基于非周期间歇控制的惯性神经网络同步_滞后同步.pdf

第41卷第2期2024年3月新疆大学学报（自然科学版中英文）Journal of Xinjiang University(Natural Science Edition in Chinese and English)Vol.41,No.2Mar.,2024Synchronization/Lag Synchronization of InertialNeural Networks under AperiodicIntermittent ControlYU Juan1，HUI Jiaojiao2(1.School of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830017,China；2.School of Information Engineering,Tarim University,Alar Xinjiang 843300,China)Abstract:The synchronization and lag synchronization of delay inertial neural networks under aperiodic intermittent controlare considered.Based on the non-reduced order method,the Lyapunov functional method and inequality technique are used toobtain the sufficient criterion for the synchronization/lag synchronization of inertial neural networks under aperiodic intermittentcontrol,which is more feasible in practical applications.Finally,the feasibility of the theoretical results is verified by numericalsimulation.Key words:inertial neural networks;synchronization/lag synchronization;aperiodic intermittent controlDOI：10.13568/ki.651094.651316.2023.09.11.0002CLC number：O175.7Document Code：AArticle ID：2096-7675(2024)02-0171-010引文格式：于娟，惠姣姣基于非周期间歇控制的惯性神经网络同步/滞后同步J新疆大学学报(自然科学版中英文)，2024，41(2)：171-180英文引文格式：YU Juan，HUI Jiaojiao Synchronization/lag synchronization of inertial neural networks under aperiodic inter-mittent controlJ Journal of Xinjiang University(Natural Science Edition in Chinese and English)，2024，41(2)：171-180基于非周期间歇控制的惯性神经网络同步/滞后同步于 娟1，惠姣姣2(1.新疆大学 数学与系统科学学院，新疆 乌鲁木齐830017；2.塔里木大学 信息工程学院，新疆 阿拉尔843300)摘要：主要研究了惯性神经网络在非周期间歇控制下的同步/滞后同步问题.基于非降阶方法,利用 Lyapunov 方法、微分不等式技巧等,得出了在非周期间歇控制下惯性神经网络同步/滞后同步的充分判据,这种方法在实际应用中更具有可行性.最后,通过数值仿真验证理论结果的可行性.关键词：惯性神经网络;同步/滞后同步;非周期间歇控制0IntroductionNeural networks(NNs)are large and complex information networks composed of numerous nerve cells.NNs modelshave been widely used in pattern recognition,signal processing,associative memory and other fields,so that makes more andmore scholars are interested in the study of its dynamic behaviors,and have made remarkable achievements1.However,it isworth noting that the main previous work of NNs focuses on the study of first-order differential states.In 1987,inertial neuralnetworks(INNs)were first proposed by Babcock et al.2,which showed the inertial feature by introducing an inductor intothe neural circuit,and the dynamic model is expressed by a second-order differential equation.The dynamic behaviors of Received Date：2023-09-11Foundation Item：This work was supported by the National Natural Science Foundation of the Peoples Republic of China“Dynamic characteristicsand synchronization control of inertial neural network models based on non-reduced order method”（61866036），“Synchronization control and topologyidentification within fixed time and preassigned time of multilayer complex network”（62263029）.Biography：YU Juan（1984），female，associate professor，research fields：synchronization control of neural networks，E-mail：xjuyjmath-172Journal of Xinjiang University(Natural Science Edition in Chinese and English)2024INNs models are more complex in compared with the standard resistor-capacitor first-order models.As far as we know,it isbecausethatINNsmodelscanbesuccessfullyappliedinvariousfields,suchasoptimizationproblems,securecommunication,image encryption and so on.Therefore,more and more scholars began to pay attention to INNs.Time delays are ubiquitousin most physical and ecological systems.Time delays INNs have also received extensive attention and achieved noteworthyresults39.These results include various types of time delays,including constant delays,proportional delays,time-varyingdelays,distributed delays,mixed time delays,generalized time delays.Wheeler et al.10first published results on INNs