Finite-Time Projective Synchronization of Fractional-Order Memristive Neural Networks with Mixed Time-Varying Delays

This paper is concerned with the finite-time projective synchronization problem of fractional-order memristive neural networks (FMNNs) with mixed time-varying delays. Firstly, under the frame of fractional-order differential inclusion and the set-valued map, several criteria are derived to ensure finite-time projective synchronization of FMNNs. Meanwhile, three properties are established to deal with different forms of the finite-time fractional differential inequation, which greatly extend some results on estimation of settling time of FMNNs. In addition to the traditional Lyapunov function with 1-norm form in Theorem 1, a more general and flexible Lyapunov function based on p-norm is constructed in Theorem 2 to analyze the finite-time projective synchronization problem, and the estimation of settling time has been verified less conservative than previous results. Finally, numerical examples are provided to demonstrate the effectiveness of the derived theoretical results.


Introduction
In the past several decades, artificial neural networks have been extensively investigated due to their wide applications in signal processing [1], combinatorial optimization [2], pattern recognition [3], associative memories [4], and so on. Memristor, first proposed by Chua in 1971 [5], was realized by Hewlett-Packard Laboratory in 2008 [6]. Compared with the resistor, the memristor possesses a memory characteristic as the synaptic characteristics of biological neurons, which is distinct from other circuit elements. As a result, by using the memristor to replace other resistors, scholars constructed the memristive neural networks (MNNs) to emulate the human brain. Moreover, as a special kind of neural networks, MNNs are nonlinear systems with statedependent switching jumps and possess more complicated properties than the traditional neural networks.
It is well known that time delays are unavoidable in neural networks. On the one hand, discrete time delays are ubiquitous because of the finite switching speed of amplifiers and the inherent information exchanging time between different neurons [7]. On the other hand, distributed time delays should also be considered in neural networks because neural networks usually have a spatial nature and the presence of a large number of parallel pathways with multifarious axon sizes and lengths [8]. Hence, these time delays, which may bring about instability, chaos, oscillation, or other poor performances of the system, should be considered when investigating the dynamical behavior for neural networks [9][10][11].
As a generalization of the ordinary differentiation and integration to arbitrary noninteger order, fractional-order calculus has long history and can be traced back to the 17th century. For a long time, the theory of fractional calculus was once considered as a purely theoretical field of mathematics and developed very slow. However, due to its infinite memory and more degrees of freedom [12], many scholars have found that fractional-order models are more accurate than integer-order ones in describing the memory and hereditary properties. In consequence, fractional-order models have been incorporated into artificial neural networks. Subsequently, fractional-order memristive neural networks (FMNNs) have attracted lots of researchers to study their dynamical behaviors [13][14][15].
Since chaos synchronization was firstly proposed in 1990 [16], synchronization control has become an important tool to control the chaos appearing in the practical applications such as security communications [17], image processing [18], and cryptography [19]. So far, many sorts of synchronization schemes have been presented such as complete synchronization [20], antisynchronization [21], phase synchronization [22], lag synchronization [23], exponential synchronization [24], and projective synchronization.
Overall, the above discussed synchronization schemes can be divided into two categories: finite-time and infinitetime synchronization. In practice, the system always expected to realize synchronization as quickly as possible. It is worth noting that the finite-time control technique can not only accelerate the convergence process but also demonstrate better interference suppression properties and robustness [25]. Consequently, the finite-time synchronization is an optimal synchronization method and has attracted lots of attention in many research studies [26][27][28][29][30][31][32][33]. For example, in [27], by using the Gronwall-Bellman integral inequality and the Volterra integral equation, the authors explored the finite-time projective synchronization problem of memristor-based delay fractional-order