# Finite-Time Stability Criteria for a Class of High-Order Fractional Cohen–Grossberg Neural Networks with Delay

Authors: Zhanying Yang; Jie Zhang; Junhao Hu; Jun Mei
Journal: Complexity
 (2020)

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## Abstract

This paper focuses on a class of delayed fractional Cohen–Grossberg neural networks with the fractional order between 1 and 2. Two kinds of criteria are developed to guarantee the finite-time stability of networks based on some analytical techniques. This method is different from those in some earlier works. Moreover, the obtained criteria are expressed as some algebraic inequalities independent of the Mittag–Leffler functions, and thus, the calculation is relatively simple in both theoretical analysis and practical applications. Finally, the feasibility and validity of obtained results are supported by the analysis of numerical simulations.


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## Body

## 1. Introduction

Neural networks have been paid much attention owing to the powerful applications in diverse fields. With the increasing requirements in practical applications, many researchers have made great efforts to develop various types of neural networks, such as Cohen–Grossberg neural networks (CGNNs) [1], cellular neural networks, bidirectional associative memory neural networks, and recurrent neural networks. As a kind of special recurrent neural network, CGNNs were firstly proposed by Cohen and Grossberg in [1]. CGNN is quite general since it includes some well-known types of neural networks, such as Hopfield neural network, cellular neural network, and shunting neural network. Nowadays, CGNNs have gained more and more interests due to their promising applications in classification, parallel computation and optimization, etc. Many researchers have made great contribution to the research on CGNNs; see [2–11] and the references therein.Nowadays, the fractional calculus has achieved significant progress in both theoretical research and practical applications. Compared with the integer-order derivative, the fractional-order derivative has some distinctive features, such as infinite memory and great freedom. Consequently, the fractional-order derivative can better characterize many systems in the real world [12–14]. In order to more accurately model the dynamics of neurons, various fractional-order neural networks (FONNs) have been generated based on the integration of the fractional calculus and neural networks. In the recent decades, the research on FONNs has undergone a prosperous development, and there have been numerous works (see [10, 15–20] and the references therein). As we all know, the successful applications of FONNs are closely associated to the dynamics of networks, among which stability has been an active topic. There have been substantial works on various types of stability, such as asymptotical stability [10], finite-time stability [21], exponential stability [22], Mittag–Leffler stability [8], and Lagrange stability [11]. Many sufficient conditions have been established to achieve the stability of systems; see [16, 18–20, 23, 24] and the references therein.Among various types of stability, finite-time stability (FTS) has aroused more interests in many fields, since many systems always operate over a limited period of time or it is necessary to focus on the behavior of systems within a limited period of time. In the existing works, there are mainly two concepts of FTS. One means that the error of any two state variables tends to zero in a limited