new methods of finite-time synchronization for a class of...

Research ArticleNew Methods of Finite-Time Synchronization for a Class ofFractional-Order Delayed Neural Networks

Weiwei Zhang,1 Jinde Cao,2,3 Ahmed Alsaedi,4 and Fuad E. Alsaadi5

1School of Mathematics and Computational Science, Anqing Normal University, Anqing 246011, China2School of Mathematics, Southeast University, Nanjing 210096, China3School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China4Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia5Department of Electrical and Computer Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Weiwei Zhang; [email protected]

Received 22 March 2017; Accepted 10 May 2017; Published 6 June 2017

Academic Editor: R. Aguilar-López

Copyright © 2017 Weiwei Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Finite-time synchronization for a class of fractional-order delayed neural networks with fractional order 𝛼, 0 < 𝛼 ≤ 1/2 and1/2 < 𝛼 < 1, is investigated in this paper. Through the use of Hölder inequality, generalized Bernoulli inequality, and inequalityskills, 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.

1. Introduction

Since the 17th century, the theory of fractional calculus wasmainly focused on the pure theoretical field of mathematics[1, 2]. In the past two decades, it has been found that thedynamical behaviors of many systems can be described bythe fractional calculus. Furthermore, fractional-ordermodelscan help exhibit the dynamical behaviors of systems. In fact,many physical systems show fractional dynamical behaviorsbecause of special properties [3–11].

Fractional-order neural networks have attracted greatattention due to their potential properties of memory andhereditary. Particularly, the dynamical analysis of fractional-order neural networks can be used to describe the dynamicalcharacteristics of neural networks. For instance, synchro-nization is considered as an important topic, as reported in[12, 13], where chaotic synchronization of fractional-orderneural networks was proposed. Mittag-Leffler stability andsynchronization of memristor-based fractional-order neuralnetworks were discussed in [14]. In addition, the resultsfor stability analysis and synchronization of fractional-ordernetworks were presented in [15–23].

However, the majority of the results were demon-strated to ensure the asymptotic stability of error systems.

Asymptotic synchronization indicates that the trajectories ofthe slave system reach to the trajectories of the master systemover the infinite horizon. In fact, it is more desirable thatthe networks can reach synchronization in a finite-time inphysical and engineering systems, achieving an optimality inconvergence time.Thus, it is necessary to study the finite-timesynchronization of neural networks. In order to achieve fastersynchronization in control systems, an effective finite-timecontrol method is utilized. Some important results on finite-time synchronization were demonstrated on integer-ordersystems [24–28]. Note that time delay [29–31] occurs inmanyphysical and engineering systems. So it is natural to studyfractional-order systems with delays. Till now few results areobtained with the consideration of the finite-time stabilityand synchronization of fractional-order neural networkswithtime delays [32, 33]. For instance, finite-time synchronizationof fractional-order memristor-based neural networks withtime delays was considered by using Laplace transform, suchas the generalized Gronwall inequality and Mittag-Lefflerfunctions, in [33].

Motivated by the above discussion, the main goal ofthis paper is to adopt new methods and obtain some newsufficient conditions that can assist master-slave systems to

HindawiMathematical Problems in EngineeringVolume 2017, Article ID 1804383, 9 pageshttps://doi.org/10.1155/2017/1804383

https://doi.org/10.1155/2017/1804383

2 Mathematical Problems in Engineering

achieve the finite-time synchronization of fractional delayedneural networks with orders 0 < 𝛼 ≤ 1/2 and 1/2 < 𝛼 < 1.

Throughout the paper, denote ‖𝑥‖ = ∑𝑛𝑖=1 |𝑥𝑖| (𝑖 =1, . . . , 𝑛) and ‖𝐴‖ = max𝑗∑𝑛𝑖=1 |𝑎𝑖𝑗| (𝑖, 𝑗 = 1, . . . , 𝑛) which arethe Euclidean vector norm and thematrix 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 integralsand derivatives. Due to the advantages of the Caputo frac-tional derivative, the definition of Caputo derivative is usedin this paper.

Definition 1 (see [1]). The fractional integral with nonintegerorder 𝛼 > 0 of a function 𝑥(𝑡) is defined by

𝐷−𝛼𝑥 (𝑡) = 1Γ (𝛼) ∫𝑡

0(𝑡 − 𝜏)𝛼−1 𝑥 (𝜏) 𝑑𝜏, (1)

where 𝑡 ≥ 0, Γ(⋅) is the Gamma function, and Γ(𝑠) =∫∞0

𝑡𝑠−1𝑒−𝑡𝑑𝑡.Definition 2 (see [1]). The Caputo derivative of fractionalorder 𝛼 of a function 𝑥(𝑡) is defined by

𝐷𝛼𝑥 (𝑡) = 1Γ (𝑛 − 𝛼) ∫𝑡

0(𝑡 − 𝜏)𝑛−𝛼−1 𝑥(𝑛) (𝜏) 𝑑𝜏, (2)

