# Stability Analysis of Fractional-Order Bidirectional Associative Memory Neural Networks with Mixed Time-Varying Delays

Authors: Zhanying Yang; Jie Zhang
Journal: Complexity
 (2019)

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## Abstract

This paper studies the stability analysis of fractional-order bidirectional associative memory neural networks with mixed time-varying delays. The orders of these systems lie in the interval1,2. Firstly, a sufficient condition is derived to ensure the finite-time stability of systems by resorting to some analytical techniques and some elementary inequalities. Next, a sufficient condition is obtained to guarantee the global asymptotic stability of systems based on the Laplace transform, the mean value theorem, the generalized Gronwall inequality, and some properties of Mittag–Leffler functions. In particular, these obtained conditions are expressed as some algebraic inequalities which can be easily calculated in practical applications. Finally, some numerical examples are given to verify the feasibility and effectiveness of the obtained main results.


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## Body

## 1. Introduction

Neural networks have drawn increasing interests due to their powerful applications in physics, mechanics, biology, information science, and sociology In order to meet the requirements in practical applications, some researchers have proposed various types of neural networks, such as cellular neural networks [1], Hopfield neural networks [2], recurrent neural networks [3], bidirectional associative memory (BAM) neural networks [4], and memristor-based neural networks [5]. In [4], Kosko first proposed bidirectional associative memory neural networks to store and invoke pattern pairs. A BAM neural network consists of two layers of associative neurons, and the neurons arranged in one layer are fully interconnected with those in the other layer, but there are no interconnections in the same layer. It has been revealed that BAM neural networks can provide potential applications in pattern recognition, signal processing, and combinatorial optimization [6, 7].In the recent decades, the fractional calculus has been paid considerable attention owing to its great development in theory and application. As is well known, fractional-order derivative is a very powerful tool to describe memory and hereditary properties of many materials and dynamical processes, and hence it can be well used to characterize a large number of systems [8–10] in many fields. In order to describe the dynamical behavior of neurons better, fractional-order neural networks have been put forward by combining the fractional calculus with neural networks. As an important type of neural network, fractional-order BAM neural network has been widely studied. In recent years, many great contributions have been made to the dynamics of systems [11–22] which is closely related to the applications in various fields. For example, the references [11, 14–20] reported some research on the global asymptotic stability, Mittag–Leffler stability, uniform stability, and finite-time stability. In [12, 21, 22], the authors investigated the adaptive synchronization, the finite-time synchronization, and the Mittag–Leffler synchronization for several kinds of fractional-order BAM neural networks.Notice that the order of systems in most works lies in the interval0,1. In the real world, it is also very significant to focus on fractional-order neural networks with the order α∈1,2 due to their successful applications. For example, for the second-order multiagent dynamics, a fractional-order observer with the order α∈1,2 can be used to capture the velocity information which is not always available [23]. In addition, the fractional-order systems with α∈1,2 have been extensively studied in mechanics, physics, and information science [24–26]. To the best of our knowledge, most results for α∈0,1 could not be directly extended to the case of α∈1,2 due to its more complicated mathematical theory. Thus, it is very attractive to carry out the study on fractional-order neural networks with α∈1,2 (see [12–14, 27–30] and the references therein). In [12, 13, 30], the authors considered the finite-time synchronization for several classes of fractional-order memristor-based neural networks with time delays. In [27], Wu et al. studied the finite-time stability for fractional-order delayed neural networks. Rakkiyappan et al. [28] reported the finite-time stability for a class of fractional-order complex-valued memristor-based neural networks with time delays. Chen et al. [29] discussed the finite-time stability for a class of fractional-order memristor-based neural networks with time delays. Recently, Xu et al. [20] considered the finite-time stability for a class of fractional-order BAM delayed neural networks. In the above works, the proofs are mainly based on the Laplace transform, the generalized Gronwall inequality and some properties of the Mittag–Leffler functions, and hence the obtained sufficient conditions are closely related to the Mittag–Leffler functions.In the aforementioned works, it is noted that only the discrete constant delays are involved in the network models. However, as revealed in [31–33], neural networks usually have a spatial nature due to the presence of a large number of parallel pathways with various axon sizes and lengths, so that there is a distribution of propagation delays over a period of time. In this situation, discrete delays cannot well characterize the neural networks since the signal propagation is no longer instantaneous. Consequently, the distributed delays should be also taken into account in the description of neural network models. In the recent decades, lots of researchers have made great efforts to the dynamics of neural networks with both discrete and distributed delays and there have been some excellent results [33–38]. Notice that these works were mainly concerned with integer-order neural networks. The research on fractional-order neural networks with discrete and distributed time delays has received little attention. In [39, 40], the authors considered the global Mittag–Leffler stability and the global exponential stability for Caputo fractional-order neural network with discrete and infinite-time distributed delays. Recently, Zhang et al. [41] investigated the asymptotic stability for a class of Riemann–Liouville fractional-order neural networks with discrete and finite-time distributed constant delays. In [42], Wu et al. studied the uniform stability of Caputo fractional-order neural networks with discrete and finite-time distributed constant delays. It is worth pointing out that the order of systems in these works lies in the interval 0,1. To the best of our knowledge, the stability analysis for the case of α∈1,2 has not been dealt within the existing literature.In this paper, we focus on a class of Caputo fractional-order BAM neural networks with discrete and distributed time-varying delays forα∈1,2. These networks can be described as(1)D0Ctαxt=−Cxt+Af1yt+Bg1yt−τt+R∫t−σtth1yθdθ+I,D0Ctαyt=−Dyt+Pf2xt+Qg2xt−τt+S∫t−σtth2xθdθ+J,where xt and yt denote the state vectors in the X-layer and Y-layer, respectively and τt and σt are the discrete and distributed time-varying delays which are continuous and bounded. The remaining notations will be specifically introduced in the next section. The main contributions of this paper include the following several aspects: (i) We consider the stability analysis for a class of fractional-order BAM neural networks with discrete and distributed time-varying delays for α∈1,2 which has not been discussed in the existing literature. (ii) For the finite-time stability problem of the system under consideration, the classical method [13, 27–30] can give a sufficient condition which is closely related to the Mittag–Leffler function. This condition cannot be easily calculated, and the settling time cannot be easily estimated. Here, we apply some analytical techniques and some elementary inequalities to investigate this problem. Under the assumption τ˙t≤μ<1 (μ is a positive constant), we derive a sufficient condition independent of the Mittag–Leffler function. It is noted that our sufficient condition can be expressed as an algebraic inequality and the estimated settling time T can be easily obtained in practical applications. (iii) To guarantee the global asymptotic stability of systems, a sufficient condition is obtained by resorting to the Laplace transform, the mean value theorem, the generalized Gronwall inequality, and some properties of Mittag–Leffler functions. Different from the above finite-time stability analysis, this does not require the assumption τ˙t≤μ<1. In addition, the obtained condition generalizes Corollary 1 in [27].The rest of this paper is organized as follows. Section2 is devoted to some preliminaries and network model. In Section 3, a sufficient condition is firstly derived to ensure the finite-time stability of systems. Next, a sufficient condition is obtained to ensure the global asymptotic stability of systems. Finally, some results related to the equilibrium point are directly given. In Section 4, some examples are provided to verify the effectiveness of the obtained main results. Finally, Section 5 gives some conclusions and presents some possible research in future.