neural networks. And Zhang and Deng studied the finite-time projective synchronization of fractional-order complex-valued memristor-based neural networks with time delays in [28]. However, we note that [26][27][28] pay attention to discussing the synchronization conditions and ignore the computation of settling time. In [29], Zhang et al. deduced the fractionalorder derivative of Lyapunov function C t 0 D α t V(t) ≤ − aV(t)− b and gave a conservative estimation of settling time with a � 0. In [30][31][32][33], the authors deduced the fractional-order derivative of Lyapunov function C t 0 D α t V(t) ≤ − aV(t) − bV β (t) and gave a conservative estimation of settling time with a � 0. And we propose a new method to estimate settling time with a ≥ 0, which is less conservative than previous results.
Among a great variety of synchronization schemes, projective synchronization, first proposed by Mainieri and Rehacek in [34], is characterized by a fact that the drive and response systems could achieve synchronization up to a scaling factor. Particularly, due to the proportional feature, projective synchronization can be used to extend binary digital to M-nary digital communication for achieving fast communication [35]. Furthermore, projective synchronization of FMNNs, which can obtain faster communication with its proportional feature, has been widely concerned and deeply studied by researchers [36][37][38][39][40]. For instance, the global projective synchronization for FMNNs is investigated via combining open-loop control with the time-delayed feedback control in [36]. And in [38], Yang et al. discussed the quasi-projective synchronization of fractional-order complex-valued neural networks, and the error bounds of quasi-projective synchronization are estimated. Based on notoriously Barbalat's lemma and the Razumikhin-type stability theorem, Zhang et al. investigated the projective synchronization of FMNNs with switching jump mismatch in [40]. However, to the best of our knowledge, there are few research studies about the finite-time projective synchronization of FMNNs with mixed time-varying delays.
Inspired by the above discussion, this paper will study the finite-time projective synchronization of FMNNs with mixed time-varying delays via applying differential inequality of the Caputo derivative and the asymptotic expansion property of Mittag-Leffler function. e main innovations and contents of this paper lie in the following aspects: (1) For fractional-order neural networks, the fractionalorder derivative of Lyapunov function V(t) can usually be reduced In the existing literature studies on finite-time synchronization of fractional-order neural networks, most pay attention to discuss the synchronization condition a ≥ 0 and ignore the computation of settling time. And a few literature studies estimate the settling time with a � 0, which will increase the conservative of the obtained result. In this paper, via applying the asymptotic expansion property of Mittag-Leffler function and the fractional-order power law inequation, we propose three properties to estimate the settling time with a ≥ 0. And the comparison result can be seen in Tables 1-3, which demonstrate our new estimation of the settling time is more applicative and accurate.
(2) First, via designing a simple time-delayed feedback controller, some sufficient conditions are derived to guarantee the finite-time projective synchronization of FMNNs. And the results can be easily extended to the complete synchronization and antisynchronization. (3) Unlike the traditional maximum absolute valuebased method to propose the synaptic weights of memristive neural networks, by introducing some transformations, FMNNs are translated to a type of fractional-order systems with uncertain parameters. (4) In addition to the traditional Lyapunov function with 1-norm form, a more general and flexible Lyapunov function based on p-norm is constructed to analyze the projective finite-time synchronization of FMNNs. e rest of this paper is organized as follows. Section 2 describes the models and introduces the preliminaries including some necessary definitions and lemmas. In Section 3, we present the main results. Simulation examples are given to confirm the validity of our results in Section 4. And conclusions follow in Section 5.
Notation: throughout this study, N, C, R, and R * denote the set of all natural numbers, complex numbers, real numbers, and nonnegative real numbers, respectively. C n ([t 0 , ∞), R) represents the family of continuous and norder differentiable function form [t 0 , ∞) to R. And sign(·) 2 Complexity denotes the sign function, where sign (x) � e symbol * involves the convolution operator. And Γ(α) is Euler's gamma function which is defined by Γ(α) � +∞ 0 s α− 1 e − s ds.

