Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 543549, 14 pages
http://dx.doi.org/10.1155/2013/543549
Research Article
Synchronization of Chaotic Delayed Fuzzy Neural Networks
under Impulsive and Stochastic Perturbations
Bing Li1,2 and Qiankun Song1
1
2
College of Science, Chongqing Jiaotong University, Chongqing 400074, China
Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China
Correspondence should be addressed to Bing Li; libingcnjy@163.com
Received 4 October 2012; Revised 3 December 2012; Accepted 14 December 2012
Academic Editor: Xiaodi Li
Copyright © 2013 B. Li and Q. Song. 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.
The synchronization problem of chaotic fuzzy cellular neural networks with mixed delays is investigated. By an impulsive
integrodifferential inequality and the Itô’s formula, some sufficient criteria to synchronize the networks under both impulsive and
stochastic perturbations are obtained. The example and simulations are given to demonstrate the efficiency and advantages of the
proposed results.
1. Introduction
Fuzzy cellular neural network (FCNN), which integrated
fuzzy logic into the structure of a traditional cellular neural
networks (CNNs) and maintained local connectivity among
cells, was first introduced by T. Yang and L. Yang [1] to deal
with some complexity, uncertainty, or vagueness in CNNs.
Lots of studies have illustrated that FCNNs are a useful
paradigm for image processing and pattern recognition [2].
So far, many important results on stability analysis and state
estimation of FCNNs have been reported (see [3–12] and the
references therein).
Recently, it has been revealed that if the network’s
parameters and time delays are appropriately chosen, then
neural networks can exhibit some complicated dynamics
and even chaotic behaviors [13, 14]. The chaotic system
exhibits unpredictable and irregular dynamics, and it has
been found in many fields. Since the drive-response concept was proposed by Pecora and Carroll [15] in 1990 for
constructing the synchronization of coupled chaotic systems,
the control and synchronization problems of chaotic systems
have been extensively investigated. In recent years, various
synchronization schemes for chaotic neural networks have
derived and demonstrated potential applications in many
areas such as secure communication, image processing and
harmonic oscillation generation; see [16–32].
Although there have been many results which can be
applied to synchronization problems of a broad class of
FCNNs [25–32], there are some disadvantages that need
attention.
(1) Synchronization procedures and schemes are rather
sensitive to the unavoidable channel disturbances which are
usually presented in two forms: impulse and random noise.
However, in [25–27], authors provided some new schemes
to synchronize the chaotic systems without considering both
impulse and random noise. In [28, 29], under the condition of
no channel disturbance, Yu et al. and Xing and Peng studied
the lag synchronization problems of FCNNs, respectively. In
[30, 31], authors studied the synchronization of impulsive
fuzzy cellular neural networks (IFCNNs) with delays. In
[32], authors derived some synchronization schemes for
FCNNs with random noise. In fact, in real system, it is more
reasonable that the two perturbations coexist simultaneously.
(2) The criteria proposed in [25–32] are valid only for
FCNNs with discrete delays. For example, in [25, 28, 30, 31],
the involved delays are constants. In [26, 27, 32], the involved
delays are time-varying delays which are continuously differentiable, and the corresponding derivatives are required to be
finite or not greater than 1. In [29], Xing and Peng provided
some new criteria on lag synchronization problem of FCNNs
but they only considered the case for bounded time-varying
delays. In fact, time delays may occur in an irregular fashion,
2
and sometimes they may be not continuously differentiable.
Besides this, distribution delays may also exist when neural
networks have a spatial extent due to the presence of a
multitude of parallel pathways with a variety of axon sizes and
lengths.
(3) Some conditions imposed on the impulsive perturbations are too strong. For instance, Feng et al. [31] required the
magnitude of jumps not to be smaller than 0 and not greater
than 2. However, the disturbance in the real environment may
be very intense.
Therefore, it is of great theoretical and practical significance to investigate synchronization problems of IFCNNs
with mixed delays and random noise. However, up to now,
to the best of our knowledge, no result for synchronization
of IFNNs with mixed delays and random noise has been
reported.
Inspired by the above discussion, this paper addresses
the exponential synchronization problem of IFCNNs with
mixed delays and random noise. Based on the properties
of nonsingular M-matrix and the It̂
𝑜’s formula, we design
some synchronization schemes with a state feedback controller to ensure the exponential synchronization control.
Our method does not resort to complicated LyapunovKrasovskii functional which is widely used. The proposed
synchronization schemes are novel and improve some of the
previous literature.
This paper is organized as follows. In Section 2, we
introduce the drive-response models and some preliminaries.
In Section 3, some synchronization criteria for FCNNs with
mixed delays are derived. In Section 4, an example and its
simulations are given to illustrate the effectiveness of theoretical results. Finally, conclusions are drawn in Section 5.
Abstract and Applied Analysis
For 𝐴, 𝐵 ∈ R𝑛×𝑛 and 𝜙 : R → R𝑛 , we denote that
󵄨 󵄨
[𝐴]+ = (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨)𝑛×𝑛 ,
𝐴 ∘ 𝐵 = (𝑎𝑖𝑗 𝑏𝑖𝑗 )𝑛×𝑛 ,
+
󵄨 󵄨𝑇
󵄨 󵄨
[𝜙] = (󵄨󵄨󵄨𝜙1 󵄨󵄨󵄨 , . . . , 󵄨󵄨󵄨𝜙𝑛 󵄨󵄨󵄨) ,
(1)
𝑇
[𝜙 (𝑡)]𝜏 = ([𝜙1 (𝑡)]𝜏 , . . . , [𝜙𝑛 (𝑡)]𝜏 ) ,
where [𝜙𝑖 (𝑡)]𝜏 = sup 𝜙𝑖 (𝑡 + 𝑠) ,
−𝜏≤𝑠≤0
𝑖 ∈ N,
and 𝐷+ 𝜙(𝑡) denotes the upper-right derivative of 𝜙(𝑡) at time
𝑡.
