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New finitetime synchronization analysis of a delayed memristive neurodynamic model
Advances in Difference Equations volumeÂ 2020, ArticleÂ number:Â 478 (2020)
Abstract
This paper presents theoretical results on the finitetime synchronization of delayed memristive neural networks (MNNs). Compared with existing ones on finitetime synchronization of discontinuous NNs, we directly regard the MNNs as a switching system, by introducing a novel analysis method, new synchronization criteria are established without employing differential inclusion theory and nonsmooth finite time convergence theorem. Finally, we give a numerical example to support the effectiveness of the theoretical results.
1 Introduction
It is well known that Chua in [1] postulated the existence of the fourth circuit element in 1971, and he named this element memristor as a contraction of memory and resistor, Chua also pointed out that the memristor can memorize its past dynamic history, such a memory characteristic makes it as a potential candidate for simulating biological synapses, and it has been shown that a simple memristive system can exhibit a plethora of complex dynamical behaviors. Until 2008, William and his research group at HewlettPackard Laboratory proclaimed that the fourth circuit element is realized by building a prototype of a solidstate memristor [2], since then, many efforts have been made devoted to the analysis and synthesis of memristive systems, see [3â€“5] and the references therein.
Especially, the socalled memristive neural networks (MNNs) are constructed by introducing resistors into artificial or biological neural networks, which greatly expand the application scope of neural networks, for example, MNNs can provide an important approach to better understand the neural processes in the human brain [3]. In addition, in the process of design and implementation of NNs, communication delays are ubiquitous in the real world owing to the finite velocity of signal switching and delivery, and it often becomes a source of some undesirable sideeffects such as oscillation and divergence in a realworld network [6]. Recently, the research on the dynamics of many types of delayed MNNs has been todayâ€™s hot topics and there are many excellent works have been reported, such as [7â€“13] and the references therein.
Synchronization, as an important nonlinear dynamical nature, has extensive applications in practical engineering fields. Under the driveâ€“response (or masterâ€“slave) framework proposed by Pecora and Carroll in [14], many theoretical results have been established on the synchronization of two MNNs, for example, the synchronization dynamics of kinds of delayed MNNs have been well studied, such as the global exponential synchronization [15], reliable asymptotic antisynchronization [16], nonfragile \(H_{\infty }\) synchronization [17], projective synchronization [18], and so on. Looking through the abovementioned literature, the trajectories of the response system can reach the trajectories of deriving system over the infinite horizon. In the application point of view, the synchronization is usually required to be realized in finite time, which is more important in some engineering processes, for instance, secure communication and artificial intelligence [19, 20]. For finitetime synchronization, it is noteworthy that the settling time depends heavily on the initial states of the system, which limits practical applications because the information of initial conditions may be hard to adjust or even impossible to estimate. To satisfy the need of fact, the fixedtime stability was originally introduced by Polyakov in [21], if it is globally finitetime stable and the settling time function is uniformly bounded for any initial values, which means that the settling or halting time is not dependent on the initial states. Nowadays, fixedtime control has been extensively applied in many areas such as power systems [22], rigid spacecraft [23], etc. According to the significant biological and engineering backgrounds of finitetime or fixedtime synchronization control, such an issue of delayed MNNs has attracted considerable attention, see [10, 24â€“30] and the references therein. It is noteworthy that for the majority of the aforementioned studies, a common method is based on the generalized finitetime convergence theorem under the Filippov differential inclusion theory, and the corresponding criteria are established via constructing Lyapunov (or Lyapunovâ€“Krasovskii) functionals.
Motivated by recent works in [31, 32] and combining with the framework for studying MNNs proposed in [33], in this paper, we further study the finitetime synchronization of a basic delayed MNN model. The main contribution of this paper lies in the following aspects.

(1)
Different from the theoretical results on dynamics of the aforementioned delayed MNNs, in this paper, the differential inclusion theory and the nonsmooth analysis techniques are abandoned, our results complement the earlier publications.

(2)
Existing finitetime synchronization results on delayed MNNs are mainly based on some generalized finitetime convergence theorem (see, e.g., [10, 24, 26, 27]), in this paper, we study the finitetime synchronization without using them, the employed approach enriches the analysis method of MNNs.

(3)
The theoretical results established in this paper are closely related to the location of the initial error states, which presents a new viewpoint of finitetime synchronization process.
