
In this paper, the synchronization problem of complex-valued memristive competitive neural networks(CMCNNs) with different time scales is investigated. Based on differential inclusions and inequality techniques, some novel sufficient conditions are derived to ensure synchronization of the drive-response systems by designing a proper controller. Finally, a numerical example is provided to illustrate the usefulness and feasibility of our results.
Citation: Yong Zhao, Shanshan Ren. Synchronization for a class of complex-valued memristor-based competitive neural networks(CMCNNs) with different time scales[J]. Electronic Research Archive, 2021, 29(5): 3323-3340. doi: 10.3934/era.2021041
[1] | Yong Zhao, Shanshan Ren . Synchronization for a class of complex-valued memristor-based competitive neural networks(CMCNNs) with different time scales. Electronic Research Archive, 2021, 29(5): 3323-3340. doi: 10.3934/era.2021041 |
[2] | Jun Guo, Yanchao Shi, Shengye Wang . Synchronization analysis of delayed quaternion-valued memristor-based neural networks by a direct analytical approach. Electronic Research Archive, 2024, 32(5): 3377-3395. doi: 10.3934/era.2024156 |
[3] | Chao Yang, Juntao Wu, Zhengyang Qiao . An improved fixed-time stabilization problem of delayed coupled memristor-based neural networks with pinning control and indefinite derivative approach. Electronic Research Archive, 2023, 31(5): 2428-2446. doi: 10.3934/era.2023123 |
[4] | Jun Guo, Yanchao Shi, Weihua Luo, Yanzhao Cheng, Shengye Wang . Exponential projective synchronization analysis for quaternion-valued memristor-based neural networks with time delays. Electronic Research Archive, 2023, 31(9): 5609-5631. doi: 10.3934/era.2023285 |
[5] | Xiangwen Yin . A review of dynamics analysis of neural networks and applications in creation psychology. Electronic Research Archive, 2023, 31(5): 2595-2625. doi: 10.3934/era.2023132 |
[6] | Shuang Liu, Tianwei Xu, Qingyun Wang . Effect analysis of pinning and impulsive selection for finite-time synchronization of delayed complex-valued neural networks. Electronic Research Archive, 2025, 33(3): 1792-1811. doi: 10.3934/era.2025081 |
[7] | Huijun Xiong, Chao Yang, Wenhao Li . Fixed-time synchronization problem of coupled delayed discontinuous neural networks via indefinite derivative method. Electronic Research Archive, 2023, 31(3): 1625-1640. doi: 10.3934/era.2023084 |
[8] | Yu-Jing Shi, Yan Ma . Finite/fixed-time synchronization for complex networks via quantized adaptive control. Electronic Research Archive, 2021, 29(2): 2047-2061. doi: 10.3934/era.2020104 |
[9] | Yilin Li, Jianwen Feng, Jingyi Wang . Mean square synchronization for stochastic delayed neural networks via pinning impulsive control. Electronic Research Archive, 2022, 30(9): 3172-3192. doi: 10.3934/era.2022161 |
[10] | Wanshun Zhao, Kelin Li, Yanchao Shi . Exponential synchronization of neural networks with mixed delays under impulsive control. Electronic Research Archive, 2024, 32(9): 5287-5305. doi: 10.3934/era.2024244 |
In this paper, the synchronization problem of complex-valued memristive competitive neural networks(CMCNNs) with different time scales is investigated. Based on differential inclusions and inequality techniques, some novel sufficient conditions are derived to ensure synchronization of the drive-response systems by designing a proper controller. Finally, a numerical example is provided to illustrate the usefulness and feasibility of our results.
Since it was first put forward by Chua[3], memristor has attracted increasing attention in recent years[19,16]. As the fourth fundamental circuit element except resistor, inductor, and capacitor in circuity, the prototype of practical memristor device was successfully developed by Hewlett-Packard Labs in 2008[17]. Moreover, memristor has many good properties, such as low power consumption, high density and good scalability. More importantly, the connection weight of the memristor is not fixed, it depends on the voltage applied in the corresponding state. Therefore, many researchers hope to use memristor as an artificial synapse to build a device similar to brain function. In view of its good charzcteristic, now memristor has been widely used to model memristor-based neural networks(MNNs)[24,6,15].
As a collective dynamical behavior, synchronization extensively exists in life, society and neural systems. Synchronization plays important role in the activity of the brain and nervous system[14]. So synchronization of memristor-based neural networks is significant[22,1]. Liao et al.[22] discussed effects of initial conditions of memristor synapses on the synchronization of the coupled memristor neural circuits. Theoretical analysis and simulations show that the memristor synapse has played an important role in the synchronization of the coupled FitzHugh-Nagumo neural circuits. Ascoli et al.[1] found that the history of the memristor plays a critical role in the synchronous oscillations in the network and enhance synchronizaiton.
In recent years, neurals networks have attracted a lot of researchers in different research areas[20,21,4,27,7,8]. It is worth mentioning that MCNNs with different time scales, which are extensions of conventional neural networks. It is a kind of unsupervised learning neural networks, which refers to the whole interconnection between input and output of the single layer neural networks[25]. MCNNs contain two types of state variables, including the aspects of long-term memory (LTM)and short-term memory(STM)[10], corresponding to the fast changes of the neural network states and the slow changes of the synapses by external stimuli, respectively. And up to now, various dynamical behaviors for competitive neural networks have been investigated[6,15,12,11,13] and have been successfully applied to control theory, signal processing, pattern recognition and optimization design and so on[25,11].