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 isdescribed by

𝐷𝛼𝑥𝑖 (𝑡) = −𝑐𝑖𝑥𝑖 (𝑡) + 𝑛∑𝑗=1

𝑎𝑖𝑗 (𝑡) 𝑓𝑗 (𝑥𝑗 (𝑡))

+ 𝑛∑𝑗=1

𝑏𝑖𝑗 (𝑡) 𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏)) + 𝐼𝑖,(3)

or equivalently

𝐷𝛼𝑥 (𝑡) = −𝐶𝑥 (𝑡) + 𝐴 (𝑡) 𝑓 (𝑥 (𝑡)) + 𝐵 (𝑡) 𝑔 (𝑥 (𝑡 − 𝜏))+ 𝐼, (4)

for 𝑡 ∈ 𝐽 = [0, 𝑇] (𝑇 > 0), where 0 < 𝛼 < 1, 𝑖 = 1, 2, . . . , 𝑛, 𝑛is the number of units in a neural network, 𝑥𝑖(𝑡) correspondsto 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)𝑇represents the external input; 𝐴(𝑡) = (𝑎𝑖𝑗(𝑡))𝑛×𝑛 and 𝐵(𝑡) =(𝑏𝑖𝑗(𝑡))𝑛×𝑛 are the connective weights matrix in the presenceand absence of delay, respectively. Functions 𝑓𝑗(𝑥𝑗(𝑡)) and𝑔𝑗(𝑥𝑗(𝑡)) denote the output of the 𝑗th unit at time 𝑡 and𝑡 − 𝜏, respectively, where 𝜏 > 0 is the transmission delayand 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 ofthe 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:

𝐷𝛼𝑦𝑖 (𝑡) = −𝑐𝑖𝑦𝑖 (𝑡) + 𝑛∑𝑗=1

𝑎𝑖𝑗 (𝑡) 𝑓𝑗 (𝑦𝑗 (𝑡))

+ 𝑛∑𝑗=1

𝑏𝑖𝑗 (𝑡) 𝑔𝑗 (𝑦𝑗 (𝑡 − 𝜏)) − 𝑢𝑖 (𝑡) + 𝐼𝑖,(5)

or equivalently

𝐷𝛼𝑦 (𝑡) = −𝐶𝑦 (𝑡) + 𝐴 (𝑡) 𝑓 (𝑦 (𝑡)) + 𝐵 (𝑡) 𝑔 (𝑦 (𝑡 − 𝜏))− 𝑈 (𝑡) + 𝐼, (6)

where 𝑦(𝑡) = (𝑦1(𝑡), . . . , 𝑦𝑛(𝑡))𝑇 ∈ 𝑅𝑛 is the state vector of thesystem response and 𝑈(𝑡) = (𝑢1(𝑡), . . . , 𝑢𝑛(𝑡))𝑇 is a suitablecontroller. The initial conditions associated with system (3)are of the form 𝑦𝑖(𝑡) = 𝜋𝑖(𝑡), 𝑡 ∈ [−𝜏, 0] (𝑖 = 1, . . . , 𝑛), where𝜋𝑖(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 withsystem (3) in a finite-time with respect to {0, 𝛿, 𝜀, 𝑇}, for asuitable designed controller𝑢𝑖(𝑡), if and only if ‖𝜋(0)−𝜓(0)‖ ≤𝛿, implying ‖𝑒𝑖(𝑡)‖ = ‖𝑦𝑖(𝑡) − 𝑥𝑖(𝑡)‖ < 𝜀, ∀𝑡 ∈ [0, 𝑇], where𝛿, 𝜖, 𝑇 are real positive numbers and 𝛿 < 𝜀.Assumption 4. The neuron activation functions 𝑓(𝑥), 𝑔(𝑥)are Lipschitz continuous, with the existence of positiveconstants𝐻, 𝐾, such that

𝑓 (𝑢) − 𝑓 (V) ≤ 𝐻 ‖𝑢 − V‖ ,𝑔 (𝑢) − 𝑔 (V) ≤ 𝐾 ‖𝑢 − V‖ , (7)for all 𝑢, V ∈ 𝑅𝑛.Assumption 5. 𝑎𝑖𝑗(𝑡) and 𝑏𝑖𝑗(𝑡) are continuous and boundedfunctions defined on 𝑅+, and let 𝐴 = sup𝑡≥0‖𝐴(𝑡)‖, 𝐵 =sup𝑡≥0‖𝐵(𝑡)‖.Lemma 6 (Hölder inequality [34]). Assume that 𝑝, 𝑞 > 1 and1/𝑝 + 1/𝑞 = 1, and if |𝑓(⋅)|𝑝, |𝑔(⋅)|𝑞 ∈ 𝐿1(𝐸), then 𝑓(⋅)𝑔(⋅) ∈𝐿1(𝐸) and

∫𝐸

𝑓 (𝑥) 𝑔 (𝑥) 𝑑𝑥≤ (∫𝐸

𝑓 (𝑥)𝑝 𝑑𝑥)1/𝑝 (∫𝐸

𝑔 (𝑥)𝑞 𝑑𝑥)1/𝑞 ,(8)

where 𝐿1(𝐸) is the Banach space of all Lebesgue measurablefunctions 𝑓 : 𝐸 → 𝑅 with ∫

𝐸|𝑓(𝑥)|𝑑𝑥 < ∞.

