˜2igψt−ϱt.
For any matrixG, we have(27)0=2ψTtG+ψ˙TtG−ψ˙t−Dβtψt+Aβtfψt+Bβtfψt−ϱt+K¯ψt−σt.
From equations (15)–(27), we have(28)ELVt≤χTtΥχt<0,where(29)χt=ψtgψtgψt−ϱtψt−σtψt−ϱtψt−ϱψ˙t∫t−ϱt−ϱtψsds∫t−ϱttψsdsψt−σ1σ∫t−σt−σtψsds1σ∫t−σttψsds∫−ϱ0∫t+θtψsdsdθ,and the elements of the matrix Υ are given in the statement of theorem.
Based on the above theorem, now we are in a position to design the gain matrix of the derived controller.Theorem 2.
The system (8) is asymptotically synchronized, if there exist matrices P>0,Q1>0,Q2>0,ℜ1>0,ℜ2>0,ℨ1>0,ℨ2>0 and matrices S,L,G, for a given scalar γ, such that the following LMI holds:(30)ΥδU1ϵV1U2εV2∗−ϵ1I000∗∗−ϵ1I00∗∗∗−ϵ2I0∗∗∗∗−ϵ2I<0,where(31)Υ1,1=2P+Q1+Q3+τ12ℜ1+ϱℜ2+ϱ24T1+ℨ1−2GDi−ℨ2+π24ℨ2−ℜ1+π24ℜ1−F1Λ1i+∑j=1NπijδPj,Υ1,2=GAi+F2Λ1i,Υ1,3=GBi,Υ1,4=GK+ℨ2−π24ℨ2,Υ1,5=ℜ1−π24ℜ1,Υ1,7=−G+Pi,Υ1,9=π241ϱℜ1,Υ1,12=−π22ℨ2,Υ2,2=−Λ1i+Q2,Υ2,5=F2Λ˜2i,Υ2,7=GAi,Υ3,3=1−μQ2,Υ3,7=γGBi,Υ4,4=2−ℨ2−π24ℨ2,Υ4,7=GK,Υ5,5=−Q1−2ℜ1+π24ℜ1,Υ5,6=ℜ1−π24ℜ1,Υ5,7=γGBi,Υ5,8=π24ℜ1,Υ5,9=π22ℜ1,Υ6,6=−Q3−ℨ2+π24ℨ2,Υ7,7=τ2ℜ1+η2ℨ2−2γG,ΥΥ8,8=−π2ℜ1,Υ9,9=−π2ℜ1,Υ10,10=−ℨ1−ℨ2−π24ℨ2,Υ10,11=π22ℨ2,Υ11,11=−π2ℨ2,Υ12,12=−π2ℨ2,Υ13,13=−T1.
Moreover, the desired gain matrices are computed byK=LG−1.Proof.
By making use ofΔKt=UFtV, LMI in (12) can be written as(32)Υi,j,w¯+U1FtV1+V1TFtU1T+U2FtV2+V2TFtU2,where(33)U1=0⋯0︸3GU0⋯0︸9,U2=0⋯0︸3GU0⋯0︸9,V1=V0⋯0︸12,V2=0⋯0︸6V0⋯0︸6.
From Lemma3, we have(34)Υi,j,w¯+ς1−1U1U1T+ς1V1TV1+ς2−1U2U2T+ς2V2TV2.
Thus, one can get(35)Υi,j,w¯U1ς1V1U2ς2V2∗−ς1I000∗∗−ς1I00∗∗∗−ς2I0∗∗∗∗−ς2I,i=1,2,…,s.Theorem 3.
The system (8) is asymptotically synchronized, if there exist matrices P>0,Q1>0,Q2>0,ℜ1>0,ℜ2>0,ℨ1>0,ℨ2>0 and matrices S,L,G, for a given scalar γ, such that the following LMI holds:(36)ΥδM1ς1N1M2ς2N2∗−ς1I000∗∗−ς1I00∗∗∗−ς2I0∗∗∗∗−ς2I<0,where(37)Υ1,1=2P+Q1+Q3+τ12ℜ1+ϱℜ2+ϱ24T1+ℨ1−2GDi−ℨ2+π24ℨ2−ℜ1+π24ℜ1−F1Λ1i+∑j=1NπijδPj,Υ1,2=GAi+F2Λ1i,Υ1,3=GBi,Υ1,4=GK+ℨ2−π24ℨ2,Υ1,5=ℜ1−π24ℜ1,Υ1,7=−G+Pi,Υ1,9=π241ϱℜ1,Υ1,12=−π22ℨ2,Υ2,2=−Λ1i+Q2,Υ2,5=F2Λ˜2i,Υ2,7=GAi,Υ3,3=1−μQ2,Υ3,7=γGBi,Υ4,4=2−ℨ2−π24ℨ2,Υ4,7=GK,Υ5,5=−Q1−2ℜ1+π24ℜ1,Υ5,6=ℜ1−π24ℜ1,Υ5,7=γGBi,Υ5,8=π24ℜ1,Υ5,9=π22ℜ1,Υ6,6=−Q3−ℨ2+π24ℨ2,Υ7,7=τ2ℜ1+η2ℨ2−2γG,Υ8,8=−π2ℜ1,Υ9,9=−π2ℜ1,Υ10,10=−ℨ1−ℨ2−π24ℨ2,Υ10,11=π22ℨ2,Υ11,11=−π2ℨ2,Υ12,12=−π2ℨ2,Υ13,13=−T1.
Moreover, the desired gain matrices are given byK=LG−1.Let us assumeΔKt=0; then, the controller in (7) will reduce to the form of sampled-data control. Then, the above theorem can be rewritten.Theorem 4.