Preliminaries.
In this section, we recall some basic definitions with respect to fractional calculus and several important lemmas used in this paper.
Definition 2 (see [42]). e Caputo fractional derivative with fractional-order α for a differentiable function where t ≥ t 0 and n is a positive integral such that n − 1 < α < n. Especially, when α ∈ (0, 1), then Definition 3 (see [41]). Mittag-− Leffler function E p,q (·) with two parameters is defined as where p > 0, q > 0, and ς ∈ C. And if q � 1, the Mittag-Leffler function with one parameter is shown as Definition 4. If there exists a nonzero constant λ ∈ R, for any solutions of the n-dimensional drive system and the response system, x i (t) and y i (t), i � 1, 2, . . . , n, with initial values x i (t 0 ) and y i (t 0 ) are such that where λ represents the projective coefficient. T * is called the settling time for synchronization. en, the response system is said to be finite-time projective synchronization to the drive system.  Table 2: e comparisons among T * 1 , T 2 , T * 3 , T 4 , and T 5 with the projective coefficient λ � 1. Lyapunov function Fractional-order differential inequation Settling time Table 3: e comparisons among T * 1 , T 2 , T * 3 , T 4 , and T 5 with the projective coefficient λ � − 1.

Lyapunov function
Fractional-order differential inequation Settling time Complexity Particularly, the drive-response system can be called finite-time complete synchronization if λ � 1 and finite-time antisynchronization if λ � − 1.
Lemma 4 (see [45]). Assume that a 1 , a 2 , . . . , a n are nonnegative constants. For two arbitrary constants p and q satisfying 0 < p < q, then n i�1 a q i Lemma 5 (see [46]). Let f(t) be a continuous function on the positive interval [t 0 , +∞) and satisfy where a is a constant and 0 < α < 1; then, Lemma 6. Let f(t) be a continuous function on the positive interval [t 0 , +∞) and satisfy where a and b are constants and 0 < α < 1; then, Proof. Let h(t) � f(t) + (b/a), and we can easily obtain By applying Lemma 5, we have □ Remark 1. For Lemma 6, Yang et al. in [38] obtain a consistent result with a > 0 and b ≥ 0. And we believe that the conditions of a > 0 and b ≥ 0 are conservative. And in Lemma 6, we consider that a and b are constants, which is more general than the results in [38].
In the following, two new fractional-order inequalities are established to propose the finite-time convergence, which play a critical role to obtain the finite-time synchronization and calculate the settling time. And we can gain three new properties.
Property 1. Let f(t) ∈ C 1 ([t 0 , +∞), R * ) be continuous and nonnegative definite. And the following inequation holds: where a > 0, b > 0, and 0 < α < 1. It can be deduced as follows: en, we can judge f(t) converges to 0 within the settling time T * 1 which can be estimated as follows: Proof. By utilizing Lemma 6, we have From Lemma 2, when β � 1, we can get Apparently, m k�1 z − k /Γ(1 − kα) > 0; then, we can judge that there exists z 0 > 0 such that Similarly, we have From the above conclusion, inequation (20) can be converted to In order to better analyze the problem, let We can easily know that the function Ω(t) is monotonically decreasing. And Definition 4, the function f(t) converges to 0 in finite time. And the settling time t * 1 can be calculated as follows: is continuous and nonnegative definite. If the following inequality holds, where a > 0, b > 0, 0 < α < 1, 0 < β ≤ 1, then it can be deduced as follows: And we can judge f(t) converges to 0 within the settling time T 2 * which can be estimated as follows: Multiplying − a on sides of the above inequation, then we can get According to the conditions of Property 2, where m is a positive integer. en, the above inequation can be written as follows: When m ≥ β, via applying Lemma 7, we can gain en, Considering m ≥ (β/(1 − β)), thus (m/β) − m − 1 ≥ 0. Once again, we apply Lemma 7 into the above inequation. And we have Combining Lemma 1, we take the fractional integral with fractional-order α from t 0 to t on sides of in equation (3); we have

