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

Authors: Meng Hui; Chen Wei; Jiao Zhang; Herbert Ho-Ching Iu; Ni Luo; Rui Yao; Lin Bai
Journal: Complexity
 (2020)

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## Abstract

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.


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## Body

## 1. 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 state-dependent 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 [9–11].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 [13–15].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 infinite-time 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 [26–33]. 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 [26–28] pay attention to discussing the synchronization conditions and ignore the computation of settling time. In [29], Zhang et al. deduced the fractional-order derivative of Lyapunov function Dt0CtαVt≤−aVt−b and gave a conservative estimation of settling time with a=0. In [30–33], the authors deduced the fractional-order derivative of Lyapunov function Dt0CtαVt≤−aVt−bVβt 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 [36–40]. 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. The main innovations and contents of this paper lie in the following aspects:(1)
For fractional-order neural networks, the fractional-order derivative of Lyapunov functionVt can usually be reduced to Dt0CtαVt≤−aVt−b or Dt0CtαVt≤−aVt−bVβt. 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 value-based 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.Table 1
The comparisons amongT1∗,T2,T3∗,T4, and T5 with the projective coefficient λ=2.
Lyapunov functionFractional-order differential inequationSettling timeTheorem1V1t=∑i=1neitDt0CtαV1t≤−φV1t−ηT1∗=1.235Remark4Dt0CtαV1t≤−ηT2=5.5778Theorem2V2t=∑i=1neitpDt0CtαV2t≤−φ¯V2t−η¯V2p−1/ptT3∗=1.9054Remark6Dt0CtαV2t≤−η¯V2p−1/ptT4=8.0421T5=6.689Table 2
The comparisons amongT1∗,T2,T3∗,T4, and T5 with the projective coefficient λ=1.
Lyapunov functionFractional-order differential inequationSettling timeTheorem1V1t=∑i=1neitDt0CtαV1t≤−φV1t−ηT1∗=1.0351Remark4Dt0CtαV1t≤−ηT2=3.5169Theorem2V2t=∑i=1neitpDt0CtαV2t≤−φ¯V2t−η¯V2p−1/ptT3∗=1.5937Remark6Dt0CtαV2t≤−η¯V2p−1/ptT4=5.1232T5=4.4965Table 3
The comparisons amongT1∗,T2,T3∗,T4, and T5 with the projective coefficient λ=−1.
Lyapunov functionFractional-order differential inequationSettling timeTheorem1V1t=∑i=1neitDt0CtαV1t≤−φV1t−ηT1∗=0.6934Remark4Dt0CtαV1t≤−ηT2=1.4814Theorem2V2t=∑i=1neitpDt0CtαV2t≤−φ¯V2t−η¯V2p−1/ptT3∗=1.3539Remark6Dt0CtαV2t≤−η¯V2p−1/ptT4=2.3539T5=2.1816The rest of this paper is organized as follows. Section2 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. Cnt0,∞,ℝ represents the family of continuous and n-order differentiable function form t0,∞ to ℝ. And sign⋅ denotes the sign function, where signx=1,ifx≥0−1,ifx<0. The symbol ∗ involves the convolution operator. And Γα is Euler’s gamma function which is defined by Γα=∫0+∞sα−1e−sds.


## 2. Preliminaries and Model Description

### 2.1. Preliminaries

In this section, we recall some basic definitions with respect to fractional calculus and several important lemmas used in this paper.Definition 1.
(see [41]). The Riemann–Liouville fractional integral with fractional-order α for an integrable function ft is defined as(1)Itαt0ft=1Γα∫t0tt−sα−1fsds,where t≥t0 and α>0.Definition 2.
(see [42]). The Caputo fractional derivative with fractional-order α for a differentiable function ft∈Cnt0,+∞,ℝ is defined as(2)Dtαt0Cft=1Γn−α∫t0tfnst−sα−n+1ds,where t≥t0 and n is a positive integral such that n−1<α<n. Especially, when α∈0,1, then(3)Dtαt0Cft=1Γ1−α∫t0tf′st−sαds.Definition 3.
(see [41]). Mittag-−Leffler function Ep,q⋅ with two parameters is defined as(4)Ep,qς=∑k=0∞ςkΓpk+q,where p>0, q>0, and ς∈ℂ. And if q=1, the Mittag-Leffler function with one parameter is shown as(5)Epς=Ep,1ς=∑k=0∞ςkΓpk+1.Definition 4.
If there exists a nonzero constantλ∈ℝ, for any solutions of the n-dimensional drive system and the response system, xit and yit,i=1,2,…,n, with initial values xit0 and yit0 are such that(6)limt⟶T∗yit−λxit=0,yit−λxit≡0,t≥T∗,where λ represents the projective coefficient. T∗ is called the settling time for synchronization. Then, the response system is said to be finite-time projective synchronization to the drive system.
Particularly, the drive-response system can be called finite-time complete synchronization ifλ=1 and finite-time antisynchronization if λ=−1.Lemma 1.
(see [41]). If ft:t0,+∞⟶ℝ, then for all t∈t0,+∞,(7)It0tαDt0Ctαft=ft−∑i=1n−1fit0i!t−t0i.
In particular, when0<α<1, it has(8)It0tαDt0Ctαft=ft−ft0.Lemma 2.
(see [43]). For all 0<α<2, β∈ℂ, m∈ℕ, and z≠0, argz<minπ,απ, then the following asymptotic formula holds:(9)Eα,βz=1αz1−βαez1α−∑k=1mz−kΓβ−kα+O1zm+1,z⟶∞.Lemma 3.
(see [44]). Let z1>0,z2>0,z3>1,z4>1, and z3−1+z4−1=1; then, the following inequality holds:(10)z1z2≤εz1z3z3+ε−z3/z4z2z4z4,where ε is an arbitrary positive constant.Lemma 4.
(see [45]). Assume that a1,a2,…,an are nonnegative constants. For two arbitrary constants p and q satisfying 0<p<q, then(11)∑i=1naiq1/q≤∑i=1naip1/p.Lemma 5.
(see [46]). Let ft be a continuous function on the positive interval t0,+∞ and satisfy(12)Dt0Ctαft≤aft,where a is a constant and 0<α<1; then,(13)ft≤ft0Eαat−t0α.Lemma 6.
Letft be a continuous function on the positive interval t0,+∞ and satisfy(14)Dt0Ctαft≤aft+b,where aandb are constants and 0<α<1; then,(15)ft≤ft0+baEαat−t0α−ba.Proof.
Letht=ft+b/a, and we can easily obtain(16)Dt0Ctαht=Dt0Ctαt≤aft+b=aht.
By applying Lemma5, we have ht≤ht0Eαat−t0α, which is equivalent to(17)ft≤ft0+baEαat−t0α−ba.Remark 1.
For Lemma6, 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].Lemma 7.
(see [30]). Suppose 0<α≤1, β≥1, and ft∈C1t0,+∞,ℝ; then,(18)Dt0Ctαfβt=Γ2−αΓ1+βΓ1+β−αfβ−1tDt0Ctαft.Lemma 8.
(see [47]). Let ft∈C1t0,+∞,ℝ be a continuous and differential function; the following inequation satisfies almost everywhere(19)Dt0Ctαft≤signftDt0Ctαft,where 0<α<1.
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.
Letft∈C1t0,+∞,ℝ∗ be continuous and nonnegative definite. And the following inequation holds:(20)Dt0Ctαft≤−aft−b,where a>0, b>0, and 0<α<1. It can be deduced as follows:(21)ft≤ft0+ba1αexp−a1/αt−t0−ba,t≥t0.
Then, we can judgeft converges to 0 within the settling time T1∗ which can be estimated as follows:(22)t1∗≤t0+a−1/αlnaft0+bbα.Proof.
By utilizing Lemma6, we have(23)ft≤ft0+baEα−at−t0α−ba.
From Lemma2, when β=1, we can get(24)Eαz=1αez1α−∑k=1mz−kΓ1−kα+O1zm+1,z⟶∞.
Based onlimz⟶∞O1/zm+1=0, we can easily gain(25)limz⟶∞Eαz⟶limz⟶∞1αez1α−∑k=1mz−kΓ1−kα.
Apparently,∑k=1mz−k/Γ1−kα>0; then, we can judge that there exists z0>0 such that(26)Eαz≤1αez1α,z>z0.
Similarly, we have(27)Eα−at−t0α≤1αexp−a1/αt−t0,t≥t0.
From the above conclusion, inequation (20) can be converted to(28)ft≤ft0+ba1αexp−a1/αt−t0−ba,t≥t0.
In order to better analyze the problem, letΩt=ft0+b/a1/αe−a1/αt−t0−b/a. We can easily know that the function Ωt is monotonically decreasing. And Ωt0=1/αft0+1/α−1b/a>1/αft0>0 and limt⟶∞Ωt⟶−b/a<0. According to the zero-point theorem, there exists t1∗∈t0,∞ such that Ωt1∗=0. Then, we can gain 0≤ft1∗≤Ωt1∗=0, so we have ft1∗=0. And when t≥t1∗, then ft≡0.
Based on Definition4, the function ft converges to 0 in finite time. And the settling time t1∗ can be calculated as follows:(29)t1∗≤t0+a−1/αlnaft0+bbα.Property 2.
Assume that functionft∈Ct0,+∞,ℝ∗ is continuous and nonnegative definite. If the following inequality holds,(30)Dt0Ctαft≤−afβt−b,where a>0, b>0, 0<α<1,0<β≤1, then it can be deduced as follows:(31)ft≤ft0+ba1/β1−β−aΓ1+1/1−β−αt−t0αΓ1+1/1−βΓ2−αΓ1+α1/1−β−ba1/β.
And we can judgeft converges to 0 within the settling time T2∗ which can be estimated as follows:(32)t2∗≤t0+f1−βt0Γ1+1/1−βΓ2−αΓ1+αaΓ1+1/1−β−α1/α.Proof.
Letgt=ft+b/a1/β>0; according to Lemma 4, we gain(33)gβt=ft+ba1/ββ≤fβt+ba.
Multiplying−a on sides of the above inequation, then we can get(34)−afβt−b≤−agβt.
According to the conditions of Property2,(35)Dt0Ctαgt=Dt0Ctαft≤−afβt−b≤−agβt.
Letwt=gβ/mt, where m is a positive integer. Then, the above inequation can be written as follows:(36)Dt0Ctαwm/βt≤−awmt.
Whenm≥β, via applying Lemma 7, we can gain(37)Γ2−αΓ1+m/βΓ1+m/β−αwm/β−1tDtαwt≤−awmt.
Then,(38)wm/β−m−1tDtαwt≤−aΓ1+m/β−αΓ2−αΓ1+m/β.
Consideringm≥β/1−β, thus m/β−m−1≥0. Once again, we apply Lemma 7 into the above inequation. And we have(39)Dtαwm/β−mt≤−aΓ1+m/β−αΓ1+m/β−mΓ1+m/βΓ1+m/β−m−α.
Combining Lemma1, we take the fractional integral with fractional-order α from t0 to t on sides of in equation (3); we have(40)wm/β−mt−wm/β−mt0≤−aΓ1+m/β−αΓ1+m/β−mt−t0αΓ1+m/βΓ1+m/β−m−αΓ1+α.
Substitutingwt=ft+b/a1/ββ/m into the above inequation, we can obtain(41)ft≤ft0+ba1/β1−β−aΓ1+m/β−αΓ1+m/β−mt−t0αΓ1+m/βΓ1+m/β−m−αΓ1+α1/1−β−ba1/β.
Apparently, the nonnegative functionft is monotonically decreasing. Hence, we can conclude that ft converges to 0 within the settling time t2∗. By simple computation, the estimation of settling time t2∗ is given by(42)t2∗≤t0+ft0+ba1/β1−β−ba1−β/βΓ1+m/βΓ1+m/β−m−αΓ1+αaΓ1+m/β−αΓ1+m/β−m1/α.
From the above analysis, the estimation of settling time is connected with a positive integerm≥β/1−β. Then, we would like to discuss the effect of m for settling time. First, we define a function Hm as follows:(43)Hm=Γ1+m/βΓ1+m/β−m⋅Γ1+m/β−m−αΓ1+m/β−α.
Based on the property of Euler’s gamma function, we have(44)Hm=1+m/β−11+m/β−2⋯1+m/β−m1+m/β−α−11+m/β−α−2⋯1+m/β−α−m=∏i=1mAi,where Ai=1+m/β−i/1+m/β−α−i,i=1,2,…,m. Obviously, Ai>1. On the contrary,(45)dAidi=α1+m/βα−i2>0.
According to the above analysis, it can be obtained thatHm is monotone increasing about m. Therefore, 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 considerm=β/1−β, we can gain a smaller estimation of settling time as follows:(46)t2∗≤t0+ft0+ba1/β1−β−ba1−β/βΓ1+1/1−βΓ1+1/1−β−β/1−β−αΓ1+αaΓ1+1/1−β−αΓ1+1/1−β−β/1−β1/α=t0+ft0+ba1/β1−β−ba1−β/βΓ1+1/1−βΓ2−αΓ1+αaΓ1+1/1−β−α1/α.
For Property2, considering b=0, we can obtain Corollary 1 as follows.Corollary 1.
Assume that functionft∈Ct0,+∞,ℝ∗ is continuous and nonnegative definite. If the following inequality holds,(47)Dt0Ctαft≤−afβt,where a>0, 0<α<1, and 0<β≤1, then it can be deduced as follows:(48)ft≤f1−βt0−aΓ1+1/1−β−αt−t0αΓ1+1/1−βΓ2−αΓ1+α1/1−β.
And we can judgeft converges to 0 within the settling time t3∗ which can be estimated as follows:(49)t3∗≤t0+f1−βt0Γ1+1/1−βΓ2−αΓ1+αaΓ1+1/1−β−α1/α.Remark 2.
Peng et al. and Wu et al. in [31, 32] proposed the estimation of settling time for the differential inequation Dt0Ctαft≤−afβt. According to the results in [31, 32], we can get another estimation of settling time:(50)t4∗≤t0+fα1+α−β−1t0Γ1+αΓ1−1+β/1+αaΓ2+αΓ1+α−1+β/1+α1/α,where 0<β<α1+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.Property 3.
Assume that functionft∈Ct0,+∞,ℝ∗ is continuous and nonnegative definite. If the following inequality holds,(51)Dt0Ctαft≤−aft−bfβt,where a>0, b>0, 0<α<1,and0<β≤1, then it can be deduced as follows:(52)f1−βt≤f1−βt0+ba1αexp−aΓ1+1/β−αΓ1+1/βΓ2−α1/αt−t0−ba,t≥t0.
And we can judgeft converges to 0 within the settling time T2∗ which can be estimated as follows:(53)t5∗≤t0+aΓ1+1/β−αΓ1+1/βΓ2−α−1/αlnaf1−βt0+bbα.Proof.
Letwt=fβ/mt, where m is a positive integer. Then, inequation (51) can be written as follows:(54)Dt0Ctαwm/βt≤−awm/βt−bwmt.
Whenm≥β, via applying Lemma 7, we can gain(55)Γ2−αΓ1+m/βΓ1+m/β−αwm/β−1tDtαwt≤−awm/βt−bwmt.
Then,(56)Γ2−αΓ1+m/βΓ1+m/β−αwm/β−m−1tDtαwt≤−awm/β−mt−b.
Consideringm≥β/1−β, thus m/β−m−1≥0. Once again, we apply Lemma 7 into the above inequation. And we have(57)Dtαwm/β−mt≤−a^wm/β−mt−b^,where(58)a^=Γ1+m/β−αΓ1+m/β−mΓ1+m/βΓ1+m/β−m−αa,b^=Γ1+m/β−αΓ1+m/β−mΓ1+m/βΓ1+m/β−m−αb.
Combining Property1, inequation (57) can be written as follows:(59)wm/β−mt≤wm/β−mt0+b^a^1αexp−a^1/αt−t0−b^a^,t≥t0.
Substitutingwt=fβ/mt and b^/a^=b/a into the above inequation, we can obtain(60)f1−βt≤f1−βt0+ba1αexp−a^1/αt−t0−ba,t≥t0.
Becauseft is a nonnegative function, we can gain ft=0 if f1−βt=0. Therefore, based on the analysis in Property 1, we can conclude that ft converges to 0 within the settling time t2∗. By simple computation, the estimation of settling time t2∗ is given by(61)t5∗≤t0+Γ1+m/β−αΓ1+m/β−m1+m/βΓ1+m/β−m−αa−1/αlnaft0+bbα.
Similar to Property1, Hm=Γ1+m/β/Γ1+m/β−m⋅Γ1+m/β−m−α/Γ1+m/β−α is monotone increasing about m.(62)t5∗≤t0+aHm−1/αlnaft0+bbα.
Therefore, it can be deduced that the settling time is monotone increasing aboutm. In order to obtain more accurate estimation of settling time, we should choose a smaller value of β/1−β.(63)f1−βt≤f1−βt0+ba1αexp−aΓ1+1/β−αΓ1+1/βΓ2−α1/αt−t0−ba,t≥t0.
And we can obtain the settling timet2∗ as follows:(64)t5∗≤t0+aΓ1+1/β−αΓ1+1/βΓ2−α−1/αlnaf1−βt0+bbα.


