Research Article Finite-Time Stability of Fractional-Order ...downloads.hindawi.com/journals/aaa/2014/634803.pdf · Finite-Time Stability of Fractional-Order BAM Neural Networks with

Research ArticleFinite-Time Stability of Fractional-Order BAM Neural Networkswith Distributed Delay

Yuping Cao1 and Chuanzhi Bai2

1 Department of Basic Courses, Lianyungang Technical College, Lianyungang, Jiangsu 222000, China2Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, China

Correspondence should be addressed to Chuanzhi Bai; [email protected]

Received 8 February 2014; Accepted 1 April 2014; Published 22 April 2014

Academic Editor: Sabri Arik

Copyright © 2014 Y. Cao and C. Bai. 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.

Based on the theory of fractional calculus, the generalized Gronwall inequality and estimates of mittag-Leffer functions, the finite-time stability of Caputo fractional-order BAM neural networks with distributed delay is investigated in this paper. An illustrativeexample is also given to demonstrate the effectiveness of the obtained result.

1. Introduction

Fractional calculus (integral and differential operations ofnoninteger order) was firstly introduced 300 years ago. Dueto lack of application background and the complexity, itdid not attract much attention for a long time. In recentdecades fractional calculus is applied to physics, appliedmathematics, and engineering [1–6]. Since the fractional-order derivative is nonlocal and has weakly singular kernels,it provides an excellent instrument for the description ofmemory and hereditary properties of dynamical processes.Nowadays, study on the complex dynamical behaviors offractional-order systems has become a very hot researchtopic.

We know that the next state of a system not onlydepends upon its current state but also upon its historyinformation. Since a model described by fractional-orderequations possesses memory, it is precise to describe thestates of neurons. Moreover, the superiority of the Caputo’sfractional derivative is that the initial conditions for fractionaldifferential equations with Caputo derivatives take on thesimilar form as those for integer-order differentiation.There-fore, it is necessary and interesting to study fractional-orderneural networks both in theory and in applications.

Recently, fractional-order neural networks have beenpresented and designed to distinguish the classical integer-order models [7–10]. Currently, some excellent results about

fractional-order neural networks have been investigated,such as Kaslik and Sivasundaram [11, 12], Zhang et al. [13],Delavari et al. [14], and Li et al. [15, 16]. On the other hand,time delay is one of the inevitable problems on the stabilityof dynamical systems in the real word [17–20]. But till now,there are few results on the problems for fractional-orderdelayed neural networks; Chen et al. [21] studied the uniformstability for a class of fractional-order neural networks withconstant delay by the analytical approach; Wu et al. [22]investigated the finite-time stability of fractional-order neuralnetworks with delay by the generalized Gronwall inequalityand estimates of Mittag-Leffler functions; Alofi et al. [23]discussed the finite-time stability of Caputo fractional-orderneural networks with distributed delay.

The integer-order bidirectional associative memory(BAM) model known as an extension of the unidirectionalautoassociator of Hopfield [24] was first introduced by Kosko[25]. This neural network has been widely studied due to itspromising potential for applications in pattern recognitionand automatic control. In recent years, integer-order BAMneural networks have been extensively studied [26–29].However, to the best of our knowledge, there is no effortbeing made in the literature to study the finite-time stabilityof fractional-order BAM neural networks so far.

Motivated by the above-mentioned works, we weredevoted to establishing the finite-time stability of Caputofractional-order BAM neural networks with distributed

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 634803, 8 pageshttp://dx.doi.org/10.1155/2014/634803

2 Abstract and Applied Analysis

delay. In this paper, we will apply Laplace transform, gen-eralized Gronwall inequality, and estimates of Mittag-Lefflerfunctions to establish the finite-time stability criterion offractional-order distributed delayed BAM neural networks.

This paper is organized as follows. In Section 2, somedefinitions and lemmas of fractional differential and integralcalculus are given and the fractional-order BAM neuralnetworks are presented. A criterion for finite-time stabilityof fractional-order BAM neural networks with distributeddelay is obtained in Section 3. Finally, the effectiveness andfeasibility of the theoretical result is shown by an example inSection 4.

2. Preliminaries

For the convenience of the reader, we first briefly recall somedefinitions of fractional calculus; formore details, see [1, 2, 5],for example.

Definition 1. The Riemann-Liouville fractional integral oforder 𝛼 > 0 of a function 𝑢 : (0,∞) → 𝑅 is given by

𝐼𝛼

0+𝑢 (𝑡) =

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑢 (𝑠) 𝑑𝑠 (1)

provided that the right side is pointwise defined on (0,∞),where Γ(⋅) is the Gamma function.

Definition 2. The Caputo fractional derivative of order 𝛾 > 0of a function 𝑢 : (0,∞) → 𝑅 can be written as

𝐶

0𝐷𝛾

𝑡𝑢 (𝑡) =

1

Γ (𝑛 − 𝛾)

∫

𝑡

0

𝑢(𝑛)

(𝑠)

(𝑡 − 𝑠)𝛾+1−𝑛

𝑑𝑠,

𝑛 − 1 < 𝛾 < 𝑛.

