Improved Results on Finite-Time Passivity and Synchronization Problem for Fractional-Order Memristor-Based Competitive Neural Networks: Interval Matrix Approach

This research paper deals with the passivity and synchronization problem of fractional-order memristor-based competitive neural networks (FOMBCNNs) for the first time. Since the FOMBCNNs’ parameters are state-dependent, FOMBCNNs may exhibit unexpected parameter mismatch when different initial conditions are chosen. Therefore, the conventional robust control scheme cannot guarantee the synchronization of FOMBCNNs. Under the framework of the Filippov solution, the drive and response FOMBCNNs are first transformed into systems with interval parameters. Then, the new sufficient criteria are obtained by linear matrix inequalities (LMIs) to ensure the passivity in finite-time criteria for FOMBCNNs with mismatched switching jumps. Further, a feedback control law is designed to ensure the finite-time synchronization of FOMBCNNs. Finally, three numerical cases are given to illustrate the usefulness of our passivity and synchronization results.


Introduction
In recent years, the research on competitive-type neural networks (CNNs) has attracted expanding consideration from mathematicians, engineers, physicists and scholars. There are two forms of state variables: short-term memory and long-term memory. They make a good application background in convex optimization, cybernetics, image recognition and associative memory. Recently, many significant results on the dynamics of different kinds of neural networks, especially recurrent neural networks [1], cellular neural networks [2], T-S fuzzy neural networks [3], BAM neural networks [4] and competitive neural networks [5][6][7][8] have been obtained, but these results are mainly discussed in integer-order cases. However, there are few results focused on CNNs with fractional-order cases, see [9][10][11][12][13].
As the fourth circuit component along with resistor, capacitor and inductor, memristor was firstly postulated by Professor Chua in 1971 [14]. A resistor describes the voltagecurrent relationship, a capacitor depicts the charge-voltage relationship and an inductor displays the flux-current relationship. Chua showed the missing flux-charge relationship, which he named memristance and is the value of a memristor. The Hewlett-Packard laboratory fabricated a practically working memristor device in 2008 [15]. Good features of memristors such as nanometer size, nonvolatility and nonlinearity make them more suitable for simulating synapses resistors in network models. If the resistor of self-feedback connections weights is instead modeled by a memristor in conventional neural networks model, then memristor-based neural networks (MNNs) can be reached. Many interesting results on the dynamics of memristor-based neural networks have recently been proposed and studied [16][17][18][19]. Currently, many researchers also claim that MNNs provide more memory storage than conventional neural networks [20]. Undoubtedly, memristor-based competitive neural networks will advantage the associative memory capability of neural networks. It is very significant to analyze the memristor-based competitive neural networks' dynamical behaviors.
As far back as the 1695s, the idea of fractional calculus [21] was discussed by Gottfried Wilhelm Leibniz. Comparing with traditional integer-order calculus, fractional calculus has unlimited memory property. Recently, fractional calculus has played a vital role in the science and engineering fields [22,23], and lots of scientific results have been reported on this topic, see [24][25][26][27][28]. At present, there is a trend to utilize fractional differential techniques to study the dynamics of networks, especially neural networks [29][30][31][32]. It should be mentioned that the memristance of the memristor has a fractional order. Therefore, it has been very common and precise to utilize fractional differential techniques for studying the dynamics of memristor-based neural network systems. Recently, an ever-increasing number of specialists have talked about memristor-based neural network systems with fractional order (MNNWFO) and some significant outcomes have been accounted for on stability [33,34], stabilization [35,36] and state estimation [37,38].
Passivity can keep a system internally stable, which helps to understand the stability of different dynamical systems and their properties. Recently, the research on passivity analysis has become a hot research subject and it has been effectively applied to different fields such as control systems, power systems and robot systems. In view of the energy theory, the systems have been described as well as their Lyapunov-related input/output information. In light of the Lyapunov theory, there have been numerous scientific results in the literature, see [39][40][41][42][43][44][45][46]. On the other hand, synchronization has already emerged as a hot research theme and some meaningful scientific results on MNNWFO have been obtained, see [47,48].
Since the life spans of machines and human beings are finite, asymptotic synchronization is inapplicable in practice. In this regard, the finite-time synchronization of nonlinear systems has been comprehensively investigated in the literature [49][50][51][52]. For example, the authors in [53] have analyzed the Mittag-Leffler synchronization in finite-time criteria Caputo fractional-order memristor-based BAM neural networks with fractional orders 0 < ξ < 1 and 1 < ξ < 2, by means of a linear feedback control law, and the generalized Gronwall inequality. On the other hand, the finite-time passivity theory can provide a powerful tool to analyze the dynamics of fractional-order neural networks. Meanwhile, to our knowledge, few published research works exist concerning this problem.
To mention a few, the authors in [54] have established the robust passivity criterion of interval-parameter-based neural networks with a Caputo fractional-order derivative via passivity theory, Lyapunov theory and LMI techniques. Unfortunately, the passivity and synchronization of FOMCNNS have not been investigated yet and this situation motivates further investigation of FOMBCNNs.
Motivated by the aforementioned issues, this paper aims at analyzing the finite-time passivity and finite-time synchronization criterion of FOMBCNNs. The novelty of this manuscript is summarized as follows. (1) As far as we know, this paper is the first attempt to address the finite-time passivity and finite-synchronization for fractional-order competitive neural networks. (2) The problem addressed in this paper is described by a class of robust analytical techniques. (3) To obtain our main results, finite-time boundedness, finite-time passivity and finite-time synchronization definitions are presented. (4) In light of these definitions, several results are established theoretically. (5) These theoretical results and techniques are improved compared to the existing passivity and synchronization results of fractional-order neural networks. The remaining structure of this research work is outlined as follows: basic results on fractional-order calculus and a description of FOMBCNN systems are formally introduced in the next section. Sections 3 and 4 describe the main results of this research paper. Section 5 yields numerical results and their simulations. Finally, Section 6 ends with conclusions.