## 2. Preliminaries and Network Model

In this section, we first recall some definitions and properties related to the Caputo derivative and Mittag–Leffler functions. Next we list some important inequalities and introduce the network model.Definition 1 (see [43]).
Letα>0 and let m be a positive integer such that m−1<α<m. For ft∈Cmt0,+∞,ℝn, the Caputo derivative of fractional order a of ft is defined by(2)Dtαt0Cft=1Γm−α∫t0tt−sm−α−1fmsds,t≥t0,where Γ⋅ is the Gamma function, i.e.,(3)Γs=∫0+∞ts−1e−tdt.Definition 2 (see [44]).
The fractional integral with noninteger orderα>0 of a function ft is defined as(4)Dt0−αft=1Γα∫t0tt−sα−1fsds,where t≥t0 and Γ⋅ is the Gamma function.Proposition 1 (see [45]).
Letα>0 and let m be a positive integer such that m−1<α<m. For xt∈Cm0,∞,ℝn, we have(5)D0−αD0Ctαxt=xt−∑k=0m−1xk0k!tk.Definition 3 (see [43]).
The Mittag–Leffler function with two parameters is defined as(6)Eα,βz=∑k=0∞zkΓkα+β,where α>0, β>0, and z∈C.
LetEαz denote the Mittag–Leffler function with one parameter, i.e., Eαz=∑k=0∞zk/Γkα+1. Obviously, we have Eα,1z=Eαz.
Letℒ be the Laplace transform operator. Then, we have(7)ℒtβ−1Eα,β−λtα=sα−βsα+λ,Res>λ1/α,ℒD0Ctαft=sαFs−∑k=0m−1sα−k−1fk0,m=α+1,where Fs denotes the Laplace transform of ft∈Cm0,+∞,ℝn.Proposition 2 (see [43]).
Let0<α<2. Suppose that µ satisfies απ/2<μ<minπ,απ. Then, for z≥0 and argz≤μ,(8)Eαz≤μ1expRez1/α+μ21+z,where μ1 and μ2 are two positive real constants.Proposition 3 (see [46]).
Let1≤α<2 and β=1,2,α. Suppose that A is a diagonal stability matrix. For any t≥0, we have(9)Eα,βAtα≤e−ωt,where −ω is the largest eigenvalue of the matrix A and ⋅ denotes any vector or induced matrix norm.
Now we present some inequalities which are crucial to our main results.Proposition 4 (generalized Bernoulli inequality [47]).
Letx<1 and x≠0. For 0<k<1, we have 1−xk<1−kx. Moreover, 1−1−xk−1<kx−1.Proposition 5 (see [48]).
Let1≤p<∞ and let ht, vt, and wt be nonnegative Lp functions on 0,T. If(10)ht≤vt+wt∫0thpsds1/p,t∈0,T,then,(11)∫0thpsds≤1−1−Wt1/p−p∫0tvpsWsds,where Wt=exp−∫0twpsds.Proposition 6 (generalized Gronwall inequality [49]).
Letq>0. Let ft be a nonnegative function locally integrable on 0≤t<T (some T≤+∞). Suppose that gt is a nonnegative, nondecreasing continuous function defined on 0≤t<T, then there exists a constant M such that gt≤M. Let wt be nonnegative and locally integrable on 0≤t<T. If wt satisfies(12)wt≤ft+gt∫0tt−sq−1wsds,on 0,T, then(13)wt≤ft+∫0t∑n=1∞gtΓqnΓnqt−snq−1fsds.
Moreover, ifft is a nondecreasing function on 0,T, then we have(14)wt≤ftEq,1gtΓqtq.
In what follows, we consider a class of Caputo fractional-order BAM neural networks with mixed time-varying delays which can be described as equation (1) or(15)D0Ctαxit=−cixit+∑j=1maijf1jyjt+∑j=1mbijg1jyjt−τt+∑j=1mrij∫t−σtth1jyjθdθ+Ii,D0Ctαyjt=−djyjt+∑i=1npjif2ixit+∑i=1nqjig2ixit−τt+∑i=1nsji∫t−σtth2ixiθdθ+Jj,where 1<α<2, i=1,2,…,n, and j=1,2,…,m. xit and yjt denote the state of neurons in the X-layer and Y-layer, respectively. The constants ci>0 and dj>0 are the self-regulating parameters of the neurons. The discrete time-varying delay τt and the distributed time-varying delay σt are continuous functions such that 0<τt≤τ¯ and 0<σt≤σ¯, where τ¯ and σ¯ are two positive constants. The constants aij, bij, and rij are the connection, the discretely delayed connection, and the distributively delayed connection. The constants pji, qji, and sji have the same meanings as aij, bij, and rij. f1j, g1j, h1j, f2i, g2i, and h2i represent the activation functions. Ii and Jj stand for the external inputs.
Besides, the activation functionsf1jx, g1jx, f2ix, and g2ix satisfy the following Lipschitz conditions:(16)f1jx−f1jy≤ξ1x−y,g1jx−g1jy≤η1x−y,h1jx−h1jy≤ζ1x−y,f2ix−f2iy≤ξ2x−y,g2ix−g2iy≤η2x−y,h2ix−h2iy≤ζ2x−y,for any x,y∈ℝ, where i=1,2,…,n and j=1,2,…,m and ξ1, ξ2, η1, η2, ζ1, and ζ2 are some positive constants.
Letκ=maxτ¯,σ¯ and let C−κ,0,ℝk denote the Banach space of k-dimensional continuous vector functions defined on the interval −κ,0. The norm of ht∈C−κ,0,ℝk is defined as ht=sup−κ≤s≤0∑i=1khis.
Letxt,yt and x¯t,y¯t be any two solutions of system (15) with different initial conditions. Let ut=xt−x¯t and vt=yt−y¯t. The initial conditions of ut,vt are given as follows:(17)uikt=φikt,vjkt=ψjkt,t∈−κ,0,k=0,1,orukt=φkt,vkt=ψkt,t∈−κ,0,k=0,1,where φkt∈C−κ,0,ℝn and ψkt∈C−κ,0,ℝm.Definition 4.
System (15) is said to be asymptotically stable if(18)ut+vt⟶0,t⟶+∞,where ut=∑i=1nuit and vt=∑j=1mvjt.Definition 5.
Suppose thatδ and ε are any positive constants such that δ<ε. If maxφ0t+ψ0t,φ1t+ψ1t<δ implies(19)ut+vt<ε,∀t∈0,T,then system (15) is said to achieve the finite-time stability with respect to δ,ε,T.


## 3. Main Results

In this section, we will consider the finite-time stability and the global asymptotic stability for a class of fractional-order BAM neural networks with mixed time-varying delays for1<α<2.