(2)

Definition 3. The Mittag-Leffler function in two parametersis defined as

𝐸𝛼,𝛽

(𝑧) =

∞

∑

𝑘=0

𝑧𝑘

Γ (𝑘𝛼 + 𝛽)

, (3)

where 𝛼 > 0, 𝛽 > 0, and 𝑧 ∈ C, where C denotes the complexplane. In particular, for 𝛽 = 1, one has

𝐸𝛼,1

(𝑧) = 𝐸𝛼(𝑧) =

∞

∑

𝑘=0

𝑧𝑘

Γ (𝑘𝛼 + 1)

. (4)

The Laplace transform of Mittag-Leffler function is

L {𝑡𝛽−1

𝐸𝛼,𝛽

(−𝜆𝑡𝛼

)} =

𝑠𝛼−𝛽

𝑠𝛼+ 𝜆

,

(R (𝑠) > |𝜆|1/𝛼

) ,

(5)

where 𝑡 and 𝑠 are, respectively, the variables in the timedomain and Laplace domain andL{⋅} stands for the Laplacetransform.

In this paper, we are interested in the finite-time stabilityof fractional-order BAM neural networks with distributeddelay by the following state equations:

𝐶

0𝐷

𝛼

𝑡𝑥𝑖(𝑡) = − 𝑐

𝑖𝑥𝑖(𝑡) +

𝑛

∑

𝑗=1

𝑏𝑖𝑗𝑓𝑗(𝑦𝑗(𝑡))

+

𝑛

∑

𝑗=1

∫

𝜏

0

𝑟𝑖𝑗(𝑠) 𝑓𝑗(𝑦𝑗(𝑡 − 𝑠)) 𝑑𝑠 + 𝐼

𝑖,

𝑡 ≥ 0,

𝐶

0𝐷

𝛽

𝑡𝑦𝑗(𝑡) = − 𝑐

𝑗𝑦𝑗(𝑡) +

𝑛

∑

𝑖=1

𝑑𝑗𝑖𝑔i (𝑥𝑖 (𝑡))

+

𝑛

∑

𝑖=1

∫

𝜏

0

𝑝𝑗𝑖(𝑠) 𝑔𝑖(𝑥𝑖(𝑡 − 𝑠)) 𝑑𝑠 + 𝐼

𝑗,

𝑖, 𝑗 = 1, . . . , 𝑛,

(6)

or in the matrix-vector notation

𝐶

0𝐷

𝛼

𝑡𝑥 (𝑡) = − 𝐶𝑥 (𝑡) + 𝐵𝑓 (𝑦 (𝑡))

+ ∫

𝜏

0

𝑅 (𝑠) 𝑓 (𝑦 (𝑡 − 𝑠)) 𝑑𝑠 + 𝐼,

𝐶

0𝐷

𝛽

𝑡𝑦 (𝑡) = − 𝐶𝑦 (𝑡) + 𝐷𝑔 (𝑥 (𝑡))

+ ∫

𝜏

0

𝑃 (𝑠) 𝑔 (𝑥 (𝑡 − 𝑠)) 𝑑𝑠 + 𝐼, 𝑡 ≥ 0,

(7)

where 1 < 𝛼, 𝛽 < 2. The model (6) is made up of two neuralfields 𝐹

𝑥and 𝐹

𝑦, where 𝑥

𝑖(𝑡) and 𝑦

𝑗(𝑡) are the activations of

the 𝑖th neuron in 𝐹𝑥and the 𝑗th neuron in 𝐹

𝑦, respectively;

(𝑥 (𝑡) , 𝑦 (𝑡)) = (𝑥1(𝑡) , . . . , 𝑥

𝑛(𝑡) , 𝑦1(𝑡) , . . . , 𝑦

𝑛(𝑡))𝑇

∈ R2𝑛

(8)

is the state vector of the network at time 𝑡; the functions

𝑓 (𝑦 (𝑡)) = (𝑓1(𝑦1(𝑡)) , 𝑓

2(𝑦2(𝑡)) , . . . , 𝑓

𝑛(𝑦𝑛(𝑡)))𝑇

,

𝑔 (𝑥 (𝑡)) = (𝑔1(𝑥1(𝑡)) , 𝑔

2(𝑥2(𝑡)) , . . . , 𝑔

𝑛(𝑥𝑛(𝑡)))𝑇

(9)

are the activation functions of the neurons at time 𝑡; 𝐶 =diag(𝑐

𝑖) is a diagonal matrix; 𝑐

𝑖> 0 represents the rate with

which the 𝑖th unit will reset its potential to the resting state inisolation when disconnected from the network and externalinputs; 𝐵 = (𝑏

𝑖𝑗)𝑛×𝑛

and𝐷 = (𝑑𝑗𝑖)𝑛×𝑛

are the feedback matrix;𝜏 > 0 denotes the maximum possible transmission delayfrom neuron to another; 𝑅 = (𝑟

𝑖𝑗)𝑛×𝑛

and 𝑃 = (𝑝𝑗𝑖)𝑛×𝑛

are the delayed feedback matrix; 𝐼 = (𝐼1, . . . , 𝐼

𝑛)𝑇 and 𝐼 =

(𝐼1, . . . , 𝐼

𝑛)𝑇 are two external bias vectors.

Let C1([−𝜏, 0],R𝑛) be the Banach space of all continu-ously differential functions over a time interval of length 𝜏,

Abstract and Applied Analysis 3

mapping the interval [−𝜏, 0] into R𝑛 with the norm definedas follows: for every 𝜑 ∈ C1([−𝜏, 0],R𝑛),

𝜑1

= max {𝜑,

𝜑}

= max{ sup𝜃∈[−𝜏,0]

𝜑 (𝜃)

, sup𝜃∈[−𝜏,0]

𝜑

(𝜃)

} .