Complexity
Substituting Apparently, the nonnegative function f(t) is monotonically decreasing. Hence, we can conclude that f(t) converges to 0 within the settling time t 2 * . By simple computation, the estimation of settling time t 2 * is given by From the above analysis, the estimation of settling time is connected with a positive integer m ≥ (β/(1 − β)). en, we would like to discuss the effect of m for settling time. First, we define a function H(m) as follows: Based on the property of Euler's gamma function, we have where According to the above analysis, it can be obtained that H(m) is monotone increasing about m. erefore, it can be deduced that the settling time is monotone increasing about m. In order to obtain more accurate estimation of settling time, we should choose a smaller value of m.
From the above analysis, if we consider m � β/(1 − β), we can gain a smaller estimation of settling time as follows: For Property 2, considering b � 0, we can obtain Corollary 1 as follows.
is continuous and nonnegative definite. If the following inequality holds, where a > 0, 0 < α < 1, and 0 < β ≤ 1, then it can be deduced as follows: And we can judge f(t) converges to 0 within the settling time t 3 * which can be estimated as follows: Remark 2. Peng et al. and Wu et al. in [31,32] proposed the estimation of settling time for the differential inequation According to the results in [31,32], we can get another estimation of settling time: where 0 < β < α(1 + α) − 1. Apparently, the results in [31,32] need a limiting condition on α and β. Compared with the above results, the estimation of settling time in Property 2 and Corollary 1 has no limits about α and β when we select a suitable value of m. Hence, the result we obtained is more general and less conservative.
is continuous and nonnegative definite. If the following inequality holds, where a > 0, b > 0, 0 < α < 1, and 0 < β ≤ 1, then it can be deduced as follows: And we can judge f(t) converges to 0 within the settling time T 2 * which can be estimated as follows: where m is a positive integer. en, inequation (51) can be written as follows: When m ≥ β, via applying Lemma 7, we can gain (55) en, (56) Once again, we apply Lemma 7 into the above inequation. And we have where Combining Property 1, inequation (57) can be written as follows: Substituting w(t) � f β/m (t) and b/a � b/a into the above inequation, we can obtain (60) Because f(t) is a nonnegative function, we can gain f(t) � 0 if f 1− β (t) � 0. erefore, based on the analysis in Property 1, we can conclude that f(t) converges to 0 within the settling time t 2 * . By simple computation, the estimation of settling time t 2 * is given by (61) erefore, it can be deduced that the settling time is monotone increasing about m. In order to obtain more accurate estimation of settling time, we should choose a smaller value of β/(1 − β).
And we can obtain the settling time t 2 * as follows: A practical integer-order memristive neural network can usually be implemented by a very largescale integration (VISI) circuit as shown in Figure 1.
According to Kirchhoff's current law, the ith subsystem with mixed delays can be described as follows: In this paper, the above integer-order FNN model is expended to fractional-order FNNs to explore the finite-time projective synchronization. And the drive system can be described by the following differential equation: where C t 0 D α t denotes the Caputo fractional derivative of order 0 < α < 1 from t 0 to t; d i ≥ 0 is the self-feedback connection weight of the ith neuron; x i (t) represents the corresponding state variable of the number of ith neurons at time t; f j (·), g j (·), and h j (·) denote the activation function between the jth-dimension of the memristor and the state variable; τ j (t) and ρ j (t), respectively, describe the discrete and distributed time-varying delays which satisfied 0 ≤ τ j (t) ≤ τ max j and 0 ≤ ρ j (t) ≤ ρ max j ; and I i represents the external input or bias, and it is a constant. Furthermore, a ij (x i (t)), b ij (x i (t)), and c ij (x i (t)) are memristive synaptic connection weights which can be given by ; C i represents the capacitors; and W aij , W bij , and W cij , respectively; denote the memductance of memristors by R aij , R bij , and R cij . And R aij describes the memristor between activation function f j (x j (t)) and x i (t); R bij denotes the memristor between activation function g j (x j (t − τ j )) and x i (t); and R cij is the memristor between activation function t t− ρ j (t) h j (x j (s))ds and x i (t). According to the characteristics of memristors and the current-voltage, we use a simplified memristor model as follows: where the switching jumps Ω i > 0 and a ij * , a * * ij , b ij * , b * * ij , c ij * , and c * * ij are all constants. And drive system (66) is supplemented with initial conditions as follows: In order to express the proof process clearly, we set some mathematical simplified forms. Let a ij ″ � max a * ij , a * * ij , ij |}, and c + ij � max |c * ij |, |c * * ij | . From the above description, we note that system (66) can be regarded as a class of the complex switching system with discontinuous right-hand side. In order to discuss the solution of the system, we have to consider them in Filippov's sense. With the theories of the set-valued map and differential inclusions [48,49], drive system (66) could be converted to the following differential inclusion: According to the set-valued map and the measurable selection theorem in [50], there exist measurable functions In order to investigate the synchronization of FMNNs, system (66) is regarded as the drive system. Similarly, the response system is described as follows: where y i (t) represents the state of the response system and u i (t) is the controller to be designed. And the initial conditions of response system (72) are given as follows: By using the set-valued map and the measurable selection theorem, there exist measurable functions h j y j (s) ds + I i + u i (t), i � 1, 2, . . . , n, t ≥ t 0 .
(74) Remark 3. e drive system and the response system are a distinct network, so Δ ij (x i (t)) probably is not equal to Δ ij (y i (t)). us, we need to select six different measurable functions δ (1) ij , Let e i (t) � y i (t) − λx i (t) denote the synchronization error. And λ represents the projective coefficient. Hence, the fractional order of the synchronization error system can be given as follows: e initial conditions associated with system (75) are written as follows: To obtain the main results expediently, we set up the following assumptions: e activation functions f j (·), g j (·), and h j (·) are Lipschitz continuous. at is, for any j ∈ N, there exist three constants L (1) j , L (2) j , and L (3) j such that where x, y ∈ R, x ≠ y.