### 2.2. Model Description

A practical integer-order memristive neural network can usually be implemented by a very large-scale integration (VISI) circuit as shown in Figure1. According to Kirchhoff’s current law, the ith subsystem with mixed delays can be described as follows:(65)dxitdt=−dixit+∑j=1naijxitfjxjt+∑j=1nbijxitgjxjt−τjt+∑j=1ncijxit∫t−ρjtthjxjsds+Ii,i=1,2,…,n,t≥t0.Figure 1
The circuit of the memristive neural network.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:(66)Dt0Ctαxit=−dixit+∑j=1naijxitfjxjt+∑j=1nbijxitgjxjt−τjt+∑j=1ncijxit∫t−ρjtthjxjsds+Ii,i=1,2,…,n,t≥t0.where Dt0Ctα denotes the Caputo fractional derivative of order 0<α<1 from t0 to t; di≥0 is the self-feedback connection weight of the ith neuron; xit represents the corresponding state variable of the number of ith neurons at time t; fj⋅, gj⋅, and hj⋅ denote the activation function between the jth-dimension of the memristor and the state variable; τjt and ρjt, respectively, describe the discrete and distributed time-varying delays which satisfied 0≤τjt≤τjmax and 0≤ρjt≤ρjmax; and Ii represents the external input or bias, and it is a constant. Furthermore, aijxit, bijxit, and cijxit are memristive synaptic connection weights which can be given by(67)aijxit=WaijCi×signij,bijxit=WbijCi×signij,cijxit=WcijCi×signij,where signij=1,i≠j−1,i=j; Ci represents the capacitors; and Waij, Wbij, and Wcij, respectively; denote the memductance of memristors by Raij, Rbij, and Rcij. And Raij describes the memristor between activation function fjxjt and xit; Rbij denotes the memristor between activation function gjxjt−τj and xit; and Rcij is the memristor between activation function ∫t−ρjtthjxjsds and xit. According to the characteristics of memristors and the current-voltage, we use a simplified memristor model as follows:(68)aijxit=aij∗,xit≤Ωi,aij∗∗,xit>Ωi,bijxit=bij∗,xit≤Ωi,bij∗∗,xit>Ωi,cijxit=cij∗,xit≤Ωi,cij∗∗,xit>Ωi,where the switching jumps Ωi>0 and aij∗ , aij∗∗, bij∗ , bij∗∗, cij∗ , and cij∗∗ are all constants. And drive system (66) is supplemented with initial conditions as follows:(69)xis=Ξis,s∈t0−τ,t0,τ=max1≤j≤nτjt,ρjt.In order to express the proof process clearly, we set some mathematical simplified forms. Letaij″=maxaij∗,aij∗∗, aij′=minaij∗,aij∗∗, bij″=maxbij∗,bij∗∗, bij′=minbij∗,bij∗∗, cij″=maxcij∗,cij∗∗, and cij′=mincij∗,cij∗∗. And a¯ij=1/2aij″+aij′, a˜ij=1/2aij″+aij′, b¯ij=1/2bij″+bij′, b˜ij=1/2bij″+bij′, c¯ij=1/2cij″+cij′, and c˜ij=1/2cij″+cij′. Also, aij+=maxaij∗,aij∗∗, bij+=maxbij∗,bij∗∗, and cij+=maxcij∗,cij∗∗.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:(70)Dt0Ctαxit∈−dixit+∑j=1na¯ij+co−a˜ij,a˜ijfjxjt+∑j=1nb¯ij+co−b˜ij,b˜ijgjxjt−τjt+∑j=1nc¯ij+co−c˜ij,c˜ij∫t−ρjtthjxjsds+Ii,i=1,2,…,n,t≥t0.According to the set-valued map and the measurable selection theorem in [50], there exist measurable functions δij1,δij2,δij3∈co−1,1 such that(71)t0CDtαxit=−dixit+∑j=1na¯ij+a˜ijδij1fjxjt+∑j=1nb¯ij+b˜ijδij2gjxjt−τjt+∑j=1nc¯ij+c˜ijδij3∫t−ρjtthjxjsds+Ii,i=1,2,…,n,t≥t0.In order to investigate the synchronization of FMNNs, system (66) is regarded as the drive system. Similarly, the response system is described as follows:(72)t0CDtαyit=−diyit+∑j=1naijyitfjyjt+∑j=1nbijyitgjyjt−τjt+∑j=1ncijyit∫t−ρjtthjyjsds+Ii+uit,i=1,2,…,n,t≥t0.where yit represents the state of the response system and uit is the controller to be designed. And the initial conditions of response system (72) are given as follows:(73)yis=Ψis,s∈t0−τ,t0,τ=max1≤j≤nτjt,ρjt.By using the set-valued map and the measurable selection theorem, there exist measurable functionsγij1,γij2,γij3∈co−1,1 such that(74)Dt0Ctαyit=−diyit+∑j=1na¯ij+a˜ijγij1fjyjt+∑j=1nb¯ij+b˜ijγij2gjyjt−τjt+∑j=1nc¯ij+c˜ijγij3∫t−ρjtthjyjsds+Ii+uit,i=1,2,…,n,t≥t0.Remark 3.
The drive system and the response system are a distinct network, soΔijxit probably is not equal to Δijyit. Thus, we need to select six different measurable functions δij1,δij2,δij3,γij1,γij2,γij3.
Leteit=yit−λxit denote the synchronization error. And λ represents the projective coefficient. Hence, the fractional order of the synchronization error system can be given as follows:(75)Dt0Ctαeit=−dieit+1−λIi+∑j=1na¯ij+a˜ijγij1fjyjt−λ∑j=1na¯ij+a˜ijδij1fjxjt+∑j=1nb¯ij+b˜ijγij2gjyjt−τjt−λ∑j=1nb¯ij+b˜ijδij2gjxjt−τjt+∑j=1nc¯ij+c˜ijγij3∫t−ρjtthjyjsds−λ∑j=1nc¯ij+c˜ijδij3∫t−ρjtthjxjsds+uit.
The initial conditions associated with system (75) are written as follows:(76)eis=Φis=Ξit−λΨis,s∈t0−τ,t0,τ=max1≤j≤nτjt,ρjt.
To obtain the main results expediently, we set up the following assumptions:Assumption 1.
The activation functionsfj⋅,gj⋅, and hj⋅ are Lipschitz continuous. That is, for any j∈ℕ, there exist three constants Lj1, Lj2, and Lj3 such that(77)fjy−fjx≤Lj1y−x,gjy−fjx≤Lj2y−x,hjy−hjx≤Lj3y−x,where x,y∈ℝ,x≠y.Assumption 2.
The activation functionsfj⋅,gj⋅, and hj⋅ are bounded. That is, for any j∈ℕ, there exist three constants Mj1, Mj2, and Mj3 such that(78)fjx≤Mj1,gjx≤Mj2,hjx≤Mj3,where x∈ℝ.
It is obvious that error system (75) can be expressed as follows:(79)Dt0Ctαeit=−dieit+1−λIi++uit+∑j=1na¯ij+a˜ijγij1fjyjt−fjλxjt+∑j=1na¯ij+a˜ijγij1fjλxjt−fjxjt+∑j=1n1−λa¯ij+γij1−λδij1a˜ijfjxjt+∑j=1nb¯ij+b˜ijγij2gjyjt−τjt−gjλxjt−τjt+∑j=1nb¯ij+b˜ijγij2gjλxjt−τjt−gjxjt−τjt+∑j=1n1−λb¯ij+γij2−λδij2b˜ijgjxjt−τjt+∑j=1nc¯ij+c˜ijγij3∫t−ρjtthjyjsds−λ∑j=1nc¯ij+c˜ijδij3∫t−ρjtthjxjsds≤−dieit+1−λIi+uit+∑j=1naij+|fjyjt−fjλxjt|+∑j=1naij+|fjλxjt−fjxjt|+∑j=1n1−λa¯ij+γij1−λδij1a˜ij|fjxjt|+∑j=1nbij+|gjyjt−τjt−gjλxjt−τjt+∑j=1nbij+|gjλxjt−τjt−gjxjt−τjt+∑j=1n1−λb¯ij+γij2−λδij2b˜ijgjxjt−τjt+∑j=1ncij+∫t−ρjtthjyjs|ds+∑j=1ncij+λ∫t−ρjtthjxjs|ds.
According to Assumption1, we can judge(80)∑j=1naij+fjyjt−fjλxjt≤∑j=1naij+Lj1ejt,∑j=1nbij+gjyjt−τjt−gjλxjt−τjt≤∑j=1nbij+Lj2ejt−τjt.
According to Assumption2, we have(81)∑j=1naij+fjλxjt−fjxjt≤∑j=1n2aij+Mj1∑j=1n1−λa¯ij+γij1−λδij1a˜ijfjxjt≤∑j=1n1−λa¯ij+1+λa˜ijMj1∑j=1nbij+gjλxjt−τjt−gjxjt−τjt≤∑j=1n2bij+Mj2∑j=1n1−λb¯ij+γij2−δij2b˜ijgjxjt−τjt≤∑j=1n1−λb¯ij+1+λb˜ijMj2∑j=1ncij+∫t−ρjtthjyjsds+∑j=1ncij+λ∫t−ρjtthjxjsds≤∑j=1n1+λcij+ρjmaxMj3.
Substituting inequations (80) and (81) into equation (79), we can obtain(82)Dt0Ctαeit≤−dieit+1−λIi+uit+∑j=1nLj1aij+ejt+Lj2bij+ejt−τjt+2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3,i=1,2,…,n,t≥t0.