We noticed that the results mentioned above have been achieved within the real number domain. However, as we all know, sometimes, it is unreasonable to deal with some problems only in the real number domain, such as symmetry detection, XOR problems, electromagnetic wave imaging and so on[23,18], which are more convenient and reasonable to deal with complex-value system. Thus, it is meaningful to study MCNNs as the generalization and extension for real-valued systems. In recent years, people have made a lot of achievements[9,26,2] in the field of complex number. Liu et al.[9] discussed global anti-synchronization of CMNNs with time delays by constructing an appropriate Lyapunov function. The proposed results of this paper are less conservative than existing literatures due to the characteristics of complex-valued memristive neural networks(CMNNs). Zhu et al.[26] investigated the synchronization of CMNNs with time delay by using the theory of the pinning control method, which control partial neurons instead of all neurons, and achieved new conclusions and progress. However, to the best of our knowledge, few scholars consider the synchronization problem of CMCNNs.
Based on the above analysis, this paper aims to investigate the synchronization problem of CMCNNs. By designing a proper controller, we achieve asymptotically stable of the error system such that achieve synchronization of the drive-response system. The contributions of this article can be summarized as follows.
(1) Different from the neural networks discussed earlier, the systems considered in this article are discussed based on complex-valued, which are an extension of the general real-valued networks.
(2) Different from asymptotic or exponential synchronization, it is shown that both the STM and LTM play a regulatory role in the systems so that the systems can show better performance.
(3) In this paper, the sufficiency of the synchronization of CMCNNs is derived by constructing a proper controller and use some inequality techniques.
The rest of the paper is organized as follows. In Section 2, some useful assumptions, definitions and lemmas needed in the paper are presented. In Section 3, a controller is designed to investigate the synchronization of CMCNNs by constructing a proper Lyapunov functional. In Section 4, a numerical example is given to illustrate the effectiveness of the obtained results. Finally, some conclusions are drawn.
In this paper, the solutions of all the systems considered below are intended in Filippov's sense[5].
{STM:ε˙zk(t)=−zk(t)+n∑l=1akl(zk(t))fl(zl(t)) +n∑l=1bkl(zk(t))fl(zl(t−τ(t)))+Hkmk(t),l=1,2,⋯,n,LTM:˙mk(t)=−mk(t)+fk(zk(t)), k=1,2,⋯,n, | (1) |
where
The initial conditions of system (1) are assumed to be
z(s)=ϕ(s),−τ≤s≤0 |
where