stability and found that INNs are more complex than various NNs.Since the inertia term is a determinant of chaos and bifurcation,many scholars have made many achievements in INNs.Theexistence of time delays lead to system instability,chaos and other complex behaviors.Among these years,some researchershave delved into the dynamical behavior of INNs with different types of delays such as exponential stability,convergence,synchronization,passivity and so on.Synchronization is a hot topic in the field of nonlinear research,and it has a broadapplication background.The concept and theoretical basis of synchronization are becoming more and more perfect.For time-delay INNs,many different types of synchronization have been found and studied1113,such as complete synchronization,lag synchronization,finite time synchronization and so on.In 11,the synchronization problem of inertial memristive NNswith time-varying delays is studied by using event-triggered control scheme and state feedback controller.INNs can not achieve synchronization by itself,it can be realized by using conventional control schemes,such asfeedback control1415,adaptive control16,impulse control17,sampled data control18,intermittent control1920,et al.Bothadaptive control and state feedback control are continuous controls requiring that the control input be continuously activated.Intermittent control is more efficient,robust,and economical than continuous control,because each cycle of this controlstrategy consists of working time and rest time,and the controller is active during each control time and stops working duringrest time.According to different control intervals,intermittent control can be divided into periodic intermittent control andaperiodic intermittent control.Periodic intermittent control requires the relative proportion of control time and rest time ineach cycle is a positive constant,which is unreasonable and relatively conservative.This is because every control periodof the intermittent control strategy needs to be changed in practical application,so it needs to be adjusted according to theactual situation.In order to overcome the limitation of periodic intermittent control,it is of great practical significance to useaperiodic intermittent control20to study the synchronization of networks both in theory and in practice.In reference 21,by directly designing the aperiodic intermittent control of INNs,the synchronization criterion of complex-valued INNs isestablished by using the Lyapunov functional theory and inequality technique.It should be noted that the synchronization results of the INNs were obtained.The method is adopted13,2223by orderreduction method,that is,the second-order system is transformed into a first-order system by variable substitution.Thismethod not only increases the dimension of INNs,but also enlarges the difficulty of theoretical analysis.In the conclusionof most order reduction methods,it is often necessary to design two controllers for the degraded first-order system in orderto obtain the synchronization criterion of the system.In fact,the more controllers there are,the more difficult it becomes toimplement in practice.Therefore,using fewer controllers,it is more practical and correct to consider the master-slave systemdirectly from the INNs itself.The aim of this paper is to solve the synchronization/lag synchronization problem of INNs with generalized time delayunder aperiodic intermittent control.The main contributions of our work are as follows:Firstly,different from the linearcontrol3,14,adaptive control16and periodic intermittent control19,22designed in the previous literature,this paper designs theaperiodic intermittent control,which can better reduce the control cost.Secondly,different