### 3.1. Finite-Time Stability Analysis

In this subsection, some sufficient conditions are derived to ensure the finite-time stability of system (15). For the time-varying delay τt, we need to make further assumptions.Assumption 1.
The discrete time-varying delayτt is a differentiable function such that(20)τ˙t≤μ<1,where µ is a constant.
In what follows, we state the main theorem. For convenience, let us first introduce some notation related to the parameters of system (15). Let γ1=max1≤i≤nci and γ2=max1≤j≤mdj. Let(21)a∗=max1≤j≤m∑i=1naij,b∗=max1≤j≤m∑i=1nbij,r∗=max1≤j≤m∑i=1nrij,p∗=max1≤i≤n∑j=1mpji,q∗=max1≤i≤n∑j=1mqji,s∗=max1≤i≤n∑j=1msji.
Let(22)λ1=maxγ2+ξ1a∗,γ1+ξ2p∗,λ2=maxη1b∗,η2q∗,λ3=maxζ1r∗,ζ2s∗.Theorem 1.
Letδ and ε be any positive constants such that δ<ε. Suppose that Assumption 1 holds. If maxφ0t+ψ0t,φ1t+ψ1t<δ and(23)1+t+β1et+2eβ22+1t1+t+β11−e−β22t1/2<εδ,∀t∈0,T,where β1=λ2/ΓαΓ2α−1e2τ¯−1/4α1−μ and β2=λ1+λ3σ¯/Γα2Γ2α−1/4α+λ2/Γα2Γ2α−1/4α1−μ, and then system (15) is finite-time stable with respect to δ,ε,T.Proof.
Letxt,yt and x¯t,y¯t be two solutions of system (15) with different initial conditions. Let ut=xt−x¯t and vt=yt−y¯t. Assume that ut,vt satisfies the initial condition (17).
Proposition 1 yields(24)uit=φi00+tφi10+1Γα∫0tt−sα−1−ciuis+∑j=1maijf1jyjs−f1jy¯js+∑j=1mbijg1jyjs−τs−g1jy¯js−τs+∑j=1mrij∫s−σssh1jyjθ−h1jy¯jθdθds.
According to the Lipschitz conditions, we obtain(25)uit≤φi00+tφi10+1Γα∫0tt−sα−1ciuis+∑j=1mξ1aijvjs+∑j=1mη1bijvjs−τs+∑j=1mζ1rij∫s−σssvjθdθds.
Similarly, we have(26)vjt≤ψi00+tψi10+1Γα∫0tt−sα−1djvjs+∑i=1nξ2pjiuis+∑i=1nη2qjiuis−τs+∑i=1nζ2sji∫s−σssuiθdθds.
Moreover, it follows that(27)ut=∑i=1nuit≤φ00+tφ10+1Γα∫0tt−sα−1∑i=1nciuis+∑j=1mξ1∑i=1naijvjs+∑j=1mη1∑i=1nbijvjs−τs+∑j=1mζ1∑i=1nrij∫s−σssvjθdθds≤φ00+tφ10+1Γα∫0tt−sα−1γ1us+ξ1a∗vs+η1b∗vs−τs+ζ1r∗∫s−σssvθdθds.
Meanwhile, we have(28)vt=∑j=1mvjt≤ψ00+tψ10+1Γα∫0tt−sα−1γ2vs+ξ2p∗us+η2q∗us−τs+ζ2s∗∫s−σssuθdθds.
Together with the estimates ofut and vt, it follows that(29)ut+vt≤φ00+ψ00+tφ10+ψ10+1Γα∫0tt−sα−1γ1+ξ2p∗us+γ2+ξ1a∗vs+η2q∗us−τs+η1b∗vs−τs+ζ1r∗∫s−σssvθdθ+ζ2s∗∫s−σssuθdθds≤φ00+ψ00+tφ10+ψ10+1Γα∫0tt−sα−1λ1us+vs+λ2us−τs+vs−τs+λ3∫s−σssuθ+vθdθds.
Making use of the Cauchy–Schwartz inequality, we obtain(30)ut+vt≤φ00+ψ00+tφ10+ψ10+λ1Γα∫0tt−s2α−1e2sds1/2∫0te−2sus+vs2ds1/2+λ2Γα∫0tt−s2α−1e2sds1/2∫0te−2sus−τs+vs−τs2ds1/2+λ3Γα∫0tt−s2α−1e2sds1/2∫0te−2s∫s−σssuθ+vθdθ2ds1/2.
For the integral∫0tt−s2α−1e2sds, we derive(31)∫0tt−s2α−1e2sds=e2t∫0ts2α−1e−2sds=2e2t4α∫02ts2α−2e−sds<2e2t4αΓ2α−1.
For the integral∫0te−2sus−τs+vs−τs2ds, we deduce(32)∫0te−2sus−τs+vs−τs2ds=∫0te−2s1−τ˙s−1us−τs+vs−τs21−τ˙sds.
Combining this with the assumption onτt, we obtain(33)∫0te−2sus−τs+vs−τs2ds≤11−μ∫0te−2sus−τs+vs−τs21−τ˙sds≤11−μ∫−τte−2ωuω+vω2dω=11−μ∫−τ0e−2sus+vs2ds+11−μ∫0te−2sus+vs2ds≤e2τ¯−121−μφ0+ψ02+11−μ∫0te−2sus+vs2ds.
Moreover, it follows that(34)∫0te−2sus−τs+vs−τs2ds1/2≤e2τ¯−121−μφ0+ψ0+11−μ∫0te−2sus+vs2ds1/2.
From the inequalities (30)–(34), we get(35)e−tut+vt≤e−tφ00+ψ00+te−tφ10+ψ10+λ1Γα2Γ2α−14α∫0te−2sus+vs2ds1/2+λ2ΓαΓ2α−1e2τ−14α1−μφ0+ψ0+λ2Γα2Γ2α−14α1−μ∫0te−2sus+vs2ds1/2+λ3Γα2Γ2α−14α∫0te−2s∫s−σssuθ+vθdθ2ds1/2.
Letρt=supt−κ≤t˜≤te−t˜ut˜+vt˜, where κ=maxτ,σ. Obviously, we have(36)e−tut+vt≤ρt.
In addition, for any fixedt, we use the mean value theorem to get(37)e−t∫t−σttuθ+vθdθ=e−tσtuβt+vβt,where t−σt<βt<t. Moreover,(38)e−t∫t−σttuθ+vθdθ=e−tσteβte−βtuβt+vβt≤σ¯ρt.