(10)

The initial conditions associated with (6) are given by

𝑥𝑖(𝜃) = 𝜑

𝑖(𝜃) , 𝑥

𝑖(𝜃) = 𝜑

𝑖(𝜃) , 𝑦

𝑗(𝜃) = 𝜓

𝑗(𝜃) ,

𝑦

𝑗(𝜃) = 𝜓

𝑗(𝜃) , 𝜃 ∈ [−𝜏, 0] ,

(11)

where 𝜑𝑖, 𝜓𝑗∈ 𝐶1

([−𝜏, 0],R).In order to obtain main result, we make the following

assumptions.

(H1) For 𝑖, 𝑗 = 1, . . . , 𝑛, the functions 𝑟𝑖𝑗(⋅) and 𝑝

𝑗𝑖(⋅) are

continuous on [0, 𝜏].

(H2) The neurons activation functions 𝑓𝑖and 𝑔

𝑗(𝑖, 𝑗 =

1, . . . , 𝑛) are bounded.

(H3) The neurons activation functions 𝑓𝑖and 𝑔

𝑗are Lips-

chitz continuous; that is, there exist positive constantsℎ𝑖, 𝑙𝑗(𝑖, 𝑗 = 1, . . . , 𝑛) such that

𝑓𝑖(𝑢) − 𝑓

𝑖(V)

≤ ℎ𝑖|𝑢 − V| ,

𝑔𝑗(𝑢) − 𝑔

𝑗(V)

≤ 𝑙𝑗|𝑢 − V| ,

∀𝑢, V ∈ R.(12)

Since the Caputo’s fractional derivative of a constant isequal to zero, the equilibriumpoint of system (6) is a constantvector (𝑥∗, 𝑦∗) = (𝑥∗

1, 𝑥∗

2, . . . , 𝑥

∗

𝑛, 𝑦∗

1, 𝑦∗

2, . . . , 𝑦

∗

𝑛)𝑇

∈ R2𝑛

which satisfies the system

𝑐𝑖𝑥∗

𝑖−

𝑛

∑

𝑗=1

𝑏𝑖𝑗𝑓𝑗(𝑦∗

𝑗) −

𝑛

∑

𝑗=1

∫

𝜏

0

𝑟𝑖𝑗(𝑠) 𝑓𝑗(𝑦∗

𝑗) 𝑑𝑠 − 𝐼

𝑖= 0,

𝑖 = 1, . . . , 𝑛,

𝑐𝑗𝑦∗

𝑗−

𝑛

∑

𝑖=1

𝑑𝑗𝑖𝑔𝑖(𝑥∗

𝑖) −

𝑛

∑

𝑖=1

∫

𝜏

0

𝑝𝑗𝑖(𝑠) 𝑔𝑖(𝑥∗

𝑖) 𝑑𝑠 − 𝐼

𝑗= 0,

𝑗 = 1, . . . , 𝑛.

(13)

By using the Schauder fixed point theorem and assumptions(H1)–(H3), it is easy to prove that the equilibrium points ofsystem (6) exist. We can shift the equilibrium point of system(6) to the origin. Denoting

(𝑢 (𝑡) , V (𝑡)) = (𝑢1(𝑡) , . . . , 𝑢

𝑛(𝑡) , V1(𝑡) , . . . , V

𝑛(𝑡))𝑇

= (𝑥1(𝑡) − 𝑥

∗

1, . . . , 𝑥

𝑛(𝑡)

−𝑥∗

𝑛, 𝑦1(𝑡) − 𝑦

∗

1, . . . , 𝑦

𝑛(𝑡) − 𝑦

∗

𝑛)𝑇

,

(14)

then system (6) can be written as

𝐶

0𝐷

𝛼

𝑡𝑢𝑖(𝑡) = − 𝑐

𝑖𝑢𝑖(𝑡) +

𝑛

∑

𝑗=1

𝑏𝑖𝑗𝐹𝑗(V𝑗(𝑡))

+

𝑛

∑

𝑗=1

∫

𝜏

0

𝑟𝑖𝑗(𝑠) 𝐹𝑗(V𝑗(𝑡 − 𝑠)) 𝑑𝑠,

𝑡 ≥ 0,

𝐶

0𝐷

𝛽

𝑡V𝑗(𝑡) = − 𝑐

𝑗V𝑗(𝑡) +

𝑛

∑

𝑖=1

𝑑𝑗𝑖𝐺𝑖(𝑢𝑖(𝑡))

+

𝑛

∑

𝑖=1

∫

𝜏

0

𝑝𝑗𝑖(𝑠) 𝐺𝑖(𝑢𝑖(𝑡 − 𝑠)) 𝑑𝑠,

𝑖, 𝑗 = 1, . . . , 𝑛,

(15)

with the initial conditions

𝑢𝑖(𝜃) = 𝜑

𝑖(𝜃) , 𝑢

𝑖(𝜃) = 𝜑

𝑖(𝜃) , V

𝑗(𝜃) = 𝜓

𝑗(𝜃) ,

V𝑗(𝜃) = 𝜓

𝑗(𝜃) , 𝜃 ∈ [−𝜏, 0] ,

(16)

where

𝐹𝑗(V𝑗(𝑡)) = 𝑓

𝑗(V𝑗(𝑡) + 𝑦

∗

𝑗) − 𝑓𝑗(𝑦∗

𝑗) ,

𝐺𝑖(𝑢𝑖(𝑡)) = 𝑔

𝑖(𝑢𝑖(𝑡) + 𝑥

∗

𝑖) − 𝑔𝑖(𝑥∗

𝑖) ,

𝜑𝑖(𝜃) = 𝜑

𝑖(𝜃) − 𝑥

∗

𝑖, 𝜓

𝑗(𝜃) = 𝜓

𝑗(𝜃) − 𝑦

∗

𝑗,

𝜃 ∈ [𝜏, 0] .