Consider IFCNNs with mixed delays as follows:
𝑛
𝑛
𝑑𝑥𝑖 (𝑡)
= −𝑐𝑖 𝑥𝑖 + ∑ 𝑎𝑖𝑗 𝑓𝑗 (𝑥𝑗 ) + ∑𝑏𝑖𝑗 𝜈𝑗 + 𝐽𝑖
𝑑𝑡
𝑗=1
𝑗=1
𝑛
𝑛
𝑗=1
𝑗=1
+ ⋀𝑇𝑖𝑗 𝜇𝑗 + ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑛
+ ⋀𝛾𝑖𝑗 ∫
𝑗=1
+∞
0
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝑠)) 𝑑𝑠
𝑛
𝑛
𝑗=1
𝑗=1
+ ⋁𝑆𝑖𝑗 𝜇𝑗 + ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑛
+ ⋁𝜃𝑖𝑗 ∫
𝑗=1
+∞
0
(2)
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝑠)) 𝑑𝑠,
𝑡 ≥ 𝑡0 , 𝑡 ≠ 𝑡𝑘 ,
Δ𝑥𝑖 (𝑡𝑘 ) = 𝑥𝑖 (𝑡𝑘+ ) − 𝑥𝑖 (𝑡𝑘− )
2. Model Description and Preliminaries
Let R𝑛 be the space of 𝑛-dimensional real column vectors,
and let R𝑚×𝑛 represent the class of 𝑚 × 𝑛 matrices with real
components. | ⋅ | denotes the Euclidean norm in R𝑛 . The
inequality “≤” (“>”) between matrices or vectors such as 𝐴 ≤
𝐵 (𝐴 > 𝐵) means that each pair of corresponding elements
of 𝐴 and 𝐵 satisfies the inequality “≤” (“>”). 𝐴 ∈ R𝑚×𝑛 is
called a nonnegative matrix if 𝐴 ≥ 0, and 𝑥 ∈ R𝑛 is called
a positive vector if 𝑥 > 0. The transpose of 𝐴 ∈ R𝑚×𝑛
or 𝑥 ∈ R𝑛 is denoted by 𝐴𝑇 or 𝑥𝑇 . Let 𝐸 denote the unit
matrix with appropriate dimensions. N := {1, 2, . . . , 𝑛}, and
N := {1, 2, . . .}, R+ := [0, +∞).
PC[𝐽, R𝑛 ] = {𝜓 : 𝐽 → R𝑛 |𝜓(𝑠) is continuous and
bounded for all but at most countable points 𝑠 ∈ 𝐽 and at these
points, 𝜓(𝑠+ ) and 𝜓(𝑠− ) exist, 𝜓(𝑠) = 𝜓(𝑠+ )}. Here, 𝐽 ⊂ R is an
interval; 𝜓(𝑠+ ) and 𝜓(𝑠− ) denote the right-hand and left-hand
limits of the function 𝜓(𝑠), respectively. Especially PC :=
PC[(−∞, 0], R𝑛 ] with the norm ‖𝜓‖ = sup−∞<𝑠≤0 |𝜓(𝑠)| for
𝜓 ∈ PC.
L𝑒 = {𝜓 : R+ → R|𝜓(𝑠) is piecewise continuous and
+∞
satisfies ∫0 |𝜓(𝑠)|𝑒𝜎0 𝑠 𝑑𝑠 < +∞ for some constant 𝜎0 > 0}.
= 𝐼𝑖𝑘 (𝑥𝑖 (𝑡𝑘− )) ,
𝑥𝑖 (𝑡0 + 𝑠) = 𝜙𝑖 (𝑠) ,
𝑘 ∈ N,
−∞ < 𝑠 ≤ 0,
where 𝑖 = 1, 2, . . . , 𝑛, 𝑛 denotes the number of units in
the neural network. 𝑥(𝑡) = (𝑥1 (𝑡), . . . , 𝑥𝑛 (𝑡))𝑇 represents
the state variable. 𝑓𝑗 (⋅) is the activation function of the 𝑗th
neuron. 𝑐𝑖 represents the passive decay rate to the state of
𝑖th neuron at time 𝑡. 𝛼𝑖𝑗 and 𝛾𝑖𝑗 are elements of the fuzzy
feedback MIN template. 𝛽𝑖𝑗 and 𝜃𝑖𝑗 are elements of the fuzzy
feedback MAX template. 𝑇𝑖𝑗 and 𝑆𝑖𝑗 are elements of fuzzy
feed-forward MIN template and fuzzy feed-forward MAX
template, respectively. 𝑎𝑖𝑗 and 𝑏𝑖𝑗 are elements of feedback
and feed-forward template, respectively. ⋀ and ⋁ denote
the fuzzy AND and fuzzy OR operations, respectively. 𝜈𝑖
and 𝐽𝑖 denote input and bias of the 𝑖th neuron, respectively.
For any 𝑖, 𝑗 ∈ N, 𝜏𝑖𝑗 (𝑡) corresponding to the transmission
delay satisfies 0 ≤ 𝜏𝑖𝑗 (𝑡) ≤ 𝜏, and 𝑘𝑖𝑗 ∈ L𝑒 is the feedback kernel. For any 𝑘 ∈ N, 𝐼𝑘 (⋅) represents the impulsive
perturbation, and 𝑡𝑘 denotes impulsive moment satisfying
𝑡𝑘 < 𝑡𝑘+1 , lim𝑘 → +∞ 𝑡𝑘 = +∞.
We make the following assumptions throughout this
paper.
Abstract and Applied Analysis
3
(𝐴 1 ) 𝑓𝑖 is globally Lipschitz continuous, that is, for any 𝑖 ∈
N, there exists nonnegative constant 𝐿 𝑖 such that
󵄨
󵄨󵄨
󵄨󵄨𝑓𝑖 (𝑢) − 𝑓𝑖 (𝑣)󵄨󵄨󵄨 ≤ 𝐿 𝑖 |𝑢 − 𝑣|
for 𝑢, 𝑣 ∈ R.