time interval, which is also regarded as a special case of Lyapunov asymptotic stability. The other is also called finite-time boundedness, which describes that the quantity related to state does not exceed a prescribed threshold in a limited time interval for a given bound on the initial value [25, 26]. It is obvious that “boundedness” is a distinctive feature of this FTS. As revealed in [25, 26], it is essentially different from the classical Lyapunov asymptotic stability. In practical applications, this kind of FTS can provide a quantitative bound related to the state, and it has made great contribution for describing the transient performance of system.In the literature, there have been some interesting results [8–11, 23, 24, 27, 28] on fractional-order Cohen–Grossberg neural networks (FOCGNNs). For example, Ke and Miao [9] investigated the FTS for a class of delayed FOCGNNs by the generalized Bellman–Gronwall inequality. In [23], Zheng et al. studied the FTS and synchronization problem of a class of memristor-based FOCGNNs. Rajivganthi et al. [24] reported the FTS for a class of BAM FOCGNNs with time delays. In these works, the proofs are mainly based on some generalized Gronwall inequalities. In [8], Wan and Wu considered the Mittag–Leffler stability of fuzzy FOCGNNs with deviating argument. Recently, Pratap et al. [10] considered the asymptotic stability and pinning synchronization for a class of delayed FOCGNNs with discontinuous activations, and Huang et al. [11] investigated the Lagrange stability and the asymptotical stability for a class of delayed FOCGNNs. The proofs are mainly based on the Lyapunov theory and some properties related to the fractional calculus. For the FONNs in the aforementioned works, notice that the fractional order α is between 0 and 1. In the real world, the fractional systems with high fractional order can appropriately describe many phenomena and have been successfully applied in physics, biology, and information science (see, for instance, [29–32]). On the other hand, some classical methods for the case α∈0,1, such as some Lyapunov methods [10] and LMI method [33], could not be directly extended to the high-order cases. Therefore, it is significant to follow through the problems on high-order FONNs.For FONNs with the fractional orderα∈1,2, there have been many excellent works on the finite-time stability or finite-time synchronization [15–17, 34–37]. The analysis is mainly based on the Laplace transform, the inverse Laplace transform, and the generalized Gronwall inequality related to the Mittag–Leffler functions. However, this method can not be directly used to deal with the FTS for FOCGNNs with α∈1,2 owing to the technical reason. In order to solve this problem, it is desired to investigate a kind of different method.In this paper, we are devoted to the FTS for a class of delayed FOCGNNs withα∈1,2. The main contributions are summarized as follows: (i) The fractional order of system considered in this paper is between 1 and 2. A criterion is derived to achieve the FTS of system. Moreover, a criterion is established to ensure the FTS for the equilibrium point of system. (ii) The proofs are based on some analytical techniques, such as the Cauchy-Schwartz inequality, the generalized Gronwall inequality, and some properties of the Caputo derivative. This method is completely different from those in some earlier works [9, 15–17, 24, 34–37]. In particular, the obtained criteria are expressed as some algebraic inequalities and hence, the calculation is relatively easy in practical applications.