akl(zk(t))={ˆakl, |zk(t)|>Tk,ˇakl, |zk(t)|≤Tk,bkl(zk(t))={ˆbkl, |zk(t)|>Tk,ˇbkl, |zk(t)|≤Tk, |
for
Remark 1. From above analysis, we know that the connection weights
In this paper, consider system (1) as drive system and corresponding response system can be described as follows;
{STM:ε˙˜zk(t)=−˜zk(t)+n∑l=1akl(˜zk(t))fl(˜zl(t)) +n∑l=1bkl(˜zk(t))fl(˜zl(t−τ(t)))+Hk˜mk(t) +uk(t),l=1,2,⋯,n,LTM:˙˜mk(t)=−˜mk(t)+fk(˜zk(t)), k=1,2,⋯,n, | (2) |
with initial condition
Before proceeding further,
uk(t)=uRk(t)+iuIk(t); |
akl(z(t))=aRkl(z(t))+iaIkl(z(t)); |
bkl(z(t))=bRkl(z(t)+ibIkl(z(t)); |
fk(zk(t))=fRk(xk(t),yk(t))+ifIk(xk(t),yk(t)), |
where
Assumption 1. Suppose that
(1)The partial derivatives of
(2) The partial derivatives
|∂fRk/∂x|≤λRRk,|∂fRk/∂y|≤λRIk, |
|∂fIk/∂x|≤λIRk,|∂fIk/∂y|≤λIIk. |
Then we have
|fRk(˜xk(t),˜yk(t))−fRk(xk(t),yk(t))|≤λRRk|˜x(t)−x(t)|+λRIk|˜y(t)−y(t)|, |
|fIk(˜xk(t),˜yk(t))−fIk(xk(t),yk(t))|≤λIRk|˜x(t)−x(t)|+λIIk|˜y(t)−y(t)|. |
Under Assumption 1, separating system (1) into real and imaginary parts as follows
{STM:ε˙xRk(t)=−xRk(t)+n∑l=1aRkl(xk(t))fRl(xl(t),yl(t)) −n∑l=1aIkl(xk(t))fIl(xl(t),yl(t))+n∑l=1bRkl(xk(t))fRl(xl(t−τ(t)),yl(t−τ(t))) −n∑l=1bIkl(xk(t))fIl(xl(t−τ(t)),yl(t−τ(t)))+HRkmRk(t) −HIkmIk(t),l=1,2,⋯,n,LTM:˙mRk(t)=−mRk(t)+fRk(xk(t),yk(t)),k=1,2,⋯,n, | (3) |
{STM:ε˙yIk(t)=−yIk(t)+n∑l=1aRkl(yk(t))fIl(xl(t),yl(t)) +n∑l=1aIkl(yk(t))fRl(xl(t),yl(t))+n∑l=1bRkl(yk(t))fIl(xl(t−τ(t)),yl(t−τ(t))) +n∑l=1bIkl(yk(t))fRl(xl(t−τ(t)),yl(t−τ(t)))+HRkmIk(t) +HIkmRk(t),l=1,2,⋯,n,LTM:˙mIk(t)=−mIk(t)+fIk(xk(t),yk(t)),k=1,2,⋯,n, | (4) |
with initial conditions
The memristive connection weights of (3) and (4) satisfy the following conditions:
aRkl(xk(t))={ˆaRkl, |xk(t)|>Tk,ˇaRkl, |xk(t)|≤Tk,aIkl(yk(t))={ˆaIkl, |yk(t)|>Tk,ˇaIkl, |yk(t)|≤Tk, |
bRkl(xk(t))={ˆbRkl, |xk(t)|>Tk,ˇbRkl, |xk(t)|≤Tk,bIkl(yk(t))={ˆbIkl, |yk(t)|>Tk,ˇbIkl, |yk(t)|≤Tk, |
where the switching jumps
Remark 2. Here, we transform a complex-valued system into two equivalent real-valued system. Similarly, the inequalities satisfied by activation function in Assumption 1 are equivalent to the Lipschitz continuity condition in the complex domain. The purpose of this process is to facilitate our discussion using the relevant theorems in the field of real numbers.
Because the memristor-based connection weights in (3) and (4) are discontinuous, then by differential inclusions feature for system with the discontinuous right-hand sides, (3) and (4) will be written as follows:
{STM:ε˙xRk(t)∈−xRk(t)+n∑l=1co[a−Rkl,a+Rkl]fRl(xl(t),yl(t))−n∑l=1co[a−Ikl,a+Ikl]fIl(xl(t),yl(t)) +n∑l=1co[b−Rkl,b+Rkl]fRl(xl(t−τ(t)),yl(t−τ(t))) −n∑l=1co[b−Ikl,b+Ikl]fIl(xl(t−τ(t)),yl(t−τ(t)))+HRkmRk(t) −HIkmIk(t)),l=1,2,⋯,n,LTM:˙mRk(t)=−mRk(t)+fRk(xk(t),yk(t)),k=1,2,⋯,n, | (5) |