The rest of this paper is arranged as follows. In Sect.Â 2, the model description and some preliminary works are presented. SectionÂ 3 discusses the finitetime synchronization of the driveâ€“response MNNs with a state feedback controller. In Sect.Â 4, a numerical example is given to substantiate the theoretical analysis. Finally, conclusions are drawn in Sect.Â 5.
2 Preliminaries
In this paper, we consider the following delayed MNN model described by Guo et al. in [33],
where \(x_{i}(t)\) denotes the neuron current activity level, \(d_{i}>0\) is the neuronal selfinhibition, \(f_{j}(\cdot )\), \(g_{j}(\cdot )\) are two activation functions, \(I_{i}\) stands for the input or bias, Î¸ represents the transmission delay, and \(a_{ij}(\cdot )\) and \(b_{ij}(\cdot )\) are dependent on the variation directions of \(f_{j}(x_{j}(t)) x_{i}(t)\) and \(g_{j}(x_{j}(t\theta ))x_{i}(t)\) along time t, respectively. More concretely, in light of the currentâ€“voltage characteristics of memristor [33], the statedependent parameters \(a_{ij}(\cdot )\) and \(b_{ij}(\cdot )\) can be specifically expressed as
in which \(a_{ij}^{*}\), \(a_{ij}^{**}\) and \(b_{ij}^{*}\), \(b_{ij}^{**}\) are different constants, \(D^{}(\cdot )\) means the left upper Diniderivation, and \(f_{ij}(t)=f_{j}(x_{j}(t))x_{i}(t)\), \(g_{ij}(t\theta )=f_{j}(x_{j}(t \theta ))x_{i}(t)\). The initial condition is equipped with \(x_{i}(t)=\varphi _{i}(t)\in C([\theta , 0], \mathbb{R})\).
In the following, we routinely regard system (2.1) as the drive system and the corresponding response system is presented as follows:
in which \(y_{i}(t)\) stands for the state variable of the ith neuron of response system and \(\mathbb{C}_{i}(t)\) corresponds to the control input. The initial condition associated with system (2.3) is given by \(y_{i}(t)=\tilde{\varphi }_{i}(t)\in C([\theta , 0], \mathbb{R})\).
Let us define the synchronization error function as , and subtract (2.3) from (2.1), we obtain the following error system:
With regard to neural networks (2.1) and (2.3), the initial condition of system (2.4) is correspondingly given as follows,
Definition 2.1
If for a suitable designed controller and any initial state , \(s\in [{ \theta }, 0]\), there is a time \(\mathbb{T}(\tilde{\boldsymbol{\varphi }},{\boldsymbol{\varphi }})\) such that
and
Then the drive systemresponse systems (2.1) and (2.3) are said to achieve finitetime synchronization. The function \(\mathbb{T}\) is called the settlingtime function.
Definition 2.2
If driveâ€“response systems (2.1) and (2.3) are finitetime synchronization and the settling time function \(\mathbb{T}(\tilde{\boldsymbol{\varphi }},{\boldsymbol{\varphi }})\) is uniformly bounded, that is, there is a constant \(\mathbb{T}_{\max }>0\) such that \(\mathbb{T}(\tilde{\boldsymbol{\varphi }},{\boldsymbol{\varphi }})\leq \mathbb{T}_{ \max }\). Then the drive system (2.1) and response system (2.3) are said to achieve fixedtime synchronization.
It is easy to see from the preknowledge that the finitetime synchronization problem in this paper is converted to finitetime stability problem of (2.4). In order to achieve this objective, further assumptions on the activations are made in the following.
Assumption 2.1
The activation functions \(f_{i}(\cdot )\) and \(g_{i}(\cdot )\) satisfy globally Lipschitz conditions and are bounded, that is, there exist positive constants \(L_{i}^{f}\), \(L_{i}^{g}\) and \(M_{i}^{f}\), \(M_{i}^{g}\) such that
and
hold for all \(u, v\in \mathbb{R}\), \(i=1,2,\ldots,n\).
3 Main results
For notational convenience, in what follows we denote \(\hat{a}_{ij}=\max \{a_{ij}^{*},a_{ij}^{**} \}\), \(\hat{b}_{ij}= \max \{b_{ij}^{*}, b_{ij}^{**} \}\), \(d_{ij}^{a}=\max \{a_{ij}^{*}, a_{ij}^{**} \}\min \{a_{ij}^{*}, a_{ij}^{**}\}\), \(d_{ij}^{b}=\max \{b_{ij}^{*}, b_{ij}^{**} \}\min \{b_{ij}^{*}, b_{ij}^{**}\}\), \(\tilde{k}_{\min }= \min_{1 \leq i\leq n}\{k_{i}\}\).