Assumption 2.
e activation functions f j (·), g j (·), and h j (·) are bounded. at is, for any j ∈ N, there exist three constants M (1) j , M (2) j , and M (3) j such that where x ∈ R.
It is obvious that error system (75) can be expressed as follows: According to Assumption 1, we can judge According to Assumption 2, we have 12 Complexity

Main Results
In this section, a feedback controller is designed for actualizing finite-time projective synchronization of fractionalorder memristor-based neural networks with mixed timevarying delays. And we choose the following feedback control strategy: where k i , h i , and q i denote the controller parameters.

e Finite-Time Projective Synchronization of FMNNs with a Lyapunov Function Based on 1-Norm
where Moreover, the settling time T 1 is estimated by Proof. Firstly, we consider the following Lyapunov function: Based on Lemma 8, the above fractional-order derivative of V(t) can be written as follows: (88) From error system (82), we have (89) Substituting the controller (83) into inequation (89), (91) erefore, in equation (90) can be rewritten as follows: where φ � min 1≤i≤n φ i , δ � min 1≤i≤n δ i , and η � n i�1 η i .
Based on the condition δ > 0 in eorem 1, we have According to the conditions of eorem 1, we can gain φ > 0, η > 0. Furthermore, by virtue of Property 1, it follows that And we can judge that the Lyapunov function V 1 (t) converges to 0, which represents drive system (66) and response system (72) actualizing finite-time projective synchronization under feedback controller (83). And the settling time T 1 which can be estimated is as follows: is completes the proof.

Remark 4. For the differential inequation
14 Complexity [29] ignored the effort of parameter φ when they calculated the settling time.
Taking the fractional integral with variable order α on both sides of the above inequation, we can gain rough Definition 1, we can calculate that V(t) converges to 0 within the settling time T 2 , and T 2 is evaluated by Apparently, the authors ignored the impact of parameter η, and let η � 0 which will increase the conservation of the settling time. Considering the parameter η > 0; the estimation of settling time T * 1 in eorem 1 is less conservative than the results in [29]. To verify that the settling time T * 1 is more accurate than T 2 , a comparison table has been given in numerical simulation.
when fractional order α � 1, the settling time t 1 is estimated by Obviously, the estimate time t 1 is consistent with the result of the integer-order system given in [51].
us, it shows the estimation of settling time T * 1 in eorem 1 is more accurate and general.