{STM:ε˙yIk(t)∈−yIk(t)+n∑l=1co[a−Rkl,a+Rkl]fIl(xl(t),yl(t))+n∑l=1co[a−Ikl,a+Ikl]fRl(xl(t),yl(t)) +n∑l=1co[b−Rkl,b+Rkl]fIl(xl(t−τ(t)),yl(t−τ(t))) +n∑l=1co[b−Ikl,b+Ikl]fRl(xl(t−τ(t)),yl(t−τ(t)))+HRkmIk(t) +HIkmRk(t),l=1,2,⋯,n,LTM:˙mIk(t)=−mIk(t)+fIk(xk(t),yk(t)),k=1,2,⋯,n, | (6) |
where
a+Rkl=max{ˆaRkl,ˇaRkl},a−Rkl=min{ˆaRkl,ˇaRkl},b+Rkl=max{ˆbRkl,ˇbRkl},b−Rkl=min{ˆbRkl,ˇbRkl}, |
a+Ikl=max{ˆaRkl,ˇaRkl},a−Ikl=min{ˆaRkl,ˇaRkl},b+Ikl=max{ˆbRkl,ˇbRkl},b−Ikl=min{ˆbRkl,ˇbRkl}. |
Or equivalently, for
{STM:ε˙xRk(t)=−xRk(t)+n∑l=1ˊaRklfRl(xl(t),yl(t)) −n∑l=1ˊaIklfIl(xl(t),yl(t))+n∑l=1ˊbRklfRl(xl(t−τ(t)),yl(t−τ(t))) −n∑l=1ˊbIklfIl(xl(t−τ(t)),yl(t−τ(t)))+HRkmRk(t) −HIkmIk(t),l=1,2,⋯,n,LTM:˙mRk(t)=−mRk(t)+fRk(xk(t),yk(t)),k=1,2,⋯,n, | (7) |
{STM:ε˙yIk(t)=−yIk(t)+n∑l=1ˊaRklfIl(xl(t),yl(t)) +n∑l=1ˊaIklfRl(xl(t),yl(t))+n∑l=1ˊbRklfIl(xl(t−τ(t)),yl(t−τ(t))) +n∑l=1ˊbIklfRl(xl(t−τ(t)),yl(t−τ(t)))+HRkmIk(t) +HIkmRk(t),l=1,2,⋯,n,LTM:˙mIk(t)=−mIk(t)+fIk(xk(t),yk(t)),k=1,2,⋯,n. | (8) |
Similar to the system (1), separating system (2) into real and imaginary parts as follows
{STM:ε˙˜xRk(t)=−˜xRk(t)+n∑l=1aRkl(˜xk(t))fRl(˜xl(t),˜yl(t)) −n∑l=1aIkl(˜xk(t))fIl(˜xl(t),˜yl(t))+n∑l=1bRkl(˜xk(t))fRl(˜xl(t−τ(t)),˜yl(t−τ(t))) −n∑l=1bIkl(˜xk(t))fIl(˜xl(t−τ(t)),˜yl(t−τ(t)))+HRk˜mRk(t) −HIk˜mIk(t)+uRk(t),l=1,2,⋯,n,LTM:˙˜mRk(t)=−˜mRk(t)+fRk(˜xk(t),˜yk(t)),k=1,2,⋯,n, | (9) |
{STM:ε˙˜yIk(t)=−˜yIk(t)+n∑l=1aRkl(˜yk(t))fIl(˜xl(t),˜yl(t)) +n∑l=1aIkl(˜yk(t))fRl(˜xl(t),˜yl(t))+n∑l=1bRkl(˜yk(t))fIl(˜xl(t−τ(t)),˜yl(t−τ(t))) +n∑l=1bIkl(˜yk(t))fRl(˜xl(t−τ(t)),˜yl(t−τ(t)))+HRk˜mIk(t) +HIk˜mRk(t)+uIk(t),l=1,2,⋯,n,LTM:˙˜mIk(t)=−˜mIk(t)+fIk(˜xk(t),˜yk(t)),k=1,2,⋯,n, | (10) |
with initial conditions
{STM:ε˙˜xRk(t)∈−˜xRk(t)+n∑l=1co[a−Rkl,a+Rkl]fRl(˜xl(t),˜yl(t))−n∑l=1co[a−Ikl,a+Ikl]fIl(˜xl(t),˜yl(t)) +n∑l=1co[b−Rkl,b+Rkl]fRl(˜xl(t−τ(t)),˜yl(t−τ(t))) −n∑l=1co[b−Ikl,b+Ikl]fIl(˜xl(t−τ(t)),˜yl(t−τ(t)))+HRk˜mRk(t) −HIk˜mIk(t)+uRk(t),l=1,2,⋯,n,LTM:˙˜mRk(t)=−˜mRk(t)+fRk(˜xk(t),˜yk(t)),k=1,2,⋯,n, | (11) |
{STM:ε˙˜yIk(t)∈−˜xIk(t)+n∑l=1co[a−Rkl,a+Rkl]fIl(˜xl(t),˜yl(t))+n∑l=1co[a−Ikl,a+Ikl]fRl(˜xl(t),˜yl(t)) +n∑l=1co[b−Rkl,b+Rkl]fIl(˜xl(t−τ(t)),˜yl(t−τ(t))) +n∑l=1co[b−Ikl,b+Ikl]fRl(˜xl(t−τ(t)),˜yl(t−τ(t)))+HRk˜mIk(t) +HIk˜mRk(t)+uIk(t),l=1,2,⋯,n,LTM:˙˜mIk(t)=−˜mIk(t)+fIk(˜xk(t),˜yk(t)),k=1,2,⋯,n. | (12) |
Or equivalently, for
{STM:ε˙˜xRk(t)=−˜xRk(t)+n∑l=1ˊaRklfRl(˜xl(t),˜yl(t)) −n∑l=1ˊaIklfIl(˜xl(t),˜yl(t))+n∑l=1ˊbRklfRl(˜xl(t−τ(t)),˜yl(t−τ(t))) −n∑l=1ˊbIklfIl(˜xl(t−τ(t)),˜yl(t−τ(t)))+HRk˜mRk(t) −HIk˜mIk(t)+uRk(t),l=1,2,⋯,n,LTM:˙˜mRk(t)=−˜mRk(t)+fRk(˜xk(t),˜yk(t)),k=1,2,⋯,n, | (13) |
{STM:ε˙˜yIk(t)=−˜xIk(t)+n∑l=1ˊaRklfIl(˜xl(t),˜yl(t)) +n∑l=1ˊaIklfRl(˜xl(t),˜yl(t))+n∑l=1ˊbRklfIl(˜xl(t−τ(t)),˜yl(t−τ(t))) +n∑l=1ˊbIklfRl(˜xl(t−τ(t)),˜yl(t−τ(t)))+HRk˜mIk(t) +HIk˜mRk(t)+uIk(t),l=1,2,⋯,n,LTM:˙˜mIk(t)=−˜mIk(t)+fIk(˜xk(t),˜yk(t)),k=1,2,⋯,n. | (14) |
Let
Assumption 2.