## 2.1. Preliminaries

In this section, we recall some basic definitions with respect to fractional calculus and several important lemmas used in this paper.Definition 1.
(see [41]). The Riemann–Liouville fractional integral with fractional-order α for an integrable function ft is defined as(1)Itαt0ft=1Γα∫t0tt−sα−1fsds,where t≥t0 and α>0.Definition 2.
(see [42]). The Caputo fractional derivative with fractional-order α for a differentiable function ft∈Cnt0,+∞,ℝ is defined as(2)Dtαt0Cft=1Γn−α∫t0tfnst−sα−n+1ds,where t≥t0 and n is a positive integral such that n−1<α<n. Especially, when α∈0,1, then(3)Dtαt0Cft=1Γ1−α∫t0tf′st−sαds.Definition 3.
(see [41]). Mittag-−Leffler function Ep,q⋅ with two parameters is defined as(4)Ep,qς=∑k=0∞ςkΓpk+q,where p>0, q>0, and ς∈ℂ. And if q=1, the Mittag-Leffler function with one parameter is shown as(5)Epς=Ep,1ς=∑k=0∞ςkΓpk+1.Definition 4.
If there exists a nonzero constantλ∈ℝ, for any solutions of the n-dimensional drive system and the response system, xit and yit,i=1,2,…,n, with initial values xit0 and yit0 are such that(6)limt⟶T∗yit−λxit=0,yit−λxit≡0,t≥T∗,where λ represents the projective coefficient. T∗ is called the settling time for synchronization. Then, the response system is said to be finite-time projective synchronization to the drive system.
Particularly, the drive-response system can be called finite-time complete synchronization ifλ=1 and finite-time antisynchronization if λ=−1.Lemma 1.
(see [41]). If ft:t0,+∞⟶ℝ, then for all t∈t0,+∞,(7)It0tαDt0Ctαft=ft−∑i=1n−1fit0i!t−t0i.
In particular, when0<α<1, it has(8)It0tαDt0Ctαft=ft−ft0.Lemma 2.
(see [43]). For all 0<α<2, β∈ℂ, m∈ℕ, and z≠0, argz<minπ,απ, then the following asymptotic formula holds:(9)Eα,βz=1αz1−βαez1α−∑k=1mz−kΓβ−kα+O1zm+1,z⟶∞.Lemma 3.
(see [44]). Let z1>0,z2>0,z3>1,z4>1, and z3−1+z4−1=1; then, the following inequality holds:(10)z1z2≤εz1z3z3+ε−z3/z4z2z4z4,where ε is an arbitrary positive constant.Lemma 4.
(see [45]). Assume that a1,a2,…,an are nonnegative constants. For two arbitrary constants p and q satisfying 0<p<q, then(11)∑i=1naiq1/q≤∑i=1naip1/p.Lemma 5.
(see [46]). Let ft be a continuous function on the positive interval t0,+∞ and satisfy(12)Dt0Ctαft≤aft,where a is a constant and 0<α<1; then,(13)ft≤ft0Eαat−t0α.Lemma 6.
Letft be a continuous function on the positive interval t0,+∞ and satisfy(14)Dt0Ctαft≤aft+b,where aandb are constants and 0<α<1; then,(15)ft≤ft0+baEαat−t0α−ba.Proof.
Letht=ft+b/a, and we can easily obtain(16)Dt0Ctαht=Dt0Ctαt≤aft+b=aht.
By applying Lemma5, we have ht≤ht0Eαat−t0α, which is equivalent to(17)ft≤ft0+baEαat−t0α−ba.Remark 1.
For Lemma6, 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].Lemma 7.
(see [30]). Suppose 0<α≤1, β≥1, and ft∈C1t0,+∞,ℝ; then,(18)Dt0Ctαfβt=Γ2−αΓ1+βΓ1+β−αfβ−1tDt0Ctαft.Lemma 8.
(see [47]). Let ft∈C1t0,+∞,ℝ be a continuous and differential function; the following inequation satisfies almost everywhere(19)Dt0Ctαft≤signftDt0Ctαft,where 0<α<1.
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.
Letft∈C1t0,+∞,ℝ∗ be continuous and nonnegative definite. And the following inequation holds:(20)Dt0Ctαft≤−aft−b,where a>0, b>0, and 0<α<1. It can be deduced as follows:(21)ft≤ft0+ba1αexp−a1/αt−t0−ba,t≥t0.
Then, we can judgeft converges to 0 within the settling time T1∗ which can be estimated as follows:(22)t1∗≤t0+a−1/αlnaft0+bbα.Proof.
By utilizing Lemma6, we have(23)ft≤ft0+baEα−at−t0α−ba.
From Lemma2, when β=1, we can get(24)Eαz=1αez1α−∑k=1mz−kΓ1−kα+O1zm+1,z⟶∞.
Based onlimz⟶∞O1/zm+1=0, we can easily gain(25)limz⟶∞Eαz⟶limz⟶∞1αez1α−∑k=1mz−kΓ1−kα.
Apparently,∑k=1mz−k/Γ1−kα>0; then, we can judge that there exists z0>0 such that(26)Eαz≤1αez1α,z>z0.
Similarly, we have(27)Eα−at−t0α≤1αexp−a1/αt−t0,t≥t0.
From the above conclusion, inequation (20) can be converted to(28)ft≤ft0+ba1αexp−a1/αt−t0−ba,t≥t0.
In order to better analyze the problem, letΩt=ft0+b/a1/αe−a1/αt−t0−b/a. We can easily know that the function Ωt is monotonically decreasing. And Ωt0=1/αft0+1/α−1b/a>1/αft0>0 and limt⟶∞Ωt⟶−b/a<0. According to the zero-point theorem, there exists t1∗∈t0,∞ such that Ωt1∗=0. Then, we can gain 0≤ft1∗≤Ωt1∗=0, so we have ft1∗=0. And when t≥t1∗, then ft≡0.
Based on Definition4, the function ft converges to 0 in finite time. And the settling time t1∗ can be calculated as follows:(29)t1∗≤t0+a−1/αlnaft0+bbα.Property 2.
Assume that functionft∈Ct0,+∞,ℝ∗ is continuous and nonnegative definite. If the following inequality holds,(30)Dt0Ctαft≤−afβt−b,where a>0, b>0, 0<α<1,0<β≤1, then it can be deduced as follows:(31)ft≤ft0+ba1/β1−β−aΓ1+1/1−β−αt−t0αΓ1+1/1−βΓ2−αΓ1+α1/1−β−ba1/β.
And we can judgeft converges to 0 within the settling time T2∗ which can be estimated as follows:(32)t2∗≤t0+f1−βt0Γ1+1/1−βΓ2−αΓ1+αaΓ1+1/1−β−α1/α.Proof.
Letgt=ft+b/a1/β>0; according to Lemma 4, we gain(33)gβt=ft+ba1/ββ≤fβt+ba.
Multiplying−a on sides of the above inequation, then we can get(34)−afβt−b≤−agβt.
According to the conditions of Property2,(35)Dt0Ctαgt=Dt0Ctαft≤−afβt−b≤−agβt.
Letwt=gβ/mt, where m is a positive integer. Then, the above inequation can be written as follows:(36)Dt0Ctαwm/βt≤−awmt.
Whenm≥β, via applying Lemma 7, we can gain(37)Γ2−αΓ1+m/βΓ1+m/β−αwm/β−1tDtαwt≤−awmt.
Then,(38)wm/β−m−1tDtαwt≤−aΓ1+m/β−αΓ2−αΓ1+m/β.
Consideringm≥β/1−β, thus m/β−m−1≥0. Once again, we apply Lemma 7 into the above inequation. And we have(39)Dtαwm/β−mt≤−aΓ1+m/β−αΓ1+m/β−mΓ1+m/βΓ1+m/β−m−α.
Combining Lemma1, we take the fractional integral with fractional-order α from t0 to t on sides of in equation (3); we have(40)wm/β−mt−wm/β−mt0≤−aΓ1+m/β−αΓ1+m/β−mt−t0αΓ1+m/βΓ1+m/β−m−αΓ1+α.
Substitutingwt=ft+b/a1/ββ/m into the above inequation, we can obtain(41)ft≤ft0+ba1/β1−β−aΓ1+m/β−αΓ1+m/β−mt−t0αΓ1+m/βΓ1+m/β−m−αΓ1+α1/1−β−ba1/β.
Apparently, the nonnegative functionft is monotonically decreasing. Hence, we can conclude that ft converges to 0 within the settling time t2∗. By simple computation, the estimation of settling time t2∗ is given by(42)t2∗≤t0+ft0+ba1/β1−β−ba1−β/βΓ1+m/βΓ1+m/β−m−αΓ1+αaΓ1+m/β−αΓ1+m/β−m1/α.
From the above analysis, the estimation of settling time is connected with a positive integerm≥β/1−β. Then, we would like to discuss the effect of m for settling time. First, we define a function Hm as follows:(43)Hm=Γ1+m/βΓ1+m/β−m⋅Γ1+m/β−m−αΓ1+m/β−α.
Based on the property of Euler’s gamma function, we have(44)Hm=1+m/β−11+m/β−2⋯1+m/β−m1+m/β−α−11+m/β−α−2⋯1+m/β−α−m=∏i=1mAi,where Ai=1+m/β−i/1+m/β−α−i,i=1,2,…,m. Obviously, Ai>1. On the contrary,(45)dAidi=α1+m/βα−i2>0.
According to the above analysis, it can be obtained thatHm is monotone increasing about m. Therefore, 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 considerm=β/1−β, we can gain a smaller estimation of settling time as follows:(46)t2∗≤t0+ft0+ba1/β1−β−ba1−β/βΓ1+1/1−βΓ1+1/1−β−β/1−β−αΓ1+αaΓ1+1/1−β−αΓ1+1/1−β−β/1−β1/α=t0+ft0+ba1/β1−β−ba1−β/βΓ1+1/1−βΓ2−αΓ1+αaΓ1+1/1−β−α1/α.
For Property2, considering b=0, we can obtain Corollary 1 as follows.Corollary 1.
Assume that functionft∈Ct0,+∞,ℝ∗ is continuous and nonnegative definite. If the following inequality holds,(47)Dt0Ctαft≤−afβt,where a>0, 0<α<1, and 0<β≤1, then it can be deduced as follows:(48)ft≤f1−βt0−aΓ1+1/1−β−αt−t0αΓ1+1/1−βΓ2−αΓ1+α1/1−β.
And we can judgeft converges to 0 within the settling time t3∗ which can be estimated as follows:(49)t3∗≤t0+f1−βt0Γ1+1/1−βΓ2−αΓ1+αaΓ1+1/1−β−α1/α.Remark 2.
Peng et al. and Wu et al. in [31, 32] proposed the estimation of settling time for the differential inequation Dt0Ctαft≤−afβt. According to the results in [31, 32], we can get another estimation of settling time:(50)t4∗≤t0+fα1+α−β−1t0Γ1+αΓ1−1+β/1+αaΓ2+αΓ1+α−1+β/1+α1/α,where 0<β<α1+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.Property 3.
Assume that functionft∈Ct0,+∞,ℝ∗ is continuous and nonnegative definite. If the following inequality holds,(51)Dt0Ctαft≤−aft−bfβt,where a>0, b>0, 0<α<1,and0<β≤1, then it can be deduced as follows:(52)f1−βt≤f1−βt0+ba1αexp−aΓ1+1/β−αΓ1+1/βΓ2−α1/αt−t0−ba,t≥t0.
And we can judgeft converges to 0 within the settling time T2∗ which can be estimated as follows:(53)t5∗≤t0+aΓ1+1/β−αΓ1+1/βΓ2−α−1/αlnaf1−βt0+bbα.Proof.
Letwt=fβ/mt, where m is a positive integer. Then, inequation (51) can be written as follows:(54)Dt0Ctαwm/βt≤−awm/βt−bwmt.
Whenm≥β, via applying Lemma 7, we can gain(55)Γ2−αΓ1+m/βΓ1+m/β−αwm/β−1tDtαwt≤−awm/βt−bwmt.
Then,(56)Γ2−αΓ1+m/βΓ1+m/β−αwm/β−m−1tDtαwt≤−awm/β−mt−b.
Consideringm≥β/1−β, thus m/β−m−1≥0. Once again, we apply Lemma 7 into the above inequation. And we have(57)Dtαwm/β−mt≤−a^wm/β−mt−b^,where(58)a^=Γ1+m/β−αΓ1+m/β−mΓ1+m/βΓ1+m/β−m−αa,b^=Γ1+m/β−αΓ1+m/β−mΓ1+m/βΓ1+m/β−m−αb.
Combining Property1, inequation (57) can be written as follows:(59)wm/β−mt≤wm/β−mt0+b^a^1αexp−a^1/αt−t0−b^a^,t≥t0.
Substitutingwt=fβ/mt and b^/a^=b/a into the above inequation, we can obtain(60)f1−βt≤f1−βt0+ba1αexp−a^1/αt−t0−ba,t≥t0.
Becauseft is a nonnegative function, we can gain ft=0 if f1−βt=0. Therefore, based on the analysis in Property 1, we can conclude that ft converges to 0 within the settling time t2∗. By simple computation, the estimation of settling time t2∗ is given by(61)t5∗≤t0+Γ1+m/β−αΓ1+m/β−m1+m/βΓ1+m/β−m−αa−1/αlnaft0+bbα.
Similar to Property1, Hm=Γ1+m/β/Γ1+m/β−m⋅Γ1+m/β−m−α/Γ1+m/β−α is monotone increasing about m.(62)t5∗≤t0+aHm−1/αlnaft0+bbα.
Therefore, it can be deduced that the settling time is monotone increasing aboutm. In order to obtain more accurate estimation of settling time, we should choose a smaller value of β/1−β.(63)f1−βt≤f1−βt0+ba1αexp−aΓ1+1/β−αΓ1+1/βΓ2−α1/αt−t0−ba,t≥t0.
And we can obtain the settling timet2∗ as follows:(64)t5∗≤t0+aΓ1+1/β−αΓ1+1/βΓ2−α−1/αlnaf1−βt0+bbα.


