Finite-Time Synchronization of Multi-Linked Memristor-Based Neural Networks With Mixed Time-Varying Delays

This paper mainly investigates the finite-time synchronization and stability issue of a class of master-slave multi-linked memristor-based neural networks (MMNNs) with mixed time-varying delays via different state-feedback controllers. Based on some synchronization analytical techniques and Lyapunov functional method, sufficient criteria are obtained to ensure that the master-slave MMNNs systems can realize finite-time synchronization under the Filippov-framework. Three different controllers are designed to synchronize the MMNNs systems, and the settling time of finite-time synchronization is estimated in advance. The correctness and the feasibility of the proposed synchronization criteria are confirmed by three simulation examples.


I. INTRODUCTION
The memristor, firstly postulated by Chua [1], is a nonlinear two-terminal electronic element indicating the relationship between flux-linkage and charge. The resistance of the memristor can be adjusted by the charge and magnetic flux. Compared with common resistor, memristor has two distinct advantages. Firstly, its resistance varies with the amount of current passing through, and remains unchanged until a reverse current is received. Secondly, one can calculate the amount of charge by measuring its resistance, thus it can act as memory resistor. Besides, the memristor exhibits switching state-dependent characteristic and has nonvolatile memory storage, and shares many similarities with synapses which form the brain neural networks. Hence, the memristor can easily achieve synaptic behavior and simulate the dynamic behavior of neuronal synapses.
The basic functions of biological brain, such as learning, memorizing and forgetting, mainly depend on the plasticity The associate editor coordinating the review of this manuscript and approving it for publication was Mou Chen . of synapses, especially the ability of changing the plasticity of synapses, which is highly correlated with the previous action potential history [2]. At the same time, memristor is a kind of nonlinear resistor with memory function, whose resistance value can be changed by controlling the change of current, this is very similar to synapses.
In order to shorten the convergence time of synchronization between master-slave MNNs systems, many effective synchronization methods have been introduced,such as pinning control, intermitent control, finite-time synchronous control. Finite-time synchronous control method was firstly introduced in 1961 [42], with which the convergence time can be calculated in advance. Therefore, it has advantages in various practical engineering applications. It is of significant importance to study the finite-time synchronization control of MNNs. There have been numerous investigations about finite-time synchronization of MNNs, such as, finite-time synchronization of fractional-order MNNs has been studied in [20], [43], finite-time synchronization of delayed MNNs has been investigated in [44]- [46], finite-time synchronization of bidirectional associative memory MNNs has been studied in [23], [43], [47], etc.
Meanwhile, as we all know, there exist time delays in various applications, owning to the hardware implementation exists in engineering applications and transmitter delays in biological neural networks, etc. Delay can affect system stability, therefore, it is of significant importance to study the MMNs involving various time delays. Multi-linked memristor-based neural networks (MMNNs) introduced in [48] can be regarded as a combination of several single-linked MNNs, with each edge having its own transmission delay. Therefore, MMNNs is more consistent with biological brain neural network and has a stronger ability in modeling various real-world applications compared with single-linked MNNs. Many scholars pay considerable attention to investigating the finte-time synchronization and stabilization of MNNs involving various time delays. And many excellent results have been reported, please see work [47], [49] and the references therein. However, there exist only few related literatures reporting the synchronization and stabilization control of MMNNs, please see [46], [48]. In [46], Qin et al. studied finite-time projective synchronization of multi-linked memristor-based neural network involving leakage delays. Therefore, the finite-time synchronzation issue on MMNNs is far from being fully studied, and there is little research on MMNNs with time-varying delays.
Motivated by the above discussions, this paper aims to investigate the finite-time synchronization problem of MMNNs involving both discrete time-varying delays and distributed time-varying delay by adopting different control strategies.
Our main contributions are: (1) Three different controllers are designed. The first one is a linear and delay-independent state-feedback controller, the second one is a non-linear adaptive and delay-independent state-feedback controller, and the last one is a nonlinear and delay-dependent state-feedback controller. Our proof procedures adopted are simple to implement in practical applications.
(2) Three different Lyapunov-Krasovskii functionals are designed. Sufficient criteria are derived to ensure that the master-slave MMNNs systems can realize finite-time synchronization under the Filippov-Framework.
(3) The correctness and the feasibility of the acquired criteria are verified by three simulation examples.
Notation. For simplicity, the following symbols are to be adopted. R, R n , R n 1 ×n 2 represent the set of real number, n-dimensional Euclidean space, and a collection of matrices with dimension of n 1 ×n 2 , respectively. The sign of T denotes the transposition operation of a vector or matrix. C([−τ, 0], R n ) represents a Banach space of continuous functions mapping the interval of [−τ, 0] into R n . co [W,Ŵ] represents the closure of convex hull formed by the matrices or numbersW andŴ. For a matrix or vector W, W 1 , W ∞ , denote its 1−norm, ∞−norm, respectively.