(1)co[a−Rkl,a+Rkl]fRl(˜xl(t),˜yl(t))−co[a−Rkl,a+Rkl]fRl(xl(t),yl(t))⊆co[a−Rkl,a+Rkl](fRl(˜xl(t),˜yl(t))−fRl(xl(t),yl(t))),(2)co[a−Ikl,a+Ikl]fIl(˜xl(t),˜yl(t))−co[a−Ikl,a+Ikl]fIl(xl(t),yl(t))⊆co[a−Ikl,a+Ikl](fIl(˜xl(t),˜yl(t))−fIl(xl(t),yl(t))),(3)co[b−Rkl,b+Rkl]fRl(˜xl(t),˜yl(t))−co[b−Rkl,b+Rkl]fRl(xl(t),yl(t))⊆co[b−Rkl,b+Rkl](fRl(˜xl(t),˜yl(t))−fRl(xl(t),yl(t))),(4)co[b−Ikl,b+Ikl]fIl(˜xl(t),˜yl(t))−co[b−Ikl,b+Ikl]fIl(xl(t),yl(t))⊆co[b−Ikl,b+Ikl](fIl(˜xl(t),˜yl(t))−fIl(xl(t),yl(t))). |
Then real and imaginary parts of the error system can be obtained by (5)-(6) and (11)-(12) as follows:
{STM:ε˙eRk(t)∈−eRk(t)+n∑l=1co[a−Rkl,a+Rkl]gRl(eRl(t),eIl(t))−n∑l=1co[a−Ikl,a+Ikl]gIl(eRl(t),eIl(t)) +n∑l=1co[b−Rkl,b+Rkl]gRl(eRl(t−τ(t)),eIl(t−τ(t))) −n∑l=1co[b−Ikl,b+Ikl]gIl(eRl(t−τ(t)),eIl(t−τ(t)))+HRkhRk(t) −HIkhIk(t)+uRk(t),l=1,2,⋯,n,LTM:˙hRk(t)=−hRk(t)+gRk(eRk(t),eIk(t)),k=1,2,⋯,n, | (15) |
{STM:ε˙eIk(t)∈−eIk(t)+n∑l=1co[a−Rkl,a+Rkl]gIl(eRl(t),eIl(t))+n∑l=1co[a−Ikl,a+Ikl]gRl(eRl(t),eIl(t)) +n∑l=1co[b−Rkl,b+Rkl]gIl(eRl(t−τ(t)),eIl(t−τ(t))) +n∑l=1co[b−Ikl,b+Ikl]gRl(eRl(t−τ(t)),eIl(t−τ(t)))+HRkhIk(t) +HIkhRk(t)+uIk(t),l=1,2,⋯,n,LTM:˙hIk(t)=−hIk(t)+gIk(eRk(t),eIk(t)),k=1,2,⋯,n, | (16) |
with initial conditions
gRl(eRl(t),eIl(t))=fRl(˜xl(t),˜yl(t))−fRl(xl(t),yl(t)); |
gIl(eRl(t),eIl(t))=fIl(˜xl(t),˜yl(t))−fIl(xl(t),yl(t)); |
gRl(eRl(t−τ(t)),eIl(t−τ(t))) |
=fRl(˜xl(t−τ(t)),˜yl(t−τ(t)))−fRl(xl(t−τ(t)),yl(t−τ(t))); |
gIl(eRl(t−τ(t)),eIl(t−τ(t)))=fIl(˜xl(t−τ(t)),˜yl(t−τ(t)))−fIl(xl(t−τ(t)),yl(t−τ(t))). |
Or equivalently, for
{STM:ε˙eRk(t)=−eRk(t)+n∑l=1aRklgRl(eRl(t),eIl(t)) −n∑l=1aIklgIl(eRl(t),eIl(t))+n∑l=1bRklgRl(eRl(t−τ(t)),eIl(t−τ(t))) −n∑l=1bIklgIl(eRl(t−τ(t)),eIl(t−τ(t)))+HRkhRk(t) −HIkhIk(t)+uRk(t),l=1,2,⋯,n,LTM:˙hRk(t)=−hRk(t)+gRk(eRk(t),eIk(t)),k=1,2,⋯,n, | (17) |
{STM:ε˙eIk(t)=−eIk(t)+n∑l=1aRklgIl(eRl(t),eIl(t)) +n∑l=1aIklgRl(eRl(t),eIl(t))+n∑l=1bRklgIl(eRl(t−τ(t)),eIl(t−τ(t))) +n∑l=1bIklgRl(eRl(t−τ(t)),eIl(t−τ(t)))+HRkhIk(t) +HIkhRk(t)+uIk(t),l=1,2,⋯,n,LTM:˙hIk(t)=−hIk(t)+gIk(eRk(t),eIk(t)),l=1,2,⋯,n. | (18) |
Lemma 2.1. For any vector
2xTy≤xTMx+yTM−1y. | (19) |
In this paper, the control inputs in (17) and (18) are taken as follows:
uR(t)=KReR(t),uI(t)=KIeI(t), | (20) |
where
eR(t)=(eR1(t),eR2(t),...,eRn(t))T,eI(t)=(eI1(t),eI2(t),...,eIn(t))T, |
KR=diag(kR1,kR2,...,kRn),KI=diag(kI1,kI2,...,kIn), |
and the controller gains to be determined.
Definition 2.2. The system (17) and (18) are asymptotically stable for any given initial conditions they satisfy:
limt→∞‖e(t)‖2=0,limt→∞‖h(t)‖2=0, |
where
e(t)=(eR(t))T,eI(t))T)T,h(t)=(hR(t))T,hI(t))T)T, |
hR(t)=(hR1(t),hR2(t),...,hRn(t))T,hI(t)=(hI1(t),hI2(t),...,hIn(t))T. |
Then drive-response systems (1) and (2) are said to be synchronized.
In the section, we present synchronization problems of CMCNNs.