## 2. Preliminaries

This section starts with recalling some necessary definitions and properties related to the Caputo derivative.Definition 1.
(see [38]). Let μ∈ℝ+,m∈ℤ+ and m−1<μ<m For θt∈Cmt0,+∞,ℝn The Caputo derivative with fractional order μ of θ is defined by(1)Dtμt0Cθt=1Γm−μ∫t0tt−rm−μ−1θmrdr,t≥t0,where Γ⋅ denotes the Gamma function, i.e.,(2)Γr=∫0+∞tr−1e−tdt.Definition 2.
(see [38]). For μ∈ℝ+the fractional order integral with order μ of a function θt is defined by(3)Dt0,t−μθt=1Γμ∫t0tt−rμ−1θrdr.Proposition 1.
(see [39]). Let μ∈ℝ+, m∈ℤ+, and m−1<μ<m. If θt∈Cmt0,+∞,ℝ, then(4)Dt0,t−μDtμt0Cθt=θt−∑k=0m−1θkt0k!tk.Next, we list two inequalities, which will play a key role in the proofs of main results.Proposition 2.
(generalized Gronwall inequality [40]). Let ut, vt and wt be nonnegative Lq functions on the interval 0,T For q∈1,+∞ if(5)ut≤vt+wt∫0tuqrdr1/q,t∈0,T.Then,(6)∫0tuqrdr≤1−1−Φt1/q−q∫0tvqrΦrdr,where Φt=exp−∫0twqrdr.Proposition 3.
(generalized Bernoulli inequality [41]). Let t<1 and t≠0. For 0<d<1. we have 1−td<1−dt. Moreover, 1−1−td−1<dt−1.In what follows, we introduce a class of delayed FOCGNNs, which can be described as(7)0CDtαxit=−pixitqixit−∑j=1naijfjxit−∑j=1nbijgjxit−τ−Ii,i=1,2,⋯,n,where 1<α<2xit∈ℝ is the state of the i-th neuron at time t. pi⋅ stands for the amplification function and qi⋅ corresponds to the behaved function. The constant τ>0 denotes the time delay. aij and bij are the connection weights. fj and gj represent the activation functions. Ii stands for the constant external input.Forht∈C−τ,0,ℝm, the norm is defined ash=sup−τ≤r≤0∑i=1mhir. Let xt and yt stand for two arbitrary solutions for network (7), and let zt=xt−yt. The initial condition is given as follows:(8)zt=φt,z′t=ψt,t∈−τ,0,where φt,ψt∈C−τ,0,ℝnDefinition 3.
Let0<δ<ε. For 1<α<2, if φ,ψ<δ implies(9)zt<ε,∀t∈0,T,where zt=∑i=1nzit, then network (7) can achieve the finite-time stability w.r.t. δ,ε,T.


## 3. Main Results

In this section, we are devoted to two kinds of finite-time stability criteria for network (7) based on some properties related to the Caputo derivative and some inequalities.