From the inequalities (35)–(38), we have(39)e−tut+vt≤λ2ΓαΓ2α−1e2τ−14α1−μφ0+ψ0+e−tφ00+ψ00+te−tφ10+ψ10+λ1Γα2Γ2α−14α∫0tρ2sds1/2+λ2Γα2Γ2α−14α1−μ∫0tρ2sds1/2+λ3σ¯Γα2Γ2α−14α∫0tρ2sds1/2.
Moreover,(40)ρt≤λ2ΓαΓ2α−1e2τ¯−14α1−μφ0+ψ0+e−tφ00+ψ00+te−tφ10+ψ10+λ1+λ3σ¯Γα2Γ2α−14α+λ2Γα2Γ2α−14α1−μ∫0tρ2sds1/2.
Hence,(41)ρt≤δβ1+e−t+te−t+β2∫0tρ2sds1/2,where β1=λ2/ΓαΓ2α−1e2τ¯−1/4α1−μ and β2=λ1+λ3σ¯/Γα2Γ2α−1/4α+λ2/Γα2Γ2α−1/4α1−μ.
In view of Proposition 5, we obtain(42)∫0tρ2sds1/2≤1−1−e−β22t1/2−1⋅∫0tδ2β1+e−s+se−s2e−β22sds1/2.
With the help of Proposition 4, we get(43)∫0tρ2sds1/2≤2eβ22t∫0tδ2e−s+se−s+β12e−β22sds1/2.
Combining this with the inequality (41), we obtain(44)ρt≤δβ1+e−t+te−t+2β2eβ22t⋅∫0tδ2e−s+se−s+β12e−β22sds1/2≤δβ1+e−t+te−t+2δeβ22t1+t+β11−e−β22t1/2=δe−t+te−t+β1+2eβ22t1+t+β11−e−β22t1/2.
Moreover, we get(45)ut+vt≤δ1+t+β1et+2eβ22+1t1+t+β11−e−β22t1/2.
According to the condition of Theorem 1, we getut+vt<ε for any t∈0,T. This indicates that system (15) is finite-time stable with respect to δ,ε,T. The proof is completed.Remark 1.
It is obvious that the left hand side of (23) increases with the increase of time t. Hence, the settling time T can be easily calculated in practical applications.
If some parameters of system (15) are assumed to satisfy some further conditions, then system (15) has a unique equilibrium point. Based on Theorem1, the finite-time stability of the equilibrium point can be ensured under the assumption of Theorem 1. In order to discuss the existence and uniqueness of the equilibrium point, the distributed time-varying delay σt is assumed to be a constant, i.e., σt=σ¯.Corollary 1.
Assume that the distributed time-varying delayσt is a constant, i.e., σt=σ¯. Let Assumption 1 hold. If the parameters of system (15) satisfy the condition in Theorem 1 and the following conditions:(46)∑i=1n1cimax1≤j≤maijξ1+bijη1+σ¯ζ1rij<1,∑j=1m1djmax1≤i≤npjiξ2+qjiη2+σ¯ζ2sji<1,then the system has a unique equilibrium point which is finite-time stable with respect to δ,ε,T.In fact, from the proof of Theorem 1, we only need to prove the existence and uniqueness of the equilibrium point. The proof mainly depends on the contraction mapping principle. For the convenience of readers, we only show some key points.Proof.
We define a mappingΘ:ℝn+m⟶ℝn+m as follows:(47)Θx,y=Θ1x1,y1,Θ2x2,y2,⋯,Θnxn,yn,Θn+1xn+1,yn+1,⋯,Θn+mxn+m,yn+mT,where(48)Θixi,yi=ci−1∑j=1maijf1jyj+∑j=1mbijg1jyj+∑j=1mrij∫t−σth1jyjdθ+Iit,Θixn+j,yn+j=dj−1∑i=1npjif2ixi+∑i=1nqjig2ixi+∑i=1nsji∫t−σth2ixidθ+Jjt.
Here,i=1,2,…,n and j=1,2,…,m.
In the following, we prove thatΘ is a contraction mapping on ℝn+m. For any two different points x,y=x1,x2,…,xn,y1,…,ym∈ℝn+m and x¯,y¯=x¯1,x¯2,…,x¯n,y¯1,…,y¯m∈ℝn+m, we have(49)Θx,y−Θx¯,y¯=∑i=1nci−1∑j=1maijf1jyj−f1jy¯j+∑j=1mbijg1jyj−g1jy¯j+∑j=1mrij∫t−σTh1jyj−h1jy¯jdθ+∑j=1mdj−1∑i=1npjif2ixi−f2ix¯i+∑i=1nqjig2ixi−g2ix¯i+∑i=1nsji∫t−σth2ixi−h2ix¯idθ.
The Lipschitz conditions lead to(50)Θx,y−Θx¯,y¯≤∑i=1n1ci∑j=1maijξ1+bijη1+σζ1rijyj−y¯j+∑j=1m1dj∑i=1npjiξ2+qjiη2+σζ2sjixi−x¯i≤∑i=1n1cimax1≤j≤maijξ1+bijη1+σζ1rij∑j=1myj−y¯i+∑j=1m1djmax1≤i≤npjiξ2+qjiη2+σζ2sji∑i=1nxi−x¯i.
Together with the condition (46), we get(51)Θx,y−Θx¯,y¯<∑j=1myj−y¯i+∑i=1nxi−x¯i=x,y−x¯,y¯.
This indicates thatΘ is a contraction mapping on ℝn+m. As a result, there exists a unique point x∗,y∗∈ℝn+m such that Θx∗,y∗=x∗,y∗, i.e.,(52)xi∗=ci−1∑j=1maijf1jyj∗+∑j=1mbijg1jyj∗+∑j=1mrij∫t−σth1jyj∗dθ+Ii,yj∗=dj−1∑i=1npjif2ixi∗+∑i=1nqjig2ixi∗+∑i=1msji∫t−σth2ixi∗dθ+Jj.
This implies that system (15) has a unique equilibrium point x∗,y∗.