## 2.2. Model Description

A practical integer-order memristive neural network can usually be implemented by a very large-scale integration (VISI) circuit as shown in Figure1. According to Kirchhoff’s current law, the ith subsystem with mixed delays can be described as follows:(65)dxitdt=−dixit+∑j=1naijxitfjxjt+∑j=1nbijxitgjxjt−τjt+∑j=1ncijxit∫t−ρjtthjxjsds+Ii,i=1,2,…,n,t≥t0.Figure 1
The circuit of the memristive neural network.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:(66)Dt0Ctαxit=−dixit+∑j=1naijxitfjxjt+∑j=1nbijxitgjxjt−τjt+∑j=1ncijxit∫t−ρjtthjxjsds+Ii,i=1,2,…,n,t≥t0.where Dt0Ctα denotes the Caputo fractional derivative of order 0<α<1 from t0 to t; di≥0 is the self-feedback connection weight of the ith neuron; xit represents the corresponding state variable of the number of ith neurons at time t; fj⋅, gj⋅, and hj⋅ denote the activation function between the jth-dimension of the memristor and the state variable; τjt and ρjt, respectively, describe the discrete and distributed time-varying delays which satisfied 0≤τjt≤τjmax and 0≤ρjt≤ρjmax; and Ii represents the external input or bias, and it is a constant. Furthermore, aijxit, bijxit, and cijxit are memristive synaptic connection weights which can be given by(67)aijxit=WaijCi×signij,bijxit=WbijCi×signij,cijxit=WcijCi×signij,where signij=1,i≠j−1,i=j; Ci represents the capacitors; and Waij, Wbij, and Wcij, respectively; denote the memductance of memristors by Raij, Rbij, and Rcij. And Raij describes the memristor between activation function fjxjt and xit; Rbij denotes the memristor between activation function gjxjt−τj and xit; and Rcij is the memristor between activation function ∫t−ρjtthjxjsds and xit. According to the characteristics of memristors and the current-voltage, we use a simplified memristor model as follows:(68)aijxit=aij∗,xit≤Ωi,aij∗∗,xit>Ωi,bijxit=bij∗,xit≤Ωi,bij∗∗,xit>Ωi,cijxit=cij∗,xit≤Ωi,cij∗∗,xit>Ωi,where the switching jumps Ωi>0 and aij∗ , aij∗∗, bij∗ , bij∗∗, cij∗ , and cij∗∗ are all constants. And drive system (66) is supplemented with initial conditions as follows:(69)xis=Ξis,s∈t0−τ,t0,τ=max1≤j≤nτjt,ρjt.In order to express the proof process clearly, we set some mathematical simplified forms. Letaij″=maxaij∗,aij∗∗, aij′=minaij∗,aij∗∗, bij″=maxbij∗,bij∗∗, bij′=minbij∗,bij∗∗, cij″=maxcij∗,cij∗∗, and cij′=mincij∗,cij∗∗. And a¯ij=1/2aij″+aij′, a˜ij=1/2aij″+aij′, b¯ij=1/2bij″+bij′, b˜ij=1/2bij″+bij′, c¯ij=1/2cij″+cij′, and c˜ij=1/2cij″+cij′. Also, aij+=maxaij∗,aij∗∗, bij+=maxbij∗,bij∗∗, and cij+=maxcij∗,cij∗∗.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:(70)Dt0Ctαxit∈−dixit+∑j=1na¯ij+co−a˜ij,a˜ijfjxjt+∑j=1nb¯ij+co−b˜ij,b˜ijgjxjt−τjt+∑j=1nc¯ij+co−c˜ij,c˜ij∫t−ρjtthjxjsds+Ii,i=1,2,…,n,t≥t0.According to the set-valued map and the measurable selection theorem in [50], there exist measurable functions δij1,δij2,δij3∈co−1,1 such that(71)t0CDtαxit=−dixit+∑j=1na¯ij+a˜ijδij1fjxjt+∑j=1nb¯ij+b˜ijδij2gjxjt−τjt+∑j=1nc¯ij+c˜ijδij3∫t−ρjtthjxjsds+Ii,i=1,2,…,n,t≥t0.In order to investigate the synchronization of FMNNs, system (66) is regarded as the drive system. Similarly, the response system is described as follows:(72)t0CDtαyit=−diyit+∑j=1naijyitfjyjt+∑j=1nbijyitgjyjt−τjt+∑j=1ncijyit∫t−ρjtthjyjsds+Ii+uit,i=1,2,…,n,t≥t0.where yit represents the state of the response system and uit is the controller to be designed. And the initial conditions of response system (72) are given as follows:(73)yis=Ψis,s∈t0−τ,t0,τ=max1≤j≤nτjt,ρjt.By using the set-valued map and the measurable selection theorem, there exist measurable functionsγij1,γij2,γij3∈co−1,1 such that(74)Dt0Ctαyit=−diyit+∑j=1na¯ij+a˜ijγij1fjyjt+∑j=1nb¯ij+b˜ijγij2gjyjt−τjt+∑j=1nc¯ij+c˜ijγij3∫t−ρjtthjyjsds+Ii+uit,i=1,2,…,n,t≥t0.Remark 3.
The drive system and the response system are a distinct network, soΔijxit probably is not equal to Δijyit. Thus, we need to select six different measurable functions δij1,δij2,δij3,γij1,γij2,γij3.
Leteit=yit−λxit denote the synchronization error. And λ represents the projective coefficient. Hence, the fractional order of the synchronization error system can be given as follows:(75)Dt0Ctαeit=−dieit+1−λIi+∑j=1na¯ij+a˜ijγij1fjyjt−λ∑j=1na¯ij+a˜ijδij1fjxjt+∑j=1nb¯ij+b˜ijγij2gjyjt−τjt−λ∑j=1nb¯ij+b˜ijδij2gjxjt−τjt+∑j=1nc¯ij+c˜ijγij3∫t−ρjtthjyjsds−λ∑j=1nc¯ij+c˜ijδij3∫t−ρjtthjxjsds+uit.
The initial conditions associated with system (75) are written as follows:(76)eis=Φis=Ξit−λΨis,s∈t0−τ,t0,τ=max1≤j≤nτjt,ρjt.
To obtain the main results expediently, we set up the following assumptions:Assumption 1.
The activation functionsfj⋅,gj⋅, and hj⋅ are Lipschitz continuous. That is, for any j∈ℕ, there exist three constants Lj1, Lj2, and Lj3 such that(77)fjy−fjx≤Lj1y−x,gjy−fjx≤Lj2y−x,hjy−hjx≤Lj3y−x,where x,y∈ℝ,x≠y.Assumption 2.
The activation functionsfj⋅,gj⋅, and hj⋅ are bounded. That is, for any j∈ℕ, there exist three constants Mj1, Mj2, and Mj3 such that(78)fjx≤Mj1,gjx≤Mj2,hjx≤Mj3,where x∈ℝ.
It is obvious that error system (75) can be expressed as follows:(79)Dt0Ctαeit=−dieit+1−λIi++uit+∑j=1na¯ij+a˜ijγij1fjyjt−fjλxjt+∑j=1na¯ij+a˜ijγij1fjλxjt−fjxjt+∑j=1n1−λa¯ij+γij1−λδij1a˜ijfjxjt+∑j=1nb¯ij+b˜ijγij2gjyjt−τjt−gjλxjt−τjt+∑j=1nb¯ij+b˜ijγij2gjλxjt−τjt−gjxjt−τjt+∑j=1n1−λb¯ij+γij2−λδij2b˜ijgjxjt−τjt+∑j=1nc¯ij+c˜ijγij3∫t−ρjtthjyjsds−λ∑j=1nc¯ij+c˜ijδij3∫t−ρjtthjxjsds≤−dieit+1−λIi+uit+∑j=1naij+|fjyjt−fjλxjt|+∑j=1naij+|fjλxjt−fjxjt|+∑j=1n1−λa¯ij+γij1−λδij1a˜ij|fjxjt|+∑j=1nbij+|gjyjt−τjt−gjλxjt−τjt+∑j=1nbij+|gjλxjt−τjt−gjxjt−τjt+∑j=1n1−λb¯ij+γij2−λδij2b˜ijgjxjt−τjt+∑j=1ncij+∫t−ρjtthjyjs|ds+∑j=1ncij+λ∫t−ρjtthjxjs|ds.
According to Assumption1, we can judge(80)∑j=1naij+fjyjt−fjλxjt≤∑j=1naij+Lj1ejt,∑j=1nbij+gjyjt−τjt−gjλxjt−τjt≤∑j=1nbij+Lj2ejt−τjt.
According to Assumption2, we have(81)∑j=1naij+fjλxjt−fjxjt≤∑j=1n2aij+Mj1∑j=1n1−λa¯ij+γij1−λδij1a˜ijfjxjt≤∑j=1n1−λa¯ij+1+λa˜ijMj1∑j=1nbij+gjλxjt−τjt−gjxjt−τjt≤∑j=1n2bij+Mj2∑j=1n1−λb¯ij+γij2−δij2b˜ijgjxjt−τjt≤∑j=1n1−λb¯ij+1+λb˜ijMj2∑j=1ncij+∫t−ρjtthjyjsds+∑j=1ncij+λ∫t−ρjtthjxjsds≤∑j=1n1+λcij+ρjmaxMj3.
Substituting inequations (80) and (81) into equation (79), we can obtain(82)Dt0Ctαeit≤−dieit+1−λIi+uit+∑j=1nLj1aij+ejt+Lj2bij+ejt−τjt+2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3,i=1,2,…,n,t≥t0.