(3)
expectation of stochastic process. 𝑈(𝑡) = (𝑈1 (𝑡), . . . , 𝑈𝑛 (𝑡))𝑇
is the state feedback controller designed by
𝑛
𝑈𝑖 (𝑡) = ∑𝑀𝑖𝑗 (𝑦𝑗 (𝑡) − 𝑥𝑗 (𝑡))
𝑗=1
(𝐴 2 ) For any 𝑘 ∈ N, there is a nonnegative constant 𝜂𝑘 such
that
󵄨
󵄨󵄨
󵄨󵄨𝑢 + 𝐼𝑖𝑘 (𝑢) − 𝑣 − 𝐼𝑖𝑘 (𝑣)󵄨󵄨󵄨 ≤ 𝜂𝑘 |𝑢 − 𝑣|
for 𝑢, 𝑣 ∈ R, 𝑖 ∈ N.
(4)
Let (2) be the drive system, and let the response system
with random noise be described by
𝑛
𝑛
𝑑𝑦𝑖 (𝑡) = [−𝑐𝑖 𝑦𝑖 + ∑ 𝑎𝑖𝑗 𝑓𝑗 (𝑦𝑗 ) + ∑ 𝑏𝑖𝑗 𝜈𝑗
𝑗=1
𝑗=1
[
𝑛
𝑛
𝑗=1
𝑗=1
+ ⋀𝛾𝑖𝑗 ∫
𝑗=1
+∞
0
𝑛
𝑗=1
𝑗=1
𝑛
𝑗=1
+∞
0
(𝐴 3 ) for 𝑖 ∈ N, there exist nonnegative constants 𝑐𝑖𝑗 , 𝑑𝑖𝑗
such that
󵄨2
󵄨
+ ∑𝑑𝑖𝑗 󵄨󵄨󵄨󵄨𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)) − 𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))󵄨󵄨󵄨󵄨 ,
where 𝜎𝑖 = (𝜎𝑖1 , . . . , 𝜎𝑖𝑛 ).
Let 𝑒(𝑡) = (𝑒1 (𝑡), . . . , 𝑒𝑛 (𝑡))𝑇 , where 𝑒𝑖 (𝑡) = 𝑦𝑖 (𝑡) − 𝑥𝑖 (𝑡), be
the synchronization error. Then, the error dynamical system
between (2) and (5) is given by
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡−𝑠)) 𝑑𝑠+𝑈𝑖 (𝑡)] 𝑑𝑡
]
𝑛
𝑛
𝑛
𝑗=1
𝑗=1
𝑛
−𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))) 𝑑𝑤𝑗 (𝑡) ,
+ ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
𝑡 ≥ 𝑡0 , 𝑡 ≠ 𝑡𝑘 ,
Δ𝑦𝑖 (𝑡𝑘 ) = 𝑦𝑖 (𝑡𝑘+ ) − 𝑦𝑖 (𝑡𝑘− ) = 𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘− )) ,
𝑛
𝑑𝑒𝑖 (𝑡) = [−𝑐𝑖 𝑒𝑖 + ∑𝑎𝑖𝑗 (𝑓𝑗 (𝑦𝑗 ) − 𝑓𝑗 (𝑥𝑗 ))
𝑗=1
[
+ ∑𝑀𝑖𝑗 𝑒𝑗 (𝑡) + ∑𝑁𝑖𝑗 𝑒𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))
𝑗=1
𝑛
𝑘 ∈ N,
− ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
−∞ < 𝑠 ≤ 0,
(5)
𝑛
+ ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
𝑇
(7)
𝑗=1
+ ∑𝜎𝑖𝑗 (𝑡, 𝑥𝑗 (𝑡) − 𝑦𝑗 (𝑡) , 𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))
𝑦𝑖 (𝑡0 + 𝑠) = 𝜓𝑖 (𝑠) ,
where 𝑀 = (𝑀𝑖𝑗 )𝑛×𝑛 , 𝑁 = (𝑁𝑖𝑗 )𝑛×𝑛 are the controller gain
matrices to be scheduled. The diffusion coefficient matrix (or
noise intensity matrix) 𝜎 : R × R𝑛 × R𝑛 → R𝑛×𝑛 satisfies
the local Lipschitz condition and the linear growth condition
(see [33]). In addition,
𝑛
+ ⋁𝑆𝑖𝑗 𝜇𝑗 + ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
+⋁𝜃𝑖𝑗 ∫
𝑗=1
𝑗=1
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠
𝑛
+ ∑ 𝑁𝑖𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)) − 𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))) ,
𝑛
󵄨
󵄨2
𝜎𝑖 𝜎𝑖𝑇 ≤ ∑𝑐𝑖𝑗 󵄨󵄨󵄨󵄨𝑥𝑗 − 𝑦𝑗 󵄨󵄨󵄨󵄨
+ 𝐽𝑖 + ⋀𝑇𝑖𝑗 𝜇𝑗 + ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑛
(6)
𝑛
where 𝑤(𝑡) = (𝑤1 (𝑡), . . . , 𝑤𝑛 (𝑡)) is an 𝑛-dimensional standard Brownian motion defined on a complete probability
space (Ω, F, P) with a natural filtration {F𝑡 }𝑡≥0 generated
by {𝑤(𝑠) : 0 ≤ 𝑠 ≤ 𝑡} and satisfying the usual
conditions (i.e., it is right continuous, and F0 contains all
P-null sets). The initial value 𝜓 = (𝜓1 (𝑠), . . . , 𝜓𝑛 (𝑠))𝑇 ∈
PC𝑏F0 [(−∞, 0], R𝑛 ] which denotes the family of all bounded