Mathematical Problems in Engineering 3

Let𝑝 = 𝑞 = 2; it reduces to the Cauchy-Schwartz inequalityas follows:

(∫𝐸

𝑓 (𝑥) 𝑔 (𝑥) 𝑑𝑥)2

≤ (∫𝐸

𝑓 (𝑥)2 𝑑𝑥)(∫𝐸

𝑔 (𝑥)2 𝑑𝑥) .(9)

Lemma 7 (generalized Bernoulli inequality [34]). If 𝑘 ∈ 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]). If 𝑥(𝑡) ∈ 𝐶𝑚[0,∞) and𝑚−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) − ∑𝑚−1𝑘=0 (𝑡𝑘/𝑘!)𝑥(𝑘)(0), 𝛼 = 𝛽 ≥ 0.

Lemma 9 (see [36]). Let 𝑢(𝑡), 𝜔(𝑡), V(𝑡), and ℎ(𝑡) be nonnega-tive continuous functions on 𝑅+ and let 𝑟 ≥ 1 be a real number.If

𝑢 (𝑡) ≤ 𝑢0 (𝑡) + 𝜔 (𝑡) (∫𝑡0V (𝑠) 𝑢𝑟 (𝑠) 𝑑𝑠)1/𝑟 , 𝑡 ∈ 𝑅+, (10)

then

∫𝑡0V (𝑠) 𝑢𝑟 (𝑠) 𝑑𝑠≤ [1 − (1 − 𝑊 (𝑡))1/𝑟]−𝑟 ∫𝑡

0V (𝑠) 𝑢𝑟0 (𝑠)𝑊 (𝑠) 𝑑𝑠,

(11)

where𝑊(𝑡) = exp(− ∫𝑡0V(𝑠)𝜔𝑟(𝑠)𝑑𝑠).

3. Finite-Time Synchronization

In this section, master-slave finite-time synchronization ofdelayed fractional-order neural networks is discussed. Theaim here is to design a suitable controller that can achievethe synchronization between the slave system and the mastersystem.

Let 𝑒𝑖(𝑡) = 𝑦𝑖(𝑡) − 𝑥𝑖(𝑡) (𝑖 = 1, 2, . . . , 𝑛) be the synchro-nization errors.

Select the linear control input functions 𝑢𝑖(𝑡) as thefollowing form:

𝑢𝑖 (𝑡) = 𝜂𝑖 (𝑦𝑖 (𝑡) − 𝑥𝑖 (𝑡)) , (12)where each 𝜂𝑖 > 0 (𝑖 = 1, . . . , 𝑛) denotes the control gain.

Then the error systems are obtained:

𝐷𝛼𝑒𝑖 (𝑡)= − (𝑐𝑖 + 𝜂𝑖) 𝑒𝑖 (𝑡)

+ 𝑛∑𝑗=1

𝑎𝑖𝑗 (𝑡) [𝑓𝑗 (𝑦𝑗 (𝑡)) − 𝑓𝑗 (𝑥𝑗 (𝑡))]

+ 𝑛∑𝑗=1

𝑏𝑖𝑗 (𝑡) [𝑔𝑗 (𝑦𝑗 (𝑡 − 𝜏)) − 𝑔𝑗 (𝑥𝑗 (𝑡 − 𝜏))] .

(13)

The vector form is as follows:

𝐷𝛼𝑒 (𝑡) = −Ω𝑒 (𝑡) + 𝐴 (𝑡) [𝑓 (𝑦 (𝑡)) − 𝑓 (𝑥 (𝑡))]+ 𝐵 (𝑡) [𝑔 (𝑦 (𝑡 − 𝜏)) − 𝑔 (𝑥 (𝑡 − 𝜏))] , (14)

where 𝑒(𝑡) = (𝑒1(𝑡), . . . , 𝑒𝑛(𝑡))𝑇 andΩ = diag(𝑐1 + 𝜂1, . . . , 𝑐𝑛 +𝜂𝑛).The initial conditions 𝑒(𝑡) of system (14) are of the follow-

ing form:

𝑒𝑖 (𝑡) = 𝜋𝑖 (𝑡) − 𝜓𝑖 (𝑡) = 𝜙𝑖 (𝑡) ,𝑡 ∈ [−𝜏, 0] (𝑖 = 1, . . . , 𝑛) . (15)

Theorem 10. When 1/2 < 𝛼 < 1, suppose that Assumptions 4and 5 hold, if

[1 + 𝑁𝑒𝑡 + 2 (1 + 𝑁) 𝑒[(𝑀+𝑁𝑒−𝜏)2+1]𝑡] < 𝜀𝛿 ,𝑡 ∈ [0, 𝑇] , (16)

where 𝑀 = (‖Ω‖ + 𝐴𝐻)√2Γ(2𝛼 − 1)/Γ(𝛼)2𝛼, 𝑁 =𝐵𝐾√2Γ(2𝛼 − 1)/Γ(𝛼)2𝛼; master system (3) is synchronizedwith slave system (5) in a finite-time with respect to {0, 𝛿, 𝜀, 𝑇}under the control (12).