## 3. Main Results

In this section, a feedback controller is designed for actualizing finite-time projective synchronization of fractional-order memristor-based neural networks with mixed time-varying delays. And we choose the following feedback control strategy:(83)uit=−kieit−signeithieit−τit+qi−1−λIi,where ki, hi, and qi denote the controller parameters.


### 3.1. The Finite-Time Projective Synchronization of FMNNs with a Lyapunov Function Based on 1-Norm

Theorem 1.
When Assumptions1 and 2 hold, drive system (66) and response system (72) achieve the finite-time projective synchronization under state feedback controller (83) if(84)φ=min1≤i≤nφi>0,δ=min1≤i≤nδi>0,η=∑i=1nηi>0,where(85)φi=di+ki−∑j=1nLi1aji+,δi=hi−∑j=1nLi2bji+,ηi=qi−∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3.
Moreover, the settling timeT1 is estimated by(86)T1∗≤t0+φ−1/αlnφV1t0+ηαη.Proof.
Firstly, we consider the following Lyapunov function:(87)V1t=∑i=1neit.
Based on Lemma8, the above fractional-order derivative of Vt can be written as follows:(88)Dt0CtαV1t≤∑i=1nsigneitt0CDtαeit.
From error system (82), we have(89)Dt0CtαV1t≤∑i=1nsigneitt0CDtαeit≤∑i=1nsigneit−dieit+1−λIi+uit+∑j=1nLj1aij+ejt+Lj2bij+ejt−τjt+2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3≤∑i=1n−dieit+signeit1−λIi+uit+∑j=1nLi1aji+eit+Li2bji+eit−τit+2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3.
Substituting the controller (83) into inequation (89),(90)Dt0CtαV1t≤∑i=1n−dieit+∑j=1nLi1aji+eit+Li2bji+eit−τit+2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3−kieit−hieit−τit−qi=−∑i=1ndi+ki−∑j=1nLi1aji+eit−∑i=1nhi−∑j=1nLi2bji+eit−τit−∑i=1nqi−∑j=1n2aij++|1−λa˜ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3.
Let(91)φi=di+ki−∑j=1nLi1aji+,δi=hi−∑j=1nLi2bji+,ηi=qi−∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3.
Therefore, in equation (90) can be rewritten as follows:(92)Dt0CtαV1t≤−∑i=1nφieit−∑i=1nδieit−τit−∑i=1nηi≤−φ∑i=1neit−δ∑i=1neit−τit−η=−φV1t−δV1t−τit−η,where φ=min1≤i≤nφi,δ=min1≤i≤nδi, and η=∑i=1nηi.
Based on the conditionδ>0 in Theorem 1, we have(93)Dt0CtαV1t≤−φV1t−η,
According to the conditions of Theorem1, we can gain φ>0,η>0. Furthermore, by virtue of Property 1, it follows that(94)V1t≤V1t0+ηφ1αexp−φ1/αt−t0−ηφ,t≥t0.
And we can judge that the Lyapunov functionV1t 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 T1 which can be estimated is as follows:(95)T1∗≤t0+φ−1/αlnφV1t0+ηηα.
This completes the proof.Remark 4.
For the differential inequationDt0CtαV1t≤−φV1t−η in fractional-order neural networks, Zhang et al. [29] ignored the effort of parameter φ when they calculated the settling time.(96)Dt0CtαV1t≤−η.
Taking the fractional integral with variable orderα on both sides of the above inequation, we can gain(97)V1t−V1t0≤−1Γα∫t0tt−sα−1ηds.
Through Definition1, we can calculate that Vt converges to 0 within the settling time T2, and T2 is evaluated by(98)T2≤t0+V1t0Γ1+αη1/α.
Apparently, the authors ignored the impact of parameterη, and let η=0 which will increase the conservation of the settling time. Considering the parameter η>0; the estimation of settling time T1∗ in Theorem 1 is less conservative than the results in [29]. To verify that the settling time T1∗ is more accurate than T2, a comparison table has been given in numerical simulation.Remark 5.
For the differential inequationDt0CtαV1t≤−φV1t−η, when fractional order α=1, the settling time t1 is estimated by(99)t1≤t0+lnφV1t0−lnηφ.
Obviously, the estimate timet1 is consistent with the result of the integer-order system given in [51]. Thus, it shows the estimation of settling time T1∗ in Theorem 1 is more accurate and general.


### 3.2. The Finite-Time Projective Synchronization of FMNNs with a Lyapunov Function Based onp-Norm

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(100)φ¯=min1≤i≤nφ¯i>0,δ¯=min1≤i≤nδ¯i>0,η¯=min1≤i≤nη¯i>0,φ¯i=Γ2−αΓ1+pΓ1+p−αdi+ki−∑j=1nLj1aij+p−1ε1p+Li1aji+pε1p−1+Lj2bij+p−1ε2p,δ¯i=Γ2−αΓ1+pΓ1+p−αhi−∑j=1nLi2bji+pε2p−1,η¯i=Γ2−αΓ1+pΓ1+p−αqi−∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3,where ε1 and ε2 are arbitrary positive constants.
Moreover, the settling timeT2 is estimated by(101)T3∗≤t0+Γ1+p/p−1−αφ¯Γ1+p/p−1Γ2−α−1/αlnφ¯V21/pt0+η¯η¯α.Proof.
Firstly, we consider the following Lyapunov function:(102)V2t=∑i=1neitp,where p is a positive integer with p>1.
Calculating the derivate ofV2t and using Lemma 7 and error system (82), one has(103)Dt0CtαV2t=Γ2−αΓ1+pΓ1+p−α∑i=1neitp−1Dt0Ctαeit.
According to Lemma8, the above fractional-order derivative of Vt can be written as follows:(104)Dt0CtαV2t≤Γ2−αΓ1+pΓ1+p−α∑i=1neitp−1signeitDt0Ctαeit≤Γ2−αΓ1+pΓ1+p−α∑i=1n−dieitp+signeit1−λIi+uiteitp−1+∑j=1nLi1aij+ejt+Li2bij+ejt−τjt+2aij++1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3eitp−1.
Substituting the feedback controller (83) into inequation (104), we can get(105)Dt0CtαV2t≤Γ2−αΓ1+pΓ1+p−α∑i=1n−dieitp+∑j=1nLj1aij+eitp−1ejt+Lj2bij+eitp−1ejt−τjt+∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3eitp−1−kieitp−hieit−τitp−qieitp−1.
By applying Lemma3, we have(106)eitp−1ejt≤p−1ε1peitp+1pε1p−1ejtp,eitp−1ejt−τjt≤p−1ε2peitp+1pε2p−1ejt−τjtp,where ε1 and ε2 are arbitrary positive constants.
Combining with inequations (105) and (106), one has(107)Dt0CtαV1t≤Γ2−αΓ1+pΓ1+p−α∑i=1n−di+kieitp−hieit−τitp−qieitp−1+∑j=1nLj1aij+p−1ε1p+Li1aji+pε1p−1+Lj2bij+p−1ε2peitp+∑j=1nLi2bji+pε2p−1eit−τitp+∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3eitp−1=−Γ2−αΓ1+pΓ1+p−α∑i=1ndi+ki−∑j=1nLj1aij+p−1ε1p+Li1aji+pε1p−1+Lj2bij+p−1ε2peitp−Γ2−αΓ1+pΓ1+p−α∑i=1nhi−∑j=1nLi2bji+pε2p−1eit−τitp−Γ2−αΓ1+pΓ1+p−α∑i=1nqi−∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3eitp−1≤−∑i=1nφ¯ieitp−∑i=1nδ¯ieit−τjtp−∑i=1nη¯ieitp−1,where(108)φ¯i=Γ2−αΓ1+pΓ1+p−αdi+ki−∑j=1nLj1aij+p−1ε1p+Li1aji+pε1p−1+Lj2bij+p−1ε2p,δ¯i=Γ2−αΓ1+pΓ1+p−αhi−∑j=1nLi2bji+pε2p−1,η¯i=Γ2−αΓ1+pΓ1+p−αqi−∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3.
Denoteφ¯=min1≤i≤nφ¯i, δ¯=min1≤i≤nδ¯i, and η¯=min1≤i≤nη¯i; one gets(109)Dt0CtαV2t≤−φ¯∑i=1neitp−δ¯∑i=1neit−τitp−η¯∑i=1neitp−1.
Applying Lemma4, we have(110)Dt0CtαV2t≤−φ¯∑i=1neitp−δ¯∑i=1neit−τitp−η¯∑i=1neitpp−1/p=−φ¯V2t−δ¯V2t−τit−η¯V2tp−1/p.
Based on the conditions of Theorem2, we know φ¯>0, δ¯>0, and η¯>0. Thus,(111)Dt0CtαV2t≤−φ¯V2t−η¯V2p−1/pt.
Hence, from Property3, we have(112)V21/pt≤V21/pt0+η¯φ¯1αexp−aΓ1+p/p−1−αΓ1+p/p−1Γ2−α1/αt−t0−η¯φ¯.
And we can judge that the Lyapunov functionV2t converges to 0, which represents drive system (66) and response system (72) achieve finite-time projective synchronization under feedback controller (83). And the settling time T3∗ can be estimated by(113)T3∗≤t0+Γ1+p/p−1−αφ¯Γ1+p/p−1Γ2−α−1/αlnφ¯V21/pt0+η¯η¯α.
The proof is accomplished.Remark 6.
For the differential inequationDt0CtαV2t≤−φ¯V2t−η¯V2p−1/pt, considering the parameter φ¯=0, then(114)Dt0CtαV2t≤−η¯V2p−1/pt,where η¯>0.
According to Corollary1, it has(115)V2t≤V21/pt0−η¯⋅aΓ1+p−αt−t0αΓ1+pΓ2−αΓ1+α1/1−β.
And we can judge that the Lyapunov functionV2t converges to 0. Furthermore, the settling time(116)T4≤t0+V21/pt0Γ1+pΓ2−αΓ1+αΓ1+p−αη¯1/α.
From Remark2, if p−1/p<α1+α−1 holds, we can obtain another estimation of settling time:(117)T5≤t0+V2α1+α+1/p−2t0Γ1+αΓ1−2p−1/p1+αη¯⋅Γ2+αΓ1+α−2p−1/p1+α1/α.
Obviously, compared withT3∗ in Theorem 2, the estimate times T4 and T5 are conservative estimations of settling time. And a comparison table about T3∗, T4, and T5 has been given in numerical simulation.