### 3.1. Stability Criterion I for Network (7)

Let us first introduce some further assumptions on the parameters of network (7) and some necessary notation.(A1) The functionpii=1,2,…,nis a continuous and bounded function such that(10)0<p¯i≤pix≤p¯i,pix−piy≤p∗x−y,forx,y∈ℝ, where pi¯, p¯i, and p∗are some positive constants.(A2) For the functionspi and qii=1,2,⋯,n, there exists θi>0 such that(11)pixqix−piyqiy≤θix−y,∀x,y∈ℝ.(A3) The functionsfj and gjj=1,2,⋯,nare bounded and satisfy the Lipschitz conditions, namely,(12)fjx≤Fj,gjx≤Gj,fjx−fjy≤ξjx−y,gjx−gjy≤ηjx−y,∀x,y∈ℝ,where Fj, Gj, ξj and ηj are positive constants.Let(13)λ1=max1≤i≤nθi+p∗Ii+p∗∑j=1naijFj+bijGj+max1≤i≤n∑i=1nξjp¯iaij,λ2=max1≤j≤n∑i=1nηjp¯ibij.Theorem 1.
Let0<δ<ε. Under the assumptions (A1)–(A3), network (7) can achieve the FTS w.r.t. δ,ε,T if maxφ,ψ<δ and(14)1+t1+2eΛ2+1t1−e−Λ2t1/2<εδ,∀t∈0,T,where Λ=λ1+λ2e−t2Γ2α−1/2αΓα.Proof.
Letxt, yt, and zt be defined as in Section 2. By Proposition 1, we obtain(15)xit−yit=xi0−yi0+xi′0−yi′0t−1Γα∫0tt−rα−1pixirqixir−piyirqiyirdr+1Γα∫0tt−rα−1pixir∑j=1naijfjxjr−piyir∑j=1naijfjyirdr+1Γα∫0tt−rα−1pixir∑j=1nbijgjxjr−τ−piyir∑j=1nbijgjyjr−τdr+1Γα∫0tt−rα−1Iipixir−piyirdr.
Using the assumptions (A1)-(A2), we have(16)xit−yit≤xi0−yi0+xi′0−yi′0t+θi+p∗IiΓα∫0tt−rα−1xir−yirdr+1Γα∫0tt−rα−1pixir∑j=1naijfjxjr−piyir∑j=1naijfjyjrdr+1Γα∫0tt−rα−1pixir∑j=1nbijgjxjr−τ−piyir∑j=1nbijgjyjr−τdr.
For the termpixir∑j=1naijfjxjr−piyir∑j=1naijfjyjr, the assumptions (A1) and (A3) lead to(17)pixir∑j=1naijfjxjr−piyir∑j=1naijfjyjr=pixir∑j=1naijfjxjr−piyir∑j=1naijfjxjr+piyir∑j=1naijfjxjr−piyir∑j=1naijfjyjr≤∑j=1np∗aijFjxir−yir+∑j=1np¯iξjaijxjr−yjr.
In the same way, we obtain(18)pixir∑j=1nbijgjxjr−τ−piyir∑j=1nbijgjyjr−τ≤∑j=1np∗bijGjxir−yir+∑j=1np¯iηjbijxjr−τ−yjr−τ.
Substituting (17) and (18) into (16), we obtain(19)zit≤zi0+zi′0t+1Γα∫0tt−rα−1θi+p∗Ii+p∗∑j=1naijFj+bijGjzirdr+1Γα∫0tt−rα−1∑j=1nξjp¯iaijzjrdr+1Γα∫0tt−rα−1∑j=1nηjp¯ibijzjr−τdr.
Consequently,(20)zt≤z0+z′0t+λ1Γα∫0tt−rα−1zrdr+λ2Γα∫0tt−rα−1zr−τdr.
With the Cauchy–Schwartz inequality, we obtain(21)zt≤z0+z′0t+λ1Γα∫0tt−r2α−1e2rdr1/2∫0te−2rzr2dr1/2+λ2Γα∫0tt−r2α−1e2rdr1/2∫0te−2rzr−τ2dr1/2.
In view of∫0tt−r2α−1e2rdr<2e2t/4αΓ2α−1, we derive(22)zte−t≤z0e−t+z′0e−t+2Γ2α−12αΓαλ1∫0te−2rzr2dr1/2+λ2e−τ∫0te−2r−τzr−τ2dr1/2.
Letϖt=supt−τ≤t≤te−tzt˜, then, for any r∈0,t, we have(23)e−rzr≤ϖr,e−r−τzr−τ≤ϖr.
Thus, inequality (22) gives(24)ϖt≤δe−t1+t+Λ∫0tϖ2rdr1/2.
With Proposition2, this yields(25)∫0tϖ2sds1/2≤δ1−1−e−Λ2t1/2−1∫0t1+r2e−2re−Λrdr1/2.
Substituting this into inequality (24), we obtain(26)ϖt≤δe−t1+t+δΛ1−1−e−Λ2t1/2−1∫0t1+r2e−2re−Λ2rdr1/2.
By virtue of Proposition3, it follows that(27)ϖt≤δ1+te−t+2δeΛ2t1+t1−e−Λ2t1/2.
Thus,(28)zt≤δ1+t1+2eΛ2+1t1−e−Λ2t1/2.
With (14), this gives zt<ε for t∈0,T, which shows that network (7) achieves the FTS w.r.t. δ,ε,T. The proof is finished.