Proof. Let 𝑒0 = 𝜙(0) be the initial condition of system (14),based on Lemma 8, the solution of system (7) in the formof the equivalent Volterra fractional integral equation is asfollows:

𝑒 (𝑡) = 𝑒0 + 𝐷−𝛼 [−Ω𝑒 (𝑡) + 𝐴 (𝑡) (𝑓 (𝑦 (𝑡)) − 𝑓 (𝑥 (𝑡)))+ 𝐵 (𝑡) (𝑔 (𝑦 (𝑡 − 𝜏)) − 𝑔 (𝑥 (𝑡 − 𝜏)))] = 𝑒0 + 1Γ (𝛼)⋅ ∫𝑡0(𝑡 − 𝑠)𝛼−1 [−Ω𝑒 (𝑠)

+ 𝐴 (𝑠) (𝑓 (𝑦 (𝑠)) − 𝑓 (𝑥 (𝑠)))+ 𝐵 (𝑠) (𝑔 (𝑦 (𝑠 − 𝜏)) − 𝑔 (𝑥 (𝑠 − 𝜏)))] 𝑑𝑠.

(17)

Taking the norm ‖ ⋅ ‖ on both sides of the above system,according to the Assumptions 4 and 5, gives the following:

‖𝑒 (𝑡)‖ ≤ 𝑒0 + 1Γ (𝛼) ∫𝑡

0(𝑡 − 𝑠)𝛼−1 [−Ω𝑒 (𝑠)

+ 𝐴 (𝑠) (𝑓 (𝑦 (𝑠)) − 𝑓 (𝑥 (𝑠)))+ 𝐵 (𝑠) (𝑔 (𝑦 (𝑠 − 𝜏)) − 𝑔 (𝑥 (𝑠 − 𝜏)))] 𝑑𝑠≤ 𝜙 (0) + 1Γ (𝛼) ∫

𝑡

0(𝑡 − 𝑠)𝛼−1 [‖Ω‖ ‖𝑒 (𝑠)‖

+ 𝐴𝐻‖𝑒 (𝑠)‖ + 𝐵𝐾 ‖𝑒 (𝑠 − 𝜏)‖] 𝑑𝑠 = 𝜙 (0)+ (‖Ω‖ + 𝐴𝐻)Γ (𝛼) ∫

𝑡

0(𝑡 − 𝑠)𝛼−1 𝑒𝑠𝑒−𝑠 ‖𝑒 (𝑠)‖ 𝑑𝑠 + 𝐵𝐾Γ (𝛼)

⋅ ∫𝑡0(𝑡 − 𝑠)𝛼−1 𝑒𝑠𝑒−𝑠 ‖𝑒 (𝑠 − 𝜏)‖ 𝑑𝑠.

(18)


By using theCauchy-Schwartz inequality, (18) gets the follow-ing:

‖𝑒 (𝑡)‖ ≤ 𝜙 (0)+ (‖Ω‖ + 𝐴𝐻)Γ (𝛼) (∫

𝑡

0(𝑡 − 𝑠)2(𝛼−1) 𝑒2𝑠𝑑𝑠)1/2

⋅ (∫𝑡0𝑒−2𝑠 ‖𝑒 (𝑠)‖2 𝑑𝑠)1/2

+ 𝐵𝐾Γ (𝛼) (∫𝑡

0(𝑡 − 𝑠)2(𝛼−1) 𝑒2𝑠𝑑𝑠)1/2

⋅ (∫𝑡0𝑒−2𝑠 ‖𝑒 (𝑠 − 𝜏)‖2)1/2 𝑑𝑠 = 𝜙 (0)

+ (∫𝑡0(𝑡 − 𝑠)2(𝛼−1) 𝑒2𝑠𝑑𝑠)1/2

⋅ [ (‖Ω‖ + 𝐴𝐻)Γ (𝛼) (∫𝑡

0𝑒−2𝑠 ‖𝑒 (𝑠)‖2 𝑑𝑠)1/2 𝑑𝑠

+ 𝐵𝐾Γ (𝛼) (∫𝑡

0𝑒−2𝑠 ‖𝑒 (𝑠 − 𝜏)‖2)1/2 𝑑𝑠] .

(19)

Note that

∫𝑡0(𝑡 − 𝑠)2(𝛼−1) 𝑒2𝑠𝑑𝑠 = ∫𝑡

0𝑧(2𝛼−2)𝑒2(𝑡−𝑧)𝑑𝑧

= 𝑒2𝑡 ∫𝑡0𝑧(2𝛼−2)𝑒−2𝑧𝑑𝑧

= 2𝑒2𝑡4𝛼 ∫2𝑡

0𝜃(2𝛼−2)𝑒−𝜃𝑑𝜃

< 2𝑒2𝑡4𝛼 Γ (2𝛼 − 1) .