from the variable transformationsof reduced order for INNs utilized78,13,23,by directly constructing Lyapunov functionals and designing aperiodic intermittentcontrol schemes for the addressed INNs,a direct analysis approach is developed in this paper to discuss the synchronizationproblem.The rest of this article is as follows.Section 2 gives some Preliminaries and Lemmas.The main conclusions are givenin Section 3.Section 4 shows numerical examples.Finally,Section 5 is the conclusion.Notations Through this paper,N stands for the set of natural numbers,N=1,2,n,nN,R,Rnstand for the sets ofall real,and the sets of all n-dimensional real,respectively.C(,0,R)denotes the Banach space composed of continuousmapping from,0 to R.No.2YU Juan，et al：Synchronization/Lag Synchronization of Inertial Neural Networks under Aperiodic Intermittent Control1731Problem Description and PreliminariesA type of INNs is considered in this paper and described as p(t)=p p(t)pp(t)+nXk=1pkfk(k(t)+nXk=1pkgk(k(tk)+Ip,pN(1)where n corresponds to the number of neurons,p(t)R represents the state of the pth neuron at time t,the second derivativeis called an inertial term of(1),p 0 and p 0,pk,pk R are connection weights related to the neurons without,withdelays,respectively.fk(k(t),gk(k(tk):RR are the activation functions of the kth neuron at time t and tk,kis theconstant delay,IpR denotes an external input on the pth neuron.The initial values of system(1)are given byp(s)=p(s),p(s)=p(s),s,0,pN(2)where=maxkNk,p(s),p(s)C(,0,R)are bounded functions.The driving system is given by system(1),the responsesystem is given as followsp(t)=pppp(t)+nXk=1pkfk(k(t)+nXk=1pkgk(k(tk)+Ip+Up(t),pN(3)where p(t)represents the state of the pth neuron at time t in the response system,the other notations are the same as system(1).The initial values of system(3)are given byp(s)=p(s),p(s)=p(s),s,0,pN,wherep(s),p(s)C(,0,R)are bounded functions.Up(t)is aperiodic intermittent controller defined byUp(t)=p(p(t)p(t)p(p(t)p(t),tltl0,lt0 and p0 represent control gain.tl,lrepresent the start time of work interval and the start timeof rest interval in the lth periodic intermittent control,respectively.Here,the initial time we denote as t0=0.Hypothesis 1For each kN,there exist real numbers Fk0,Gk0 such that for any,R,|fk()fk()|Lfk|,|gk()gk()|Lgk|.Definition 1The INNs system(1)and system(3)can be achieved lag synchronization if limt|p(t)p(t)|=0under some suitable control inputs Up(t)with pN.When=0,the INNs system(1)and system(3)are synchronized undersome suitable control inputs Up(t).Hypothesis 2There exist two contants 0+,such thatinflNltl=,suplNtl+1tl=.Remark 1According to 24,the significance of Hypothesis 2 is to ensure that each rest width is not greater than.When the rest width is equal to 0,intermittent control becomes continuous control,which is not the objective discussed inthis paper.174Journal of Xinjiang University(Natural Science Edition in Chinese and English)2024Hypothesis 3For each pN,there exist some positive constants p,p,psuch thatp0,p0,4pp2p.wherep=p?1+pp+12nXk=1(|pk|Lfk+|pk|Lgk)?,p=p+p(1+2pppp),p=p+p(pp+12nXk=1(|pk|Lfk+|pk|Lgk)+nXk=1k(|kp|Lfp+|kp|Lgpe2p).Lemma 125Let(t)=(tl)/(ttl),t l,tl+1,obviously,(t)is strictly monotonically increasing and(t)(tl+1l)/(tl+1tl).If=limsupltl+1ltl+1tl,then 0 0 such that 2 0,then the neural system(1)and system(3)are synchronized based on the aperiodic intermittent controller(4).ProofFor t0,construct the following Lyapunov functionalV(t)=12nXp=1e2tp2p(t)+12nXp=1p?p(t)+p(t)?2e2t+nXp=1nXk=1p|pk|Lgke2kZttk2k(s)e2sds.When tltl,calculating the derivative of V(t)along error systemV(t)=e2tnXp=1n?(p+p)p(p+p)?2p(t)+?p+p(1+2pppp)?p(t)p(t)+p(1+pp)2p(t)+p?p(t)+p(t)?nXk=1pkfk(k(t)+nXk=1pk gk(k(tk)?o+nXp=1nXk=1p|pk|Lgke2k?2k(t)e2t2k(tk)e2(tk)?