Theorem 3.1. Under Assumptions 1-2, the systems (1) and (2) can be asymptotically synchronized with control inputs (20), if there exist constants
TR=1ε[2I−2KR−2ARλRR−ARλRI(AR)T+2AIλIR+AIλII(AI)T−r1(BRλRR)(BRλRR)T−r∗1(BRλRI)(BRλRI)T−r2(BIλIR)(BIλIR)T−r∗2(BIλII)(BIλII)T−r3HR(HR)T+r4HI(HI)T−λIRI−λRRI]−r9λRRI−r0λIRI−P, |
TI=1ε[2I−2KI−λRII+λIII−ARλIR(AR)T−2ARλII−AIλRR(AI)T−2AIλRI−r5(BRλIR)(BRλIR)T−r∗5(BRλII)(BRλII)T−r6(BIλRR)(BIλRR)T−r∗6(BIλRI)(BIλRI)T−r7HR(HR)T−r8HI(HI)T]−r∗9λRII−r∗0λIII−Q. |
Proof. Consider the following Lyapunov function
V(t,e(t),h(t))=(eR(t))TeR(t)+(eI(t))TeI(t)+(hR(t))ThR(t)+(hI(t))ThI(t)+∫tt−τ(t)(eR(s))TPeR(s)ds+∫tt−τ(t)eI(s))TQeI(s)ds | (21) |
Then, calculating the time derivative of
˙V(t,e(t),h(t))≤2(eR(t))T˙eR(t)+2(eI(t))T˙eI(t)+2(hR(t))T˙hR(t)+2(hI(t))T˙hI(t)+(eR(t))TPeR(t)−(eR(t−τ(t)))TPeR(t−τ(t))(1−γ)+(eI(t))TQeI(t)−(eI(t−τ(t)))TQeI(t−τ(t))(1−γ) | (22) |
According to Assumption 1-2 and Lemma 2.1, we have
2(eR(t))T˙eR(t)=2(eR(t))T1ε[−eR(t)+ARgR(e(t))−AIgI(e(t))+BRgR(e(t−τ(t)))−BIgI(e(t−τ(t)))+HRhR(t)−HIhI(t)+uR(t)] | (23) |
2(eR(t))TARgR(e(t))≤2(eR(t))TAR(λRReR(t)+λRIeI(t))=2(eR(t))TARλRReR(t)+2(eR(t))TARλRIeI(t)≤2(eR(t))TARλRReR(t)+(eR(t))TARλRI(AR)TeR(t) +(eI(t))TλRIeI(t) | (24) |
2(eR(t))TAIgI(e(t))≤2(eR(t))TAI(λIReR(t)+λIIeI(t))=2(eR(t))TAIλIReR(t)+2(eR(t))TAIλIIeI(t)≤2(eR(t))TAIλIReR(t)+(eR(t))TAIλII(AI)TeR(t) +(eI(t))TλIIeI(t) | (25) |
2(eR(t))TBRgR(e(t−τ(t)))≤(eR(t))T(BRλRR)M1(BRλRR)TeR(t)+(eR(t−τ(t)))TM−11eR(t−τ(t))+(eR(t))T(BRλRI)M∗1(BRλRI)TeR(t)+(eI(t−τ(t)))TM∗−11eI(t−τ(t)) | (26) |
Choose
M1=(r100r1),M∗1=(r∗100r∗1), |
So (26) can be simplified as
2(eR(t))TBRgR(e(t−τ(t)))≤r1(eR(t))T(BRλRR)(BRλRR)TeR(t)+1r1(eR(t−τ(t)))TeR(t−τ(t))+r∗1(eR(t))T(BRλRI)(BRλRI)TeR(t)+1r∗1(eI(t−τ(t)))TeI(t−τ(t)) |
(Note: The following inequalities use the same method.)
2(eR(t))TBIgI(e(t−τ(t)))≤r2(eR(t))T(BIλIR)(BIλIR)TeR(t)+1r2(eR(t−τ(t)))TeR(t−τ(t))+r∗2(eR(t))T(BIλII)(BIλII)TeR(t)+1r∗2(eI(t−τ(t)))TeI(t−τ(t)) | (27) |
2(eR(t))THRhR(t)≤r3(eR(t))THR(HR)TeR(t)+1r3(hR(t))ThR(t) | (28) |
2(eR(t))THIhI(t)≤r4(eR(t))THI(HI)TeR(t)+1r4(hI(t))ThI(t) | (29) |
Similarly,
2(eI(t))T˙eI(t)=2(eI(t))T1ε[−eI(t)+ARgI(e(t))+AIgR(e(t))+BRgI(e(t−τ(t)))+BIgR(e(t−τ(t)))+HRhI(t)+HIhR(t)+uI(t)] | (30) |
2(eI(t))TARgI(e(t))≤2(eI(t))TAR(λIReR(t)+λIIeI(t))=2(eI(t))TARλIReR(t)+2(eI(t))TARλIIeI(t)≤(eI(t))TARλIR(AR)TeI(t)+(eR(t))TλIReR(t) +2(eI(t))TARλIIeI(t) | (31) |
2(eI(t))TAIgR(e(t))≤2(eI(t))TAI(λRReR(t)+λRIeI(t))=2(eI(t))TAIλRReR(t)+2(eI(t))TAIλRIeI(t)≤(eI(t))TAIλRR(AI)TeI(t)+(eR(t))TλRReR(t) +2(eI(t))TAIλRIeI(t) | (32) |
2(eI(t))TBRgI(e(t−τ(t)))≤r5(eI(t))T(BRλIR)(BRλIR)TeI(t)+1r5(eR(t−τ(t)))TeR(t−τ(t))+r∗5(eI(t))T(BRλII)(BRλII)TeI(t)+1r∗5(eI(t−τ(t)))TeI(t−τ(t)) | (33) |