II. PRELIMINARIES AND MMNNs MODELS A. DESCRIPTION OF MMNNs SYSTEMS
In this paper, we focus on studing the master-slave synchronization problem of multi-linked memristor-based neural networks (MMNNs) with mixed time-varying delays consisting of n nodes, and there exist m different links between any two nodes of the MMNNs systems. The master-slave MMNNs systems can be described by the following two differential equations. The master MMNNs system is described aṡ The corresponding slave MMNNs system can be expressed asẏ where z k (t), y k (t) represent state variables of the kth neuron of the master-slave MMNNs systems (1) and (2), respectively; γ k represents the rate of neuron self-inhibition; h(·), g(·) and f (·) stand for activation functions without and with time-varying delays; τ l (t) denotes discrete time-varying delay, l = 1, 2, . . . , m, which satisfies 0 ≤ τ l (t) ≤ τ ; (t) represents distributed delay satisfying 0 ≤ (t) ≤ M ; µ i (t) is the controller to be determined later; J k (t) is the external input; w ks (·), v (l)ks (·), and c ks (·) represent memristive connection weights which satisfy where k = 1, 2, . . . , n. Here, C k denotes the value of the kth capacitor, M ks , M * (l)ks , M * * ks denote the memductances of memristors R ks , R * (l)ks , R * * ks , respectively. Moreover, R ks represents the memristor between h s (z s (t)) and z k (t), or the memristor between h s (y s (t)) and y k (t), R * (l)ks represents the memristor between g s (z s (t−τ l (t)) and z k (t), or the memristor between g s (y s (t−τ l (t)) and y k (t), R * * ks represents the memristor between t t− (t) f s (z s (x)) dx and y k (t), or the memristor between t t− (t) f s (y s (x)) dx and z k (t). J k (t) represents the external input. According to the characteristics of the memristors, we set the connection weights as follows: where k, s = 1, 2, . . . , n, l = 1, 2, . . . , m, x k (t) represents z k (t) or y k (t), and k is a positive constant, denoting the switching jumps thresholds. The invariant value means unchanged, w * ks , w * * ks , v * (l)ks , v * * (l)ks , c * ks and c * * ks are known constants.
The synchronization issue between MMNNs systems (1) and (2) can be transformed into the corresponding stability problem, the error system between MMNNs systems (1) and (2) is defined as e k (t) = y k (t)−z k (t), then we can geṫ for k = 1, 2, . . . , n. Or, where The results of this paper are based on one definition, two important lemmas and three assumptions, as follows: Assumption 1: The functions h(u), g(u) and f (u) are assumed to be bounded. Then for any u ∈ R n , there always exist positive constants ξ h k , ξ Assumption 3: (t) and τ l (t), l = 1, 2, . . . , m are continuously differential functions, which satisfy 0 ≤˙ (t) ≤ D < 1, 0 ≤τ l (t) ≤ τ D < 1. Here D , τ D are two known positive constants.