F0 -measurable and PC-valued random variables 𝜓 with the
𝑝
norm ‖𝜓‖F = sup−∞<𝑠≤0 E|𝜓(𝑠)|𝑝 , where E denotes the
𝑛
− ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
𝑛
+ ⋀𝛾𝑖𝑗 ∫
𝑗=1
𝑛
0
− ⋀𝛾𝑖𝑗 ∫
𝑗=1
+∞
+∞
0
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝑠)) 𝑑𝑠
4
Abstract and Applied Analysis
𝑛
+∞
+ ⋁𝜃𝑖𝑗 ∫
𝑗=1
0
𝑛
+∞
−⋁𝜃𝑖𝑗 ∫
0
𝑗=1
Lemma 4 (see [1]). Let 𝛼𝑖𝑗 , 𝛽𝑖𝑗 ∈ R and 𝑥, 𝑦 ∈ R𝑛 be the two
states of the system (2). Then, one has
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠
󵄨󵄨
󵄨󵄨 𝑛
𝑛
󵄨
󵄨󵄨
󵄨󵄨⋀𝛼 𝑓 (𝑥 ) − ⋀𝛼 𝑓 (𝑦 )󵄨󵄨󵄨
󵄨󵄨
𝑖𝑗 𝑗
𝑗
𝑖𝑗 𝑗
𝑗 󵄨󵄨
󵄨󵄨
󵄨󵄨𝑗=1
𝑗=1
󵄨
󵄨
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝑠)) 𝑑𝑠] 𝑑𝑡
]
𝑛
󵄨 󵄨󵄨
󵄨
≤ ∑ 󵄨󵄨󵄨󵄨𝛼𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑓𝑗 (𝑥𝑗 ) − 𝑓𝑗 (𝑦𝑗 )󵄨󵄨󵄨󵄨 ,
𝑛
+ ∑𝜎𝑖𝑗 (𝑡, 𝑒𝑗 (𝑡) , 𝑒𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))) 𝑑𝑤𝑗 (𝑡) ,
𝑗=1
𝑗=1
󵄨󵄨 𝑛
󵄨󵄨
𝑛
󵄨󵄨
󵄨
󵄨󵄨⋁𝛽 𝑓 (𝑥 ) − ⋁𝛽 𝑓 (𝑦 )󵄨󵄨󵄨
󵄨󵄨
𝑖𝑗 𝑗
𝑗
𝑖𝑗 𝑗
𝑗 󵄨󵄨
󵄨󵄨𝑗=1
󵄨󵄨
𝑗=1
󵄨
󵄨
𝑡 ≥ 𝑡0 , 𝑡 ≠ 𝑡𝑘 ,
Δ𝑒𝑖 (𝑡𝑘 ) = 𝑒𝑖 (𝑡𝑘+ ) − 𝑒𝑖 (𝑡𝑘− )
= 𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘− )) − 𝐼𝑖𝑘 (𝑥𝑖 (𝑡𝑘− )) ,
𝑒𝑖 (𝑡0 + 𝑠) = 𝜓𝑖 (𝑠) − 𝜙𝑖 (𝑠) ,
𝑛
󵄨 󵄨󵄨
󵄨
≤ ∑ 󵄨󵄨󵄨󵄨𝛽𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑓𝑗 (𝑥𝑗 ) − 𝑓𝑗 (𝑦𝑗 )󵄨󵄨󵄨󵄨 .
𝑘 ∈ N,
𝑗=1
−∞ < 𝑠 ≤ 0.
(8)
For convenience, we use the following notations: 𝐷1 =
diag{−𝑐1 , . . . , −𝑐𝑛 }, 𝐿 = diag{𝐿 1 , . . . , 𝐿 𝑛 }, 𝐾(𝑠) = (𝑘𝑖𝑗 (𝑠))𝑛×𝑛 ,
𝐴 = (𝑎𝑖𝑗 )𝑛×𝑛 , 𝑀 = (𝑀𝑖𝑗 )𝑛×𝑛 with 𝑀𝑖𝑖 = 𝑀𝑖𝑖 , 𝑀𝑖𝑗 = |𝑀𝑖𝑗 | for
𝑖 ≠ 𝑗, 𝛼 = (𝛼𝑖𝑗 )𝑛×𝑛 , 𝛽 = (𝛽𝑖𝑗 )𝑛×𝑛 , Γ = (𝛾𝑖𝑗 )𝑛×𝑛 , Θ = (𝜃𝑖𝑗 )𝑛×𝑛 ,
𝐶 = (𝑐𝑖𝑗 )𝑛×𝑛 , and 𝐷 = (𝑑𝑖𝑗 )𝑛×𝑛 .
The following definition and lemmas will be employed.
Definition 1. The systems (2) and (5) are called to be globally
exponentially synchronized in 𝑝-moment, if there exist positive constants 𝜆, 𝐾 such that
󵄩𝑝
󵄩
E|𝑒 (𝑡)|𝑝 ≤ 𝐾󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩F 𝑒−𝜆(𝑡−𝑡0 ) ,
𝑡 ≥ 𝑡0 .
(9)
It is said especially to be globally exponentially synchronized
in mean square when 𝑝 = 2.
For any nonsingular M-matrix 𝐴 (see [34]), we define
that
Lemma 5 (see [36]). For the integer 𝑝 ≥ 2 and 𝑥 =
(𝑥1 , . . . , 𝑥𝑛 )𝑇 ∈ R𝑛 , there exists a positive constant 𝑒𝑝 (𝑛) such
that
𝑝/2
𝑛
𝑒𝑝 (𝑛) (∑𝑥𝑖2 )
𝑖=1
𝑛
󵄨 󵄨𝑝
≤ ∑󵄨󵄨󵄨𝑥𝑖 󵄨󵄨󵄨 .
𝑛
M𝐴 = {𝑧 ∈ R | 𝐴𝑧 > 0, 𝑧 > 0} .
(10)
Lemma 2 (see [35]). For a nonsingular M-matrix 𝐴, M𝐴 is a
nonempty cone without conical surface.
Lemma 3 (see [36]). For 𝑥𝑖 ≥ 0, 𝛼𝑖 > 0, and
Lemma 6. For 𝑘 ∈ N, assume that 𝑣 = (𝑣1 (𝑡), . . . , 𝑣𝑛 (𝑡))𝑇 ∈
PC[(−∞, +∞), R𝑛 ] satisfies
𝐷+ 𝑣 (𝑡) ≤ 𝐴 0 𝑣 (𝑡) + 𝑃𝑣 (𝑡) + 𝑄 [𝑣 (𝑡)]𝜏
+∫
+∞
0
𝑣 (𝑡𝑘+ )
≤
𝜌𝑘 𝑣 (𝑡𝑘− ) ,
𝑣𝑡0 ∈ PC,
Υ (𝑠) 𝑣 (𝑡 − 𝑠) 𝑑𝑠,
𝑛
𝛼
∏𝑥𝑖 𝑖
𝑖=1
(14)
where 𝑣𝑡0 = 𝑣 (𝑡0 + 𝑠) ,
𝑠 ∈ (−∞, 0] ,
in which
(𝐶1 ) 𝐴 0 = diag{𝑎1 , . . . , 𝑎𝑛 }, 𝑃 = (𝑝𝑖𝑗 )𝑛×𝑛 with 𝑝𝑖𝑗 ≥ 0 for
𝑖 ≠ 𝑗, 𝑄 = (𝑞𝑖𝑗 )𝑛×𝑛 ≥ 0, Υ(𝑠) = (𝜐𝑖𝑗 (𝑠))𝑛×𝑛 ≥ 0, and
𝜐𝑖𝑗 ∈ L𝑒 , 𝑖, 𝑗 ∈ N.