## 3.1. The Finite-Time Projective Synchronization of FMNNs with a Lyapunov Function Based on 1-Norm

Theorem 1.
When Assumptions1 and 2 hold, drive system (66) and response system (72) achieve the finite-time projective synchronization under state feedback controller (83) if(84)φ=min1≤i≤nφi>0,δ=min1≤i≤nδi>0,η=∑i=1nηi>0,where(85)φi=di+ki−∑j=1nLi1aji+,δi=hi−∑j=1nLi2bji+,ηi=qi−∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3.
Moreover, the settling timeT1 is estimated by(86)T1∗≤t0+φ−1/αlnφV1t0+ηαη.Proof.
Firstly, we consider the following Lyapunov function:(87)V1t=∑i=1neit.
Based on Lemma8, the above fractional-order derivative of Vt can be written as follows:(88)Dt0CtαV1t≤∑i=1nsigneitt0CDtαeit.
From error system (82), we have(89)Dt0CtαV1t≤∑i=1nsigneitt0CDtαeit≤∑i=1nsigneit−dieit+1−λIi+uit+∑j=1nLj1aij+ejt+Lj2bij+ejt−τjt+2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3≤∑i=1n−dieit+signeit1−λIi+uit+∑j=1nLi1aji+eit+Li2bji+eit−τit+2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3.
Substituting the controller (83) into inequation (89),(90)Dt0CtαV1t≤∑i=1n−dieit+∑j=1nLi1aji+eit+Li2bji+eit−τit+2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3−kieit−hieit−τit−qi=−∑i=1ndi+ki−∑j=1nLi1aji+eit−∑i=1nhi−∑j=1nLi2bji+eit−τit−∑i=1nqi−∑j=1n2aij++|1−λa˜ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3.
Let(91)φi=di+ki−∑j=1nLi1aji+,δi=hi−∑j=1nLi2bji+,ηi=qi−∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3.
Therefore, in equation (90) can be rewritten as follows:(92)Dt0CtαV1t≤−∑i=1nφieit−∑i=1nδieit−τit−∑i=1nηi≤−φ∑i=1neit−δ∑i=1neit−τit−η=−φV1t−δV1t−τit−η,where φ=min1≤i≤nφi,δ=min1≤i≤nδi, and η=∑i=1nηi.
Based on the conditionδ>0 in Theorem 1, we have(93)Dt0CtαV1t≤−φV1t−η,
According to the conditions of Theorem1, we can gain φ>0,η>0. Furthermore, by virtue of Property 1, it follows that(94)V1t≤V1t0+ηφ1αexp−φ1/αt−t0−ηφ,t≥t0.
And we can judge that the Lyapunov functionV1t 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 T1 which can be estimated is as follows:(95)T1∗≤t0+φ−1/αlnφV1t0+ηηα.
This completes the proof.Remark 4.
For the differential inequationDt0CtαV1t≤−φV1t−η in fractional-order neural networks, Zhang et al. [29] ignored the effort of parameter φ when they calculated the settling time.(96)Dt0CtαV1t≤−η.
Taking the fractional integral with variable orderα on both sides of the above inequation, we can gain(97)V1t−V1t0≤−1Γα∫t0tt−sα−1ηds.
Through Definition1, we can calculate that Vt converges to 0 within the settling time T2, and T2 is evaluated by(98)T2≤t0+V1t0Γ1+αη1/α.
Apparently, the authors ignored the impact of parameterη, and let η=0 which will increase the conservation of the settling time. Considering the parameter η>0; the estimation of settling time T1∗ in Theorem 1 is less conservative than the results in [29]. To verify that the settling time T1∗ is more accurate than T2, a comparison table has been given in numerical simulation.Remark 5.
For the differential inequationDt0CtαV1t≤−φV1t−η, when fractional order α=1, the settling time t1 is estimated by(99)t1≤t0+lnφV1t0−lnηφ.
Obviously, the estimate timet1 is consistent with the result of the integer-order system given in [51]. Thus, it shows the estimation of settling time T1∗ in Theorem 1 is more accurate and general.


## 3.2. The Finite-Time Projective Synchronization of FMNNs with a Lyapunov Function Based onp-Norm

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(100)φ¯=min1≤i≤nφ¯i>0,δ¯=min1≤i≤nδ¯i>0,η¯=min1≤i≤nη¯i>0,φ¯i=Γ2−αΓ1+pΓ1+p−αdi+ki−∑j=1nLj1aij+p−1ε1p+Li1aji+pε1p−1+Lj2bij+p−1ε2p,δ¯i=Γ2−αΓ1+pΓ1+p−αhi−∑j=1nLi2bji+pε2p−1,η¯i=Γ2−αΓ1+pΓ1+p−αqi−∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3,where ε1 and ε2 are arbitrary positive constants.
Moreover, the settling timeT2 is estimated by(101)T3∗≤t0+Γ1+p/p−1−αφ¯Γ1+p/p−1Γ2−α−1/αlnφ¯V21/pt0+η¯η¯α.Proof.
Firstly, we consider the following Lyapunov function:(102)V2t=∑i=1neitp,where p is a positive integer with p>1.
Calculating the derivate ofV2t and using Lemma 7 and error system (82), one has(103)Dt0CtαV2t=Γ2−αΓ1+pΓ1+p−α∑i=1neitp−1Dt0Ctαeit.
According to Lemma8, the above fractional-order derivative of Vt can be written as follows:(104)Dt0CtαV2t≤Γ2−αΓ1+pΓ1+p−α∑i=1neitp−1signeitDt0Ctαeit≤Γ2−αΓ1+pΓ1+p−α∑i=1n−dieitp+signeit1−λIi+uiteitp−1+∑j=1nLi1aij+ejt+Li2bij+ejt−τjt+2aij++1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3eitp−1.
Substituting the feedback controller (83) into inequation (104), we can get(105)Dt0CtαV2t≤Γ2−αΓ1+pΓ1+p−α∑i=1n−dieitp+∑j=1nLj1aij+eitp−1ejt+Lj2bij+eitp−1ejt−τjt+∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3eitp−1−kieitp−hieit−τitp−qieitp−1.
By applying Lemma3, we have(106)eitp−1ejt≤p−1ε1peitp+1pε1p−1ejtp,eitp−1ejt−τjt≤p−1ε2peitp+1pε2p−1ejt−τjtp,where ε1 and ε2 are arbitrary positive constants.
Combining with inequations (105) and (106), one has(107)Dt0CtαV1t≤Γ2−αΓ1+pΓ1+p−α∑i=1n−di+kieitp−hieit−τitp−qieitp−1+∑j=1nLj1aij+p−1ε1p+Li1aji+pε1p−1+Lj2bij+p−1ε2peitp+∑j=1nLi2bji+pε2p−1eit−τitp+∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3eitp−1=−Γ2−αΓ1+pΓ1+p−α∑i=1ndi+ki−∑j=1nLj1aij+p−1ε1p+Li1aji+pε1p−1+Lj2bij+p−1ε2peitp−Γ2−αΓ1+pΓ1+p−α∑i=1nhi−∑j=1nLi2bji+pε2p−1eit−τitp−Γ2−αΓ1+pΓ1+p−α∑i=1nqi−∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3eitp−1≤−∑i=1nφ¯ieitp−∑i=1nδ¯ieit−τjtp−∑i=1nη¯ieitp−1,where(108)φ¯i=Γ2−αΓ1+pΓ1+p−αdi+ki−∑j=1nLj1aij+p−1ε1p+Li1aji+pε1p−1+Lj2bij+p−1ε2p,δ¯i=Γ2−αΓ1+pΓ1+p−αhi−∑j=1nLi2bji+pε2p−1,η¯i=Γ2−αΓ1+pΓ1+p−αqi−∑j=1n2aij++|1−λa¯ij+1+λa˜ijMj1+2bij++1−λb¯ij+1+λb˜ijMj2+1+λcij+ρjmaxMj3.
Denoteφ¯=min1≤i≤nφ¯i, δ¯=min1≤i≤nδ¯i, and η¯=min1≤i≤nη¯i; one gets(109)Dt0CtαV2t≤−φ¯∑i=1neitp−δ¯∑i=1neit−τitp−η¯∑i=1neitp−1.
Applying Lemma4, we have(110)Dt0CtαV2t≤−φ¯∑i=1neitp−δ¯∑i=1neit−τitp−η¯∑i=1neitpp−1/p=−φ¯V2t−δ¯V2t−τit−η¯V2tp−1/p.
Based on the conditions of Theorem2, we know φ¯>0, δ¯>0, and η¯>0. Thus,(111)Dt0CtαV2t≤−φ¯V2t−η¯V2p−1/pt.
Hence, from Property3, we have(112)V21/pt≤V21/pt0+η¯φ¯1αexp−aΓ1+p/p−1−αΓ1+p/p−1Γ2−α1/αt−t0−η¯φ¯.
And we can judge that the Lyapunov functionV2t converges to 0, which represents drive system (66) and response system (72) achieve finite-time projective synchronization under feedback controller (83). And the settling time T3∗ can be estimated by(113)T3∗≤t0+Γ1+p/p−1−αφ¯Γ1+p/p−1Γ2−α−1/αlnφ¯V21/pt0+η¯η¯α.
The proof is accomplished.Remark 6.
For the differential inequationDt0CtαV2t≤−φ¯V2t−η¯V2p−1/pt, considering the parameter φ¯=0, then(114)Dt0CtαV2t≤−η¯V2p−1/pt,where η¯>0.
According to Corollary1, it has(115)V2t≤V21/pt0−η¯⋅aΓ1+p−αt−t0αΓ1+pΓ2−αΓ1+α1/1−β.
And we can judge that the Lyapunov functionV2t converges to 0. Furthermore, the settling time(116)T4≤t0+V21/pt0Γ1+pΓ2−αΓ1+αΓ1+p−αη¯1/α.
From Remark2, if p−1/p<α1+α−1 holds, we can obtain another estimation of settling time:(117)T5≤t0+V2α1+α+1/p−2t0Γ1+αΓ1−2p−1/p1+αη¯⋅Γ2+αΓ1+α−2p−1/p1+α1/α.
Obviously, compared withT3∗ in Theorem 2, the estimate times T4 and T5 are conservative estimations of settling time. And a comparison table about T3∗, T4, and T5 has been given in numerical simulation.