(20)

Noting that 𝑒(𝑡) = 𝜙(𝑡) (𝑡 ∈ [−𝜏, 0]) and ‖𝜙‖ =sup𝜃∈[−𝜏,0]‖𝜙(𝜃)‖, one obtains

∫𝑡0𝑒−2𝑠 ‖𝑒 (𝑠 − 𝜏)‖2 𝑑𝑠 ≤ 𝑒−2𝜏 ∫𝑡

−𝜏𝑒−2𝑢 ‖𝑒 (𝑢)‖2 𝑑𝑢

= 𝑒−2𝜏 [∫0−𝜏

𝑒−2𝑢 ‖𝑒 (𝑢)‖2 𝑑𝑢 + ∫𝑡0𝑒−2𝑢 ‖𝑒 (𝑢)‖2 𝑑𝑢]

≤ 𝜙2 + 𝑒−2𝜏 ∫𝑡0𝑒−2𝑠 ‖𝑒 (𝑠)‖2 𝑑𝑠.

(21)

Then

‖𝑒 (𝑡)‖ ≤ 𝜙 (0)+ 𝑒𝑡√2Γ (2𝛼 − 1)2𝛼 [(‖Ω‖ + 𝐴𝐻)Γ (𝛼) (∫

𝑡

0𝑒−2𝑠 ‖𝑒 (𝑠)‖2 𝑑𝑠)1/2

+ 𝐵𝐾Γ (𝛼) (𝜙2 + 𝑒−2𝜏 ∫𝑡

0𝑒−2𝑠 ‖𝑒 (𝑠)‖2)1/2 𝑑𝑠] < (𝜙

+ 𝑒𝑡√2Γ (2𝛼 − 1)𝐵𝐾2𝛼Γ (𝛼) 𝜙)+ 𝑒𝑡 [√2Γ (2𝛼 − 1) (‖Ω‖ + 𝐴𝐻)2𝛼Γ (𝛼)+ √2Γ (2𝛼 − 1)𝐵𝐾2𝛼Γ (𝛼) 𝑒−𝜏](∫

𝑡

0𝑒−2𝑠 ‖𝑒 (𝑠)‖2 𝑑𝑠)1/2 𝑑𝑠.

(22)

Let 𝑀 = √2Γ(2𝛼 − 1)(‖Ω‖ + 𝐴𝐻)/2𝛼Γ(𝛼) and 𝑁 =√2Γ(2𝛼 − 1)𝐵𝐾/2𝛼Γ(𝛼).Then

‖𝑒 (𝑡)‖ 𝑒−𝑡≤ (𝜙 𝑒−𝑡 + 𝑁𝜙)

+ (𝑀 + 𝑁𝑒−𝜏) (∫𝑡0𝑒−2𝑠 ‖𝑒 (𝑠)‖2 𝑑𝑠)1/2 𝑑𝑠.

(23)

According to Lemmas 7 and 9, one has

‖𝑒 (𝑡)‖ 𝑒−𝑡 ≤ (𝜙 𝑒−𝑡 + 𝑁𝜙) + (𝑀 + 𝑁𝑒−𝜏)⋅ [(1 − (1 − 𝑒−(𝑀+𝑁𝑒−𝜏)2𝑡)1/2)−2

⋅ ∫𝑡0(𝜙 𝑒−𝑠 + 𝑁𝜙)2 𝑒−(𝑀+𝑁𝑒−𝜏)2𝑠𝑑𝑠]

1/2

≤ (𝜙 𝑒−𝑡 + 𝑁𝜙) + (𝑀 + 𝑁𝑒−𝜏) 2𝑒(𝑀+𝑁𝑒−𝜏)2𝑡⋅ [∫𝑡0(𝜙 𝑒−𝑠 + 𝑁𝜙)2 𝑒−(𝑀+𝑁𝑒−𝜏)2𝑠𝑑𝑠]

1/2

≤ (𝜙 𝑒−𝑡 + 𝑁𝜙) + 2 (1 + 𝑁) 𝜙 𝑒(𝑀+𝑁𝑒−𝜏)2𝑡= [𝑒−𝑡 + 𝑁 + 2 (1 + 𝑁) 𝑒(𝑀+𝑁𝑒−𝜏)2𝑡] 𝜙 .

(24)

Therefore

‖𝑒 (𝑡)‖ ≤ [1 + 𝑁𝑒𝑡 + 2 (1 + 𝑁) 𝑒[(𝑀+𝑁𝑒−𝜏)2+1]𝑡] 𝜙 . (25)Hence, if (16) is satisfied and ‖𝜙‖ < 𝛿, then ‖𝑒(𝑡)‖ < 𝜀; mastersystem (3) is synchronized with slave system (5) in a finite-time.

