# New Methods of Finite-Time Synchronization for a Class of Fractional-Order Delayed Neural Networks

Authors: Weiwei Zhang; Jinde Cao; Ahmed Alsaedi; Fuad E. Alsaadi
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Finite-time synchronization for a class of fractional-order delayed neural networks with fractional orderα, 0<α≤1/2 and 1/2<α<1, is investigated in this paper. Through the use of Hölder inequality, generalized Bernoulli inequality, and inequality skills, two sufficient conditions are considered to ensure synchronization of fractional-order delayed neural networks in a finite-time interval. Numerical example is given to verify the feasibility of the theoretical results.


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## Body

## 1. Introduction

Since the 17th century, the theory of fractional calculus was mainly focused on the pure theoretical field of mathematics [1, 2]. In the past two decades, it has been found that the dynamical behaviors of many systems can be described by the fractional calculus. Furthermore, fractional-order models can help exhibit the dynamical behaviors of systems. In fact, many physical systems show fractional dynamical behaviors because of special properties [3–11].Fractional-order neural networks have attracted great attention due to their potential properties of memory and hereditary. Particularly, the dynamical analysis of fractional-order neural networks can be used to describe the dynamical characteristics of neural networks. For instance, synchronization is considered as an important topic, as reported in [12, 13], where chaotic synchronization of fractional-order neural networks was proposed. Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks were discussed in [14]. In addition, the results for stability analysis and synchronization of fractional-order networks were presented in [15–23].However, the majority of the results were demonstrated to ensure the asymptotic stability of error systems. Asymptotic synchronization indicates that the trajectories of the slave system reach to the trajectories of the master system over the infinite horizon. In fact, it is more desirable that the networks can reach synchronization in a finite-time in physical and engineering systems, achieving an optimality in convergence time. Thus, it is necessary to study the finite-time synchronization of neural networks. In order to achieve faster synchronization in control systems, an effective finite-time control method is utilized. Some important results on finite-time synchronization were demonstrated on integer-order systems [24–28]. Note that time delay [29–31] occurs in many physical and engineering systems. So it is natural to study fractional-order systems with delays. Till now few results are obtained with the consideration of the finite-time stability and synchronization of fractional-order neural networks with time delays [32, 33]. For instance, finite-time synchronization of fractional-order memristor-based neural networks with time delays was considered by using Laplace transform, such as the generalized Gronwall inequality and Mittag-Leffler functions, in [33].Motivated by the above discussion, the main goal of this paper is to adopt new methods and obtain some new sufficient conditions that can assist master-slave systems to achieve the finite-time synchronization of fractional delayed neural networks with orders0<α≤1/2 and 1/2<α<1.Throughout the paper, denotex=∑i=1n‍xi(i=1,…,n) and A=maxj∑i=1n‍aij(i,j=1,…,n) which are the Euclidean vector norm and the matrix norm, respectively; xi and aij are the element of the vector x and the matrix A.