## 4. 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:(118)t0CDt0.98xit=−dixit+∑j=12aijxitfjxjt+∑j=12bijxitgjxjt−τjt+∑j=12cijxit∫t−ρjtthjxjsds+Ii,t0CDt0.98yit=−diyit+∑j=12aijyitfjyjt+∑j=12bijyitgjyjt−τjt+∑j=12cijyit∫t−ρjtthjyjsds+Ii+uit,where i=1,2. And the values of the memristors are given as follows:(119)a11x1t=0.3,x1t≤1,0.2,x1t>1,a12x1t=−0.4,x1t≤1,−0.22,x1t>1,a21x2t=−0.9,x2t≤1,−0.8,x2t>1,a22x2t=0.5,x2t≤1,0.8,x2t>1,b11x1t=−0.6,x1t≤1,−0.2,x1t>1,b12x1t=0.5,x1t≤1,0.5,x1t>1,b21x2t=0.1,x2t≤1,0.4,x2t>1,`b22x2t=−0.4,x2t≤1,−0.3,x2t>1,c11x1t=−0.1,x1t≤1,−0.4,x1t>1,c12x1t=0.7,x1t≤1,0.8,x1t>1,c21x2t=0.4,x2t≤1,0.3,x2t>1,c22x2t=−0.85,x2t≤1,−0.95,x2t>1.For drive-response system (118), we select the system parameters as follows: I1=I2=0.1, d1=d2=0.1, τ1t=et/et+1,τ2t=0.8∗et/et+1,ρ1t=0.5cost+0.5,andρ2t=0.4sint+0.4. Hence, we can obtain that τ1max=1,τ2max=0.8,ρ1max=1,ρ2max=0.8,andτ=maxj=1,2τjt,ρjt=1.And the initial values xs=−1.5,2.5T,ys=2.5,−0.5T,∀s∈−1,0.The activations are adopted as follows:(120)fjxjt=gjxjt=hjxjt=1−e−xjt1+e−xjt,j=1,2.It is easy to check that Assumption1 and 2 are satisfied. By simple calculation, Lj1=Lj2=Lj3=Mj1=Mj2=Mj3=1, for j=1,2.


### 4.1. Projective Synchronization

First, we choose the projective coefficientλ=2. We can get the initial values Φ1s=y1s−2x1s=5.5, Φ2s=y2s−2x2s=−5.5, V1t0=11, and V2t0=60.5.According to Theorem1, it can be calculated as follows:(121)φ1=k1−1.1,φ2=k2−1.1,δ1=h1−1,δ2=h2−0.9,η1=q1−8.9,η2=q2−10.38.And we choosek1=3.1,k2=3.1,h1=1,h2=0.9,q1=9.9,andq2=11.38 which make φ=2andη=2.Then, we have a finite-time fractional-order differential inequation as follows:(122)Dt0CtαV1t≤−2V1t−2.Based on the result in Theorem1, drive-response system (118) could achieve synchronization in finite time. And we can calculate the setting time T1∗ as follows:(123)T1∗≤t0+φ−1/αlnφV1t0+ηαη≤2−1/0.98ln2×11+20.98×2=1.235.To verify Remark4, we can calculate T2 to demonstrate that T1∗ is less conservative than T2. It follows that(124)T2≤t0+V1t0Γ1+αη1/α=11⋅Γ1.9821/α=5.5778.On the contrary, we selectp=2 and ε1=ε2=1 for Theorem 2. And it is easy to calculate(125)φ¯1=1.9608k1−1.4,φ¯2=1.9608k2−1.85;δ¯1=1.9608h1−0.5,δ¯2=1.9608h2−0.45;η¯1=1.9608q1−8.9,η¯2=1.9608q2−10.38.Under the controller parameter,k1=3.1,k2=3.1,h1=1,h2=0.9,q1=9.9,andq2=11.38 which lead to φ¯=2.451andη¯=1.9608.Hence, we have another finite-time fractional-order differential inequation as follows:(126)Dt0CtαV2t≤−2.451V2t−1.9608V21/2t.Based on the result in Theorem2, we can estimate the setting time T3 as follows:(127)T3∗≤t0+Γ1+p/p−1−αφ¯Γ1+p/p−1Γ2−α−1/αlnφ¯V21/pt0+η¯η¯α=2.451Γ2.02Γ3Γ1.02−1/0.98ln2.451×60.5+1.96081.9608×0.98=1.9054.To verify Remark6, we can calculate T4 to demonstrate that T3∗ is less conservative than T4. It follows that(128)T4≤t0+V21pt0Γ1+pΓ2−αΓ1+αΓ1+p−αη¯1/α=60.5⋅Γ3Γ1.02Γ1.981.9608Γ2.021/0.98=8.0421.Consideringp=2andα=0.98, p−1/p<α1+α−1 holds. Thus, we have(129)T5≤t0+V2α1+α+1/p−2t0Γ1+αΓ1−2p−1/p1+αη¯⋅Γ2+αΓ1+α−2p−1/p1+α1/α=60.50.4404⋅Γ1.98Γ0.24241.9608⋅Γ2.98Γ1.22241/0.98=6.689.Therefore, drive-response system (118) can achieve finite-time projective synchronization under controller (83). And the state trajectories and error curves of FMCGNNs are as shown in Figures 2–4, which demonstrate the effectiveness of Theorems 1 and 2.Figure 2
The first dimensional state trajectories of the drive system and the response system when the projective coefficientλ=2.Figure 3
The second dimensional state trajectories of the drive system and the response system when the projective coefficientλ=2.Figure 4
The synchronization error evolution when the projective coefficientλ=2.


### 4.2. Complete Synchronization

First, we choose the projective coefficientλ=1. We can get Φ1s=y1s−2x1s=4, Φ2s=y2s−2x2s=−3, V1t0=7, and V2t0=25.According to Theorem1, it can be calculated as follows:(130)φ1=k1−1.1,φ2=k2−1.1;δ1=h1−1,δ2=h2−0.9;η1=q1−6.36,η2=q2−8.12.And we choosek1=3.1,k2=3.1,h1=1,h2=0.9,q1=7.36,andq2=9.12 which make φ=2andη=2.Then, we have a finite-time fractional-order differential inequation as follows:(131)Dt0CtαV1t≤−2V1t−2.Based on the result in Theorem1, drive-response system (118) could achieve synchronization in finite time. And we can calculate the setting time T1∗ as follows:(132)T1∗≤t0+φ−1/αlnφV1t0+ηαη≤2−1/0.98ln2×7+20.98×2=1.0351.To verify Remark4, we can calculate T2 to demonstrate that T1∗ is less conservative than T2. It follows that(133)T2≤t0+V1t0Γ1+αη1/α=7⋅Γ1.9821/α=3.5169.On the contrary, we selectp=2 and ε1=ε2=1 for Theorem 2. And it is easy to calculate(134)φ¯1=1.9608k1−1.4,φ¯2=1.9608k2−1.85;δ¯1=1.9608h1−0.5,δ¯2=1.9608h2−0.45;η¯1=1.9608q1−6.36,η¯2=1.9608q2−8.12.Under the controller parameter,k1=3.1,k2=3.1,h1=1,h2=0.9,q1=7.36,andq2=9.12 which lead to φ¯=2.451andη¯=1.9608.Hence, we have another finite-time fractional-order differential inequation as follows:(135)Dt0CtαV2t≤−2.451V2t−1.9608V21/2t.Based on the result in Theorem2, we can estimate the setting time T3 as follows:(136)T3∗≤t0+Γ1+p/p−1−αφ¯Γ1+p/p−1Γ2−α−1/αlnφ¯V21/pt0+η¯η¯α=2.451Γ2.02Γ3Γ1.02−1/0.98ln2.451×25+1.96081.9608×0.98=1.5937.To verify Remark6, we can calculate T4 to demonstrate that T3∗ is less conservative than T4. It follows that(137)T4≤t0+V21/pt0Γ1+pΓ2−αΓ1+αΓ1+p−αη¯1/α=25⋅Γ3Γ1.02Γ1.981.9608Γ2.021/0.98=5.1232.Consideringp=2andα=0.98, p−1/p<α1+α−1 holds. Thus, we have(138)T5≤t0+V2α1+α+1/p−2t0Γ1+αΓ1−2p−1/p1+αη¯⋅Γ2+αΓ1+α−2p−1/p1+α1/α=250.4404⋅Γ1.98Γ0.24241.9608⋅Γ2.98Γ1.22241/0.98=4.4965.Therefore, drive-response system (118) can achieve finite-time complete synchronization under controller (83). And the state trajectories and error curves of FMCGNNs are as shown in Figures 5–7, which demonstrate the effectiveness of Theorems 1 and 2.Figure 5
The first dimensional state trajectories of the drive system and the response system when the projective coefficientλ=1.Figure 6
The second dimensional state trajectories of the drive system and the response system when the projective coefficientλ=1.Figure 7
The synchronization error evolution when the projective coefficientλ=1.


### 4.3. Antisynchronization

First, we choose the projective coefficientλ=−1. We can get Φ1s=y1s−2x1s=1, Φ2s=y2s−2x2s=2, V1t0=3, and V2t0=5.According to Theorem1, it can be calculated as follows:(139)φ1=k1−1.1,φ2=k2−1.1;δ1=h1−1,δ2=h2−0.9;η1=q1−9.08,η2=q2−11.12.And we choosek1=3.1,k2=3.1,h1=1,h2=0.9,q1=10.08,andq2=12.12 which make φ=2andη=2.Then, we have a finite-time fractional-order differential inequation as follows:(140)Dt0CtαV1t≤−2V1t−2.Based on the result in Theorem1, drive-response system (118) could achieve synchronization in finite time. And we can calculate the setting time T1∗ as follows:(141)T1∗≤t0+φ−1/αlnφV1t0+ηαη≤2−1/0.98ln2×3+20.98×2=0.6934.To verify Remark4, we can calculate T2 to demonstrate that T1∗ is less conservative than T2. It follows that(142)T2≤t0+V1t0Γ1+αη1/α=3⋅Γ1.9821/α=1.4814.On the contrary, we selectp=2 and ε1=ε2=1 for Theorem 2. And it is easy to calculate(143)φ¯1=1.9608k1−1.4,φ¯2=1.9608k2−1.85;δ¯1=1.9608h1−0.5,δ¯2=1.9608h2−0.45;η¯1=1.9608q1−9.08,η¯2=1.9608q2−11.12.Under the controller parameter,k1=3.1,k2=3.1,h1=1,h2=0.9,q1=10.08,andq2=12.12 which lead to φ¯=2.451andη¯=1.9608.Hence, we have another finite-time fractional-order differential inequation as follows:(144)Dt0CtαV2t≤−2.451V2t−1.9608V21/2t.Based on the result in Theorem2, we can estimate the setting time T3 as follows:(145)T3∗≤t0+Γ1+p/p−1−αφ¯Γ1+p/p−1Γ2−α−1/αlnφ¯V21/pt0+η¯η¯α=2.451Γ2.02Γ3Γ1.02−1/0.98ln2.451×5+1.96081.9608×0.98=1.3539.To verify Remark6, we can calculate T4 to demonstrate that T3∗ is less conservative than T4. It follows that(146)T4≤t0+V21/pt0Γ1+pΓ2−αΓ1+αΓ1+p−αη¯1/α=5⋅Γ3Γ1.02Γ1.981.9608Γ2.021/0.98=2.2539.Consideringp=2andα=0.98, p−1/p<α1+α−1 holds. Thus, we have(147)T5≤t0+V2α1+α+1/p−2t0Γ1+αΓ1−2p−1/p1+αη¯⋅Γ2+αΓ1+α−2p−1/p1+α1/α=50.4404⋅Γ1.98Γ0.24241.9608⋅Γ2.98Γ1.22241/0.98=2.1816.Therefore, drive-response system (118) can achieve finite-time antisynchronization under controller (83). And the state trajectories and error curves of FMCGNNs are as shown in Figures 8–10, which demonstrate the effectiveness of Theorems 1 and 2.Figure 8
The first dimensional state trajectories of the drive system and the response system when the projective coefficientλ=−1.Figure 9
The second dimensional state trajectories of the drive system and the response system when the projective coefficientλ=−1.Figure 10
The synchronization error evolution when the projective coefficientλ=−1.