(17)

Similarly, by using thematrix-vector notation, system (15) canbe expressed as

𝐶

0𝐷

𝛼

𝑡𝑢 (𝑡) = − 𝐶𝑢 (𝑡) + 𝐵𝐹 (V (𝑡))

+ ∫

𝜏

0

𝑅 (𝑠) 𝐹 (V (𝑡 − 𝑠)) 𝑑𝑠,

𝑡 ≥ 0,

𝐶

0𝐷

𝛽

𝑡V (𝑡) = − 𝐶V (𝑡) + 𝐷𝐺 (𝑢 (𝑡))

+ ∫

𝜏

0

𝑃 (𝑠) 𝐺 (𝑢 (𝑡 − 𝑠)) 𝑑𝑠,

𝑡 ≥ 0,

(18)

with the initial condition

𝑢 (𝜃) = 𝜑 (𝜃) , 𝑢

(𝜃) = 𝜑

(𝜃) , V (𝜃) = 𝜓 (𝜃) ,

V (𝜃) = 𝜓 (𝜃) , 𝜃 ∈ [−𝜏, 0] ,(19)

where

𝐹 (V (𝑡)) = (𝐹1(V1(𝑡)) , 𝐹

2(V2(𝑡)) , . . . , 𝐹

𝑛(V𝑛(𝑡)))𝑇

,

𝐺 (𝑢 (𝑡)) = (𝐺1(𝑢1(𝑡)) , 𝐺

2(𝑢2(𝑡)) , . . . , 𝐺

𝑛(𝑢𝑛(𝑡)))𝑇

.

(20)


Define the functions as follows:

ℎ𝑖(𝑡) =

{{

{{

{

𝐹𝑖(V𝑖(𝑡))

V𝑖(𝑡)

, V𝑖(𝑡) ̸= 0,

0, V𝑖(𝑡) = 0,

𝑙𝑗(𝑡) =

{{

{{

{

𝐺𝑗(𝑢𝑗(𝑡))

𝑢𝑗(𝑡)

, 𝑢𝑗(𝑡) ̸= 0,

0, 𝑢𝑗(𝑡) = 0,

(21)

where 𝑖, 𝑗 = 1, . . . , 𝑛. From assumption (H3), we can obtain|ℎ𝑖(𝑡)| ≤ ℎ

𝑖, |𝑙𝑗(𝑡)| ≤ 𝑙

𝑗. By (21), we have

𝐹𝑖(V𝑖(𝑡)) = ℎ

𝑖(𝑡) V𝑖(𝑡) , 𝐺

𝑗(𝑢𝑗(𝑡)) = 𝑙

𝑗(𝑡) 𝑢𝑗(𝑡) ,

𝑖, 𝑗 = 1, . . . , 𝑛.

(22)

Thus, system (18) can be furtherwritten as the following form:

𝐶

0𝐷

𝛼

𝑡𝑢 (𝑡) = − 𝐶𝑢 (𝑡) + 𝐵𝐻 (𝑡) V (𝑡)

+ ∫

𝜏

0

𝑅 (𝑠)𝐻 (𝑡 − 𝑠) V (𝑡 − 𝑠) 𝑑𝑠,

𝑡 ≥ 0,

𝐶

0𝐷

𝛽

𝑡V (𝑡) = − 𝐶V (𝑡) + 𝐷𝐿 (𝑡) 𝑢 (𝑡)

+ ∫

𝜏

0

𝑃 (𝑠) 𝐿 (𝑡 − 𝑠) 𝑢 (𝑡 − 𝑠) 𝑑𝑠,

𝑡 ≥ 0,

(23)

where𝐻(𝑡) = diag{ℎ𝑖(𝑡)}, 𝐿(𝑡) = diag{𝑙

𝑗(𝑡)}.

Definition 4. System (23) with the initial condition (19) isfinite-time stable with respect to {𝛿, 𝜀, 𝑡

0, 𝐽}, 𝛿 < 𝜀, if and only

if(𝜑, 𝜓)

1

:=𝜑1

+𝜓1

< 𝛿 (24)

implies

‖(𝑢 (𝑡) , V (𝑡))‖ = ‖𝑢 (𝑡)‖ + ‖V (𝑡)‖ < 𝜀, ∀𝑡 ∈ 𝐽, (25)

where 𝛿 is a positive real number and 𝜀 > 0, 𝛿 < 𝜀, 𝑡0denotes

the initial time of observation of the system, and 𝐽 denotestime interval 𝐽 = [𝑡

0, 𝑡0+ 𝑇).

A technical result about norm upper-bounding functionof the matrix function 𝐸

𝛼,𝛽is given in [30] as follows.