We are now in a position to state our main results as follows.
Theorem 3.1
Let AssumptionÂ 2.1hold, if the error system (2.4) is controlled with the control law
where the control parameters \(\alpha _{i}\), \(\beta _{i}\), \(k_{i}\)are positive constants satisfying
and
then the response system (2.3) can in finiteâ€“time synchronize with the drive system (2.1).
In order to prove the main results, we first establish the following two lemmas.
Lemma 3.2
Let AssumptionÂ 2.1hold and conditions (3.1) and (3.2) be satisfied. Then, for each of system (2.4) with , it would finiteâ€“timely cross the hyperplane with .
Proof
Observe from (3.1), (3.2) and \(0<\mu <1\) that
and
Firstly, we obtain from the continuity argument and (3.1) that there exists a sufficiently small Îµ satisfying
Set
One can easily see that , \(i=1,2,\ldots,n\), and we shall discuss it into the following two cases:
Case I: , \(i=1,2,\ldots,n\). We know from the continuity argument that there exists a constant \(\sigma >0\) such that , \(i=1,2,\ldots,n\), and , for \(s\in (t, t+\sigma )\).
Case II: If there exists an index \(i_{0}\) and a time \(t_{0}\geq 0\) such that , then we have
Note from
that
Then we deduce from (3.4), (3.5) and (3.6) that
Therefore, there must exist a constant \(\sigma >0\) such that , and , for \(s\in (t_{0}, t_{0}+\sigma )\).
We conclude from the above two cases that is nonincreasing and , \(t\geq 0\), which means that
Therefore, as time t increases, \(\sup_{t\theta \leq s\leq t} (\max_{i=1,2,\ldots,n}e_{i}(s) )\) would be less than 1. Denote the first time satisfying \(\sup_{t\theta \leq s\leq t} (\max_{i=1,2,\ldots,n}e_{i}(s) )=1\) as \(\mathbb{T}_{1}\), then we have
That is to say, every error function would cross the hyperplane
and the time it takes is no more than \(\mathbb{T}_{1}\). The proof of LemmaÂ 3.2 is complete.â€ƒâ–¡
Lemma 3.3
Let AssumptionÂ 2.1and conditions (3.1)â€“(3.2) be satisfied, if
and
hold for all \(i=1,2,\ldots,n\), then for each of system (2.4) with
would fixedâ€“timely flow to 0.
Proof
Define
It is easy to see that
and if there exist an index \(i_{1}\) and a time \(t_{1}\geq 0\) such that , then one has
Notice that
Elementary calculation from (3.9) produces
which, together with (3.8), leads to
which means that there exists some \(\varsigma >0\) such that holds for all \(s\in (t_{1}, t_{1}+\varsigma )\).
Therefore, we obtain from the above discussions that
which reduces to
and hence one can easily deduce that, as time t increases, would flow toÂ 0. Denote \(\mathbb{T}_{2}\) as the time such that , we then have
which implies that the timeâ€“taken for each from 1 to 0 is no more than \(\frac{1}{\tilde{k}_{\min }(1\mu )}\), \(i=1,2,\ldots,n\). The proof of LemmaÂ 3.3 is complete.â€ƒâ–¡
Based on Lemmas 3.2 and 3.3, we are ready to prove TheoremÂ 3.1.
Proof of TheoremÂ 3.1
For every solution of error system (2.4), we treat it into two cases according to the location of initial error function:
Case I: .
In this case, we obtain from LemmaÂ 3.3 that each error state component \(e_{i}(t)\), \(i=1,2,\ldots,n\), would flow to 0, and the timeâ€“taken is no more than \(\frac{1}{\tilde{k}_{\min }(1\mu )}\). In other words, the driveâ€“response system (2.1) and (2.3) under control law (3.1) achieve fixedtime synchronization.
Case II: .
In this case, we conclude from LemmaÂ 3.2 that would flow to 1 with a finite time, which is no more than , and then in a similar manner to that carried out in LemmaÂ 3.3, we see that each would continue to flow 0 in fixed time, which is no more than \(\frac{1}{\tilde{k}_{\min }(1\mu )}\). In short, as time t increases, each \(e_{i}(t)\) would finally achieve 0 in finite time \(\mathbb{T}_{\mathrm{total}}\) with .â€ƒâ–¡
Remark 3.1
The proof of the previous results provides a new perspective for better understanding the finitetime synchronization of MNNs. That is to say, if the absolute value of initial error function is less than or equal to 1, then each error function will achieve 0 in a fixed time; if the absolute value of initial error function is greater than 1, then each error function will firstly from the initial function to 1 in a finite time, and then further reach 0 in a fixed time. When the control parameter \(\mu >1\), whether there is such a mechanism process to realize fixedtime synchronization is another issue worth studying and discussing.