(𝐶2 ) Π = −(𝐴 0 + 𝑃 + 𝑄 + ∫0
M-matrix.
Υ(𝑠)𝑑𝑠) is a nonsingular
Then, there must exist 𝑧 = (𝑧1 , . . . , 𝑧𝑛 )𝑇 ∈ MΠ and 𝜆 ∈ (0, 𝜎0 ]
such that
𝑘−1
𝑣 (𝑡) ≤ (∏󰜚𝑗 ) 𝑧𝑒−𝜆(𝑡−𝑡0 ) ,
𝑡𝑘−1 ≤ 𝑡 < 𝑡𝑘 , 𝑘 ∈ N,
(15)
provided that the initial value 𝑣𝑡0 satisfies
𝑛
≤ ∑𝛼𝑖 𝑥𝑖 .
𝑡 ≥ 𝑡0 , 𝑡 ≠ 𝑡𝑘 ,
𝜌𝑘 ≥ 0,
𝑗=0
𝛼𝑖 = 1,
(13)
𝑖=1
+∞
∑𝑛𝑖=1
(12)
(11)
𝑖=1
The sign of equality holds if and only if 𝑥𝑖 = 𝑥𝑗 for all 𝑖, 𝑗 ∈ N.
𝑣 (𝑡) ≤ 𝑧𝑒−𝜆(𝑡−𝑡0 ) ,
−∞ < 𝑡 ≤ 𝑡0 ,
(16)
where 󰜚0 = 1, 󰜚𝑘 = max{1, 𝜌𝑘 }, and 𝑧, 𝜆 can be determined by
(𝜆𝐸 + 𝐴 0 + 𝑃 + 𝑄𝑒𝜆𝜏 + ∫
+∞
0
Υ (𝑠) 𝑒𝜆𝑠 𝑑𝑠) 𝑧 < 0.
(17)
Abstract and Applied Analysis
5
Proof. By condition (𝐶2 ) and Lemma 2, we can find 𝑧̃ =
(̃𝑧1 , . . . , 𝑧̃𝑛 )𝑇 ∈ MΠ such that Π̃𝑧 > 0, namely, (𝐴 0 + 𝑃 +
+∞
𝑄 + ∫0 Υ(𝑠)𝑑𝑠) 𝑧̃ < 0. By the continuity, there must be some
positive constant 𝜆 ∈ (0, 𝜎0 ] satisfying
(𝜆𝐸 + 𝐴 0 + 𝑃 + 𝑄𝑒𝜆𝜏 + ∫
+∞
0
Υ (𝑠) 𝑒𝜆𝑠 𝑑𝑠) 𝑧̃ < 0.
(18)
𝑣𝑖 (𝑡) < (1 + 𝜖) 𝑤𝑖 (𝑡) ,
𝑖 ∈ N.
(19)
If inequality (19) is not true, then there must exist some 𝑚 ∈
N and 𝑡∗ ∈ (𝑡0 , 𝑡1 ) such that
∗
𝑣𝑚 (𝑡 ) = (1 + 𝜖) 𝑤𝑚 (𝑡 ) , 𝑣𝑖 (𝑡) < (1 + 𝜖) 𝑤𝑖 (𝑡) ,
𝑡 ∈ (−∞, 𝑡∗ ) , 𝑖 ∈ N,
󸀠
𝐷+ 𝑣𝑚 (𝑡∗ ) ≥ (1 + 𝜖) 𝑤𝑚
(𝑡∗ ) .
(20)
(21)
𝜐𝑚𝑗 (𝑠) 𝑣𝑗 (𝑡∗ − 𝑠) 𝑑𝑠
−𝑡0 )
+ (1 + 𝜖) 𝑒−𝜆(𝑡
∗
∗
−𝑡0 )
−𝑡0 )
𝑛
+∞
×∑∫
𝑗=1 0
∗
𝑛
∑𝑝𝑚𝑗 𝑧𝑗
(22)
Recalling 𝜌𝑘 ≥ 1, it follows from (24) and (25) that
𝑘
𝑣𝑖 (𝑡) ≤ ( ∏ 󰜚𝑚 ) 𝑤𝑖 (𝑡) ,
𝑡 ∈ (−∞, 𝑡𝑘 ] , 𝑖 ∈ N.
𝑚=0
(26)
Repeating the proof similar to (19) can yield
𝑘
𝑣𝑖 (𝑡) ≤ ( ∏ 󰜚𝑚 ) 𝑤𝑖 (𝑡) ,
𝑡 ∈ [𝑡𝑘 , 𝑡𝑘+1 ) , 𝑖 ∈ N.
(27)
By the mathematical induction, we derive that for any 𝑘 ∈ N,
𝑘−1
𝑣𝑖 (𝑡) ≤ (∏󰜚𝑗 ) 𝑤𝑖 (𝑡)
(28)
𝑡 ∈ [𝑡𝑘−1 , 𝑡𝑘 ) , 𝑖 ∈ N.