### 3.2. Asymptotic Stability Analysis

In this subsection, some sufficient conditions are derived to ensure the global asymptotic stability of system (15). Let a∗, b∗, r∗, p∗, q∗, and s∗ be defined as in (21).Theorem 2.
Letγ=min1≤i≤n,1≤j≤mci,dj. If(53)γ>λ1+λ2eγτ¯+λ3σ¯eγσ¯Γα1/α,where λ1=maxξ1a∗,ξ2p∗, λ2=maxη1b∗,η2q∗, and λ3=maxζ1r∗,ζ2s∗, then system (15) is globally asymptotically stable.Proof.
Letxt,yt and x¯t,y¯t be any two solutions of system (15) with different initial conditions. Let ut=xt−x¯t and vt=yt−y¯t. Assume that ut,vt satisfies the initial condition (17).
Making use of the Laplace transform and the inverse Laplace transform, we derive(54)uit=φi00Eα−citα+tφi10Eα,2−citα+∫0tt−sα−1Eα,α−cit−sα∑j=1maijf1jyjs−f1jy¯js+∑j=1mbijg1jyjs−τs−g1jy¯js−τs+∑j=1mrij∫s−σssh1jyjθ−h1jy¯jθdθds.
In the same way, it follows that(55)vjt=ψj00Eα−djtα+tψj10Eα,2−djtα+∫0tt−sα−1Eα,α−djt−sα∑i=1npjif2ixis−f2ix¯is+∑i=1nqjig2ixis−τs−g2ix¯is−τs+∑i=1nsji∫s−σssh2ixiθ−h2ix¯iθdθds.
According to Proposition 3, we get(56)uit≤φi00e−cit+tφi10e−cit+∫0tt−sα−1e−cit−s∑j=1mξ1aijvjs+∑j=1mη1bijvjs−τs+∑j=1mζ1rij∫s−σssvjθdθds,(57)vjt≤ψi00e−djt+tψi10e−djt+∫0tt−sα−1e−djt−s∑i=1nξ2pjiuis+∑i=1nη2qjiuis−τs+∑i=1nζ2sji∫s−σssuiθdθds.
Letγ1=min1≤i≤nci, a∗=max1≤j≤m∑i=1naij, b∗=max1≤j≤m∑i=1nbij, and r∗=max1≤j≤m∑i=1nrij. From the inequality (56), we obtain(58)ut=∑i=1nuit≤e−γ1tφ00+te−γ1tφ10+∫0tt−sα−1e−γ1t−s∑j=1mξ1∑i=1naijvjs+∑j=1mη1∑i=1nbijvjs−τs+∑j=1mζ1∑i=1nrij∫s−σssvjθdθds≤e−γ1tφ00+te−γ1tφ10+∫0tt−sα−1e−γ1t−sξ1a∗vs+η1b∗vs−τs+ζ1r∗∫s−σssvθdθds.
From the inequality (57), we have(59)vt≤e−γ2tψ00+te−γ2tψ10+∫0tt−sα−1e−γ2t−sξ2p∗us+η2q∗us−τs+ζ2s∗∫s−σssuθdθds,where γ2=min1≤j≤mdj, p∗=min1≤i≤n∑j=1mpji, q∗=min1≤i≤n∑j=1mqji, and s∗=min1≤i≤n∑j=1msji.
Obviously, we haveγ≤γ1 and γ≤γ2. Together with the estimates of ut and vt, it follows that(60)eγtut+vt≤φ00+ψ00+tφ10+ψ10+∫0tt−sα−1eγsξ1a∗vs+η1b∗vs−τs+ζ1r∗∫s−σssvθdθ+ξ2p∗us+η2q∗us−τs+ζ2s∗∫s−σssuθdθds.
Letρt=supt−κ≤t˜≤teγt˜ut˜+vt˜, where κ=maxτ¯,σ¯. Obviously, we have(61)eγtut+vt≤ρt,eγt−τtut−τt+vt−τt≤ρt.
In addition, for any fixedt, we use the mean value theorem to get(62)eγt∫t−σttuθ+vθdθ=eγtσuβt+vβt,where t−σt<βt<t. Moreover,(63)eγt∫t−σtuθ+vθdθ=eγt−βtσeγβtuβt+vβt≤eγσσ¯ρt.
Together with the inequalities (60) and (61), we have(64)eγtut+vt≤φ00+ψ00+tφ10+ψ10+∫0tt−sα−1λ1ρs+λ2eγτsρs+λ3σ¯eγσ¯ρsds≤φ00+ψ00+tφ10+ψ10+λ1+λ2eγτ¯+λ3σ¯eγσ¯∫0tt−sα−1ρsds,where λ1=maxξ1a∗,ξ2p∗, λ2=maxη1b∗,η2q∗, and λ3=maxζ1r∗,ζ2s∗. Thus,(65)ρt≤φ00+ψ00+tφ10+ψ10+λ1+λ2eγτ¯+λ3σ¯eγσ¯∫0tt−sα−1ρsds.
Owing to Proposition 6, we obtain(66)ρt≤φ00+ψ00+tφ10+ψ10Eαλ1+λ2eγτ¯+λ3σ¯eγσ¯Γαtα.
Moreover, we have(67)ut+vt≤e−γtφ00+ψ00+tφ10+ψ10⋅Eαλ1+λ2eγτ¯+λ3σ¯eγσ¯Γαtα.
With the help of Proposition 2, we get(68)ut+vt≤e−γtφ00+ψ00+tφ10+ψ10⋅μ1eλ1+λ2eγτ¯+λ3σ¯eγσ¯Γα1/αt+μ21+λ1+λ2eγτ¯+λ3σ¯eγσ¯Γαtα,where μ1 and μ2 are two positive constants. If γ>λ1+λ2eγτ¯+λ3σ¯eγσ¯Γα1/α, then(69)xt+yt⟶0,t⟶+∞.
This indicates that system (15) is globally asymptotically stable. The proof is completed.
Based on Corollary 1 and Theorem 2, we can easily obtain the following result related to the equilibrium point.Corollary 2.
Assume that the distributed time-varying delayσt is a constant, i.e., σt=σ¯. If the parameters of system (15) satisfy the conditions (46) and (53), then the system has a unique equilibrium point which is globally asymptotically stable.Remark 2.
Proposition 3 plays a key role in the proof of Theorem 2. Notice that the letterα in Proposition 3 satisfies α∈1,2. For α∈0,1, the inequality (9) in Proposition 3 should be changed to(70)Eα,βAtα≤Me−ωt,where M is a positive constant. It is worth pointing out that M cannot be definitely given [46]. Hence, for α∈0,1, if we follow the proof of Theorem 2, we will obtain a condition depending on M, which cannot be easily verified.Remark 3.
In [42], Wu et al. investigated the quasi-uniform stability of a class of fractional-order neural networks with discrete and finite-time distributed constant delays for α∈0,1. In this paper, we focus on a class of fractional-order BAM neural networks with discrete and finite-time distributed time-varying delays for α∈1,2. Two sufficient conditions (see Theorem 1 and Theorem 2) are obtained to ensure the finite-time stability and the global asymptotic stability of systems, respectively.Remark 4.
Forα∈0,1, Tyagi et al. [39] considered the uniform stability and the Mittag–Leffler stability of fractional-order complex-valued neural networks with discrete and continuously distributed constant delays; Srivastava et al. [40] studied the exponential stability of a class of fractional-order impulsive neural networks with discrete time-varying delay and distributed delay. These two kinds of mixed time delays are completely different from that considered in this paper.Remark 5.
In the existing literature, there have been some works [14, 20, 27–30] on the finite-time stability of fractional-order neural networks with the orders satisfying 1<α<2. The obtained sufficient conditions in these works are some inequalities related to the Mittag–Leffler functions whose proofs mainly depend on the Laplace transform, the generalized Gronwall inequality, and some inequalities related to Mittag–Leffler functions. Following this method, we can also get a sufficient condition ensuring the finite-time stability of system (15), i.e.,(71)e−γt1+tEαλ1+λ2eγτ¯+λ3σ¯eγσ¯Γαtα<εδ,∀t∈0,T,where γ, λ1, λ2, and λ3 are defined as in Subsection 3.2. Obviously, this condition is closely related to the Mittag–Leffler function, and hence it cannot be easily calculated. It is also noted that the settling time T1 cannot be easily estimated. In order to overcome this problem, we use some analytical techniques and some elementary inequalities to investigate the finite-time stability of system (15). We derive a sufficient condition (see Theorem 1) which can be expressed as an algebraic inequality. In particular, the estimate of the settling time T2 can be easily obtained in practical applications. In addition, the size relationship between T1 and T2 cannot be obtained from the theoretical perspective under the same parameters.Remark 6.
When the distributed time-varying delay vanishes and the discrete time-varying delay is constant, i.e.,σ¯=0 and σt=σ¯, Theorem 2 is reduced to Corollary 1 in [27].Remark 7.
In [15], Wu et al. reported the global Mittag–Leffler stabilization of fractional-order BAM neural networks via linear feedback control or partial feedback control. For a class of uncertain fractional-order nonlinear systems, the robust stabilization by proposing a kind of fractional-order sliding mode controller is considered. In [50], Yin et al. discussed the robust stability for a class of uncertain fractional-order nonlinear systems with external disturbance by using an adaptive sliding mode technique based on a fractional-order switching type control law. Notice that the order of systems in these works lie in the interval 0,1. In future, we will consider the stabilization problem of system (15) by designing a kind of fractional-order sliding mode controller. In addition, we will also investigate some applications of fractional-order systems with α∈1,2 in electrical engineering based on a recent research.