The system (8) is asymptotically synchronized, if there exist matrices P>0,Q1>0,Q2>0,ℜ1>0,ℜ2>0,ℨ1>0,ℨ2>0 and matrices S,L,G, for a given scalar γ, such that the following LMI holds:(38)Υδ<0,where(39)Υ1,1=2P+Q1+Q3+τ12ℜ1+ϱℜ2+ϱ24T1+ℨ1−2GD−ℨ2+π24ℨ2−ℜ1+π24ℜ1−F1Λ1i+∑j=1NπijδPj,Υ1,2=GA+F2Λ1i,Υ1,3=GB,Υ1,4=GK+ℨ2−π24ℨ2,Υ1,5=ℜ1−π24ℜ1,Υ1,7=−G+P,Υ1,9=π241ϱℜ1,Υ1,12=−π22ℨ2,Υ2,2=−Λ1i+Q2,Υ2,5=F2Λ˜2i,Υ2,7=GA,Υ3,3=1−μQ2,Υ3,7=γGB,Υ4,4=2−ℨ2−π24ℨ2,Υ4,7=GK,Υ5,5=−Q1−2ℜ1+π24ℜ1,Υ5,6=ℜ1−π24ℜ1,Υ5,7=γGB,Υ5,8=π24ℜ1,Υ5,9=π22ℜ1,Υ6,6=−Q3−ℨ2+π24ℨ2,Υ7,7=τ2ℜ1+η2ℨ2−2γG,Υ8,8=−π2ℜ1,Υ9,9=−π2ℜ1,Υ10,10=−ℨ1−ℨ2−π24ℨ2,Υ10,11=π22ℨ2,Υ11,11=−π2ℨ2,Υ12,12=−π2ℨ2,Υ13,13=−T1.
Along with, the controller gain matrices are defined asG=KL−1.Remark 1.
In the literature, one can find many control methods for achieving synchronization of neural networks such as pinning control, feedback control, adaptive control, impulsive control, and sampled-data control. Different from the previous literature, in this work, we employed the novel control technique, namely, nonfragile sampled-data control which includes the benefits of both control techniques for synchronization of SMJNNs. Moreover, switching topology or jump connection often arises in a network due to link failures or new development. In Markovian jumping systems, the jump time is exponentially distributed and also irrelevant to sojourn time which leads some restrictions in the utilization of the network. Thus, semi-Markovian jumping parameters which generalizes Markovian jumping parameters are also taken into account. This shows the novelty of the work.Remark 2.
It is worth noting that fractional calculus has a nearly identical background to traditional calculus. Its applications in physics and engineering, on the other hand, are a relatively new source of interest. Fractional-order models would be more suitable for describing memory and inherited properties of different materials than conventional integer-order models. Many known structures that exhibit fractional dynamics have been found to be useful in interdisciplinary fields such as viscoelasticity, dielectric polarization, electromagnetic waves, and complex system quantum evolution. Due to this reason, fractional-order system has been one of the most promising research topics in recent times. In our future work, the qualitative behaviors of fractional-order systems will be considered.