(5)According to Hypothesis 1 and the fact that xy(x2+y2)/2,one has,nXp=1nXk=1p?pk p(t)fk(k(t)?nXp=1nXk=1pLfk|pk|p(t)|k(t)|12nXp=1nXk=1?k|kp|Lfp2p(t)+p|pk|Lfk 2p(t)?(6)No.2YU Juan，et al：Synchronization/Lag Synchronization of Inertial Neural Networks under Aperiodic Intermittent Control175Similarly,nXp=1nXk=1p?pk p(t)gk(k(tk)?12nXp=1nXk=1p|pk|Lgk?2k(tk)+2p(t)?(7)nXp=1nXk=1p?pkp(t)fk(k(t)?12nXp=1nXk=1?k|kp|Lfp+p|pk|Lfk?2p(t)(8)nXp=1nXk=1p?pkp(t)gk(k(tk)?12nXp=1nXk=1p|pk|Lgk?2k(tk)+2p(t)?(9)Submit(6)(9)into(5),V(t)e2tnXp=1n?p+p(pp+12nXk=1(|pk|Lfk+|pk|Lgk)+nXk=1k(|kp|Lfp+|kp|Lgpe2p)?2p(t)+?p+p(1+2pppp)?p(t)p(t)+p?1+pp+12nXk=1(|pk|Lfk+|pk|Lgk)?2p(t)o=e2tnXp=1np 2p(t)+pp(t)p(t)+p2p(t)oDenote=pN:p=0.It is evident from Hypothesis 3 that p=0 for p.On the other hand,note that p0,p0and 2p4ppin Hypothesis 3,we haveV(t)e2tnnXpNpn p(t)+p2pp(t)o2+nXpNnp2p4po2p(t)o0(10)Thus,V(t)V(tl),tltl(11)When lttl+1,V(t)e2tnXp=1n?p+p(pp+12nXk=1(|pk|Lfk+|pk|Lgk)+nXk=1k(|kp|Lfp+|kp|Lgpe2k)?2p(t)+?p+p(1+2pppp)?p(t)p(t)+p?1+pp+12nXk=1(|pk|Lfk+|pk|Lgk)?2p(t)+pp 2p(t)+pp2p(t)+p(p+p)p(t)p(t)onXp=1pp(p(t)+p(t)2V(t)(12)176Journal of Xinjiang University(Natural Science Edition in Chinese and English)2024where p=maxp,p,=max1pnp,furthermore,V(t)V(l)e(tl),lttl+1(13)According to(11),when 0t0,V(t)V(0),on the grounds of(13),when 0tt1,V(t)V(0)e(t0)V(0)e(t0),in a similar way,when t1t1,V(t)V(t1)V(0)e(t10),when 1tt2,V(t)V(1)e(t1)V(0)e(t10)+(t1).Next,we prove the following inequality by mathematical inductionV(t)V(0)ePlm=1(tmm1),tltlV(t)V(0)e?Plm=1(tmm1)+(tl)?,lttl+1(14)Assumption when tl1tl1,V(t)V(0)ePl1m=1(tmm1),when l1ttl,V(t)V(0)e?Pl1m=1(tmm1)+(tl1)?,when tltl,V(t)V(tl)V()ePlm=1(tmm1),when lttl+1,V(t)V(l)e(tl)V()e?Plm=1(tmm1)+(tl)?.This shows that the inequality(14)is true.On the basis of(14)and Lemma 1,when tltl,one hasV(t)V(0)ePlm=1(tmm1)=V(0)ePlm=1tmm1tmtm1(tmtm1)V(0)ePlm=1(tmtm1)V(0)e(t).when lt 0 and T 0 are two fixed constants,aperiodicallyintermittent control reduces to periodical control as a special case.In this case,the proportion of rest span is a fixed constant=/T.So,the following corollary is presented.Corollary 1Under Hypotheses 13,if there exits a constant 0 such that 2 0,then the neural system(1)and system(3)are synchronized based on the periodic intermittent control.Remark 2In the literature 22,the synchronization problem of inertial NNs under periodic intermittent control isstudied by using the reduced order method.Compared with the literature,we use the non-reduced order method and studythe aperiodic intermittent control,and the research results are more general.Next,it will replace the Hypothesis 3 with the following assumption and get an important corollary.Hypothesis 4For any pN,there exists a non-negative constant p,such thatp+p1pp,p1p+12nXk=1(|pk|Lfk+|pk|Lgk),p12nXk=1(|pk|Lfk+|pk|Lgk)+nXk=1kp(|kp|Lfp+|kp|Lgpe2k)p.Corollary 2Based on Hypotheses 1,2 and 4,INNs models(1)and(3)achieved synchronization under aperiodicintermittent controller(4).Next,aperiodic intermittent control Up(t)takes the following formUp(t)=p(p(t)p(t)p(p(t)p(t),+tlt+l0,+lt0 and p0 represent control gain.tl+,l+represent the start time of work interval and the starttime of rest interval in the lth periodic intermittent control,respectively.0 is the lag delay.Let p(t)=p(t)p(t)be the synchronization error,the error system is easily described by p(t)=p p(t)pp(t)+nXk=1pkfk(k(t)+nXk=1pk gk(k(tk)p(p(t)p(p(t),+tlt+l,pN,p(t)=p p(t)pp(t)+nXk=1pkfk(k(t)+nXk=1pk gk(k(tk),+lt+tl+1,pN,wherefk(k(t)=fk(k(t)fk(k(t),gk(tk)=gk(k(tk)gk(k(tk).Theorem 2Under Hypotheses 13,if there exists a constant 0 such that 20,then system(1)and system(3)are lag synchronized based on the aperiodic intermittent controller(15).3Numerical ExampleConsider the