2(eI(t))TBIgR(e(t−τ(t)))≤r6(eI(t))T(BIλRR)(BIλRR)TeI(t)+1r6(eR(t−τ(t)))TeR(t−τ(t))+r∗6(eI(t))T(BIλRI)(BIλRI)TeI(t)+1r∗6(eI(t−τ(t)))TeI(t−τ(t)) | (34) |
2(eI(t))THRhI(t)≤r7(eI(t))THR(HR)TeI(t)+1r7(hI(t))ThI(t) | (35) |
2(eI(t))THIhR(t)≤r8(eI(t))THI(HI)TeI(t)+1r8(hR(t))ThR(t) | (36) |
2(hR(t))T˙hR(t)=2(hR(t))T(−hR(t)+gR(e(t))) | (37) |
2(hR(t))TgR(e(t))≤2(hR(t))T(λRReR(t)+λRIeI(t))≤1r9(hR(t))ThR(t)+r9λRR(eR(t))TeR(t)+1r∗9(hR(t))ThR(t)+r∗9λRI(eI(t))TeI(t) | (38) |
2(hI(t))T˙hI(t)=2(hI(t))T(−hI(t)+gI(e(t))) | (39) |
2(hI(t))TgI(e(t))≤2(hI(t))T(λIReR(t)+λIIeI(t))≤1r0(hI(t))ThI(t)+r0λIR(eR(t))TeR(t)+1r∗0(hI(t))ThI(t)+r∗0λII(eI(t))TeI(t) | (40) |
Substituting(23)-(40) to (22) yield
˙V(t)≤−(eR(t))T{1ε[2I−2KR−2ARλRR−ARλRI(AR)T+2AIλIR+AIλII(AI)T−r1(BRλRR)(BRλRR)T−r∗1(BRλRI)(BRλRI)T−r2(BIλIR)(BIλIR)T−r∗2(BIλII)(BIλII)T−r3HR(HR)T+r4HI(HI)T−λIRI−λRRI]−r9λRRI−r0λIRI−P}eR(t)−(eI(t))T{1ε[2I−2KI−λRII+λIII−ARλIR(AR)T−2ARλII−AIλRR(AI)T−2AIλRI−r5(BRλIR)(BRλIR)T−r∗5(BRλII)(BRλII)T−r6(BIλRR)(BIλRR)T−r∗6(BIλRI)(BIλRI)T−r7HR(HR)T−r8HI(HI)T]−r∗9λRII−r∗0λIII−Q}eI(t)+(hR(t))T[−2I+Iεr3+Iεr8+Ir9+Ir∗9]hR(t)+(hI(t))T[−2I+Iεr4+Iεr7+Ir0+Ir∗0]hI(t)+(eR(t−τ(t)))T[−Iεr1−Iεr2+Iεr5+Iεr6−(1−γ)P]eR(t−τ(t))+(eI(t−τ(t)))T[−Iεr∗1−Iεr∗2+Iεr∗5+Iεr∗6−(1−γ)Q]eI(t−τ(t)) | (41) |
where
It is easy to know that there are real numbers
1εr3+1r8+1r9+1r∗9−2<0,1εr4+1r7+1r0+1r∗0−2<0. | (42) |
Letting
−Iεr1−Iεr2+Iεr5+Iεr6=(1−γ)P,−Iεr∗1−Iεr∗2+Iεr∗5+Iεr∗6=(1−γ)Q,λR=min{λmin(TR),2−1εr3+1r8+1r9+1r∗9},λI=min{λmin(TI),2−1εr4+1r7+1r0+1r∗0}. | (43) |
From (41)-(43), it can be seen that
˙V(t)≤−λR(‖eR(t)‖2+‖hR(t)‖2)−λI(‖eI(t)‖2+‖hI(t)‖2). | (44) |
Moreover, in (44), the equality holds if and only if
limt→∞‖e(t)‖2=0,limt→∞‖h(t)‖2=0. |
According to Definition 2.2, the trivial solution of system (17) and (18) are asymptotically stable. We can conclude that the neural networks (1) and (2) can be synchronized with control inputs (20). The proof is complete.
Remark 3. From Theorem 3.1, we can see that the existence of the variable delay affects the value of the matrix
Corollary 1. Under Assumptions 1-2, and the controllers (20) when
TR=1ε[2I−2KR−2ARλRR−ARλRI(AR)T+2AIλIR+AIλII(AI)T−r1(BRλRR)(BRλRR)T−r∗1(BRλRI)(BRλRI)T−r2(BIλIR)(BIλIR)T−r∗2(BIλII)(BIλII)T−r3HR(HR)T+r4HI(HI)T−λIRI−λRRI]−r9λRRI−r0λIRI−P, |
TI=1ε[2I−2KI−λRII+λIII−ARλIR(AR)T−2ARλII−AIλRR(AI)T−2AIλRI−r5(BRλIR)(BRλIR)T−r∗5(BRλII)(BRλII)T−r6(BIλRR)(BIλRR)T−r∗6(BIλRI)(BIλRI)T−r7HR(HR)T−r8HI(HI)T]−r∗9λRII−r∗0λIII−Q. |
Proof. The proof process is similar to Theorem 3.1 by taking
−Iεr1−Iεr2+Iεr5+Iεr6=P, |
−Iεr∗1−Iεr∗2+Iεr∗5+Iεr∗6=Q, |
and is omitted here.