Remark 3.2
With the differential inclusion theory and nonsmooth finite (or fixed)time convergence theorem, the researchers studied the finite (or fixed)time synchronization of MNNs (see, e.g., [10, 24, 26, 27] and the references therein). Different from the method employed in those works, in this paper, we directly study the finitetime synchronization of the delayed MNNs (2.1) without using the theory of differential equations with discontinuous righthand sides. Therefore, the theoretical results established in this paper enrich the already existing finitetime synchronization methods.
Remark 3.3
Different from some existing finitetime controllers with time delays in such as [24, 30, 34], the designed control law in TheoremÂ 3.1 depends only on the current states at time t, it does not involve any information on the past states, which is much easier to be verified and realized in practice. Therefore, the designed finitetime control scheme is some less conservative. On the other hand, to realize the finitetime synchronization of discontinuous NNs, some useful Lyapunov functions or Lyapunovâ€“Krasovskii functions are constructed based on the nonsmooth finitetime convergence theorem together differential inequality techniques; see, e.g., [20, 28, 30, 34]. In this paper, we investigate the finitetime synchronization problem of the considered delayed system by some mathematical analysis techniques via constructing different Lyapunov functions.
4 Numerical simulations
In this section, we will present a numerical example to illustrate the obtained theoretical results. For convenience, we denote \(f_{ij}(t)=a_{j}(x_{j}(t))x_{i}(t)\) and \(g_{ij}(t\theta )=b_{j}(x_{j}(t\theta ))x_{i}(t)\).
Example 4.1
Consider a twoneuron memristive neural network model as follows:
where
and
We can verify that the activation functions satisfy AssumptionÂ 2.1 with \(L_{i}^{f}=L_{i}^{g}=1\), \(M_{i}^{f}=M_{i}^{g}=1\). Moreover, the response system is described by
where the activation functions and system parameters are the same as that in system (4.1), and the controllers are designed as follows:
It follows from simple computations that conditions (3.1) and (3.2) are satisfied. Therefore, we conclude from TheoremÂ 3.1 that the finitetime synchronization between system (4.1) and system (4.2) is achieved. FiguresÂ 1â€“2 show the simulation results with the initial conditions \(x_{1}(s)=8.5\), \(x_{2}(s)=2.5\), \(y_{1}(s)=4.5\), \(y_{2}(s)=3.4\), \(s\in [4.5, 0]\). Specifically, Fig.Â 1 shows the trajectories of the state evolution of system (4.1) and system (4.2), we can observe that the state of networks (4.2) in finite time synchronizes with system (4.1). FigureÂ 2 shows the finitetime synchronization between system (4.1) and system (4.2), it is readily seen that the state evolution error approaches zero quickly as time goes.
5 Conclusion
This paper performed a finitetime synchronization analysis of delayed MNNs based on the previous works [31â€“33]. Different from the existing works, we turn to the synchronization analysis by discussing the MNNs directly. Some new criteria ensuring finitetime synchronization of delayed MNNs were established by designing the suitable controller and constructing some novel Lyapunov functions. It is worth mentioning that the presented methodology herein without employing the differential inclusion theory and nonsmooth finite time convergence theorem, which are usually used to handle the finitetime synchronization problem. Finally, a numerical example is presented to substantiate the results. The future work will focus on the investigation of the finitetime synchronization of MNNs with mixed timevarying delays or leakage delays or impulse disturbance.
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The authors are thankful to the editor and anonymous reviewers for their insightful comments and suggestions, which strengthened our manuscript.
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This work was jointly supported by the National Natural Science Foundation of China (11801008), Key Program of University Natural Science Research Fund of Anhui Province (KJ2018A0082), Key Program of Scientific Research Fund for Young Teachers of AUST (QN2017209).
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Ren, D., Yao, A. New finitetime synchronization analysis of a delayed memristive neurodynamic model. Adv Differ Equ 2020, 478 (2020). https://doi.org/10.1186/s1366202002929x
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DOI: https://doi.org/10.1186/s1366202002929x