Lemma 5. If 𝛼 ≥ 1, then, for 𝛽 = 1, 2, 𝛼, one has𝐸𝛼,𝛽

(𝐴𝑡𝛼

)

≤

𝑒𝐴𝑡𝛼

, 𝑡 ≥ 0. (26)

Moreover, if 𝐴 is a diagonal stability matrix, then𝐸𝛼,𝛽

(𝐴𝑡𝛼

)

≤ 𝑒−𝜔𝑡

, 𝑡 ≥ 0, (27)

where −𝜔 (𝜔 > 0) is the largest eigenvalue of the diagonalstability matrix 𝐴.

Lemma 6 (see [31]). Let 𝑢(𝑡), 𝑎(𝑡) be nonnegative and localintegrable on [0, 𝑇)(𝑇 ≤ +∞), and let 𝑔 be a nonnegative,nondecreasing continuous function defined on [0, 𝑇), 𝑔(𝑡) ≤𝑀, and let𝑀 be a real constant, 𝛼 > 0, with

𝑢 (𝑡) ≤ 𝑎 (𝑡) + 𝑔 (𝑡) ∫

t

0

(𝑡 − 𝑠)𝛼−1

𝑢 (𝑠) 𝑑𝑠, 𝑡 ∈ [0, 𝑇) . (28)

Then

𝑢 (𝑡) ≤ 𝑎 (𝑡) + ∫

𝑡

0

[

∞

∑

𝑛=1

(𝑔 (𝑡) Γ (𝛼))𝑛

Γ (𝑛𝛼)

(𝑡 − 𝑠)𝑛𝛼−1

𝑎 (𝑠)] 𝑑𝑠,

𝑡 ∈ [0, 𝑇) .

(29)

Moreover, if 𝑎(𝑡) is a nondecreasing function on [0, 𝑇), then

𝑢 (𝑡) ≤ 𝑎 (𝑡) 𝐸𝛼,1

(𝑔 (𝑡) Γ (𝛼) 𝑡𝛼

) , 𝑡 ∈ [0, 𝑇) . (30)

3. Main Result

We first give a key lemma in the proof of our main result asfollows.

Lemma 7. Let 𝑢(𝑡), V(𝑡) be nonnegative and local integrableon [0, 𝑇) (𝑇 ≤ +∞), and let 𝑎

1(𝑡), 𝑎2(𝑡) be nonnegative,

nondecreasing and local integrable on [0, 𝑇), and let 𝑏1, 𝑏2be

two positive constants, 𝛼, 𝛽 > 1, with

𝑢 (𝑡) ≤ 𝑎1(𝑡) + 𝑏

1∫

𝑡

0

(𝑡 − 𝑠)𝛼−1V (𝑠) 𝑑𝑠, 𝑡 ∈ [0, 𝑇) , (31)

V (𝑡) ≤ 𝑎2(𝑡) + 𝑏

2∫

𝑡

0

(𝑡 − 𝑠)𝛽−1

𝑢 (𝑠) 𝑑𝑠, 𝑡 ∈ [0, 𝑇) . (32)

Then

𝑢 (𝑡) ≤ (𝑎1(𝑡) + 𝑏

1∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑎2(𝑠) 𝑑𝑠)

× 𝐸𝛼+𝛽

(𝑏1𝑏2Γ (𝛼) Γ (𝛽) 𝑡

𝛼+𝛽

) , 𝑡 ∈ [0, 𝑇) ,

V (𝑡) ≤ (𝑎2(𝑡) + 𝑏

2∫

𝑡

0

(𝑡 − 𝑠)𝛽−1


× 𝐸𝛼+𝛽

(𝑏1𝑏2Γ (𝛼) Γ (𝛽) 𝑡

𝛼+𝛽

) , 𝑡 ∈ [0, 𝑇) .

(33)

Proof. Substituting (32) into (31), we obtain

𝑢 (𝑡) ≤ 𝑎1(𝑡) + 𝑏

1∫

𝑡

0

(𝑡 − 𝑠)𝛼−1V (𝑠) 𝑑𝑠

≤ 𝑎1(𝑡) + 𝑏

1∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑎2(𝑠) 𝑑𝑠

+ 𝑏1𝑏2∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

∫

𝑠

0

(𝑠 − 𝜉)𝛽−1

𝑢 (𝜉) 𝑑𝜉.

(34)


Changing the order of integration in the above doubleintegral, we obtain

𝑢 (𝑡) ≤ 𝑎1(𝑡) + 𝑏

1∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


+ 𝑏1𝑏2∫

𝑡

0

𝑢 (𝜉) 𝑑𝜉∫

𝑡

𝜉

(𝑡 − 𝑠)𝛼−1

(𝑠 − 𝜉)𝛽−1

𝑑𝑠

= 𝑎1(𝑡) + 𝑏

1∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


+ 𝑏1𝑏2∫

𝑡

0

Γ (𝛼) Γ (𝛽)

Γ (𝛼 + 𝛽)

(𝑡 − 𝜉)𝛼+𝛽−1

𝑢 (𝜉) 𝑑𝜉.

(35)

Let 𝑎(𝑡) = 𝑎1(𝑡) + 𝑏

1∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑎2(𝑠)𝑑𝑠, 𝑔(𝑡) =

𝑏1𝑏2((Γ(𝛼)Γ(𝛽))/Γ(𝛼 + 𝛽)); then 𝑎(𝑡) is a nonnegative, non-

decreasing, and local integrable function and 𝑔(𝑡) is anonnegative, nondecreasing continuous function. Thus, byLemma 6 (30), one has

𝑢 (𝑡) ≤ (𝑎1(𝑡) + 𝑏

1∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


× 𝐸𝛼+𝛽

(𝑏1𝑏2Γ (𝛼) Γ (𝛽) 𝑡

𝛼+𝛽

) .