### 3.2. Stability Criterion II for Network (7)

In this section, we discuss the finite-time stability of equilibrium point for network (7). For the parameters of network (7), some further hypotheses [9] are given as follows:(H1) The functionpi⋅i=1,2,⋯,n is a continuous and bounded function such that 0<p¯i≤pi⋅≤p¯i on ℝ, where p¯i and p¯i are two positive constants.(H2) The functionqi⋅i=1,2,⋯,n is a monotonic differentiable function such that 0<q¯i≤qi′⋅≤q¯i on ℝ, where q¯i and q¯i are two positive constants.(H3) The functionsfj and gjj=1,2,⋯,n satisfy the Lipschitz conditions:(29)fjx−fjy≤ξjx−y,gjx−gjy≤ηjx−y,∀x,y∈ℝ,whereξj and ηj are two positive constants.(H4) Fori,j=1,2,⋯,n,aij,bij,q¯i,ξj and ηj satisfy the following condition:(30)∑i=1n1q¯imax1≤j≤na∗ξj+b∗ηj<1,where a∗=max1≤j≤n∑i=1naij,b∗=max1≤j≤n∑i=1nbij.By an argument similar to that in [9, 24], the assumptions (H1)–(H4) can guarantee the existence and uniqueness of equilibrium point for network (7). In what follows, we will concentrate on the finite-time stability for the equilibrium point x∗ Now, we introduce some notation. Let(31)ρ1=max1≤i≤np¯iq¯i+max1≤j≤na∗ξj∑i=1np¯i,ρ2=max1≤j≤nb∗ηj∑i=1np¯i,ϑ=ρ1+ρ2e−τ2Γ2α−12αΓα.For network (7), let xt represent an arbitrary solution with the initial conditions: xt=ϕ0t,x′tt∈−τ,0, where ϕ0t,ϕ1t∈C−τ,0,ℝn. Let ϕ−x∗=maxϕ0t−x∗,ϕ1t−x∗′.Theorem 2.
Let0<δ<ε.Under the assumptions (H1)–(H4), the unique equilibrium point x∗ of network (7) achieves the FTS w.r.t. δ,ε,T, if φ−x∗<δ and(32)1+t1+2eϑ2+1t1−e−ϑ2t1/2<εδ,∀t∈0,T.Proof.
Sincex∗ is the equilibrium point for system (7), we have(33)qixi∗−∑j=1naijfjxi∗−∑j=1nbijgjxi∗−Ii=0,i=1,2,⋯,n.
Based on (7) and (33), we use Proposition 1 to obtain(34)xit−xi∗=xi0−xi∗+xi′0t−1Γα∫0tt−rα−1pixiqixir−qixi∗dr+1Γα∫0tt−rα−1pixi∑j=1naijfjxjr−fjxj∗dr+1Γα∫0tt−rα−1pixi∑j=1nbijgjxjr−τ−gjxj∗dr.
For the termqixir−qixi∗, applying Lagrange’s mean value theorem, it follows that(35)qixir−qixi∗≤q¯ixir−xi∗.
Letuit=xit−xi∗. Obviously, equation (34) leads to(36)uit≤ui0+ui′0t+1Γα∫0tt−rα−1p¯iq¯iuir+∑j=1np¯iaijξjujr+∑j=1np¯ibijηjujr−τds.
Following the treatment similar to that of (20), we can obtain inequality (32).Remark 1.
Whenpixit=1 and qixit=cixitci>0, network (7) is reduced to that in [34]. The corresponding results can be easily derived from those in this paper.Remark 2.
Whenα=1, network (7) is reduced to an integer-order one. The corresponding finite-time criteria can be easily obtained by repeating the Proofs of Theorems 1 and 2.Remark 3.
Forα∈0,1, Ke and Miao [9] studied the FTS of equilibrium point for a class of delayed FOCGNNs based on the generalized Bellman–Gronwall inequality; Rajivganthi et al. [24] considered the FTS for a class of BAM FOCGNNs with delay by resorting to some inequalities; Zheng et al. [23] reported the FTS for a class of memristor-based FOCGNNs with delay based on a kind of Gronwall’s inequality. It seems to us that these methods can not be directly extended to the case of α∈1,2.Remark 4.
In the literature, there have been many works [15–17, 34–37] on the finite-time stability or finite-time synchronization for FONNs with α∈1,2.The obtained sufficient conditions are some inequalities involving the Mittag–Leffler functions. The proofs are mainly based on the Laplace transform, the inverse Laplace transform, and the generalized Gronwall–Bellman inequalities related to the Mittag–Leffler functions. However, this method is not applicable to network (7) owing to the technical reason. In the present paper, a kind of different method was used to discuss the FTS of network (7). More precisely, two kinds of finite-time criteria were obtained based on some properties of the Caputo derivative and some inequalities. Especially, these two criteria are expressed as some algebraic inequalities independent of the Mittag–Leffler functions. Therefore, the verification is relatively easy in practical applications.