Corollary 2. Under Assumptions 1-2, and the controllers (20) when
TR=1ε[2I−2KR−2(AR+BR)λRR−(AR+BR)λRI+2(AI+BI)λIR+(AI+BI)λII−r3HR(HR)T+r4HI(HI)T−(AR+BR)λIR−(AI+BI)λRR]−r9λRRI−r0λIRI, |
TI=1ε[2I−2KI−(AR+BR)λRI+(AI+BI)λII−(AR+BR)λIR−2(AR+BR)λII−(AI+BI)λRR−2(AI+BI)λRI−r7HR(HR)T−r8HI(HI)T]−r∗9λRII−r∗0λIII. |
Proof. The proof process is similar to Theorem 3.1 and is omitted here.
In the section, a numerical example is given to demonstrate the validity of the above results. Consider the following memristor-based complex-valued competitive neural networks:
{STM:ε˙zk(t)=−zk(t)+2∑l=1akl(zk(t))fl(zl(t)) +2∑l=1bkl(zk(t))fl(zl(t−τ(t)))+Hkmk(t),l=1,2LTM:˙mk(t)=−mk(t)+fk(zk(t)), k=1,2, | (45) |
where
aR11(x1(t))={−1.5,|x1|<1,−0.4,|x1|≥1,aR12(x1(t))={−0.2,|x1|<1,−1.0,|x1|≥1, |
aR21(x2(t))={−0.7,|x2|<1,−1.4,|x2|≥1,aR22(x2(t))={−2.0,|x2|<1,−0.8,|x2|≥1, |
aI11(y1(t))={−1.5,|y1|<1,−0.06,|y1|≥1,aI12(y1(t))={−1.5,|y1|<1,−0.28,|y1|≥1, |
aI21(y2(t))={−0.6,|y2|<1,−0.25,|y2|≥1,aI22(y2(t))={−1.0,|y2|<1,−1.6,|y2|≥1, |
bR11(x1(t))={0.2,|x1|<1,1.0,|x1|≥1,bR12(x1(t))={−3.0,|x1|<1,−0.5,|x1|≥1, |
bR21(x2(t))={−1.6,|x2|<1,−2.0,|x2|≥1,bR22(x2(t))={−1.0,|x2|<1,−1.25,|x2|≥1, |
bI11(y1(t))={1.2,|y1|<1,0.6,|y1|≥1,bI12(y1(t))={1.25,|y1|<1,0.13,|y1|≥1, |
bI21(y2(t))={1.4,|y2|<1,0.03,|y2|≥1,bI22(y2(t))={−1.5,|y2|<1,−0.7,|y2|≥1, |
with initial values
{STM:˙εk(t)=−˜zk(t)+2∑l=1akl(˜zk(t))fl(˜zl(t))+2∑l=1bkl(˜zk(t))fl(˜zl(t−τ(t)))+Hk˜mk(t),l=1,2LTM:˜mk(t)=−˜mk(t)+fk(˜zk(t)),k=1,2 | (46) |
with initial values
Let
When the controllers is the form of (20), then by simple computation we get
AR=(−1.5−0.2−0.7−2),AI=(−1.5−1.5−0.6−1), |
BR=(0.23−1.6−1),BI=(−1.21.251.4−1.5). |
And choose
r1=r2=12,r3=2,r4=3,r5=r6=13,r7=r8=2,r9=3,r0=2, |
r∗1=r∗2=13,r∗5=r∗6=14,r∗9=3,r∗0=2. |
so from Theorem 3.1, we have
TR=(5.854.537.647.72),TI=(4.473.963.074.41). |
Then, all the conditions of Theorem 3.1 are satisfied. Under the controllers (20), the synchronization errors of real parts and imaginary parts are depicted by Figure 1. Therefore, according to Theorem 3.1, the systems (45) and (46) are synchronized.
This paper is concerned with synchronization of CMCNNs with different time scales. Firstly, we improved the model: (1) improved the ordinary neural network model to CMCNNs with different time scales; (2) extended the a common real-valued system to a complex-valued system. Then, we achieved the synchronization problem of the drive and response systems by designing a proper controller. In theory, the control design is operable and can be easily realized. Moreover, our results are more general and extend the previously known results. Finally, the effectiveness of our results has been demonstrated by Section 4. In further research, the main results of this paper can be extended to no time-delay for the feedback controller. We will also explore more dynamical behaviors of CMCNNs, for example, finite-time synchronization, fixed-time synchronization and anti-synchronization.
This work was supported by the National Natural Science Foundation of China (No.11972115, No.12062004) and Talent Special Projects of School-level Scientific Research Programs under Guangdong Polytechnic Normal University (No.2021SDKYA004).
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