(36)

Similarly, we get

V (𝑡) ≤ (𝑎2(𝑡) + 𝑏

2∫

𝑡

0

(𝑡 − 𝑠)𝛽−1


× 𝐸𝛼+𝛽

(𝑏1𝑏2Γ (𝛼) Γ (𝛽) 𝑡

𝛼+𝛽

) .

(37)

For convenience, let

𝑅 = sup0≤𝑠≤𝜏

{‖𝑅 (𝑠)‖} , 𝑃 = sup0≤𝑠≤𝜏

{‖𝑃 (𝑠)‖} ,

ℎ = max1≤𝑖≤𝑛

{ℎ𝑖} , 𝑙 = max

1≤𝑗≤𝑛

{𝑙𝑗} ,

Θ (𝑡) := max{ℎ𝛼

𝑡𝛼

(1 +

𝑡

𝛼 + 1

) (𝜇 (𝐵) + 𝑅𝜏𝑒𝛾𝜏

) ,

𝑙

𝛽

𝑡𝛽

(1 +

𝑡

𝛽 + 1

) (𝜇 (𝐷) + 𝑃𝜏𝑒𝛾𝜏

)} ,

(38)

where −𝛾 is the largest eigenvalue of the diagonal stabilitymatrix −𝐶 and 𝜇(⋅) denotes the largest singular value ofmatrix (⋅).

In the following, sufficient conditions for finite-timestability of fractional-order BAM neural networks with dis-tributed delay are derived.

Theorem 8. Let 1 < 𝛼, 𝛽 < 2. If system (23) satisfies (H1)–(H3) with the initial condition (19), and

𝑒−𝛾𝑡

((1 + 𝑡) + Θ (𝑡)) 𝐸𝛼+𝛽

× [ℎ𝑙 (𝜇 (𝐵) + 𝑅𝜏𝑒𝛾𝜏


) Γ (𝛼) Γ (𝛽) 𝑡𝛼+𝛽

]

<

𝜀

𝛿

,

(39)

where 𝑡 ∈ 𝐽 = [0, 𝑇), then system (23) is finite-time stable withrespect to {𝛿, 𝜀, 0, 𝐽}, 𝛿 < 𝜀.

Proof. By Laplace transform and inverse Laplace transform,system (23) is equivalent to

𝑢 (𝑡) = 𝐸𝛼(−𝐶𝑡𝛼

) 𝜑 (0) + 𝑡𝐸𝛼,2

(−𝐶𝑡𝛼

) 𝜑

(0)

+ ∫

𝑡

0

(𝑡 − 𝑠 )𝛼−1

𝐸𝛼,𝛼

(−𝐶𝑡𝛼

)

× [𝐵𝐻 (𝑠) V (𝑠)

+ ∫

𝜏

0

𝑅 (𝜂)𝐻 (𝑠 − 𝜂) V (𝑠 − 𝜂) 𝑑𝜂] 𝑑𝑠,

(40)

V (𝑡) = 𝐸𝛽(−𝐶𝑡𝛽

) 𝜓 (0) + 𝑡𝐸𝛽,2

(−𝐶𝑡𝛽

) 𝜓

(0)

+ ∫

𝑡

0

(𝑡 − 𝑠)𝛽−1

𝐸𝛽,𝛽

(−𝐶𝑡𝛽

)

× [𝐷𝐿 (𝑠) 𝑢 (𝑠)

+ ∫

𝜏

0

𝑃 (𝜂) 𝐿 (𝑠 − 𝜂) 𝑢 (𝑠 − 𝜂) 𝑑𝜂] 𝑑𝑠.

(41)

From (40), (41), and Lemma 5, we obtain

‖𝑢 (𝑡)‖ ≤ (𝜑+

𝜑𝑡) 𝑒−𝛾𝑡

+ ∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑒−𝛾(𝑡−𝑠)

×

𝐵𝐻 (𝑠) V (𝑠)

+ ∫

𝜏

0

𝑅 (𝜂)𝐻 (𝑠 − 𝜂) V (𝑠 − 𝜂) 𝑑𝜂

𝑑𝑠,

(42)


‖V (𝑡)‖ ≤ (𝜓+

𝜓𝑡) 𝑒−𝛾𝑡

+ ∫

𝑡

0

(𝑡 − 𝑠)𝛽−1

𝑒−𝛾(𝑡−𝑠)

×

𝐷𝐿 (𝑠) 𝑢 (𝑠)

+ ∫

𝜏

0

𝑃 (𝜂) 𝐿 (𝑠 − 𝜂) 𝑢 (𝑠 − 𝜂) 𝑑𝜂

𝑑𝑠.