Theorem 11. When 0 < 𝛼 ≤ 1/2, suppose that Assumptions 4and 5 hold, if

1 + 𝑁𝑒𝑡 + 𝑞 (1 + 𝑁) 𝑒[(𝑀+𝑁𝑒−𝜏)𝑞+1]𝑡 < 𝜀𝛿 , 𝑡 ∈ [0, 𝑇] , (26)


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; thenmaster system (3) is synchronized with slave system (5) in afinite-time with respect to {0, 𝛿, 𝜀, 𝑇} under control (12).Proof. Similar to Theorem 10, we can obtain the followingestimation:

‖𝑒 (𝑡)‖ ≤ 𝜙 (0)+ (‖Ω‖ + 𝐴𝐻)Γ (𝛼) ∫

𝑡

0(𝑡 − 𝑠)𝛼−1 𝑒𝑠𝑒−𝑠 ‖𝑒 (𝑠)‖ 𝑑𝑠

+ 𝐵𝐾Γ (𝛼) ∫𝑡

0(𝑡 − 𝑠)𝛼−1 𝑒𝑠𝑒−𝑠 ‖𝑒 (𝑠 − 𝜏)‖ 𝑑𝑠.

(27)

Let𝑝 = 1+𝛼, 𝑞 = 1+1/𝛼; obviously,𝑝, 𝑞 > 1 and 1/𝑝+1/𝑞 = 1,and following the Hölder inequality, we get

‖𝑒 (𝑡)‖ ≤ 𝜙 (0)+ (‖Ω‖ + 𝐴𝐻)Γ (𝛼) [∫

𝑡

0(𝑡 − 𝑠)𝑝(𝛼−1) 𝑒𝑝𝑠𝑑𝑠]1/𝑝

⋅ [∫𝑡0𝑒−𝑞𝑠 ‖𝑒 (𝑠)‖𝑞 𝑑𝑠]1/𝑞

+ 𝐵𝐾Γ (𝛼) [∫𝑡

0(𝑡 − 𝑠)𝑝(𝛼−1) 𝑒𝑝𝑠𝑑𝑠]1/𝑝

⋅ [∫𝑡0𝑒−𝑞𝑠 ‖𝑒 (𝑠 − 𝜏)‖𝑞 𝑑𝑠]1/𝑞 = 𝜙 (0)

+ [∫𝑡0(𝑡 − 𝑠)𝑝(𝛼−1) 𝑒𝑝𝑠𝑑𝑠]1/𝑝

⋅ [ (‖Ω‖ + 𝐴𝐻)Γ (𝛼) (∫𝑡

0𝑒−𝑞𝑠 ‖𝑒 (𝑠)‖𝑞 𝑑𝑠)1/𝑞

+ 𝐵𝐾Γ (𝛼) (∫𝑡

0𝑒−𝑞𝑠 ‖𝑒 (𝑠 − 𝜏)‖𝑞 𝑑𝑠)1/𝑞] .

(28)

Since

∫𝑡0(𝑡 − 𝑠)𝑝(𝛼−1) 𝑒𝑝𝑠𝑑𝑠 = 𝑒𝑝𝑡 ∫𝑡

0𝑧𝑝(𝛼−1)𝑒−𝑝𝑧𝑑𝑧

= 𝑒𝑝𝑡𝑝𝑝(𝛼−1)+1 ∫𝑝𝑡

0𝑢𝑝(𝛼−1)𝑒−𝑢𝑑𝑢 ≤ 𝑒𝑝𝑡𝑝𝛼2 Γ (𝛼2) ,

∫𝑡0𝑒−𝑞𝑠 ‖𝑒 (𝑠 − 𝜏)‖𝑞 𝑑𝑠 ≤ 𝑒−𝑞𝜏 ∫𝑡

−𝜏𝑒−𝑞𝑢 ‖𝑒 (𝑢)‖𝑞 𝑑𝑢

= 𝑒−𝑞𝜏 [∫0−𝜏

𝑒−𝑞𝑢 ‖𝑒 (𝑢)‖𝑞 𝑑𝑢 + ∫𝑡0𝑒−𝑞𝑢 ‖𝑒 (𝑢)‖𝑞 𝑑𝑢]

≤ 𝜙𝑞 + 𝑒−𝑞𝜏 ∫𝑡0𝑒−𝑞𝑠 ‖𝑒 (𝑠)‖𝑞 𝑑𝑠.

(29)

then

‖𝑒 (𝑡)‖ ≤ 𝜙 (0) + [Γ (𝛼2)

𝑝𝛼2 ]1/𝑝

⋅ 𝑒𝑡 [(‖Ω‖ + A𝐻)Γ (𝛼) (∫𝑡

0𝑒−𝑞𝑠 ‖𝑒 (𝑠)‖𝑞 𝑑𝑠)1/𝑞

+ 𝐵𝐾Γ (𝛼) (𝜙𝑞 + 𝑒−𝑞𝜏 ∫𝑡

0𝑒−𝑞𝑠 ‖𝑒 (𝑠)‖𝑞 𝑑𝑠)1/𝑞]

≤ [[𝜙 + 𝐵𝐾( Γ (𝛼

2)Γ𝑝 (𝛼) 𝑝𝛼2 )

1/𝑝

𝑒𝑡 𝜙]]+ 𝑒𝑡 [

[(‖Ω‖ + 𝐴𝐻)( Γ (𝛼2)Γ𝑝 (𝛼) 𝑝𝛼2 )

1/𝑝

+ 𝐵𝐾( Γ (𝛼2)Γ𝑝 (𝛼) 𝑝𝛼2 )1/𝑝

𝑒−𝜏]]

⋅ (∫𝑡0𝑒−𝑞𝑠 ‖𝑒 (𝑠)‖𝑞 𝑑𝑠)1/𝑞 .