III. MAIN RESULTS
In this paper, three different controllers are designed in this section to realize finite-time synchronization between the MMNNs systems (1) and (2). Meanwhile, based on some synchronization analytical techniques, sufficient criteria are derived to ensure that the synchronization can be acquired in finite time.
Calculating the derivative of V 1 (t) along the trajectory of (8) leads tȯ Obviously, From Assumption 2, we can conclude that Similarly, we can conclude and Based on Assumption 1, we can obtain sign T (e(t))(Ẁ (t)−Ẃ (t))h(y(t)) = n k=1 n s=1 sign(e k (t))(ẁ ks (t)−ẃ ks (t))h s (y s (t)) Similar to (18), we conclude that Similarly, On the other hand, we have where we define Then we can geṫ According to Assumptions 2 and 3, we can calculate the derivatives of V 2 (t) and V 3 (t), and obtaiṅ Therefore, according to (22), (23), and (24), we can get thaṫ We can easily conclude thaṫ wherē According to Theorem 1, it concludes thaṫ There are mainly two cases concerning the value of n k=1 χ k .

Case 1:
Only when e(t) ≡ 0, which means that e k (t) = 0, k = 1, 2, . . . , n, we can get n k=1 χ k = 0. According to Definition 2.3, we can conclude that the MMNNs (2) is synchronized with the MMNNs (1). Case 2: if e(t) = 0, which means that there exists at least one item e * (t) satisfying e * (t) = 0, then we conclude that n k=1 χ k ≥ 1. We can conclude from (29) thaṫ In this case, taking integral operation on both sides of the inequality (28) between regions (0, t), where t > 0, we further conclude that The inequality V(t) ≥ 0 always holds according to the definition of V(t). SinceV(t) < 0, it can be concluded that when t = t * 1 ≥ V(0) , V(t) ≡ 0, which means e k (t) ≡ 0, thus χ k = (sign(e k (t))) 2 ≡ 0. Therefore, we can infer that n k=1 χ k ≡ 0. Further, we obtain . According to Definition 1, we can conclude that the slave MMNNs system (2) can synchronize with the master MMNNs system (1) under the controller (9) during the settling time, given by t * 1 = V(0) . The proof of Theorem 1 is finished. Corollary 1: For master-slave MMNNs systems (1) and (2), if there exists only one link between any two nodes of the systems, which means m = 1, the MMNNs systems degenerate into single-linked MNNs systems. Under this condition, if and ζ meet the following requirements: then MNNs systems (1) and (2) with m = 1 can gain finte-time synchronization within a calculated time, which can be calculated in advance by utilizing (10) with m = 1. Proof: The proof process is similar to that of Theorem 1, hence, we omit the proof process.
Remark 1: Corollary 1 can be viewed as a simple case of Theorem 1 since there is only one link between any two nodes of the MMNNs systems, or, in other words, they are MNNs systems which have been studied in [49], [54]. The exponential synchronization of delayed MNNs was investigated in [24].

B. NON-LINEAR ADAPTIVE AND DELAY-INDEPENDENT STATE-FEEDBACK CONTROLLER
In this subsection, we design an adaptive and non-linear statefeedback controller to realize the finite-time synchronization between MMNNs systems (1) and (2), the following controller is utilized for k = 1, 2, . . . , n, * k (t), ζ * k (t) are time-varying parameters. The adaptive laws of * k (t), ζ * k (t) are designed as follows:˙ * . . , n are unknown positive constants to be determined later.
Remark 2: The controller (30) also appears in Ref. [18], [37], [45], etc. Compared with the regular feedback controller (i.e. µ(t) = −ke(t)), the controller (30) has two adaptive control parameters and is more flexible to use. We can also find a similar form of controller (30) in Refs. [15], [34], [41], [55], etc. However, these research results either deal with asymptotic control, or deal with different networks models. In comparison, the model in this paper is more general. In addition, adaptive control can modify its own characteristics to adapt to the dynamic characteristics of objects. Therefore, here we choose adaptive control.
Proof: Under the controller (30), we construct the following Lyapunov-Krasovskii functional where Based on the Chain rule introduced in [53], along the trajectory of (8), the derivative of V 1 (t) can be calculated aṡ V 1 (t) = sign T (e(t))ė(t)

+Ẃ (t)H (e(t))+
From Assumption 2, we can get that and According to Assumption 1, we otain Similar to (38), we can get And we have On the other hand, we have Similarly, we calculate the derivatives of V 2 (t), V 3 (t), V 4 (t) as follows: Then from (34)-(44), we can obtaiṅ wherē According to Theorem 2, we get the conclusion as followṡ Apparently, the inequality (46) is similarity to (27), so the proving procedure behind is quite similar, so we omit the following proof process.
Remark 3: The synchronization criteria provided in Theorems 1 and 2 are both in the numerical form. We can further consider utilizing the linear matrix inequality (LMI) method to calculate them and study the synchronization problem of the delayed nonlinear system with which the synchronization criteria are much less conservative.