## 3.1. Stability Criterion I for Network (7)

Let us first introduce some further assumptions on the parameters of network (7) and some necessary notation.(A1) The functionpii=1,2,…,nis a continuous and bounded function such that(10)0<p¯i≤pix≤p¯i,pix−piy≤p∗x−y,forx,y∈ℝ, where pi¯, p¯i, and p∗are some positive constants.(A2) For the functionspi and qii=1,2,⋯,n, there exists θi>0 such that(11)pixqix−piyqiy≤θix−y,∀x,y∈ℝ.(A3) The functionsfj and gjj=1,2,⋯,nare bounded and satisfy the Lipschitz conditions, namely,(12)fjx≤Fj,gjx≤Gj,fjx−fjy≤ξjx−y,gjx−gjy≤ηjx−y,∀x,y∈ℝ,where Fj, Gj, ξj and ηj are positive constants.Let(13)λ1=max1≤i≤nθi+p∗Ii+p∗∑j=1naijFj+bijGj+max1≤i≤n∑i=1nξjp¯iaij,λ2=max1≤j≤n∑i=1nηjp¯ibij.Theorem 1.
Let0<δ<ε. Under the assumptions (A1)–(A3), network (7) can achieve the FTS w.r.t. δ,ε,T if maxφ,ψ<δ and(14)1+t1+2eΛ2+1t1−e−Λ2t1/2<εδ,∀t∈0,T,where Λ=λ1+λ2e−t2Γ2α−1/2αΓα.Proof.
Letxt, yt, and zt be defined as in Section 2. By Proposition 1, we obtain(15)xit−yit=xi0−yi0+xi′0−yi′0t−1Γα∫0tt−rα−1pixirqixir−piyirqiyirdr+1Γα∫0tt−rα−1pixir∑j=1naijfjxjr−piyir∑j=1naijfjyirdr+1Γα∫0tt−rα−1pixir∑j=1nbijgjxjr−τ−piyir∑j=1nbijgjyjr−τdr+1Γα∫0tt−rα−1Iipixir−piyirdr.
Using the assumptions (A1)-(A2), we have(16)xit−yit≤xi0−yi0+xi′0−yi′0t+θi+p∗IiΓα∫0tt−rα−1xir−yirdr+1Γα∫0tt−rα−1pixir∑j=1naijfjxjr−piyir∑j=1naijfjyjrdr+1Γα∫0tt−rα−1pixir∑j=1nbijgjxjr−τ−piyir∑j=1nbijgjyjr−τdr.
For the termpixir∑j=1naijfjxjr−piyir∑j=1naijfjyjr, the assumptions (A1) and (A3) lead to(17)pixir∑j=1naijfjxjr−piyir∑j=1naijfjyjr=pixir∑j=1naijfjxjr−piyir∑j=1naijfjxjr+piyir∑j=1naijfjxjr−piyir∑j=1naijfjyjr≤∑j=1np∗aijFjxir−yir+∑j=1np¯iξjaijxjr−yjr.
In the same way, we obtain(18)pixir∑j=1nbijgjxjr−τ−piyir∑j=1nbijgjyjr−τ≤∑j=1np∗bijGjxir−yir+∑j=1np¯iηjbijxjr−τ−yjr−τ.
Substituting (17) and (18) into (16), we obtain(19)zit≤zi0+zi′0t+1Γα∫0tt−rα−1θi+p∗Ii+p∗∑j=1naijFj+bijGjzirdr+1Γα∫0tt−rα−1∑j=1nξjp¯iaijzjrdr+1Γα∫0tt−rα−1∑j=1nηjp¯ibijzjr−τdr.
Consequently,(20)zt≤z0+z′0t+λ1Γα∫0tt−rα−1zrdr+λ2Γα∫0tt−rα−1zr−τdr.
With the Cauchy–Schwartz inequality, we obtain(21)zt≤z0+z′0t+λ1Γα∫0tt−r2α−1e2rdr1/2∫0te−2rzr2dr1/2+λ2Γα∫0tt−r2α−1e2rdr1/2∫0te−2rzr−τ2dr1/2.
In view of∫0tt−r2α−1e2rdr<2e2t/4αΓ2α−1, we derive(22)zte−t≤z0e−t+z′0e−t+2Γ2α−12αΓαλ1∫0te−2rzr2dr1/2+λ2e−τ∫0te−2r−τzr−τ2dr1/2.
Letϖt=supt−τ≤t≤te−tzt˜, then, for any r∈0,t, we have(23)e−rzr≤ϖr,e−r−τzr−τ≤ϖr.
Thus, inequality (22) gives(24)ϖt≤δe−t1+t+Λ∫0tϖ2rdr1/2.
With Proposition2, this yields(25)∫0tϖ2sds1/2≤δ1−1−e−Λ2t1/2−1∫0t1+r2e−2re−Λrdr1/2.
Substituting this into inequality (24), we obtain(26)ϖt≤δe−t1+t+δΛ1−1−e−Λ2t1/2−1∫0t1+r2e−2re−Λ2rdr1/2.
By virtue of Proposition3, it follows that(27)ϖt≤δ1+te−t+2δeΛ2t1+t1−e−Λ2t1/2.
Thus,(28)zt≤δ1+t1+2eΛ2+1t1−e−Λ2t1/2.
With (14), this gives zt<ε for t∈0,T, which shows that network (7) achieves the FTS w.r.t. δ,ε,T. The proof is finished.