(30)

Let 𝑀 = (‖Ω‖ + 𝐴𝐻)(Γ(𝛼2)/Γ𝑝(𝛼)𝑝𝛼2)1/𝑝, 𝑁 = 𝐵𝐾(Γ(𝛼2)/Γ𝑝(𝛼)𝑝𝛼2)1/𝑝. Then‖𝑒 (𝑡)‖ 𝑒−𝑡

≤ (𝜙 𝑒−𝑡 + 𝑁𝜙)+ (𝑀 + 𝑁𝑒−𝜏) (∫𝑡

0𝑒−𝑞𝑠 ‖𝑒 (𝑠)‖𝑞 𝑑𝑠)1/𝑞 𝑑𝑠.

(31)

According to Lemmas 7 and 9, one has

‖𝑒 (𝑡)‖ 𝑒−𝑡 ≤ (𝜙 𝑒−𝑡 + 𝑁𝜙) + (𝑀 + 𝑁𝑒−𝜏)⋅ [(1 − (1 − 𝑒−(𝑀+𝑁𝑒−𝜏)𝑞𝑡)1/𝑞)−𝑞

⋅ ∫𝑡0(𝜙 𝑒−𝑠 + 𝑁𝜙)𝑞 𝑒−(𝑀+𝑁𝑒−𝜏)𝑞𝑠𝑑𝑠]

1/𝑞

≤ (𝜙 𝑒−𝑡 + 𝑁𝜙) + (𝑀 + 𝑁𝑒−𝜏) 𝑞𝑒(𝑀+𝑁𝑒−𝜏)𝑞𝑡⋅ [∫𝑡0(𝜙 𝑒−𝑠 + 𝑁𝜙)𝑞 𝑒−(𝑀+𝑁𝑒−𝜏)𝑞𝑠𝑑𝑠]

1/𝑞


≤ (𝜙 𝑒−𝑡 + 𝑁𝜙) + 𝑞 (1 + 𝑁) 𝜙 𝑒(𝑀+𝑁𝑒−𝜏)𝑞𝑡= [𝑒−𝑡 + 𝑁 + 𝑞 (1 + 𝑁) 𝑒(𝑀+𝑁𝑒−𝜏)𝑞𝑡] 𝜙 .

(32)

Therefore

‖𝑒 (𝑡)‖ ≤ [1 + 𝑁𝑒𝑡 + 𝑞 (1 + 𝑁) 𝑒[(𝑀+𝑁𝑒−𝜏)𝑞+1]𝑡] 𝜙 . (33)Hence, if (26) is satisfied and ‖𝜙‖ < 𝛿, then ‖𝑒(𝑡)‖ < 𝜀; mastersystem (3) is synchronized with slave system (5) in a finite-time.

Remark 12. References [24–28] discussed the finite-timesynchronization of neural networks, which only consideredinteger-order systems.

Remark 13. Generally, time-delayed differential modelsunavoidably exist in neural networks [37]. However, quite afew researches investigated the finite-time synchronization ofneural networks with delay. Hence, it is foremost importantand necessary to consider finite-time synchronization offractional-order delayed neural networks.

Remark 14. In this paper, Assumptions 4 and 5 are general,not too strict; many functions can satisfy the assumptions; forexample, [15, 17, 29] considered these assumptions too.

Remark 15. In [33], the authors discussed the finite-timesynchronization of fractional-order memristor-based neuralnetworks with time delays by using Laplace transform, gen-eralized Gronwall’s inequality and Mittag-Leffler functions,and the results showed Mittag-Leffler functions. In thispaper, employing Lemma 4 proposed in [35] and Bernoulliinequality can obtain results that only include exponentialfunctions, given that the form is simpler and the calculationis easier.

Remark 16. InTheorem 10, from (12) and (14) and inequality(16), it is obvious that the convergence time 𝑇 is proportionalto the inverse of the control gain 𝜂𝑖. Therefore, a greater 𝜂𝑖results in the shorter convergence time 𝑇, and therefore 𝜂𝑖should be selected in accordance with the convergence time𝑇 to be short.Remark 17. In [32], the authors discussed finite-time stabilityof fractional-order neural networks with delay by utilizingGronwall inequality. Unlike the previous work [32], in thispaper, we discussed the finite-time synchronization for a classof fractional-order delayed neural networks by employingLemma 4 proposed in [35], Bernoulli inequality, and inequal-ity skills; two new delay-dependent sufficient conditions havebeen established.