## 2. Preliminaries and Model Description

There are some definitions of the fractional-order integrals and derivatives. Due to the advantages of the Caputo fractional derivative, the definition of Caputo derivative is used in this paper.Definition 1 (see [1]).
The fractional integral with noninteger orderα>0 of a function x(t) is defined by(1)D-αxt=1Γα∫0tt-τα-1xτdτ,where t≥0, Γ(·) is the Gamma function, and Γ(s)=∫0∞‍ts-1e-tdt.Definition 2 (see [1]).
The Caputo derivative of fractional orderα of a function x(t) is defined by(2)Dαxt=1Γn-α∫0tt-τn-α-1xnτdτ,where t≥0; n-1<α<n∈Z+.
In this paper, we consider a class of fractional-order neural networks with time delay as master system, which is described by(3)Dαxit=-cixit+∑j=1naijtfjxjt+∑j=1nbijtgjxjt-τ+Ii,or equivalently(4)Dαxt=-Cxt+Atfxt+Btgxt-τ+I,for t∈J=[0,T](T>0), where 0<α<1,i=1,2,…,n,n is the number of units in a neural network, xi(t) corresponds to the state of the ith unit at time t and denotes x(t)=(x1(t),…,xn(t))T∈Rn, and C=diag(ci>0) is the self-regulating parameter of the neurons. I=(I1,I2,…,In)T represents the external input; A(t)=(aij(t))n×n and B(t)=(bij(t))n×n are the connective weights matrix in the presence and absence of delay, respectively. Functions fj(xj(t)) and gj(xj(t)) denote the output of the jth unit at time t and t-τ, respectively, where τ>0 is the transmission delay and denotes f(x(t))=(f1(x1(t)),…,fn(xn(t)))T, g(x(t))=(g1(x1(t)),…,gn(xn(t)))T.
The initial conditions associated with system (1) are of the form xi(t)=ψi(t),t∈[-τ,0](i=1,…,n), where ψi(t) denotes the real-valued continuous function defined on [-τ,0], with the norm given by ψ=sups∈[-τ,0]ψ(s).The slave system is given:(5)Dαyit=-ciyit+∑j=1naijtfjyjt+∑j=1nbijtgjyjt-τ-uit+Ii,or equivalently(6)Dαyt=-Cyt+Atfyt+Btgyt-τ-Ut+I,where y(t)=(y1(t),…,yn(t))T∈Rn is the state vector of the system response and U(t)=(u1(t),…,un(t))T is a suitable controller. The initial conditions associated with system (3) are of the form yi(t)=πi(t),t∈[-τ,0](i=1,…,n), where πi(t) denotes the real-valued continuous function defined on [-τ,0], with the norm given by π=sups∈[-τ,0]π(s).For generalities, the following definition, assumptions, and lemmas are presented.Definition 3.
System (1) is said to be synchronized with system (3) in a finite-time with respect to {0,δ,ε,T}, for a suitable designed controller ui(t), if and only if π(0)-ψ(0)≤δ, implying ei(t)=yi(t)-xi(t)<ε, ∀t∈[0,T], where δ,ϵ,T are real positive numbers and δ<ε.Assumption 4.
The neuron activation functionsf(x),g(x) are Lipschitz continuous, with the existence of positive constants H,K, such that(7)fu-fv≤Hu-v,gu-gv≤Ku-v,for all u,v∈Rn.Assumption 5.
a
i
j
(
t
) and bij(t) are continuous and bounded functions defined on R+, and let A=supt≥0A(t),B=supt≥0B(t).Lemma 6 (Hölder inequality [34]).
Assume thatp,q>1 and 1/p+1/q=1, and if |f(·)|p,|g(·)|q∈L1(E), then f(·)g(·)∈L1(E) and(8)∫Efxgxdx≤∫Efxpdx1/p∫Egxqdx1/q,where L1(E) is the Banach space of all Lebesgue measurable functions f:E→R with ∫E‍fxdx<∞.
Letp=q=2; it reduces to the Cauchy-Schwartz inequality as follows:(9)∫Efxgxdx2≤∫Efx2dx∫Egx2dx.Lemma 7 (generalized Bernoulli inequality [34]).
Ifk∈R+, x<1 and x≠0, then, for 0<k<1, (1-x)k<1-kx, or (1-(1-x)k)-1<(kx)-1.Lemma 8 (see [35]).
Ifx(t)∈Cm[0,∞) and m-1<α<m∈z+, then(1)
D-αD-βx(t)=D-(α+β)x(t),α,β≥0,(2)
DαD-βx(t)=x(t), α=β≥0,(3)
D-αDβx(t)=x(t)-∑k=0m-1(tk/k!)x(k)(0), α=β≥0.Lemma 9 (see [36]).
Letu(t),ω(t),v(t), and h(t) be nonnegative continuous functions on R+ and let r≥1 be a real number. If(10)ut≤u0t+ωt∫0tvsursds1/r,t∈R+,then(11)∫0tvsursds≤1-1-Wt1/r-r∫0tvsu0rsWsds,where W(t)=exp⁡(-∫0t‍v(s)ωr(s)ds).