## 3.2. Stability Criterion II for Network (7)

In this section, we discuss the finite-time stability of equilibrium point for network (7). For the parameters of network (7), some further hypotheses [9] are given as follows:(H1) The functionpi⋅i=1,2,⋯,n is a continuous and bounded function such that 0<p¯i≤pi⋅≤p¯i on ℝ, where p¯i and p¯i are two positive constants.(H2) The functionqi⋅i=1,2,⋯,n is a monotonic differentiable function such that 0<q¯i≤qi′⋅≤q¯i on ℝ, where q¯i and q¯i are two positive constants.(H3) The functionsfj and gjj=1,2,⋯,n satisfy the Lipschitz conditions:(29)fjx−fjy≤ξjx−y,gjx−gjy≤ηjx−y,∀x,y∈ℝ,whereξj and ηj are two positive constants.(H4) Fori,j=1,2,⋯,n,aij,bij,q¯i,ξj and ηj satisfy the following condition:(30)∑i=1n1q¯imax1≤j≤na∗ξj+b∗ηj<1,where a∗=max1≤j≤n∑i=1naij,b∗=max1≤j≤n∑i=1nbij.By an argument similar to that in [9, 24], the assumptions (H1)–(H4) can guarantee the existence and uniqueness of equilibrium point for network (7). In what follows, we will concentrate on the finite-time stability for the equilibrium point x∗ Now, we introduce some notation. Let(31)ρ1=max1≤i≤np¯iq¯i+max1≤j≤na∗ξj∑i=1np¯i,ρ2=max1≤j≤nb∗ηj∑i=1np¯i,ϑ=ρ1+ρ2e−τ2Γ2α−12αΓα.For network (7), let xt represent an arbitrary solution with the initial conditions: xt=ϕ0t,x′tt∈−τ,0, where ϕ0t,ϕ1t∈C−τ,0,ℝn. Let ϕ−x∗=maxϕ0t−x∗,ϕ1t−x∗′.Theorem 2.
Let0<δ<ε.Under the assumptions (H1)–(H4), the unique equilibrium point x∗ of network (7) achieves the FTS w.r.t. δ,ε,T, if φ−x∗<δ and(32)1+t1+2eϑ2+1t1−e−ϑ2t1/2<εδ,∀t∈0,T.Proof.
Sincex∗ is the equilibrium point for system (7), we have(33)qixi∗−∑j=1naijfjxi∗−∑j=1nbijgjxi∗−Ii=0,i=1,2,⋯,n.
Based on (7) and (33), we use Proposition 1 to obtain(34)xit−xi∗=xi0−xi∗+xi′0t−1Γα∫0tt−rα−1pixiqixir−qixi∗dr+1Γα∫0tt−rα−1pixi∑j=1naijfjxjr−fjxj∗dr+1Γα∫0tt−rα−1pixi∑j=1nbijgjxjr−τ−gjxj∗dr.
For the termqixir−qixi∗, applying Lagrange’s mean value theorem, it follows that(35)qixir−qixi∗≤q¯ixir−xi∗.
Letuit=xit−xi∗. Obviously, equation (34) leads to(36)uit≤ui0+ui′0t+1Γα∫0tt−rα−1p¯iq¯iuir+∑j=1np¯iaijξjujr+∑j=1np¯ibijηjujr−τds.
Following the treatment similar to that of (20), we can obtain inequality (32).Remark 1.
Whenpixit=1 and qixit=cixitci>0, network (7) is reduced to that in [34]. The corresponding results can be easily derived from those in this paper.Remark 2.
Whenα=1, network (7) is reduced to an integer-order one. The corresponding finite-time criteria can be easily obtained by repeating the Proofs of Theorems 1 and 2.Remark 3.
Forα∈0,1, Ke and Miao [9] studied the FTS of equilibrium point for a class of delayed FOCGNNs based on the generalized Bellman–Gronwall inequality; Rajivganthi et al. [24] considered the FTS for a class of BAM FOCGNNs with delay by resorting to some inequalities; Zheng et al. [23] reported the FTS for a class of memristor-based FOCGNNs with delay based on a kind of Gronwall’s inequality. It seems to us that these methods can not be directly extended to the case of α∈1,2.Remark 4.
In the literature, there have been many works [15–17, 34–37] on the finite-time stability or finite-time synchronization for FONNs with α∈1,2.The obtained sufficient conditions are some inequalities involving the Mittag–Leffler functions. The proofs are mainly based on the Laplace transform, the inverse Laplace transform, and the generalized Gronwall–Bellman inequalities related to the Mittag–Leffler functions. However, this method is not applicable to network (7) owing to the technical reason. In the present paper, a kind of different method was used to discuss the FTS of network (7). More precisely, two kinds of finite-time criteria were obtained based on some properties of the Caputo derivative and some inequalities. Especially, these two criteria are expressed as some algebraic inequalities independent of the Mittag–Leffler functions. Therefore, the verification is relatively easy in practical applications.