Remark 4:
The design of controller (47) is derived from the structure of the MMNNs model. It contains more free parameters, which is conducive to achieve the synchronization effect of the master-slave MMNNs systems (2) and (1).
Theorem 3: Suppose that the Assumptions 1 -3 hold, and there exist n×n-matrices P, Q l , H, and an appropriately designed vector X , such that where l = 1, 2, . . . , m, L h , L g , L f , h , g and f are described in Assumption 1. Then the MMNNs system (2) can be synchronized with (1) under the controller (47) during the settling time t * 3 , given by where V(0) = 1 2 e T (0)e(0). Proof: Construct the following Lyapunov functional V(t) Computing the derivative of V(t) along the trajectory of (8) leads tȯ We can easily conclude From Assumption 2, it follows Similarly, we can get And we have
Hence, the proof of Theorem 3 is completed. Corollary 2: For master-slave MMNNs systems (2) and (1), if m = 1, the MMNNs systems degenerate into single-linked MNNs systems. Under this condition, if P, Q 1 , H and X meet the following requirements: then MNNs systems (1) and (2) with m = 1 can gain finte-time synchronization within a calculated time, which can be calculated by using (48) with m = 1.
Proof: The proof process is similar to that of Theorem 3, hence, we omit the proof process here.
Remark 5: Corollary 2 can be viewed as a special case of Theorem 3, since there is only one link between any two nodes of the MMNNs systems.
Remark 6: Theorem 1 adopts a linear and delay-independent state-feedback controller, Theorem 2 uses a non-linear adaptive and delay-dependent controller, Theorem 3 utilizes a non-linear and delay-independent state-feedback controllers. The former two controllers adopted are simpler, but they need more assumptions in the proof procedures. The last controller adopted is more complex, while it requires fewer hypothetical prerequisites. Hence, the last one is more conservative.
Remark 7: Because different controllers are adopted in deriving Theorems 1, 2, and 3, different synchronization criteria are obtained. From these criteria, we can conclude that because the form of controller (47) is more complex, so the criterion of Theorem 3 is simpler than those of Theorems 1 and 2.
Remark 8: The controllers (9), (30) and (47) contain discontinuous terms sign(e i (t)) or sign(e(t)) which might lead to chatting phenomenon, which largely exists in practical engineering application. Therefore, we must take effective measures to avoid this situation. In this paper, we mainly use an approximate value e i (t) |e i (t)|+ to replace sign(e i (t)), in which is a known small enough positive constant. However, this replacement is not made in the process of theoretical derivation in this paper, because it may cause confusion.
Remark 9: Xiaoyang Liu et al. studied the finite-time synchronization problem in Refs. [56], [57] and obtained some interesting results. In Ref. [56], the problem of finite-time consensus of multi-agent systems was investigated, and a centralized switching consensus protocol was designed to realize the finite-time consensus. The finite-time synchronization problem of nonlinear coupled neural networks was investigated by designing a new switching pinning controller in Ref. [57]. The above results used switching control, and in this paper we use adaptive state-feedback control. In addition, the research models are different.
Next we give the following three simulation examples to verify the effectiveness of Theorems 1, 2, and 3.
Example 1: This simulation is conducted to verify the effectiveness of Theorem 1 (see III-A). When there are no controller exerted on the MMNNs system (62), the state curves of z 1 (t), y 1 (t), and z 2 (t), y 2 (t) are shown in Figs. 1 and 2, respectively. Fig. 3 shows the error curves of e 1 (t) and e 2 (t) between the MMNNs (61) and (62).
Remark 10: Comparing Figs. 6 with 3, we can obtain that MMNNs systems (62) and (61) realize finite-time synchronization during t * 1 since the error curves of e 1 (t), e 2 (t) tend to zero in 0.5 second as shown in Fig. 6, which verify the correctness of the criterion proposed in Theorem 1.