## 3. Finite-Time Synchronization

In this section, master-slave finite-time synchronization of delayed fractional-order neural networks is discussed. The aim here is to design a suitable controller that can achieve the synchronization between the slave system and the master system.Letei(t)=yi(t)-xi(t)(i=1,2,…,n) be the synchronization errors.Select the linear control input functionsui(t) as the following form:(12)uit=ηiyit-xit,where each ηi>0(i=1,…,n) denotes the control gain.Then the error systems are obtained:(13)Dαeit=-ci+ηieit+∑j=1naijtfjyjt-fjxjt+∑j=1nbijtgjyjt-τ-gjxjt-τ.The vector form is as follows:(14)Dαet=-Ωet+Atfyt-fxt+Btgyt-τ-gxt-τ,where e(t)=(e1(t),…,en(t))T and Ω=diag(c1+η1,…,cn+ηn).The initial conditionse(t) of system (14) are of the following form: (15)eit=πit-ψit=ϕit,t∈-τ,0i=1,…,n.Theorem 10.
When1/2<α<1, suppose that Assumptions 4 and 5 hold, if(16)1+Net+21+NeM+Ne-τ2+1t<εδ,t∈0,T,where M=(Ω+AH)2Γ(2α-1)/Γ(α)2α, N=BK2Γ(2α-1)/Γ(α)2α; master system (3) is synchronized with slave system (5) in a finite-time with respect to {0,δ,ε,T} under the control (12).Proof.
Lete0=ϕ(0) be the initial condition of system (14), based on Lemma 8, the solution of system (7) in the form of the equivalent Volterra fractional integral equation is as follows:(17)et=e0+D-α-Ωet+Atfyt-fxt+Btgyt-τ-gxt-τ=e0+1Γα∫0tt-sα-1-Ωes+Asfys-fxs+Bsgys-τ-gxs-τds.Taking the norm · on both sides of the above system, according to the Assumptions 4 and 5, gives the following:(18)et≤e0+1Γα∫0tt-sα-1-Ωes+Asfys-fxs+Bsgys-τ-gxs-τds≤ϕ0+1Γα∫0tt-sα-1Ωes+AHes+BKes-τds=ϕ0+Ω+AHΓα∫0tt-sα-1ese-sesds+BKΓα∫0tt-sα-1ese-ses-τds.By using the Cauchy-Schwartz inequality, (18) gets the following:(19)et≤ϕ0+Ω+AHΓα∫0tt-s2α-1e2sds1/2∫0te-2ses2ds1/2+BKΓα∫0tt-s2α-1e2sds1/2∫0te-2ses-τ21/2ds=ϕ0+∫0tt-s2α-1e2sds1/2Ω+AHΓα∫0te-2ses2ds1/2ds+BKΓα∫0te-2ses-τ21/2ds.Note that(20)∫0tt-s2α-1e2sds=∫0tz2α-2e2t-zdz=e2t∫0tz2α-2e-2zdz=2e2t4α∫02tθ2α-2e-θdθ<2e2t4αΓ2α-1.Noting that e(t)=ϕ(t)(t∈[-τ,0]) and ϕ=supθ∈[-τ,0]ϕ(θ), one obtains(21)∫0te-2ses-τ2ds≤e-2τ∫-τte-2ueu2du=e-2τ∫-τ0e-2ueu2du+∫0te-2ueu2du≤ϕ2+e-2τ∫0te-2ses2ds.Then(22)et≤ϕ0+et2Γ2α-12αΩ+AHΓα∫0te-2ses2ds1/2+BKΓαϕ2+e-2τ∫0te-2ses21/2ds<ϕ+et2Γ2α-1BK2αΓαϕ+et2Γ2α-1Ω+AH2αΓα+2Γ2α-1BK2αΓαe-τ∫0te-2ses2ds1/2ds.Let M=2Γ(2α-1)(Ω+AH)/2αΓ(α) and N=2Γ(2α-1)BK/2αΓ(α).
Then(23)ete-t≤ϕe-t+Nϕ+M+Ne-τ∫0te-2ses2ds1/2ds.According to Lemmas 7 and 9, one has(24)ete-t≤ϕe-t+Nϕ+M+Ne-τ1-1-e-M+Ne-τ2t1/2-2·∫0tϕe-s+Nϕ2e-M+Ne-τ2sds1/2≤ϕe-t+Nϕ+M+Ne-τ2eM+Ne-τ2t·∫0tϕe-s+Nϕ2e-M+Ne-τ2sds1/2≤ϕe-t+Nϕ+21+NϕeM+Ne-τ2t=e-t+N+21+NeM+Ne-τ2tϕ.Therefore(25)et≤1+Net+21+NeM+Ne-τ2+1tϕ.Hence, if (16) is satisfied and ϕ<δ, then e(t)<ε; master system (3) is synchronized with slave system (5) in a finite-time.Theorem 11.
When0<α≤1/2, suppose that Assumptions 4 and 5 hold, if(26)1+N¯et+q1+N¯eM¯+N¯e-τq+1t<εδ,t∈0,T,where M¯=(Ω+AH)[Γ(α2)/Γp(α)pα2]1/p, N¯=BK[Γ(α2)/Γp(α)pα2]1/p and p=1+α, q=1+1/α; then master system (3) is synchronized with slave system (5) in a finite-time with respect to {0,δ,ε,T} under control (12).Proof.