## 4. Numerical Simulations

In this section, two examples are presented to illustrate the effectiveness of two criteria.Example 1.
Consider the following FOCGNN model:(37)0CDt1.8xit=−pixitqixit−∑j=13aijfjxjt−∑j=13bijgjxjt−0.1−Ii,i=1,2,3.
Here,pixit=0.2sinxit+0.3 and qixit=0.8cosxit for i=1,2,3,a11=0.027, a12=0.008, a13=0.029, a21=0.018, a22=0.017, a23=0.005, a31=0.003, a32=0.029, a33=0.029, b11=0.029, b12=0.004, b13=0.024, b21=0.015, b22=0.013, b23=0.029, b31=0.024, b32=0.028, b33=0.02, fjxjt=0.05xjt+1−xjt−1, gjxjt−0.1=0.05xjt−0.1+1−xjt−0.1−1, I1=0.0478, I2=−0.014, and I3=0.081.
Obviously,0.1≤p1x1≤0.5, 0.1≤p2x2≤0.5, 0.1≤p3x3≤0.5, p∗=0.2, θ1=θ2=θ3=0.56, Fj=Gj=ξj=ηj=0.1. Moreover, we obtain λ1=0.5815, λ2=0.011, and Λ=0.218
Letxt and yt be two solutions of network (37) with the initial conditions:(38)xt=1.12−0.007t,1.56+0.002t,−1.53+0.01tT,x′t=−0.007,0.002,0.01T,yt=1.125−0.001t,1.566+0.006t,−1.525+0.02tT,y′t=−0.001,0.006,0.02T,for t∈−0.1,0. The time curves for xt and yt are shown in Figure 1.
Based on the initial conditions,δ is taken as δ=0.02. Let ε=1. Inequality (14) gives the settling time Ts=2.7424. The time response of xt−yt is depicted in Figure 2. Obviously, xt−yt<1 holds for any t∈0,2.7424 which coincides with the result of Theorem 1.Figure 1
The time response forxt and yt.
(a)(b)(c)Figure 2
The time response for the error betweenxt and yt.Example 2.
Consider the following FOCGNN model:(39)0CDt1.6xit=−pixitqixit−∑j=13aijfjxjt−∑j=13bijgjxjt−0.1−Ii,for i=1,2,3, where pixit=0.1cosxit+0.2i=1,2,3, q1x1t=0.25x1t, q2x2t=0.4x2t, q3x3t=0.5x3t, a11=0.04, a12=0.05, a13=−0.01, a21=0.04, a22=−0.03, a23=−0.03, a31=−0.06, a32=0.05, a33=0.05, b11=−0.05, b12=0.05, b13=0.09, b21=0.08, b22=−0.024, b23=−0.02, b31=0.05, b32=−0.04, b33=0.04, fjxjt=0.025xjt+1−xjt−1, gjxjt−0.1=0.025xjt−0.1+1−xjt−0.1−1,I1=0.042,I2=0.061,andI3=0.039.
Obviously,(40)0.1≤p1x1≤0.3,0.1≤p2x2≤0.3,0.1≤p3x3≤0.3,dq1x1dx1=0.25,dq2x2dx2=0.4,dq3x3dx3=0.5.
From the above data, we takeξj=ηj=0.05j=1,2,3, a∗=0.14, and b∗=0.18. Moreover,(41)∑i=1n1q¯imax1≤j≤na∗ξj+b∗ηj=0.136<1.
This indicates that network (39) has a unique equilibrium point x∗. Based on (33), we have(42)0.25x1∗+0.01f1x1∗−0.1f2x2∗−0.08f3x3∗−0.042=0,0.4x2∗−0.12f1x1∗+0.054f2x2∗+0.05f3x3∗−0.061=0,0.5x3∗+0.01f1x1∗−0.01f2x2∗−0.09f3x3∗−0.039=0.
This gives the equilibrium pointx∗=0.17,0.15,0.08T.
Letxt and yt be two solutions with the initial conditions:(43)xt=0.15+0.0015t,0.16−0.001t,0.1−0.001tT,x′t=0.0015,−0.001,−0.001T,yt=0.18−0.001t,0.13+0.0012t,0.007+0.0012tT,y′t=−0.001,0.0012,0.0012T,for any t∈−0.1,0. The time curves are depicted in Figure 3. Moreover, the time evolution for xt−x∗ and yt−x∗ is shown in Figure 4.
Based on the above data, it follows thatρ1=0.1563, ρ2=0.0081 and ϑ=0.0897. We take δ=0.05. Let ε=1. The condition (32) gives the settling time Ts=2.694. From Figure 4, it can be checked that xt−x∗<1 and yt−x∗<1 hold for t∈0,2.694. This fact is consistent with Theorem 2.Figure 3
The time response forxt and yt.
(a)(b)(c)Figure 4
The time response forxt−x∗ and yt−x∗.


## 5. Conclusions

In the recent decade, many efforts have been made to the research on FONNs with the fractional order between 1 and 2. The methods are mainly based on the Laplace transform, the inverse Laplace transform, and the generalized Gronwall inequality related to the Mittag–Leffler functions. However, these methods do not work well for the considered FOCGNNs owing to the technical reason. In this paper, the finite-time stability criteria were derived based on the analytic techniques and some inequalities. This kind of method is completely different from the above ones. In particular, the obtained criteria are expressed as some algebraic inequalities independent of the Mittag–Leffler functions and thus, they can be easily verified in practical applications. In the future work, we will investigate the finite-time guaranteed cost control for FONNs with high fractional order. It seems to us that some new techniques would be developed to deal with this problem.



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