## 4. Numerical Simulation

This section displays two numerical examples to highlight the advantages of the theoretical results.Example 1.
Consider the NNs with(40)D1=1.2000.7,D2=1.1000.6,A1=−1.2−2.3−2.1−1.4,A2=1.5−1.145.0−2.6,B1=3.2−5.33.14.1,B2=2.1−1.25.15−1.05.
The nonlinear activation functions are taken as(41)f1ω=f2ω=tanhωt,with F1+=F2+=1 and F1−=F2−=0. Then,(42)F1=0000,F2=0.5000.5.
The transition rates are assumed to beα11δ∈2.2,1.8,α22δ∈−1.9,−1.5. From that, we have α11,1=−2.2,α11,2=−1.8,α22,1=−1.9, and α22,2=−1.5.
By resolving the LMIs obtained in Theorem1 and by using the help of Matlab LMI toolbox with the parameters ϱ=0.2, the gain matrix is attained as(43)K=10−3×−0.26540.39540.36080.2513.
Figure1 displays the chaotic nature of the master system and Figure 2 shows the chaotic behavior of the slave system. By applying the designed nonfragile sampled-data control technique, the phase portraits of the error system are shown in Figure 3.Figure 1
Chaotic behavior of the master system.Figure 2
Chaotic behavior of slave system.Figure 3
State response of the error system.Example 2.
Consider the NNs with(44)D1=1.2001.8,D2=1001.5,A1=−3.7+π44.13.0−4.8+π4,A2=−3.7+π44.13.0−4.8+π4B1=−0.12π4−3.61.8−3.12π4,B2=−0.12π4−3.61.8−3.12π4.
The nonlinear activation function and transition rates are taken as in the previous example. From the LMIs in Theorem4 with the parameters ς1=ς2=0.2,ϱ=0.5, one can get the gain matrix(45)K=0.0351000.0552.
In Figure4, the chaotic nature of the master system is presented, and Figure 5 shows the chaotic behavior of the slave system. By making use of the nonfragile sampled-data control technique, the state trajectories of the error system are shown in Figure 6.Figure 4
Chaotic nature of the master system.Figure 5
Chaotic nature of the slave system.Figure 6
State trajectories of the error system.


## 5. Conclusion

In this work, the synchronization of SMJNNs has been analyzed through the hybrid control technique, namely, nonfragile sampled-data control. Some criteria that ensure the synchronization of investigated SMJNNs with and without uncertainties in the control technique have been derived in the form of LMIs using Lyapunov stability theory and Writinger-based integral inequality approaches. The acquired LMIs are solved with Matlab LMI toolbox. Last, numerical simulations are granted to validate the designed controllers. Quaternion-valued neural network is the generalization of complex-valued neural networks, in which state, connection weight, and activation function are all quaternion numbers. Compared to real-valued NNs and complex-valued NNs, quaternion-valued NNs show significant advantages in multidimensional data processing. Recently, fractional-order quaternion-valued neural networks have gained great attention among the researchers due to its applications in many fields such as attitude control, image processing, computer graphics, prediction of three-dimensional wind processing, and so on. Due to its growing applications, it is important to analyze the qualitative behaviors of the fractional-order quaternion-valued NN. This will be our future work.



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