(43)

Let 𝑈(𝑡) = sup𝜃∈[𝑡−𝜏,𝑡]

‖𝑢(𝜃)‖𝑒𝛾𝜃, and 𝑉(𝑡) =

sup𝜃∈[𝑡−𝜏,𝑡]

‖V(𝜃)‖𝑒𝛾𝜃; then

‖𝑢 (𝑠)‖ 𝑒𝛾𝑠

≤ 𝑈 (𝑠) , ‖𝑢 (𝑠 − 𝜏)‖ 𝑒𝛾(𝑠−𝜏)

≤ 𝑈 (𝑠) , (44)

‖V (𝑠)‖ 𝑒𝛾𝑠 ≤ 𝑉 (𝑠) , ‖V (𝑠 − 𝜏)‖ 𝑒𝛾(𝑠−𝜏) ≤ 𝑉 (𝑠) . (45)

Thus, we have by (42) and (44) that

‖𝑢 (𝑡)‖ 𝑒𝛾𝑡

≤𝜑+

𝜑𝑡

+ ∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

× [ℎ𝜇 (𝐵) ‖V (𝑠)‖ 𝑒𝛾𝑠

+∫

𝜏

0

ℎ𝑅V (𝑠 − 𝜂)

𝑒𝛾(𝑠−𝜂)

𝑒𝛾𝜂

𝑑𝜂] 𝑑𝑠

≤𝜑+

𝜑𝑡

+ ∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

ℎ [𝜇 (𝐵) + 𝑅𝜏𝑒𝛾𝜏

] 𝑉 (𝑠) 𝑑𝑠,

(46)

where 𝜇(𝐵) denotes the largest singular value of matrix 𝐵.Similarly, by (43) and (45), we get

‖V (𝑡)‖ 𝑒𝛾𝑡

≤𝜓+

𝜓𝑡

+ ∫

𝑡

0

(𝑡 − 𝑠)𝛽−1

× [𝑙𝜇 (𝐷) ‖𝑥 (𝑠)‖ 𝑒𝛾𝑠

+ ∫

𝜏

0

𝑙𝑃𝑥 (𝑠 − 𝜂)

𝑒𝛾(𝑠−𝜂)

𝑒𝛾𝜂

𝑑𝜂] 𝑑𝑠

≤𝜓+

𝜓𝑡

+ ∫

𝑡

0

(𝑡 − 𝑠)𝛽−1

𝑙 [𝜇 (𝐷) + 𝑃𝜏𝑒𝛾𝜏

] 𝑈 (𝑠) 𝑑𝑠.

(47)

Hence, by (46) and (47), we have

𝑈 (𝑡) ≤𝜑1

(1 + 𝑡)

+ ∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

ℎ [𝜇 (𝐵) + 𝑅𝜏𝑒𝛾𝜏

] 𝑉 (𝑠) 𝑑𝑠,

𝑉 (𝑡) ≤𝜓1

(1 + 𝑡)

+ ∫

𝑡

0

(𝑡 − 𝑠)𝛽−1

𝑙 [𝜇 (𝐷) + 𝑃𝜏𝑒𝛾𝜏

] 𝑈 (𝑠) 𝑑𝑠.

(48)

Set

𝑎1(𝑡) =

𝜑1

(1 + 𝑡) , 𝑎2(𝑡) =

𝜓1

(1 + 𝑡) ,

𝑏1= ℎ (𝜇 (𝐵) + 𝑅𝜏𝑒

𝛾𝜏

) , 𝑏2= 𝑙 (𝜇 (𝐷) + 𝑃𝜏𝑒

𝛾𝜏

) .

(49)

By simple computation, we have

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


≤𝜓1

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

(1 + 𝑠) 𝑑𝑠

=

𝜓1

𝛼

𝑡𝛼

(1 +

𝑡

𝛼 + 1

) ,

∫

𝑡

0

(𝑡 − 𝑠)𝛽−1


≤𝜑1

∫

𝑡

0

(𝑡 − 𝑠)𝛽−1

(1 + 𝑠) 𝑑𝑠

=

𝜑1

𝛽

𝑡𝛽

(1 +

𝑡

𝛽 + 1

) .

(50)

It follows from (48)–(50) and Lemma 7 that

𝑈 (𝑡) ≤ [(1 + 𝑡)𝜑1

+

𝜓1

𝛼

𝑡𝛼

×(1 +

𝑡

𝛼 + 1

) ℎ (𝜇 (𝐵) + 𝑅𝜏𝑒𝛾𝜏

) ]

⋅ 𝐸𝛼+𝛽

[ℎl (𝜇 (𝐵) + 𝑅𝜏𝑒𝛾𝜏)

× (𝜇 (𝐷) + 𝑃𝜏𝑒𝛾𝜏

) Γ (𝛼) Γ (𝛽) 𝑡𝛼+𝛽

] ,

𝑉 (𝑡) ≤ [(1 + 𝑡)𝜓1

+

𝜑1

𝛽

𝑡𝛽

×(1 +

𝑡

𝛽 + 1

) 𝑙 (𝜇 (𝐷) + 𝑃𝜏𝑒𝛾𝜏

) ]

⋅ 𝐸𝛼+𝛽

[ℎ𝑙 (𝜇 (𝐵) + 𝑅𝜏𝑒𝛾𝜏

)


) Γ (𝛼) Γ (𝛽) 𝑡𝛼+𝛽

] .

(51)


By (51), we obtain

‖(𝑢 (𝑡) , V (𝑡))‖

= ‖𝑢 (𝑡)‖ + ‖V (𝑡)‖

≤ 𝑒−𝛾𝑡

(𝜑, 𝜓)

1

((1 + 𝑡) + Θ (𝑡))

× 𝐸𝛼+𝛽


)


) Γ (𝛼) Γ (𝛽) 𝑡𝛼+𝛽

] .