System Description and Preliminaries
For a real matrix U, Λ min (U) and Λ max (U) signify the maximal and the minimal eigenvalue of U, respectively. The superscript T indicates the matrix transposition. For all u = 0, the matrix H is positive definite if u T Hu > 0. H > H means H − H > 0. I is the identity matrix. The symmetric term in a matrix is displayed by . K(U) stands for the closure of the convex hull of U. For β(t) = (β 1 (t), ..., β n (t)) T ∈ R n , we denote

Preliminaries
This section comprises the rudimentary definitions and lemmas, which are further employed in the subsequent section.
where Γ(·) is the gamma function.

Lemma 2.
For 0 < ξ < 1, function p(t) is continuous and is defined on [0, +∞), then there exist constants 1 > 0 and 2 ≥ 0 such that Lemma 3. Let p(t) ∈ R n be a continuously differentiable vector-valued function, then for any where H ∈ R n×n is a positive definite symmetric matrix.

Model Description
We consider a fractional-order memristor-based competitive neural network (FOM-BCNN) in this manuscript: where C 0 D ξ t signifies the Caputo derivative with order 0 < ξ < 1, u j (t) represents the state variables; the current activity level is represented by p j ; a j and b j signifies the disposable scaling positive scalars; g k (·) represents the neuron activations; δ ji signifies the synaptic efficiency; the weights of the external stimulus is denoted by s j ; ζ = (ζ 1 , ..., ζ n ) T represents the constant external stimulus; x j (t) and y j (t) are disturbing input vectors; π f and θ f are known scalars; the synaptic memristor-based connection weights satisfy q jk u j (t) = q jk , |u j (t)| ≤ I j q jk , |u j (t)| > I j , in which I j > 0 are switching jumps andq jk > 0,q jk > 0 are constants. Define w j (t) = ∑ m i=1 δ ji (t)ζ i = ζδ T j (t), j = 1, 2, ..., n, where δ j (·) = [δ j1 (·), ..., δ jm (·)] T . Then, the transformation system (1) can be written as where |ζ i | 2 = ζ 2 1 + ... + ζ 2 m is scalar. Without loss of generality, the input stimulus vector is supposed to be normalized with unit magnitude |ζ i | 2 = 1. Then, the above equation can be written as Remark 1. The memristor-based connection weights of the system are changed based on the system state. Therefore, FOMBCNN (1) can be regarded as a state-dependent switching system, which is a special case of the dynamics of competitive neural networks.
Define q + jk = max q jk ,q jk , q − jk = min q jk ,q jk ,q jk = 1 2 q + jk + q − jk and q jk = 1 2 q + jk − q − jk . Then, FOMBCNN (1) can be written in the following form: where Based on Filippov's theory [55] and some transformation techniques, it can be obtained from (4) that According to the measurable selection theorem [56], there exists a measurable function Throughout this manuscript, we make the following assumptions. (A1). The nonlinear feedback function g k is bounded and satisfies: where l k > 0 (k = 1, 2, ..., n) are scalars.
then the following inequality is fulfilled: where η > 0 is a constant.