## 3.1. Finite-Time Stability Analysis

In this subsection, some sufficient conditions are derived to ensure the finite-time stability of system (15). For the time-varying delay τt, we need to make further assumptions.Assumption 1.
The discrete time-varying delayτt is a differentiable function such that(20)τ˙t≤μ<1,where µ is a constant.
In what follows, we state the main theorem. For convenience, let us first introduce some notation related to the parameters of system (15). Let γ1=max1≤i≤nci and γ2=max1≤j≤mdj. Let(21)a∗=max1≤j≤m∑i=1naij,b∗=max1≤j≤m∑i=1nbij,r∗=max1≤j≤m∑i=1nrij,p∗=max1≤i≤n∑j=1mpji,q∗=max1≤i≤n∑j=1mqji,s∗=max1≤i≤n∑j=1msji.
Let(22)λ1=maxγ2+ξ1a∗,γ1+ξ2p∗,λ2=maxη1b∗,η2q∗,λ3=maxζ1r∗,ζ2s∗.Theorem 1.
Letδ and ε be any positive constants such that δ<ε. Suppose that Assumption 1 holds. If maxφ0t+ψ0t,φ1t+ψ1t<δ and(23)1+t+β1et+2eβ22+1t1+t+β11−e−β22t1/2<εδ,∀t∈0,T,where β1=λ2/ΓαΓ2α−1e2τ¯−1/4α1−μ and β2=λ1+λ3σ¯/Γα2Γ2α−1/4α+λ2/Γα2Γ2α−1/4α1−μ, and then system (15) is finite-time stable with respect to δ,ε,T.Proof.
Letxt,yt and x¯t,y¯t be two solutions of system (15) with different initial conditions. Let ut=xt−x¯t and vt=yt−y¯t. Assume that ut,vt satisfies the initial condition (17).
Proposition 1 yields(24)uit=φi00+tφi10+1Γα∫0tt−sα−1−ciuis+∑j=1maijf1jyjs−f1jy¯js+∑j=1mbijg1jyjs−τs−g1jy¯js−τs+∑j=1mrij∫s−σssh1jyjθ−h1jy¯jθdθds.
According to the Lipschitz conditions, we obtain(25)uit≤φi00+tφi10+1Γα∫0tt−sα−1ciuis+∑j=1mξ1aijvjs+∑j=1mη1bijvjs−τs+∑j=1mζ1rij∫s−σssvjθdθds.
Similarly, we have(26)vjt≤ψi00+tψi10+1Γα∫0tt−sα−1djvjs+∑i=1nξ2pjiuis+∑i=1nη2qjiuis−τs+∑i=1nζ2sji∫s−σssuiθdθds.
Moreover, it follows that(27)ut=∑i=1nuit≤φ00+tφ10+1Γα∫0tt−sα−1∑i=1nciuis+∑j=1mξ1∑i=1naijvjs+∑j=1mη1∑i=1nbijvjs−τs+∑j=1mζ1∑i=1nrij∫s−σssvjθdθds≤φ00+tφ10+1Γα∫0tt−sα−1γ1us+ξ1a∗vs+η1b∗vs−τs+ζ1r∗∫s−σssvθdθds.
Meanwhile, we have(28)vt=∑j=1mvjt≤ψ00+tψ10+1Γα∫0tt−sα−1γ2vs+ξ2p∗us+η2q∗us−τs+ζ2s∗∫s−σssuθdθds.
Together with the estimates ofut and vt, it follows that(29)ut+vt≤φ00+ψ00+tφ10+ψ10+1Γα∫0tt−sα−1γ1+ξ2p∗us+γ2+ξ1a∗vs+η2q∗us−τs+η1b∗vs−τs+ζ1r∗∫s−σssvθdθ+ζ2s∗∫s−σssuθdθds≤φ00+ψ00+tφ10+ψ10+1Γα∫0tt−sα−1λ1us+vs+λ2us−τs+vs−τs+λ3∫s−σssuθ+vθdθds.
Making use of the Cauchy–Schwartz inequality, we obtain(30)ut+vt≤φ00+ψ00+tφ10+ψ10+λ1Γα∫0tt−s2α−1e2sds1/2∫0te−2sus+vs2ds1/2+λ2Γα∫0tt−s2α−1e2sds1/2∫0te−2sus−τs+vs−τs2ds1/2+λ3Γα∫0tt−s2α−1e2sds1/2∫0te−2s∫s−σssuθ+vθdθ2ds1/2.
For the integral∫0tt−s2α−1e2sds, we derive(31)∫0tt−s2α−1e2sds=e2t∫0ts2α−1e−2sds=2e2t4α∫02ts2α−2e−sds<2e2t4αΓ2α−1.
For the integral∫0te−2sus−τs+vs−τs2ds, we deduce(32)∫0te−2sus−τs+vs−τs2ds=∫0te−2s1−τ˙s−1us−τs+vs−τs21−τ˙sds.
Combining this with the assumption onτt, we obtain(33)∫0te−2sus−τs+vs−τs2ds≤11−μ∫0te−2sus−τs+vs−τs21−τ˙sds≤11−μ∫−τte−2ωuω+vω2dω=11−μ∫−τ0e−2sus+vs2ds+11−μ∫0te−2sus+vs2ds≤e2τ¯−121−μφ0+ψ02+11−μ∫0te−2sus+vs2ds.
Moreover, it follows that(34)∫0te−2sus−τs+vs−τs2ds1/2≤e2τ¯−121−μφ0+ψ0+11−μ∫0te−2sus+vs2ds1/2.
From the inequalities (30)–(34), we get(35)e−tut+vt≤e−tφ00+ψ00+te−tφ10+ψ10+λ1Γα2Γ2α−14α∫0te−2sus+vs2ds1/2+λ2ΓαΓ2α−1e2τ−14α1−μφ0+ψ0+λ2Γα2Γ2α−14α1−μ∫0te−2sus+vs2ds1/2+λ3Γα2Γ2α−14α∫0te−2s∫s−σssuθ+vθdθ2ds1/2.
Letρt=supt−κ≤t˜≤te−t˜ut˜+vt˜, where κ=maxτ,σ. Obviously, we have(36)e−tut+vt≤ρt.
In addition, for any fixedt, we use the mean value theorem to get(37)e−t∫t−σttuθ+vθdθ=e−tσtuβt+vβt,where t−σt<βt<t. Moreover,(38)e−t∫t−σttuθ+vθdθ=e−tσteβte−βtuβt+vβt≤σ¯ρt.
From the inequalities (35)–(38), we have(39)e−tut+vt≤λ2ΓαΓ2α−1e2τ−14α1−μφ0+ψ0+e−tφ00+ψ00+te−tφ10+ψ10+λ1Γα2Γ2α−14α∫0tρ2sds1/2+λ2Γα2Γ2α−14α1−μ∫0tρ2sds1/2+λ3σ¯Γα2Γ2α−14α∫0tρ2sds1/2.
Moreover,(40)ρt≤λ2ΓαΓ2α−1e2τ¯−14α1−μφ0+ψ0+e−tφ00+ψ00+te−tφ10+ψ10+λ1+λ3σ¯Γα2Γ2α−14α+λ2Γα2Γ2α−14α1−μ∫0tρ2sds1/2.
Hence,(41)ρt≤δβ1+e−t+te−t+β2∫0tρ2sds1/2,where β1=λ2/ΓαΓ2α−1e2τ¯−1/4α1−μ and β2=λ1+λ3σ¯/Γα2Γ2α−1/4α+λ2/Γα2Γ2α−1/4α1−μ.
In view of Proposition 5, we obtain(42)∫0tρ2sds1/2≤1−1−e−β22t1/2−1⋅∫0tδ2β1+e−s+se−s2e−β22sds1/2.
With the help of Proposition 4, we get(43)∫0tρ2sds1/2≤2eβ22t∫0tδ2e−s+se−s+β12e−β22sds1/2.
Combining this with the inequality (41), we obtain(44)ρt≤δβ1+e−t+te−t+2β2eβ22t⋅∫0tδ2e−s+se−s+β12e−β22sds1/2≤δβ1+e−t+te−t+2δeβ22t1+t+β11−e−β22t1/2=δe−t+te−t+β1+2eβ22t1+t+β11−e−β22t1/2.
Moreover, we get(45)ut+vt≤δ1+t+β1et+2eβ22+1t1+t+β11−e−β22t1/2.
According to the condition of Theorem 1, we getut+vt<ε for any t∈0,T. This indicates that system (15) is finite-time stable with respect to δ,ε,T. The proof is completed.Remark 1.
It is obvious that the left hand side of (23) increases with the increase of time t. Hence, the settling time T can be easily calculated in practical applications.
If some parameters of system (15) are assumed to satisfy some further conditions, then system (15) has a unique equilibrium point. Based on Theorem1, the finite-time stability of the equilibrium point can be ensured under the assumption of Theorem 1. In order to discuss the existence and uniqueness of the equilibrium point, the distributed time-varying delay σt is assumed to be a constant, i.e., σt=σ¯.Corollary 1.
Assume that the distributed time-varying delayσt is a constant, i.e., σt=σ¯. Let Assumption 1 hold. If the parameters of system (15) satisfy the condition in Theorem 1 and the following conditions:(46)∑i=1n1cimax1≤j≤maijξ1+bijη1+σ¯ζ1rij<1,∑j=1m1djmax1≤i≤npjiξ2+qjiη2+σ¯ζ2sji<1,then the system has a unique equilibrium point which is finite-time stable with respect to δ,ε,T.In fact, from the proof of Theorem 1, we only need to prove the existence and uniqueness of the equilibrium point. The proof mainly depends on the contraction mapping principle. For the convenience of readers, we only show some key points.Proof.
We define a mappingΘ:ℝn+m⟶ℝn+m as follows:(47)Θx,y=Θ1x1,y1,Θ2x2,y2,⋯,Θnxn,yn,Θn+1xn+1,yn+1,⋯,Θn+mxn+m,yn+mT,where(48)Θixi,yi=ci−1∑j=1maijf1jyj+∑j=1mbijg1jyj+∑j=1mrij∫t−σth1jyjdθ+Iit,Θixn+j,yn+j=dj−1∑i=1npjif2ixi+∑i=1nqjig2ixi+∑i=1nsji∫t−σth2ixidθ+Jjt.
Here,i=1,2,…,n and j=1,2,…,m.
In the following, we prove thatΘ is a contraction mapping on ℝn+m. For any two different points x,y=x1,x2,…,xn,y1,…,ym∈ℝn+m and x¯,y¯=x¯1,x¯2,…,x¯n,y¯1,…,y¯m∈ℝn+m, we have(49)Θx,y−Θx¯,y¯=∑i=1nci−1∑j=1maijf1jyj−f1jy¯j+∑j=1mbijg1jyj−g1jy¯j+∑j=1mrij∫t−σTh1jyj−h1jy¯jdθ+∑j=1mdj−1∑i=1npjif2ixi−f2ix¯i+∑i=1nqjig2ixi−g2ix¯i+∑i=1nsji∫t−σth2ixi−h2ix¯idθ.
The Lipschitz conditions lead to(50)Θx,y−Θx¯,y¯≤∑i=1n1ci∑j=1maijξ1+bijη1+σζ1rijyj−y¯j+∑j=1m1dj∑i=1npjiξ2+qjiη2+σζ2sjixi−x¯i≤∑i=1n1cimax1≤j≤maijξ1+bijη1+σζ1rij∑j=1myj−y¯i+∑j=1m1djmax1≤i≤npjiξ2+qjiη2+σζ2sji∑i=1nxi−x¯i.
Together with the condition (46), we get(51)Θx,y−Θx¯,y¯<∑j=1myj−y¯i+∑i=1nxi−x¯i=x,y−x¯,y¯.
This indicates thatΘ is a contraction mapping on ℝn+m. As a result, there exists a unique point x∗,y∗∈ℝn+m such that Θx∗,y∗=x∗,y∗, i.e.,(52)xi∗=ci−1∑j=1maijf1jyj∗+∑j=1mbijg1jyj∗+∑j=1mrij∫t−σth1jyj∗dθ+Ii,yj∗=dj−1∑i=1npjif2ixi∗+∑i=1nqjig2ixi∗+∑i=1msji∫t−σth2ixi∗dθ+Jj.
This implies that system (15) has a unique equilibrium point x∗,y∗.