(52)

Thus, if condition (39) is satisfied and ‖(𝜑, 𝜓)‖1

< 𝛿, then‖(𝑢(𝑡), V(𝑡))‖ < 𝜀, 𝑡 ∈ 𝐽; that is, system (23) is finite-timestable. This completes the proof.

4. An Illustrative Example

In this section, we give an example to illustrate the effective-ness of our main result.

Consider the following two-state Caputo fractional BAMtype neural networks model with distributed delay

𝐶

0𝐷

𝛼

𝑡𝑥1(𝑡) = − 0.7𝑥

1(𝑡) − 0.2𝑓

1(𝑦1(𝑡)) + 0.1𝑓

2(𝑦2(𝑡))

+ ∫

𝜏

0

𝑠3/2

𝑓1(𝑦1(𝑡 − 𝑠)) 𝑑𝑠

+ ∫

𝜏

0

𝑠𝑓2(𝑦2(𝑡 − 𝑠)) 𝑑𝑠,

𝐶

0𝐷

𝛼

𝑡𝑥2(𝑡) = − 0.6𝑥

2(𝑡) + 0.3𝑓

1(𝑦1(𝑡)) + 0.2𝑓

2(𝑦2(𝑡))

+ ∫

𝜏

0

𝑠𝑓1(𝑦1(𝑡 − 𝑠)) 𝑑𝑠

− ∫

𝜏

0

𝑠3/2

𝑓2(𝑦2(𝑡 − 𝑠)) 𝑑𝑠,

𝐶

0𝐷

𝛽

𝑡𝑦1(𝑡) = − 0.7𝑦

1(𝑡) + 0.4𝑔

1(𝑥1(𝑡)) + 0.2𝑔

2(𝑥2(𝑡))

− ∫

𝜏

0

𝑠𝑔1(𝑥1(𝑡 − 𝑠)) 𝑑𝑠

+ ∫

𝜏

0

𝑠2

𝑔2(𝑥2(𝑡 − 𝑠)) 𝑑𝑠,

𝐶

0𝐷

𝛽

𝑡𝑦2(𝑡) = − 0.6𝑦

2(𝑡) + 0.1𝑔

1(𝑥1(𝑡)) − 0.3𝑔

2(𝑥2(𝑡))

+ ∫

𝜏

0

𝑠2

𝑔1(𝑥1(𝑡 − 𝑠)) 𝑑𝑠

+ ∫

𝜏

0

𝑠𝑔2(𝑥2(𝑡 − 𝑠)) 𝑑𝑠

(53)

with the initial condition

𝑥 (𝑡) = 𝜑 (𝑡) =

1

15

sin 𝑡, 𝑥 (𝑡) = 𝜑 (𝑡) = 115

cos 𝑡,

𝑡 ∈ [−𝜏, 0] ,

𝑦 (𝑡) = 𝜓 (𝑡) =

1

15

cos 𝑡, 𝑦 (𝑡) = 𝜓 (𝑡) = − 115

sin 𝑡,

𝑡 ∈ [−𝜏, 0] ,

(54)

where 𝛼 = 1.2, 𝛽 = 1.3, 𝜏 = 0.2, and𝑓𝑗(𝑥𝑗) = 𝑔

𝑗(𝑥𝑗) =

(1/2)(|𝑥𝑗+ 1| − |𝑥

𝑗− 1|), 𝑗 = 1, 2. It is easy to know that

(𝑥∗

1, 𝑥∗

2, 𝑦∗

1, 𝑦∗

2)𝑇

= (0, 0, 0, 0)𝑇 is an equilibrium point of

system (53). Since ‖(𝜑, 𝜓)‖1

= 1/15 < 0.07, we may let𝛿 = 0.07. Take

𝑡0= 0, 𝐽 = [0, 30) , 𝜀 = 1,

𝐶 = [

−0.7 0

0 −0.6] , 𝐵 = [

−0.2 0.1

0.3 0.2] ,

𝐷 = [

0.4 0.2

0.1 −0.3] , 𝑅 (𝑠) = [

𝑠3/2

𝑠

𝑠 −𝑠3/2

] ,

𝑃 (𝑠) = [

−𝑠 𝑠2

𝑠2

𝑠

] .

(55)

It is easy to check that

ℎ = 𝑙 = 1, 𝛾 = 0.6, 𝜇 (𝐵) = 0.3828,

𝜇 (𝐷) = 0.4515, 𝑅 = 0.2894, 𝑃 = 0.24,

Θ (𝑡) = max {0.1697𝑡1.2 (𝑡 + 2.2) , 0.1691𝑡1.3 (𝑡 + 2.3)} ,

𝐸𝛼+𝛽



) Γ (𝛼) Γ (𝛽) 𝑡𝛼+𝛽

]

= 𝐸2.5

(0.1867𝑡2.5

) .

(56)

From condition (41) of Theorem 8, we can get

𝑒−0.6𝑡

[ (1 + 𝑡)

+max {0.1697𝑡1.2 (𝑡 + 2.2) ,

0.1691𝑡1.3

(𝑡 + 2.3)}]

× 𝐸2.5

(0.1867𝑡2.5

) <

1

0.07

.

(57)

We can obtain that the estimated time of finite-time stabilityis 𝑇 ≈ 23.78. Hence, system (53) is finite-time stable withrespect to {0.07, 1, 0, [0, 30)}.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.


Acknowledgments

This work is supported by the Natural Science Foundationof Jiangsu Province (BK2011407) and the Natural ScienceFoundation of China (11271364 and 10771212).

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