for all Φ T Φ ≤ M , if and only if there exists some β > 0 such that

Finite-Time Passivity
In this section, we demonstrate the finite-time passivity criterion of FOMBCNN (1), which is equivalent to the FOMBCNN system (6).
and for all x(t) ∈ R n and y(t) ∈ R n Assumption (A2) is held.
where H 1 , H 2 are symmetric positive definite matrices, if the following relationship holds: Under the zero initial values, there exists a constant υ > 0 such that if there exist symmetric positive definite matrices H 1 , a positive diagonal matrix H 2 and positive constants α i , i = 1, 2, 3, 4, 5 and β > 0 such that where Proof. Define the following functional for FOMBCNN (9) as According to Lemma 3, we have where L = diag{l 1 , ..., l n } and According to the Schur complement Lemma, (18) is equivalent to LMIs (12) and (17) is equivalent to the following LMIs ( Based on Lemma 5 in (20), there exists β > 0 such that Φ < 0 is equivalent to which can be rewritten as Based on the Schur Complement Lemma, expression (22) can be rearranged as condition (11). Therefore, from conditions (11) and (12), we have Taking the fractional integration of (23) from t 0 to t (t 0 ≤ t ≤ T σ ) and based on Lemma 1, one can obtain . (24) On the other hand, and Combining (24)-(26), one obtains and Proof. Based on previous Theorem 3, one can easily obtain In view of Lemma 6, there exists ρ > 0 such that Then, from (29)-(32), we have Now, we set In view of Lemma 1, we have due to W(u(t), w(t)) ≥ 0. Combining (34) and (35), we obtain Λ < 0. Hence, which means that the main FOMBCNN system (1) is passive in finite time with respect to σ 1 , σ 2 , T σ , H 1 , H 2 , λ 1 , λ 2 .
Remark 2. When x j (t) = y j (t) = q u (t) = q w (t) = 0, the asymptotic stability of the FOMBCNN model (1) can directly be obtained from Theorem 3. As a result, the passivity investigation is a high impact level of stability for FOMBCNNs.

Remark 3.
We now discuss a special case of system (1). Especially, when y(t) = 0, q w (t) = 0 and F = I, then model (9) is reduced to the FOMBCNN discussed in [60] Based on Theorem 1, the following corollary can be obtained for the system model (35). and Remark 4. Generally, the maximum absolute value method is an effective tool to study the dynamics of FOMBCNNs. Because the obtained sufficient conditions are all considered in the form of the maximum absolute value of the memristor-based synaptic connection strengths, i.e., max{|q jk |, |q jk |}, when we choose these kinds of conditions (max{|q jk |, |q jk |}), we lose half of the information. However, in this paper, the proposed system is changed from a fractional-order memristor-based competitive neural network into an interval parameter system. To overcome this issue, in this present work, we have transformed the memristor-based connection weights into interval parameters, which reduce more information losses. Therefore, the interval parameter approach is more effective than the maximum absolute value method.