Theorem 7. Assume that (𝐴 1 )–(𝐴 3 ) hold and
+∞
̃ = −(𝐷0 + 𝑃 + 𝑄 + ∫ Υ(𝑠)𝑑𝑠) is a
(𝐴4 ) for 𝑝 ≥ 2, 𝐷
0
nonsingular M-matrix, where 𝑃 = [𝐴]+ 𝐿 + 𝑀 + (𝑝 −
1)𝐶 := (𝑝𝑖𝑗 )𝑛×𝑛 , 𝑄 = ([𝛼]+ +[𝛽]+ )𝐿+[𝑁]+ +(𝑝−1)𝐷 :=
(𝑞𝑖𝑗 )𝑛×𝑛 , Υ(𝑠) = ([Γ]+ 𝐿+[Θ]+ 𝐿)∘[𝐾(𝑠)]+ := (𝜐𝑖𝑗 (𝑠))𝑛×𝑛 ,
𝐷0 = diag{𝑑1 , . . . , 𝑑𝑛 } with
𝑛
∑𝑞𝑚𝑗 𝑧𝑗 𝑒𝜆𝜏
−𝑡0 )
𝑛
󵄨 󵄨 󵄨 󵄨 󵄨 󵄨
𝑑𝑖 = − 𝑝𝑐𝑖 + (𝑝 − 1) ∑ [( 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑖𝑗 󵄨󵄨󵄨󵄨
𝜐𝑚𝑗 (𝑠) 𝑒𝜆𝑠 𝑑𝑠𝑧𝑗
𝑗=1
< (1 + 𝜖) 𝑒−𝜆(𝑡
∗
−𝑡0 )
(−𝜆𝑧𝑚 )
+∫
which contradicts (21). Therefore, (19) holds. Letting 𝜖 → 0+
in (19), we get
𝑡 ∈ [𝑡0 , 𝑡1 ) , 𝑖 ∈ N.
+∞
0
󸀠
= (1 + 𝜖) 𝑤𝑚
(𝑡∗ ) ,
𝑣𝑖 (𝑡) ≤ 𝑤𝑖 (𝑡) = 󰜚0 𝑤𝑖 (𝑡) ,
𝑖 ∈ N.
𝑚=0
In this section, by using Lemma 6, we will obtain some
sufficient criteria to synchronize the drive-response systems
(2) and (5).
𝑗=1
+ (1 + 𝜖) 𝑒−𝜆(𝑡
≤ ( ∏ 󰜚𝑚 ) 𝑤𝑖 (𝑡𝑘 ) ,
(25)
3. Exponential Synchronization
𝑎𝑚 𝑧𝑚
𝑗=1
+ (1 + 𝜖) 𝑒−𝜆(𝑡
𝑘
The proof is completed.
𝑗=1
∗
𝑚=0
𝑗=0
+ ∑ 𝑞𝑚𝑗 [𝑣𝑗 (𝑡∗ )]𝜏
≤ (1 + 𝜖) 𝑒−𝜆(𝑡
𝑣𝑖 (𝑡𝑘 ) ≤ 𝜌𝑘 𝑣𝑖 (𝑡𝑘− ) ≤ 󰜚𝑘 ( ∏ 󰜚𝑚 ) 𝑤𝑖 (𝑡𝑘 )
≤ (∏󰜚𝑗 ) 𝑧𝑖 𝑒−𝜆(𝑡−𝑡0 ) ,
𝑛
𝑗=1 0
For 𝑡 = 𝑡𝑘 , from (14) and (24), we have
𝑘−1
𝑗=1
+∞
(24)
𝑗=0
𝑛
𝐷+ 𝑣𝑚 (𝑡∗ ) ≤ 𝑎𝑚 𝑣𝑚 (𝑡∗ ) + ∑𝑝𝑚𝑗 𝑣𝑗 (𝑡∗ )
𝑛
𝑡 ∈ [𝑡𝜈−1 , 𝑡𝜈 ) , 𝑖 ∈ N.
𝑚=0
𝑚=0
On the other hand, (14) together with (17) and (20) leads to
+∑∫
𝜈−1
𝑣𝑖 (𝑡) ≤ ( ∏ 󰜚𝑚 ) 𝑤𝑖 (𝑡) ,
𝑘−1
Noting that 𝑣𝑡0 ∈ PC, we can find a constant 𝐵 > 0 such that
‖𝑣𝑡0 ‖ ≤ 𝐵. Denote that 𝑧 := (𝐵/min𝑖∈N 𝑧̃𝑖 )̃𝑧 = (𝑧1 , . . . , 𝑧𝑛 )𝑇 .
Obviously, 𝑧 and 𝜆 satisfy (16) and (17).
Let 𝑤𝑖 (𝑡) := 𝑧𝑖 𝑒−𝜆(𝑡−𝑡0 ) for 𝑡 ∈ R, 𝑖 ∈ N. For any small
enough 𝜖 > 0, (16) implies that 𝑣𝑖 (𝑡) ≤ 𝑤𝑖 (𝑡) < (1 + 𝜖)𝑤𝑖 (𝑡), 𝑡 ∈
(−∞, 𝑡0 ]. Next, we claim that for any 𝑡 ∈ [𝑡0 , 𝑡1 ),
∗
Suppose that for 𝜈 = 1, 2, . . . , 𝑘, the following inequalities
hold
(23)
+
󵄨
󵄨 󵄨 󵄨 󵄨 󵄨
(󵄨󵄨󵄨󵄨𝛾𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜃𝑖𝑗 󵄨󵄨󵄨󵄨) 󵄨󵄨󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨󵄨 𝑑𝑠)𝐿 𝑗
𝑝−2
󵄨 󵄨
(𝑐𝑖𝑗 + 𝑑𝑖𝑗 )+𝑀𝑖𝑗 + 󵄨󵄨󵄨󵄨𝑁𝑖𝑗 󵄨󵄨󵄨󵄨] ,
2
𝑖 ∈ N,
(29)
6
Abstract and Applied Analysis
(𝐴5 ) the impulsive perturbations satisfy
ln 𝜁𝑘
< 𝜆,
𝑡
𝑘∈N 𝑘 − 𝑡𝑘−1
sup
(𝜆𝐸 + 𝐷0 + 𝑃 + 𝑄𝑒
+∫
0
×∫
𝜆𝑠
Υ (𝑠) 𝑒 𝑑𝑠) 𝑧 < 0,
(31)
+∞
𝑛
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠 − ⋀𝛾𝑖𝑗
𝑗=1
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝑠)) 𝑑𝑠
𝑛
+∞
+ ⋁𝜃𝑖𝑗 ∫
̃ = −(𝐷0 +𝑃+𝑄+∫+∞ Υ(𝑠)𝑑𝑠) is a nonsingular
Proof. Since 𝐷
0
M-matrix, by Lemma 2 and the continuity, there must be a
constant vector 𝑧 = (𝑧1 , . . . , 𝑧𝑛 )𝑇 ∈ M𝐷̃ and a constant 𝜆 ∈
(0, 𝜎0 ] such that (31) holds.