## 3.2. Asymptotic Stability Analysis

In this subsection, some sufficient conditions are derived to ensure the global asymptotic stability of system (15). Let a∗, b∗, r∗, p∗, q∗, and s∗ be defined as in (21).Theorem 2.
Letγ=min1≤i≤n,1≤j≤mci,dj. If(53)γ>λ1+λ2eγτ¯+λ3σ¯eγσ¯Γα1/α,where λ1=maxξ1a∗,ξ2p∗, λ2=maxη1b∗,η2q∗, and λ3=maxζ1r∗,ζ2s∗, then system (15) is globally asymptotically stable.Proof.
Letxt,yt and x¯t,y¯t be any two solutions of system (15) with different initial conditions. Let ut=xt−x¯t and vt=yt−y¯t. Assume that ut,vt satisfies the initial condition (17).
Making use of the Laplace transform and the inverse Laplace transform, we derive(54)uit=φi00Eα−citα+tφi10Eα,2−citα+∫0tt−sα−1Eα,α−cit−sα∑j=1maijf1jyjs−f1jy¯js+∑j=1mbijg1jyjs−τs−g1jy¯js−τs+∑j=1mrij∫s−σssh1jyjθ−h1jy¯jθdθds.
In the same way, it follows that(55)vjt=ψj00Eα−djtα+tψj10Eα,2−djtα+∫0tt−sα−1Eα,α−djt−sα∑i=1npjif2ixis−f2ix¯is+∑i=1nqjig2ixis−τs−g2ix¯is−τs+∑i=1nsji∫s−σssh2ixiθ−h2ix¯iθdθds.
According to Proposition 3, we get(56)uit≤φi00e−cit+tφi10e−cit+∫0tt−sα−1e−cit−s∑j=1mξ1aijvjs+∑j=1mη1bijvjs−τs+∑j=1mζ1rij∫s−σssvjθdθds,(57)vjt≤ψi00e−djt+tψi10e−djt+∫0tt−sα−1e−djt−s∑i=1nξ2pjiuis+∑i=1nη2qjiuis−τs+∑i=1nζ2sji∫s−σssuiθdθds.
Letγ1=min1≤i≤nci, a∗=max1≤j≤m∑i=1naij, b∗=max1≤j≤m∑i=1nbij, and r∗=max1≤j≤m∑i=1nrij. From the inequality (56), we obtain(58)ut=∑i=1nuit≤e−γ1tφ00+te−γ1tφ10+∫0tt−sα−1e−γ1t−s∑j=1mξ1∑i=1naijvjs+∑j=1mη1∑i=1nbijvjs−τs+∑j=1mζ1∑i=1nrij∫s−σssvjθdθds≤e−γ1tφ00+te−γ1tφ10+∫0tt−sα−1e−γ1t−sξ1a∗vs+η1b∗vs−τs+ζ1r∗∫s−σssvθdθds.
From the inequality (57), we have(59)vt≤e−γ2tψ00+te−γ2tψ10+∫0tt−sα−1e−γ2t−sξ2p∗us+η2q∗us−τs+ζ2s∗∫s−σssuθdθds,where γ2=min1≤j≤mdj, p∗=min1≤i≤n∑j=1mpji, q∗=min1≤i≤n∑j=1mqji, and s∗=min1≤i≤n∑j=1msji.
Obviously, we haveγ≤γ1 and γ≤γ2. Together with the estimates of ut and vt, it follows that(60)eγtut+vt≤φ00+ψ00+tφ10+ψ10+∫0tt−sα−1eγsξ1a∗vs+η1b∗vs−τs+ζ1r∗∫s−σssvθdθ+ξ2p∗us+η2q∗us−τs+ζ2s∗∫s−σssuθdθds.
Letρt=supt−κ≤t˜≤teγt˜ut˜+vt˜, where κ=maxτ¯,σ¯. Obviously, we have(61)eγtut+vt≤ρt,eγt−τtut−τt+vt−τt≤ρt.
In addition, for any fixedt, we use the mean value theorem to get(62)eγt∫t−σttuθ+vθdθ=eγtσuβt+vβt,where t−σt<βt<t. Moreover,(63)eγt∫t−σtuθ+vθdθ=eγt−βtσeγβtuβt+vβt≤eγσσ¯ρt.
Together with the inequalities (60) and (61), we have(64)eγtut+vt≤φ00+ψ00+tφ10+ψ10+∫0tt−sα−1λ1ρs+λ2eγτsρs+λ3σ¯eγσ¯ρsds≤φ00+ψ00+tφ10+ψ10+λ1+λ2eγτ¯+λ3σ¯eγσ¯∫0tt−sα−1ρsds,where λ1=maxξ1a∗,ξ2p∗, λ2=maxη1b∗,η2q∗, and λ3=maxζ1r∗,ζ2s∗. Thus,(65)ρt≤φ00+ψ00+tφ10+ψ10+λ1+λ2eγτ¯+λ3σ¯eγσ¯∫0tt−sα−1ρsds.
Owing to Proposition 6, we obtain(66)ρt≤φ00+ψ00+tφ10+ψ10Eαλ1+λ2eγτ¯+λ3σ¯eγσ¯Γαtα.
Moreover, we have(67)ut+vt≤e−γtφ00+ψ00+tφ10+ψ10⋅Eαλ1+λ2eγτ¯+λ3σ¯eγσ¯Γαtα.
With the help of Proposition 2, we get(68)ut+vt≤e−γtφ00+ψ00+tφ10+ψ10⋅μ1eλ1+λ2eγτ¯+λ3σ¯eγσ¯Γα1/αt+μ21+λ1+λ2eγτ¯+λ3σ¯eγσ¯Γαtα,where μ1 and μ2 are two positive constants. If γ>λ1+λ2eγτ¯+λ3σ¯eγσ¯Γα1/α, then(69)xt+yt⟶0,t⟶+∞.
This indicates that system (15) is globally asymptotically stable. The proof is completed.
Based on Corollary 1 and Theorem 2, we can easily obtain the following result related to the equilibrium point.Corollary 2.
Assume that the distributed time-varying delayσt is a constant, i.e., σt=σ¯. If the parameters of system (15) satisfy the conditions (46) and (53), then the system has a unique equilibrium point which is globally asymptotically stable.Remark 2.
Proposition 3 plays a key role in the proof of Theorem 2. Notice that the letterα in Proposition 3 satisfies α∈1,2. For α∈0,1, the inequality (9) in Proposition 3 should be changed to(70)Eα,βAtα≤Me−ωt,where M is a positive constant. It is worth pointing out that M cannot be definitely given [46]. Hence, for α∈0,1, if we follow the proof of Theorem 2, we will obtain a condition depending on M, which cannot be easily verified.Remark 3.
In [42], Wu et al. investigated the quasi-uniform stability of a class of fractional-order neural networks with discrete and finite-time distributed constant delays for α∈0,1. In this paper, we focus on a class of fractional-order BAM neural networks with discrete and finite-time distributed time-varying delays for α∈1,2. Two sufficient conditions (see Theorem 1 and Theorem 2) are obtained to ensure the finite-time stability and the global asymptotic stability of systems, respectively.Remark 4.
Forα∈0,1, Tyagi et al. [39] considered the uniform stability and the Mittag–Leffler stability of fractional-order complex-valued neural networks with discrete and continuously distributed constant delays; Srivastava et al. [40] studied the exponential stability of a class of fractional-order impulsive neural networks with discrete time-varying delay and distributed delay. These two kinds of mixed time delays are completely different from that considered in this paper.Remark 5.
In the existing literature, there have been some works [14, 20, 27–30] on the finite-time stability of fractional-order neural networks with the orders satisfying 1<α<2. The obtained sufficient conditions in these works are some inequalities related to the Mittag–Leffler functions whose proofs mainly depend on the Laplace transform, the generalized Gronwall inequality, and some inequalities related to Mittag–Leffler functions. Following this method, we can also get a sufficient condition ensuring the finite-time stability of system (15), i.e.,(71)e−γt1+tEαλ1+λ2eγτ¯+λ3σ¯eγσ¯Γαtα<εδ,∀t∈0,T,where γ, λ1, λ2, and λ3 are defined as in Subsection 3.2. Obviously, this condition is closely related to the Mittag–Leffler function, and hence it cannot be easily calculated. It is also noted that the settling time T1 cannot be easily estimated. In order to overcome this problem, we use some analytical techniques and some elementary inequalities to investigate the finite-time stability of system (15). We derive a sufficient condition (see Theorem 1) which can be expressed as an algebraic inequality. In particular, the estimate of the settling time T2 can be easily obtained in practical applications. In addition, the size relationship between T1 and T2 cannot be obtained from the theoretical perspective under the same parameters.Remark 6.
When the distributed time-varying delay vanishes and the discrete time-varying delay is constant, i.e.,σ¯=0 and σt=σ¯, Theorem 2 is reduced to Corollary 1 in [27].Remark 7.
In [15], Wu et al. reported the global Mittag–Leffler stabilization of fractional-order BAM neural networks via linear feedback control or partial feedback control. For a class of uncertain fractional-order nonlinear systems, the robust stabilization by proposing a kind of fractional-order sliding mode controller is considered. In [50], Yin et al. discussed the robust stability for a class of uncertain fractional-order nonlinear systems with external disturbance by using an adaptive sliding mode technique based on a fractional-order switching type control law. Notice that the order of systems in these works lie in the interval 0,1. In future, we will consider the stabilization problem of system (15) by designing a kind of fractional-order sliding mode controller. In addition, we will also investigate some applications of fractional-order systems with α∈1,2 in electrical engineering based on a recent research.