Example 2: This simulation is conducted to verify Theorem 2 (see III-B). According to Theorem 2, we can   get ω * 1 ≥ 16.61, ω * 2 ≥ 17.36, ζ * 1 > 19.69, and ζ * 2 > 20.81, so we choose ω * 1 = 17, ω * 2 = 18, ζ * 1 = 20 and ζ * 2 = 21, respectively. Calculating the settling time and get t * 2 = 754.58. The state trajectory curves of z 1 (t), y 1 (t), z 2 (t), and y 2 (t) are described by Figs. 7 and 8, respectively.    Fig. 9 describes the trajectory curves of the errors e 1 (t) and e 2 (t). By analyzing Fig. 9, we conclude that MMNNs systems (62) and (61) realize synchronization during t * 2 since the error curves of e 1 (t), e 2 (t) tend to zero in 0.5 second as shown in Fig. 9, that verify the correctness of the method proposed in Theorem 2.  Remark 11: Besides, the settling time t * 2 is so big because we choose relatively small control gains ζ * 1 = 20, ζ * 2 = 21, and obtain * = 0.19, which makes the settling time V(0) * really big. But in real-world engineering applications, the synchronization time is very short which we can conclude from Fig. 9. According to Theorem 2, we conclude that t * 2 is strongly associated with the model parameters and initial conditions of MMNNs systems, and in most of the time, it is really hard to implent because gaining all parameters in adance is very hard. Example 3: This simulation is carried out to verify the effectiveness of Theorem 3 (see III-C).
According to the synchronization criterion proposed in Theorem 3, we set the following controller parameters:   Furthermore, we choose κ = 1, ρ = 0.68, and calculate the settling time and get t * 3 = 9.246. The state trajectory curves of z 1 (t), y 1 (t), and z 2 (t), y 2 (t) are shown in Figs. 10 and 11, respectivey. Fig. 12 describes the trajectory curves of the errors e 1 (t), e 2 (t) between the MMNNs systems (61) and (62).
Remark 12: Comparing Figs. 12 with 3, we conclude that MMNNs systems (62) and (61) realize synchronization during t * 3 , since the error curves e 1 (t), e 2 (t) tend to zero in Fig. 12, which verifies the correctness of the method proposed in Theorem 3. Comparing Figs. 12 and 6, we can easily conclude that the more complex the controller is, the longer time needed to obtain synchronization from analyzing the results of Figs. 12 and 6. However, the controller (47) needs more constraints while the controller (63) is much less conservative. In the actual engineering environment, it is very hard to find the appropriate activation functions which satisfy all the constraints, and in the neural networks, it is very hard to mimic the actual activation functions. Hence, we have to choose between the time cost and the conservativeness.
Remark 13: Comparing Figs. 12, 9, 6 with 3, we arrival at a conclusion that although MMNNs systems (62) and (61) can obtain finite-time synchronization in 15 seconds without  any controller, the convergence time is too long to satisfy the requirements of real-world applications, such as secure communication. When the controllers (63), (30), and (47) are added to the MMNNs system (62), the MMNN systems (62) and (61) can achieve synchronization within 0.5 second, which greately shorten the convergence time and could be applied in secure communication.

V. CONCLUSION AND FUTURE WORK
Based on the concept of master-slave, in our paper, the synchronization and stabilization problem of the MMNNs systems is investigated over a finite time interval. We design three different state-feedback controllers to obtain the finite-time synchronization between master and slave MMNNs systems. The first one is a linear and delay-dependent state-feedback controller, the second one is a non-linear adaptive and delay-independent state-feedback controller, and the last one is non-linear and delay-dependent state-feedback controller. By utilizing the proposed controllers, sufficient criteria are derived to ensure the finite-time synchronization of the proposed MMNNs systems. Three different numerical simulations are presented to validate the correctness and the feasibility of the obtained theoretical criteria. Our future work will focus on the investigation of the finite-time synchronization and stability of the multi-linked memristor-based bidirectional associative memory (BAM) neural networks with mixed time-varying delays.