Remark 18. Different from integer-order delayed systems, it isdifficult to construct Lyapunov functions for fractional-ordernonlinear systemswith time delay; results have been obtainedusing different techniques from approaches used in the areaof finite synchronization. By employing inequality skills, new

Time (s)

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e 1

0 5 10 15 20 25 30

Figure 1: The errors state of 𝑒1 (order 𝛼 = 0.4).

sufficient conditions ensuring finite-time synchronization arederived.

4. Numerical Simulations

Consider the following two-dimensional delayed fractional-order Hopfield neural networks:

𝐷𝛼𝑥 (𝑡) = −𝐶𝑥 (𝑡) + 𝐴 (𝑡) 𝑓 (𝑥 (𝑡))+ 𝐵 (𝑡) 𝑔 (𝑥 (𝑡 − 𝜏)) + 𝐼, (34)

for 𝑡 ∈ 𝐽 = [0, 15], where 𝑥(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡))𝑇, 𝛼 = 0.4 or𝛼 = 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(𝑥1(𝑡)),tanh(𝑥2(𝑡)))𝑇, because of (tanh(𝑥)) = 1 − (tanh(𝑥))2 ≤ 1,according to LagrangeTheorem; clearly,𝑓(𝑥) and𝑔(𝑥) satisfyAssumption 4 with 𝐻 = 𝐾 = 1, 𝐴(𝑡) = ( 0.2 −0.10.1 0.2 ), 𝐵(𝑡) =( −0.5 −0.1−0.2 −0.5 ), 𝐶 = ( 0.1 00 0.1 ). So, according to Assumption 5, 𝐴 =sup𝑡≥0‖𝐴(𝑡)‖ = 0.3, 𝐵 = sup𝑡≥0‖𝐵(𝑡)‖ = 0.7, ‖𝐶‖ = 0.1.Choose the initial values of master system and slave systemas follows: 𝑥1(0) = 4, 𝑥2(0) = 2, 𝑦1(0) = 3, 𝑦2(0) = 1.

When 𝛼 = 0.4, take 𝛿 = 0.01, 𝜖 = 1, 𝜏 = 0.1 andselect the control gain 𝜂1 = 4, 𝜂2 = 5; apparently, thecondition of Theorem 11 is satisfied. It could be verified that𝑀 = 8.0815, 𝑁 = 1.0674, and the estimated time of finite-time synchronization is 𝑇 = 0.7099. Synchronization errorsbetween master and slave systems are shown in Figures 1 and2. It is clearly seen that the synchronization errors convergeto zero, indicating that master system and slave system aresynchronized in a finite-time. For comparison purposes, thecurves of the state variable of the master system and the slavesystem are shown in Figures 3 and 4.

When 𝛼 = 0.7, take 𝛿 = 0.01, 𝜖 = 1, 𝜏 = 0.1 and selectthe control gain 𝜂1 = 2, 𝜂2 = 4; apparently, the condition ofTheorem 10 is satisfied. It is easy to obtain𝑀 = 4.2950, 𝑁 =0.6992, and the estimated time of finite-time synchronizationis 𝑇 = 0.5937. Synchronization errors between master andslave systems are shown in Figures 5 and 6. It can be seen thatthe synchronization errors converge to zero, confirming thatmaster system and slave system are synchronized in a finite-time. Also, for comparison purposes, the curves of the state


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Time (s)0 5 10 15 20 25 30


Sync

hron

izat

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traj

ecto

ries

0

0.5

1

1.5

2

2.5

3

3.5

4

x1y1

Time (s)0 5 10 15 20 25 30

Figure 3: The synchronization trajectories of 𝑥1 and 𝑦1 (order 𝛼 =0.4).

Sync

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traj

ecto

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x2y2

Time (s)0 5 10 15 20 25 30


variable of the master system and the slave system are shownin Figures 7 and 8.

Time (s)0 5 10 15 20 25 30

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Time (s)0 5 10 15 20 25 30

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Time (s)0 5 10 15 20 25 30

x1y1

Sync

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0

0.5

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2.5

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3.5

4


5. Conclusions

In this paper, finite-time synchronization for a classfractional-order delayed neural networks with order 𝛼,0 < 𝛼 ≤ 1/2 and 1/2 < 𝛼 < 1, was discussed. Some newsufficient conditions are derived to ensure the finite-timesynchronization for this class of fractional-order systems.


Time (s)0 5 10 15 20 25 30

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Numerical example is presented to verify the effectivenessof the theoretical results. The proposed methods are noveland solve well the finite-time synchronization of fractional-order delayed neural networks. We would like to pointout that it is possible to extend the methods to otherfractional-order models, such as fractional-order delayedneutral-type neural networks and fractional neural networkswith incommensurate. These issues will be further worthdiscussing.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (Grant no. 11571016), the Nat-ural Science Foundation of Anhui Province (Grant no.1608085MA14), and the Natural Science Foundation of theHigher Education Institutions of Anhui Province (Grant no.KJ2015A152).

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