Similar to Theorem10, we can obtain the following estimation:(27)et≤ϕ0+Ω+AHΓα∫0tt-sα-1ese-sesds+BKΓα∫0tt-sα-1ese-ses-τds. Let p=1+α, q=1+1/α; obviously, p,q>1 and 1/p+1/q=1, and following the Hölder inequality, we get(28)et≤ϕ0+Ω+AHΓα∫0tt-spα-1epsds1/p∫0te-qsesqds1/q+BKΓα∫0tt-spα-1epsds1/p∫0te-qses-τqds1/q=ϕ0+∫0tt-spα-1epsds1/pΩ+AHΓα∫0te-qsesqds1/q+BKΓα∫0te-qses-τqds1/q.Since(29)∫0tt-spα-1epsds=ept∫0tzpα-1e-pzdz=eptppα-1+1∫0ptupα-1e-udu≤eptpα2Γα2,∫0te-qses-τqds≤e-qτ∫-τte-queuqdu=e-qτ∫-τ0e-queuqdu+∫0te-queuqdu≤ϕq+e-qτ∫0te-qsesqds.then(30)et≤ϕ0+Γα2pα21/petΩ+AHΓα∫0te-qsesqds1/q+BKΓαϕq+e-qτ∫0te-qsesqds1/q≤ϕ+BKΓα2Γpαpα21/petϕ+etΩ+AHΓα2Γpαpα21/p+BKΓα2Γpαpα21/pe-τ∫0te-qsesqds1/q. Let M¯=(Ω+AH)(Γ(α2)/Γp(α)pα2)1/p, N¯=BK(Γ(α2)/Γp(α)pα2)1/p. Then(31)ete-t≤ϕe-t+N¯ϕ+M¯+N¯e-τ∫0te-qsesqds1/qds. According to Lemmas 7 and 9, one has(32)ete-t≤ϕe-t+N¯ϕ+M¯+N¯e-τ1-1-e-M¯+N¯e-τqt1/q-q·∫0tϕe-s+N¯ϕqe-M¯+N¯e-τqsds1/q≤ϕe-t+N¯ϕ+M¯+N¯e-τqeM¯+N¯e-τqt·∫0tϕe-s+N¯ϕqe-M¯+N¯e-τqsds1/q≤ϕe-t+N¯ϕ+q1+N¯ϕeM¯+N¯e-τqt=e-t+N¯+q1+N¯eM+Ne-τqtϕ.Therefore(33)et≤1+N¯et+q1+N¯eM¯+N¯e-τq+1tϕ.Hence, if (26) is satisfied and ϕ<δ, then e(t)<ε; master system (3) is synchronized with slave system (5) in a finite-time.Remark 12.
References [24–28] discussed the finite-time synchronization of neural networks, which only considered integer-order systems.Remark 13.
Generally, time-delayed differential models unavoidably exist in neural networks [37]. However, quite a few researches investigated the finite-time synchronization of neural networks with delay. Hence, it is foremost important and necessary to consider finite-time synchronization of fractional-order delayed neural networks.Remark 14.
In this paper, Assumptions4 and 5 are general, not too strict; many functions can satisfy the assumptions; for example, [15, 17, 29] considered these assumptions too.Remark 15.
In [33], the authors discussed the finite-time synchronization of fractional-order memristor-based neural networks with time delays by using Laplace transform, generalized Gronwall’s inequality and Mittag-Leffler functions, and the results showed Mittag-Leffler functions. In this paper, employing Lemma 4 proposed in [35] and Bernoulli inequality can obtain results that only include exponential functions, given that the form is simpler and the calculation is easier.Remark 16.
In Theorem10, from (12) and (14) and inequality (16), it is obvious that the convergence time T is proportional to the inverse of the control gain ηi. Therefore, a greater ηi results in the shorter convergence time T, and therefore ηi should be selected in accordance with the convergence time T to be short.Remark 17.
In [32], the authors discussed finite-time stability of fractional-order neural networks with delay by utilizing Gronwall inequality. Unlike the previous work [32], in this paper, we discussed the finite-time synchronization for a class of fractional-order delayed neural networks by employing Lemma 4 proposed in [35], Bernoulli inequality, and inequality skills; two new delay-dependent sufficient conditions have been established.Remark 18.
Different from integer-order delayed systems, it is difficult to construct Lyapunov functions for fractional-order nonlinear systems with time delay; results have been obtained using different techniques from approaches used in the area of finite synchronization. By employing inequality skills, new sufficient conditions ensuring finite-time synchronization are derived.