We denote by 𝑒 = (𝑒1 , . . . , 𝑒𝑛 )𝑇 the solution of error
dynamical system (8) with the initial value 𝜓 − 𝜙 ∈
PC𝑏F0 [(−∞, 0], R𝑛 ] and let
𝑇
𝑉 (𝑒) = (𝑉1 (𝑒) , . . . , 𝑉𝑛 (𝑒)) ,
+∞
0
Then, drive-response systems (2) and (5) are globally exponential synchronization in 𝑝-moment.
𝑗=1
0
𝑛
+∞
−⋁𝜃𝑖𝑗 ∫
𝑗=1
0
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝑠)) 𝑑𝑠]
]
1
󵄨𝑝−2
󵄨
+ 𝑝 (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 𝜎𝑖 𝜎𝑖𝑇 .
2
(34)
By (𝐴 1 ) and Lemma 4, we have
(32)
󵄨 󵄨𝑝
𝑉𝑖 (𝑒) = 󵄨󵄨󵄨𝑒𝑖 󵄨󵄨󵄨 , 𝑖 ∈ N.
Calculating the time derivative of 𝑉𝑖 (𝑒(𝑡)) along the trajectory
of error system (8) and by the Itô’s formula [33], we get for any
𝑘 ∈ N,
𝜕𝑉 (𝑒)
𝑑𝑉𝑖 (𝑒 (𝑡)) = L𝑉𝑖 (𝑒 (𝑡)) 𝑑𝑡 + 𝑖 𝜎𝑑𝑤 (𝑡) ,
𝜕𝑒
𝑡 ≥ 𝑡0 , 𝑡 ≠ 𝑡𝑘 ,
(33)
where L𝑉𝑖 (𝑒(𝑡)) is given by
󵄨𝑝
󵄨𝑝−1
󵄨
󵄨
L𝑉𝑖 (𝑒 (𝑡)) ≤ − 𝑝𝑐𝑖 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 + 𝑝 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑛
󵄨
󵄨 󵄨 󵄨
󵄨
󵄨𝑝−1
× ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 + 𝑝󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑗=1
𝑛
󵄨
󵄨
󵄨
󵄨𝑝−1
× ∑ 𝑀𝑖𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 +𝑝 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑗=1
𝑛
󵄨
󵄨 󵄨󵄨
× ∑ 󵄨󵄨󵄨󵄨𝑁𝑖𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))󵄨󵄨󵄨󵄨
󵄨
󵄨𝑝−2
L𝑉𝑖 (𝑒 (𝑡)) = 𝑝󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 𝑒𝑖 (𝑡)
𝑗=1
𝑛
󵄨
󵄨𝑝−1 󵄨 󵄨
+ 𝑝 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨󵄨𝛼𝑖𝑗 󵄨󵄨󵄨󵄨
𝑛
× [−𝑐𝑖 𝑒𝑖 + ∑ 𝑎𝑖𝑗
𝑗=1
[
𝑗=1
𝑛
× (𝑓𝑗 (𝑦𝑗 ) − 𝑓𝑗 (𝑥𝑗 )) + ∑ 𝑀𝑖𝑗 𝑒𝑗 (𝑡)
𝑗=1
𝑛
+ ∑𝑁𝑖𝑗 𝑒𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))
𝑗=1
𝑛
+ ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
𝑛
− ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
+ ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
𝑗=1
0
for a given 𝑧 ∈ M𝐷̃.
𝑛
𝑗=1
×∫
𝑝
+∞
𝑛
(30)
where 𝜁𝑘 = max{1, 𝜂𝑘 }, and 𝜆 ∈ (0, 𝜎0 ] is determined by
𝜆𝜏
𝑛
− ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))) + ⋀𝛾𝑖𝑗
󵄨
󵄨
󵄨
󵄨𝑝−1
× 𝐿 𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))󵄨󵄨󵄨󵄨 + 𝑝 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑛
󵄨 󵄨 󵄨
󵄨
󵄨
󵄨𝑝−1
× ∑ 󵄨󵄨󵄨󵄨𝛽𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))󵄨󵄨󵄨󵄨 + 𝑝 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑗=1
𝑛
󵄨 󵄨
× ∑ 󵄨󵄨󵄨󵄨𝛾𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ∫
𝑗=1
+∞
0
󵄨
󵄨󵄨
󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨 |(𝑡 − 𝑠)| 𝑑𝑠
󵄨
󵄨
𝑛
+∞
󵄨
󵄨
󵄨
󵄨𝑝−1 󵄨 󵄨
+ 𝑝 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨󵄨𝜃𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ∫ 󵄨󵄨󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨󵄨
𝑗=1
0
1
󵄨
󵄨
󵄨𝑝−2
󵄨
× 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨 𝑑𝑠 + 𝑝 (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 𝜎𝑖 𝜎𝑖𝑇
2
󵄨𝑝
󵄨
:= −𝑝𝑐𝑖 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 + 𝐼1 + 𝐼2 + 𝐼3 + 𝐼4 + 𝐼5 + 𝐼6 + 𝐼7 + 𝐼8 .
(35)
Abstract and Applied Analysis
7
𝑛
󵄨𝑝
󵄨
+ (𝑝 − 1) ∑ 𝑐𝑖𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨
Using Lemma 3 and (𝐴 3 ), it is easy to get
𝑗=1
+
𝑛
󵄨 󵄨
󵄨𝑝
󵄨
𝐼1 ≤ ( ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑗=1
(𝑝 − 1) (𝑝 − 2) 𝑛
󵄨𝑝
󵄨
( ∑ 𝑑𝑖𝑗 ) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
2
𝑗=1
𝑛
󵄨𝑝
󵄨
+ (𝑝 − 1) ∑ 𝑑𝑖𝑗 [󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 ] .