## 4.1. Projective Synchronization

First, we choose the projective coefficientλ=2. We can get the initial values Φ1s=y1s−2x1s=5.5, Φ2s=y2s−2x2s=−5.5, V1t0=11, and V2t0=60.5.According to Theorem1, it can be calculated as follows:(121)φ1=k1−1.1,φ2=k2−1.1,δ1=h1−1,δ2=h2−0.9,η1=q1−8.9,η2=q2−10.38.And we choosek1=3.1,k2=3.1,h1=1,h2=0.9,q1=9.9,andq2=11.38 which make φ=2andη=2.Then, we have a finite-time fractional-order differential inequation as follows:(122)Dt0CtαV1t≤−2V1t−2.Based on the result in Theorem1, drive-response system (118) could achieve synchronization in finite time. And we can calculate the setting time T1∗ as follows:(123)T1∗≤t0+φ−1/αlnφV1t0+ηαη≤2−1/0.98ln2×11+20.98×2=1.235.To verify Remark4, we can calculate T2 to demonstrate that T1∗ is less conservative than T2. It follows that(124)T2≤t0+V1t0Γ1+αη1/α=11⋅Γ1.9821/α=5.5778.On the contrary, we selectp=2 and ε1=ε2=1 for Theorem 2. And it is easy to calculate(125)φ¯1=1.9608k1−1.4,φ¯2=1.9608k2−1.85;δ¯1=1.9608h1−0.5,δ¯2=1.9608h2−0.45;η¯1=1.9608q1−8.9,η¯2=1.9608q2−10.38.Under the controller parameter,k1=3.1,k2=3.1,h1=1,h2=0.9,q1=9.9,andq2=11.38 which lead to φ¯=2.451andη¯=1.9608.Hence, we have another finite-time fractional-order differential inequation as follows:(126)Dt0CtαV2t≤−2.451V2t−1.9608V21/2t.Based on the result in Theorem2, we can estimate the setting time T3 as follows:(127)T3∗≤t0+Γ1+p/p−1−αφ¯Γ1+p/p−1Γ2−α−1/αlnφ¯V21/pt0+η¯η¯α=2.451Γ2.02Γ3Γ1.02−1/0.98ln2.451×60.5+1.96081.9608×0.98=1.9054.To verify Remark6, we can calculate T4 to demonstrate that T3∗ is less conservative than T4. It follows that(128)T4≤t0+V21pt0Γ1+pΓ2−αΓ1+αΓ1+p−αη¯1/α=60.5⋅Γ3Γ1.02Γ1.981.9608Γ2.021/0.98=8.0421.Consideringp=2andα=0.98, p−1/p<α1+α−1 holds. Thus, we have(129)T5≤t0+V2α1+α+1/p−2t0Γ1+αΓ1−2p−1/p1+αη¯⋅Γ2+αΓ1+α−2p−1/p1+α1/α=60.50.4404⋅Γ1.98Γ0.24241.9608⋅Γ2.98Γ1.22241/0.98=6.689.Therefore, drive-response system (118) can achieve finite-time projective synchronization under controller (83). And the state trajectories and error curves of FMCGNNs are as shown in Figures 2–4, which demonstrate the effectiveness of Theorems 1 and 2.Figure 2
The first dimensional state trajectories of the drive system and the response system when the projective coefficientλ=2.Figure 3
The second dimensional state trajectories of the drive system and the response system when the projective coefficientλ=2.Figure 4
The synchronization error evolution when the projective coefficientλ=2.


## 4.2. Complete Synchronization

First, we choose the projective coefficientλ=1. We can get Φ1s=y1s−2x1s=4, Φ2s=y2s−2x2s=−3, V1t0=7, and V2t0=25.According to Theorem1, it can be calculated as follows:(130)φ1=k1−1.1,φ2=k2−1.1;δ1=h1−1,δ2=h2−0.9;η1=q1−6.36,η2=q2−8.12.And we choosek1=3.1,k2=3.1,h1=1,h2=0.9,q1=7.36,andq2=9.12 which make φ=2andη=2.Then, we have a finite-time fractional-order differential inequation as follows:(131)Dt0CtαV1t≤−2V1t−2.Based on the result in Theorem1, drive-response system (118) could achieve synchronization in finite time. And we can calculate the setting time T1∗ as follows:(132)T1∗≤t0+φ−1/αlnφV1t0+ηαη≤2−1/0.98ln2×7+20.98×2=1.0351.To verify Remark4, we can calculate T2 to demonstrate that T1∗ is less conservative than T2. It follows that(133)T2≤t0+V1t0Γ1+αη1/α=7⋅Γ1.9821/α=3.5169.On the contrary, we selectp=2 and ε1=ε2=1 for Theorem 2. And it is easy to calculate(134)φ¯1=1.9608k1−1.4,φ¯2=1.9608k2−1.85;δ¯1=1.9608h1−0.5,δ¯2=1.9608h2−0.45;η¯1=1.9608q1−6.36,η¯2=1.9608q2−8.12.Under the controller parameter,k1=3.1,k2=3.1,h1=1,h2=0.9,q1=7.36,andq2=9.12 which lead to φ¯=2.451andη¯=1.9608.Hence, we have another finite-time fractional-order differential inequation as follows:(135)Dt0CtαV2t≤−2.451V2t−1.9608V21/2t.Based on the result in Theorem2, we can estimate the setting time T3 as follows:(136)T3∗≤t0+Γ1+p/p−1−αφ¯Γ1+p/p−1Γ2−α−1/αlnφ¯V21/pt0+η¯η¯α=2.451Γ2.02Γ3Γ1.02−1/0.98ln2.451×25+1.96081.9608×0.98=1.5937.To verify Remark6, we can calculate T4 to demonstrate that T3∗ is less conservative than T4. It follows that(137)T4≤t0+V21/pt0Γ1+pΓ2−αΓ1+αΓ1+p−αη¯1/α=25⋅Γ3Γ1.02Γ1.981.9608Γ2.021/0.98=5.1232.Consideringp=2andα=0.98, p−1/p<α1+α−1 holds. Thus, we have(138)T5≤t0+V2α1+α+1/p−2t0Γ1+αΓ1−2p−1/p1+αη¯⋅Γ2+αΓ1+α−2p−1/p1+α1/α=250.4404⋅Γ1.98Γ0.24241.9608⋅Γ2.98Γ1.22241/0.98=4.4965.Therefore, drive-response system (118) can achieve finite-time complete synchronization under controller (83). And the state trajectories and error curves of FMCGNNs are as shown in Figures 5–7, which demonstrate the effectiveness of Theorems 1 and 2.Figure 5
The first dimensional state trajectories of the drive system and the response system when the projective coefficientλ=1.Figure 6
The second dimensional state trajectories of the drive system and the response system when the projective coefficientλ=1.Figure 7
The synchronization error evolution when the projective coefficientλ=1.


## 4.3. Antisynchronization

First, we choose the projective coefficientλ=−1. We can get Φ1s=y1s−2x1s=1, Φ2s=y2s−2x2s=2, V1t0=3, and V2t0=5.According to Theorem1, it can be calculated as follows:(139)φ1=k1−1.1,φ2=k2−1.1;δ1=h1−1,δ2=h2−0.9;η1=q1−9.08,η2=q2−11.12.And we choosek1=3.1,k2=3.1,h1=1,h2=0.9,q1=10.08,andq2=12.12 which make φ=2andη=2.Then, we have a finite-time fractional-order differential inequation as follows:(140)Dt0CtαV1t≤−2V1t−2.Based on the result in Theorem1, drive-response system (118) could achieve synchronization in finite time. And we can calculate the setting time T1∗ as follows:(141)T1∗≤t0+φ−1/αlnφV1t0+ηαη≤2−1/0.98ln2×3+20.98×2=0.6934.To verify Remark4, we can calculate T2 to demonstrate that T1∗ is less conservative than T2. It follows that(142)T2≤t0+V1t0Γ1+αη1/α=3⋅Γ1.9821/α=1.4814.On the contrary, we selectp=2 and ε1=ε2=1 for Theorem 2. And it is easy to calculate(143)φ¯1=1.9608k1−1.4,φ¯2=1.9608k2−1.85;δ¯1=1.9608h1−0.5,δ¯2=1.9608h2−0.45;η¯1=1.9608q1−9.08,η¯2=1.9608q2−11.12.Under the controller parameter,k1=3.1,k2=3.1,h1=1,h2=0.9,q1=10.08,andq2=12.12 which lead to φ¯=2.451andη¯=1.9608.Hence, we have another finite-time fractional-order differential inequation as follows:(144)Dt0CtαV2t≤−2.451V2t−1.9608V21/2t.Based on the result in Theorem2, we can estimate the setting time T3 as follows:(145)T3∗≤t0+Γ1+p/p−1−αφ¯Γ1+p/p−1Γ2−α−1/αlnφ¯V21/pt0+η¯η¯α=2.451Γ2.02Γ3Γ1.02−1/0.98ln2.451×5+1.96081.9608×0.98=1.3539.To verify Remark6, we can calculate T4 to demonstrate that T3∗ is less conservative than T4. It follows that(146)T4≤t0+V21/pt0Γ1+pΓ2−αΓ1+αΓ1+p−αη¯1/α=5⋅Γ3Γ1.02Γ1.981.9608Γ2.021/0.98=2.2539.Consideringp=2andα=0.98, p−1/p<α1+α−1 holds. Thus, we have(147)T5≤t0+V2α1+α+1/p−2t0Γ1+αΓ1−2p−1/p1+αη¯⋅Γ2+αΓ1+α−2p−1/p1+α1/α=50.4404⋅Γ1.98Γ0.24241.9608⋅Γ2.98Γ1.22241/0.98=2.1816.Therefore, drive-response system (118) can achieve finite-time antisynchronization under controller (83). And the state trajectories and error curves of FMCGNNs are as shown in Figures 8–10, which demonstrate the effectiveness of Theorems 1 and 2.Figure 8
The first dimensional state trajectories of the drive system and the response system when the projective coefficientλ=−1.Figure 9
The second dimensional state trajectories of the drive system and the response system when the projective coefficientλ=−1.Figure 10
The synchronization error evolution when the projective coefficientλ=−1.


## 5. 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 time-varying delays.



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