## 4. Numerical Simulations

Consider the following two-dimensional delayed fractional-order Hopfield neural networks:(34)Dαxt=-Cxt+Atfxt+Btgxt-τ+I,for t∈J=[0,15], where x(t)=(x1(t),x2(t))T, α=0.4 or α=0.7, τ=0.1, I=(0,0)T, and fj(xj(t))=gj(xj(t))=tanh(xj(t)),(j=1,2), denoting tanh(x(t))=(tanh⁡(x1t),tanh⁡(x2(t)))T, because of tanh⁡x′=1-(tanh⁡(x))2≤1, according to Lagrange Theorem; clearly, f(x) and g(x) satisfy Assumption 4 with H=K=1, A(t)=0.2-0.10.10.2, B(t)=-0.5-0.1-0.2-0.5, C=0.1000.1. So, according to Assumption 5, A=supt≥0A(t)=0.3, B=supt≥0B(t)=0.7, C=0.1. Choose the initial values of master system and slave system as follows: x1(0)=4, x2(0)=2, y1(0)=3, y2(0)=1.Whenα=0.4, take δ=0.01, ϵ=1, τ=0.1 and select the control gain η1=4, η2=5; apparently, the condition of Theorem 11 is satisfied. It could be verified that M¯=8.0815, N¯=1.0674, and the estimated time of finite-time synchronization is T=0.7099. Synchronization errors between master and slave systems are shown in Figures 1 and 2. It is clearly seen that the synchronization errors converge to zero, indicating that master system and slave system are synchronized in a finite-time. For comparison purposes, the curves of the state variable of the master system and the slave system are shown in Figures 3 and 4.Figure 1
The errors state ofe1 (order α=0.4).Figure 2
The errors state ofe2 (order α=0.4).Figure 3
The synchronization trajectories ofx1 and y1 (order α=0.4).Figure 4
The synchronization trajectories ofx2 and y2 (order α=0.4).Whenα=0.7, take δ=0.01, ϵ=1, τ=0.1 and select the control gain η1=2, η2=4; apparently, the condition of Theorem 10 is satisfied. It is easy to obtain M=4.2950,N=0.6992, and the estimated time of finite-time synchronization is T=0.5937. Synchronization errors between master and slave systems are shown in Figures 5 and 6. It can be seen that the synchronization errors converge to zero, confirming that master system and slave system are synchronized in a finite-time. Also, for comparison purposes, the curves of the state variable of the master system and the slave system are shown in Figures 7 and 8.Figure 5
The errors state ofe1 (order α=0.7).Figure 6
The errors state ofe2 (order α=0.7).Figure 7
The synchronization trajectories ofx1 and y1 (order α=0.7).Figure 8
The synchronization trajectories ofx2 and y2 (order α=0.7).


## 5. Conclusions

In this paper, finite-time synchronization for a class fractional-order delayed neural networks with orderα, 0<α≤1/2 and 1/2<α<1, was discussed. Some new sufficient conditions are derived to ensure the finite-time synchronization for this class of fractional-order systems. Numerical example is presented to verify the effectiveness of the theoretical results. The proposed methods are novel and solve well the finite-time synchronization of fractional-order delayed neural networks. We would like to point out that it is possible to extend the methods to other fractional-order models, such as fractional-order delayed neutral-type neural networks and fractional neural networks with incommensurate. These issues will be further worth discussing.



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