## 4. Numerical Simulations

In this section, some numerical examples are provided to illustrate the effectiveness of our main results.Consider a class of fractional-order BAM neural networks with mixed time-varying delays which can be described as follows:(72)D0Ctαxt=−Cxt+Af1yt+Bg1yt−τt+R∫t−σtth1yθdθ+I,D0Ctαyt=−Dyt+Pf2xt+Qg2xt−τt+S∫t−σtth2xθdθ+J.Example 1.
For system (72), the parameters are given as follows: α=1.8, x=x1,x2T, y=y1,y2T, C=diag0.03,0.04, D=diag0.05,0.03, τt=0.1−0.001e−t, σt=0.1−0.001e−2t, f1yt=siny1t,siny2t, g1yt−τ=1/2siny1t−τt,1/2siny2t−τt, h1yt=tanhy1t,tanhy2t, f2xt=sinx1t,sinx2t, g2xt−τ=1/2sinx1t−τ,1/2sinx2t−τ, h2yt=tanhx1t,tanhx2t, I=0.25,0.3T, J=0.15,0.25T, and(73)A=0.020.020.030.05,B=−0.0250.03−0.0270.019,R=−0.040.03−0.0350.05,P=−0.025−0.035−0.045−0.02,Q=−0.042−0.037−0.026−0.03,S=−0.02−0.03−0.04−0.05.
A simple calculation gives the following data:τ¯=σ¯=0.1, μ=0.001, ξ1=ξ2=ζ1=ζ2=1, and η1=η2=1/2. Let xt,yt and x¯t,y¯t be two solutions of systems with the following initial conditions:(74)xt=4.26−0.15t,3.79−0.25tT,x′t=−0.15,−0.25T,yt=3.36−0.125t,4.28−0.1tT,y′t=−0.125,−0.1T,x¯t=4.29−0.14t,3.75−0.22tT,x′¯t=−0.14,−0.22T,y¯t=3.6−0.12t,4.3−0.13tT,y′¯t=−0.12,−0.13T,for any t∈−0.1,0. The state trajectories are shown in Figure 1, and the trajectories of the error are depicted in Figure 2.
By calculating, it follows thatλ1=0.12, λ2=0.034, λ3=0.08, β1=0.006, and β2=0.085. Let us choose ε=1. According to the initial data, we take δ=0.15>maxφ0t+ψ0t,φ1t+ψ1t. Moreover, from Theorem 1, the estimated settling time T=2.8609 can be easily obtained. The norm of error is depicted in Figure 2 which indicates the correctness of Theorem 1. On the other hand, by the sufficient condition (71) based on the existing technique, the estimated settling time T=2.7611. This implies the effectiveness of Theorem 1.Figure 1
State trajectories of the system for two initial conditions in Example 1.
(a)
(b)
(c)
(d)Figure 2
(a) State trajectories of error system in Example 1. (b) Time evolution for the norm of error in Example 1.
(a)
(b)Example 2.
For the system (72), the parameters are given as follows: α=1.6, x=x1,x2T, y=y1,y2T, C=diag2.6,3.4, D=diag3,3.2, τt=0.1−0.001e−t, σt=0.1−0.001e−2t, f1yt=siny1t,siny2t, g1yt−τ=1/2cosy1t−τt,1/2cosy2t−τt, h1yt=tanhy1t,tanhy2t, f2xt=sinx1t,sinx2t, g2xt−τ=1/2cosx1t−τ,1/2cosx2t−τ, h2yt=tanhx1t,tanhx2t, I=3.2,0.7T, J=0.2,0.4T, and(75)A=−0.5−0.7−0.6−0.8,B=−1.04−0.92−0.74−0.8,R=5465,P=−0.7−1−0.9−0.8,Q=−1.22−0.8−0.64−1.1,S=4556.
Obviously,τ¯=σ¯=0.1, ξ1=ξ2=ζ1=ζ2=1, and η1=η2=1/2. Let xt,yt and x¯t,y¯t be two solutions of this system with the following initial conditions:(76)xt=2.2−0.2t,2.5−0.3tT,x′t=−0.2,−0.3T,yt=2.6−0.25t,2.3−0.1tT,y′t=−0.25,−0.1T,x¯t=3.6−0.4t,1.2−0.3tT,x′¯t=−0.4,−0.3T,y¯t=3.6−0.3t,3.5−22tT,y′¯t=−0.3,−0.22T,for any t∈−0.1,0.
By calculating, it follows thatγ=2.6, λ1=1.8, λ2=0.95, and λ3=11. Moreover, λ1+λ2eγτ¯+λ3σ¯eγσ¯Γα1/α=2.3724. Obviously, γ>λ1+λ2eγτ¯+λ3σ¯eγσ¯Γα1/α. Thus, the condition of Theorem 2 is satisfied. The state trajectories and the norm of error are depicted in Figures 3 and 4, which indicates the feasibility of Theorem 1.Figure 3
State trajectories of the system for two initial conditions in Example 2.
(a)
(b)
(c)
(d)Figure 4
Time evolution for the norm of error in Example 2.Example 3.
For system (72), the parameters are given as follows: α=1.95, x=x1,x2T, y=y1,y2T, C=diag0.06,0.05, D=diag0.04,0.02, τt=0.1−0.001e−t, σt=0.1−0.001e−3t, f1yt=cosy1t,cosy2tT, g1yt−τ=1/2cosy1t−τt,1/2cosy2t−τtT, h1yt=tanhy1t,tanhy2tT, f2xt=cosx1t,cosx2tT, g2xt−τ=1/2cosx1t−τ,1/2cosx2t−τT, h2yt=tanhx1t,tanhx2tT, I=0.2,0.2T, J=0.1,0.05T, and(77)A=−0.04−0.06−0.05−0.04,B=−0.04−0.055−0.052−0.045,R=−0.03−0.035−0.04−0.035,P=0.040.060.040.075,Q=−0.04−0.06−0.04−0.03,S=0.030.060.040.06.
Obviously,τ¯=σ¯=0.01, μ=0.001, ξ1=ξ2=ζ1=ζ2=1, and η1=η2=1/2. Let xt,yt, x¯t,y¯t, and x^t,y^t be three solutions of systems with the following initial conditions:(78)xt=2.075−0.06t,2.185−0.06tT,x′t=−0.06,−0.06T,yt=1.05−0.065t,−1.125−0.055tT,y′t=−0.065,−0.055T,x¯t=2.1−0.0625t,2.21−0.07tT,x′¯t=−0.0625,−0.07T,y¯t=1.1−0.075t,−1.15−0.065tT,y′¯t=−0.075,−0.065T,x^t=2.05−0.075t,2.16−0.075tT,x′^t=−0.075,−0.075Ty^t=1.075−0.08t,−1.09−0.075tT,x′^t=−0.08,−0.075T,for any t∈−0.1,0. The state trajectories and the trajectories of errors are depicted in Figures 5 and 6, respectively.
By calculating, we easily get the following data:λ1=0.195, λ2=0.05, λ3=0.12, β1=0.008, and β2=0.13.
Let us takeε=1. For any two of the above three solutions, the initial conditions allow us to take the values of δ as follows:(1)
Forxt,yt and x¯t,y¯t, the value of δ is taken as δ=0.125. Moreover, the estimated settling time T1=5.1137 can easily obtained from Theorem 1.(2)
Forx¯t,y¯t and x^t,y^t, the value of δ is taken as δ=0.11. Theorem 1 gives the estimate of settling time T2=5.4696.(3)
Forxt,yt and x^t,y^t, the value of δ is taken as δ=0.185. Based on Theorem 1, the estimated settling time T3=3.7477 can easily derived.
For the above three cases, the norms of errors are depicted in Figure7 which indicates the correctness of Theorem 1.
In addition, by the sufficient condition (71) based on the existing technique, the estimated settling time is 2.6256, 2.7558, and 2.1774 for the above three cases. This implies the effectiveness of Theorem 1.Figure 5
State trajectories of the system for three initial conditions in Example 3.
(a)
(b)
(c)
(d)Figure 6
State trajectories of error systems for the system in Example 3.
(a)
(b)
(c)
(d)Figure 7
Time evolutions for the norms of errors in Example 3.Example 4.
For the system (72), the parameters are given as follows: α=1.75,x=x1,x2T, y=y1,y2,y3T,C=diag2.6,2.3,D=diag2.2,2.8,3.4, τt=0.1−0.001e−t,σt=0.1−0.001e−2t,f1yt=cosy1t,cosy2t,cosy3tT,g1yt−τ=1/2cosy1t−τt,1/2cosy2t−τt,1/2cosy3t−τtT,h1yt=tanhy1t,tanhy2t,tanhy3tT,f2xt=cosx1t,cosx2tT,g2xt−τ=1/2cosx1t−τ,1/2cosx2t−τT,h2yt=tanhx1t,tanhx2tT,I=4.8,1T,J=5.7,4.2,6.4T, and(79)A=0.60.7−0.90.90.840.6,B=−0.5−0.6−0.8−0.9−0.74−0.6,R=4−3−3.54−3.2−2,P=0.6−0.50.40.8−1.20.7,Q=−0.5−0.6−0.6−0.4−0.4−0.6,S=−2−32−2−43.
An immediate calculation gives the following parameters:τ¯=σ¯=0.01, ξ1=ξ2=ζ1=ζ2=1, and η1=η2=1/2. Let xt,yt, x¯t,y¯t, and x^t,y^t be three solutions of this system with the following initial conditions:(80)xt=1.2−0.9t,1.5−1.1tT,x′t=−0.9,−1.1T,yt=1.6−0.7t,1.3−1.2t,2.3−0.8tT,y′t=−0.7,−1.2,−0.8T,x¯t=2.7−0.7t,3.1−0.9tT,x′¯t=−0.7,−0.9T,y¯t=3.6−0.9t,2.5−1.4t,3.5−tT,y′¯t=−0.9,−1.4,−1T,x^t=2.1−0.6t,2.25−0.8tT,x′^t=−0.6,−0.8T,y^t=2.6−0.9t,1.9−1.2t,2.8−1.2tT,y′^t=−0.9,−1.2,−1.2T,for any t∈−0.1,0.
By calculating, it follows thatγ=2.2, λ1=2.2, λ2=0.8, and λ3=8. Moreover, λ1+λ2eγτ¯+λ3σ¯eγσ¯Γα1/α=2.1619. Obviously, γ>λ1+λ2eγτ¯+λ3σ¯eγσ¯Γα1/α. Thus, the condition of Theorem 2 is satisfied. For the above three solutions, the time evolutions and the norms of errors are shown in Figures 8 and 9, respectively. These indicate the feasibility of Theorem 2.Figure 8
State trajectories of the system for three initial conditions in Example 4.
(a)
(b)
(c)
(d)
(e)Figure 9
Time evolutions for the norms of errors in Example 4.


## 5. Conclusions

This paper investigated the stability of fractional-order BAM neural networks with discrete and finite-time distributed time-varying delays. The order of systems is between 1 and 2. Firstly, to ensure the finite-time stability of systems, a sufficient condition was derived based on some analytical techniques and some elementary inequalities which are completely different from those in some earlier works. Unlike those existing conditions for fractional-order neural networks with discrete constant delay, our sufficient condition is not related to the Mittag–Leffler function. It is worth noting that this condition is expressed as an algebraic inequality, and the settling time can be easily estimated in practical applications. Next, a sufficient condition was obtained to guarantee the global asymptotic stability of systems based on the Laplace transform, the mean value theorem, the generalized Gronwall inequality, and some properties of Mittag–Leffler functions. Different from the above finite-time stability analysis, this does not require the additional assumption on the distributed time-varying delay. Moreover, this obtained condition generalizes that in the earlier work. In addition, two conditions were directly obtained to ensure the stability of the equilibrium point. Finally, some numerical examples were provided to verify the effectiveness of our main results. In future, for fractional-order systems withα∈1,2, we will discuss the stabilization problem based on fractional-order sliding mode control and investigate some practical applications in electrical engineering. This may be a big challenge for us since the mathematical theory for α∈1,2 is more complicated than that for α∈0,1.



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