Theorem 2. Under the state feedback controller (83) and
Assumptions 1 and 2, drive system (66) and response system (72) achieve the finite-time projective synchronization if where ε 1 and ε 2 are arbitrary positive constants. Moreover, the settling time T 2 is estimated by Proof. Firstly, we consider the following Lyapunov function: where p is a positive integer with p > 1.
Calculating the derivate of V 2 (t) and using Lemma 7 and error system (82), one has According to Lemma 8, the above fractional-order derivative of V(t) can be written as follows: Substituting the feedback controller (83) into inequation (104), we can get By applying Lemma 3, we have where ε 1 and ε 2 are arbitrary positive constants. Combining with inequations (105) and (106), one has 16 Complexity where Denote φ � min 1≤i≤n φ i , δ � min 1≤i≤n δ i , and η � min 1≤i≤n η i ; one gets Applying Lemma 4, we have Based on the conditions of eorem 2, we know φ > 0, δ > 0, and η > 0. us,

Numerical Simulation
In this section, several numerical examples are given to illustrate the effectiveness of the obtained results. We consider the following two-dimensional fractional-order memristive neural networks with mixed time-varying delays: where i � 1, 2. And the values of the memristors are given as follows: 18 Complexity For drive-response system (118), we select the system parameters as follows: I 1 � I 2 � 0.1, d 1 � d 2 � 0.1, τ 1 (t) � e t /(e t + 1), τ 2 (t) � 0.8 * e t /(e t + 1), ρ 1 (t) � 0.5 cos(t)+ 0.5, and ρ 2 (t) � 0.4 sin(t) + 0.4. Hence, we can obtain that τ max e activations are adopted as follows: It is easy to check that Assumption 1 and 2 are satisfied. By simple calculation,
en, we have a finite-time fractional-order differential inequation as follows: Based on the result in eorem 1, drive-response system (118) could achieve synchronization in finite time. And we can calculate the setting time T * 1 as follows: To verify Remark 4, we can calculate T 2 to demonstrate that T * 1 is less conservative than T 2 . It follows that On the contrary, we select p � 2 and ε 1 � ε 2 � 1 for eorem 2. And it is easy to calculate Under the controller parameter, k 1 � 3.1, k 2 � 3.1, h 1 � 1, h 2 � 0.9, q 1 � 9.9, and q 2 � 11.38 which lead to φ � 2.451 and η � 1.9608.
Hence, we have another finite-time fractional-order differential inequation as follows: Based on the result in eorem 2, we can estimate the setting time T 3 as follows:
en, we have a finite-time fractional-order differential inequation as follows: Based on the result in eorem 1, drive-response system (118) could achieve synchronization in finite time. And we can calculate the setting time T * 1 as follows: To verify Remark 4, we can calculate T 2 to demonstrate that T * 1 is less conservative than T 2 . It follows that On the contrary, we select p � 2 and ε 1 � ε 2 � 1 for eorem 2. And it is easy to calculate Under the controller parameter, k 1 � 3.1, k 2 � 3.1, h 1 � 1, h 2 � 0.9, q 1 � 7.36, and q 2 � 9.12 which lead to φ � 2.451 and η � 1.9608.
Hence, we have another finite-time fractional-order differential inequation as follows: Based on the result in eorem 2, we can estimate the setting time T 3 as follows: 20 Complexity    To verify Remark 6, we can calculate T 4 to demonstrate that T * 3 is less conservative than T 4 . It follows that Considering p � 2 and α � 0.98, (p − 1)/p < α(1 + α) − 1 holds. us, we have
Hence, we have another finite-time fractional-order differential inequation as follows: Based on the result in eorem 2, we can estimate the setting time T 3 as follows: 22 Complexity

Conclusion
In this paper, we have extensively discussed the finite-time projective synchronization for a class of FMNNs which are translated to a type of fractional-order systems with time delays and uncertain parameters. By means of applying the differential inequality of the Caputo derivative and the asymptotic expansion property of Mittag-Leffler function, three useful properties are proposed which play a vital role in calculating the settling time of finite-time synchronization. Besides the time-delayed feedback control utilized in this paper, other control approaches, such as impulsive control, intermittent control, and pinning control, can also be applied in the future work for studying projective synchronization of memristive neural networks with mixed timevarying delays.

Data Availability
All data included in this study are available upon request by contact with the corresponding author.

Conflicts of Interest
e authors declare that they have no conflicts of interest.