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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
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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,
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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)
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4 Abstract and Applied Analysis
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
𝑎1(𝑠) 𝑑𝑠)
× 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)
-
Abstract and Applied Analysis 5
Changing the order of integration in the above doubleintegral,
we obtain
𝑢 (𝑡) ≤ 𝑎1(𝑡) + 𝑏
1∫
𝑡
0
(𝑡 − 𝑠)𝛼−1
𝑎2(𝑠) 𝑑𝑠
+ 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
𝑎2(𝑠) 𝑑𝑠
+ 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
𝑎2(𝑠) 𝑑𝑠)
× 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
𝑎1(𝑠) 𝑑𝑠)
× 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
) (𝜇 (𝐷) + 𝑃𝜏𝑒𝛾𝜏
) 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)
-
6 Abstract and Applied Analysis
‖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
𝑎2(𝑠) 𝑑𝑠
≤𝜓1
∫
𝑡
0
(𝑡 − 𝑠)𝛼−1
(1 + 𝑠) 𝑑𝑠
=
𝜓1
𝛼
𝑡𝛼
(1 +
𝑡
𝛼 + 1
) ,
∫
𝑡
0
(𝑡 − 𝑠)𝛽−1
𝑎1(𝑠) 𝑑𝑠
≤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
)
× (𝜇 (𝐷) + 𝑃𝜏𝑒𝛾𝜏
) Γ (𝛼) Γ (𝛽) 𝑡𝛼+𝛽
] .
(51)
-
Abstract and Applied Analysis 7
By (51), we obtain
‖(𝑢 (𝑡) , V (𝑡))‖
= ‖𝑢 (𝑡)‖ + ‖V (𝑡)‖
≤ 𝑒−𝛾𝑡
(𝜑, 𝜓)
1
((1 + 𝑡) + Θ (𝑡))
× 𝐸𝛼+𝛽
[ℎ𝑙 (𝜇 (𝐵) + 𝑅𝜏𝑒𝛾𝜏
)
× (𝜇 (𝐷) + 𝑃𝜏𝑒𝛾𝜏
) 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
[ℎ𝑙 (𝜇 (𝐵) + 𝑅𝜏𝑒𝛾𝜏
) (𝜇 (𝐷) + 𝑃𝜏𝑒𝛾𝜏
) 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.
-
8 Abstract and Applied Analysis
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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