Finite-Time Synchronization
Let x j (t) = y j (t) = 0 in FOMBCNN (1), then we have Here, all the parameters are similar to the FOMBCNN system (1). Similar to FOM-BCNN (4), the above system can be written as follows: whereq jk , ∆ jk u j (t) are already defined in (4). Based on Filippov's theory [55] and some transformation techniques, it can be obtained from (39) that Based on the measurable selection theorem [56], there exists a measurable function The response FOMBCNN with control inputs is denoted by: where the memristor-based connection weights q jk ũ j (t) are already defined in Section 2, h 1j (t), h 2j (t) are suitable controllers to be designed. Similar to FOMBCNN (38), the above response system (41) can be written as follows: where Based on Filippov's theory [55] and some transformation techniques, it can be obtained from (42) that Based on the measurable selection theorem [56], there exists a measurable functioñ The following finite-time synchronization definition is very important role to achieve the finite time synchronization criteria.
Denote e j (t) =ũ j (t) − u j (t) and z j (t) =w j (t) − w j (t). One has from (40) and (43) that where ζ k e k (t) = g k ũ k (t) − g k u k (t) .

Remark 5.
When the memristor-based connection weightsq kl =q kl , which means the connection weights are implemented only by a resistor, then the presented results are also still valid for the robust passivity and finite-time synchronization of fractional-order competitive neural networks, while these conservative results are not yet studied in the literature.

Remark 7.
Suppose that δ ji (t) = 0 in (1), the proposed system model is also still true for finitetime passivity and finite-time synchronization of fractional-order memristor based neural networks, and these results have not yet been studied in existing research works.

Remark 8.
In the existing literature, there are several results on synchronization analysis of memristive neural networks with switching jumps mismatch parameters. Specifically, Yang et al. [63] studied the asymptotic and finite-time synchronization problem of integer-order memristive neural networks, while the authors in [64] investigated the exponential synchronization problem of integer-order time-delayed memristive neural networks. Compared with the above-mentioned results, our criteria guarantee the finite-time passivity and finite-time synchronization of fractional-order memristive competitive neural networks. Moreover, the systems discussed in [63,64] are special cases of our stability results when δ ij (t) = x j (t) = y j (t) = τ(t) = 0 and ς = 1.

Numerical Results
Here, two numerical examples are given to validate the advantages of the obtained results. Example 1. Considering the following three-dimensional FOMBCNN: where The measured output vector of model (71) is assumed to be: We note that Assumptions (A1) and (A2) satisfy L = diag{1.5, 1.5, 1.5}. By standard computation, we get Let σ 1 = 2, σ 2 = 10, T σ = 3 and H 1 = H 2 = I. Then, it is clear that the given LMIs (14) and (15) of Theorem 3 are feasible, and these solutions are given as follows:

Conclusions
Based on a robust control method, both the passivity and synchronization criterion of FOMBCNNs were investigated in this manuscript. Through a key role of fractional order properties, finite-time stability theory and fractional-order Lyapunov functional, some novel sufficient criterion was derived to ensure the designed FOMBCNN is finite-time passive. Furthermore, a finite-time discontinuous feedback control law was designed to achieve synchronization in finite time for FOMBCNNs and we also evaluated the upper bound of the settling time. The feasibility and advantages of the obtained finite-time passivity and finite-time synchronization were illustrated in two numerical computer simulations. In the future, the global Mittag-Leffler synchronization, projective synchronization and quasi-synchronization problems will be considered for FOMBCNNs via nonfragile control [63,65], delayed impulsive control [66], quantized control [67], and quantized intermittent control [68].