𝑛
󵄨𝑝
󵄨 󵄨 󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 ,
𝜏
𝑗=1
(36)
𝑗=1
𝑛
󵄨𝑝
󵄨
𝐼2 ≤ ( ∑𝑀𝑖𝑗 ) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
Thus, we have
𝑗=1
𝑛
󵄨 󵄨
󵄨𝑝
󵄨𝑝
󵄨
󵄨
L𝑉𝑖 (𝑒 (𝑡)) ≤ − 𝑝𝑐𝑖 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 + ( ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑛
󵄨𝑝
󵄨
+ ∑ 𝑀𝑖𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 ,
𝑗=1
𝑗=1
𝑛
󵄨𝑝
󵄨 󵄨 󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨
𝑛
󵄨 󵄨
󵄨
󵄨𝑝
𝐼3 ≤ ( ∑ 󵄨󵄨󵄨󵄨𝑁𝑖𝑗 󵄨󵄨󵄨󵄨) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑗=1
𝑗=1
𝑛
󵄨𝑝
󵄨
+ ( ∑𝑀𝑖𝑗 ) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑛
󵄨𝑝
󵄨 󵄨 󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑁𝑖𝑗 󵄨󵄨󵄨󵄨 [󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 ] ,
𝜏
𝑗=1
𝑗=1
𝑛
󵄨𝑝
󵄨
+ ∑𝑀𝑖𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨
𝑛
󵄨 󵄨
󵄨𝑝
󵄨
𝐼4 ≤ ( ∑ 󵄨󵄨󵄨󵄨𝛼𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑗=1
𝑗=1
𝑛
󵄨 󵄨
󵄨
󵄨𝑝
+ ( ∑ 󵄨󵄨󵄨󵄨𝑁𝑖𝑗 󵄨󵄨󵄨󵄨) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑛
󵄨𝑝
󵄨 󵄨
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝛼𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 [󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 ] ,
𝜏
𝑗=1
𝑗=1
𝑛
󵄨𝑝
󵄨 󵄨 󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑁𝑖𝑗 󵄨󵄨󵄨󵄨 [󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 ]
𝑛
󵄨 󵄨
󵄨𝑝
󵄨
𝐼5 ≤ ( ∑ 󵄨󵄨󵄨󵄨𝛽𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑗=1
𝑛
󵄨 󵄨
󵄨𝑝
󵄨
+ ( ∑ 󵄨󵄨󵄨󵄨𝛼𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑛
󵄨𝑝
󵄨 󵄨
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝛽𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 [󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 ] ,
𝑛
+∞
󵄨 󵄨
𝐼6 ≤ ( ∑ 󵄨󵄨󵄨󵄨𝛾𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ∫
𝑗=1
0
𝑛
󵄨 󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝛾𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ∫
𝑗=1
𝑛
+∞
0
+∞
󵄨 󵄨
𝐼7 ≤ ( ∑ 󵄨󵄨󵄨󵄨𝜃𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ∫
𝑗=1
𝑗=1
𝜏
𝑗=1
0
󵄨
󵄨󵄨
󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨 𝑑𝑠) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨󵄨𝑝
󵄨
󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑗 (𝑡 − 𝑠)󵄨󵄨 𝑑𝑠,
󵄨󵄨
󵄨
󵄨
󵄨󵄨𝑝
󵄨
󵄨󵄨
󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨 𝑑𝑠) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨󵄨𝑝
󵄨
󵄨
𝑛
+∞
󵄨󵄨
󵄨𝑝
󵄨 󵄨
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝜃𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ∫ 󵄨󵄨󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨 𝑑𝑠,
0
𝑛
󵄨𝑝
󵄨 󵄨
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝛼𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 [󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 ]
𝑗=1
𝑛
1
󵄨2
󵄨
󵄨𝑝−2
󵄨
𝐼8 ≤ 𝑝 (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 ∑𝑐𝑖𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨
2
𝑗=1
𝜏
𝑛
󵄨 󵄨
󵄨𝑝
󵄨
+ ( ∑ 󵄨󵄨󵄨󵄨𝛽𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
𝑗=1
𝑛
󵄨𝑝
󵄨 󵄨
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝛽𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 [󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 ]
𝜏
𝑗=1
𝑛
󵄨 󵄨
+ ( ∑ 󵄨󵄨󵄨󵄨𝛾𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ∫
+∞
0
𝑗=1
𝑗=1
≤
𝜏
𝑗=1
󵄨
󵄨󵄨
󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨 𝑑𝑠) (𝑝 − 1) 󵄨󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨󵄨𝑝
󵄨
󵄨
𝑛
+∞
󵄨󵄨
󵄨𝑝
󵄨 󵄨
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝛾𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ∫ 󵄨󵄨󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨 𝑑𝑠
𝑗=1
0
𝑛
1
󵄨2
󵄨
󵄨𝑝−2
󵄨
+ 𝑝 (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 ∑𝑑𝑖𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))󵄨󵄨󵄨󵄨
2
𝑗=1
𝑛
+∞
󵄨
󵄨 󵄨
󵄨
󵄨
󵄨𝑝
+ ( ∑ 󵄨󵄨󵄨󵄨𝜃𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ∫ 󵄨󵄨󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨󵄨 𝑑𝑠) (𝑝 − 1) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
(𝑝 − 1) (𝑝 − 2) 𝑛
󵄨𝑝
󵄨
( ∑ 𝑐𝑖𝑗 ) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨
2
𝑗=1
󵄨 󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝜃𝑖𝑗 󵄨󵄨󵄨󵄨 𝐿 𝑗 ∫
0
𝑗=1
𝑛
𝑗=1
+∞
0
󵄨󵄨
󵄨𝑝
󵄨󵄨
󵄨󵄨𝑘𝑖𝑗 (𝑠)󵄨󵄨󵄨 󵄨󵄨󵄨𝑒𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨 𝑑𝑠
󵄨󵄨
󵄨
󵄨
8
Abstract and Applied Analysis
(𝑝 − 1) (𝑝 − 2) 𝑛
󵄨𝑝
󵄨
( ∑ 𝑐𝑖𝑗 ) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 + (𝑝 − 1)
2
𝑗=1
+
8
𝑛
󵄨𝑝 (𝑝 − 1) (𝑝 − 2)
󵄨
× ∑𝑐𝑖𝑗 󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 +
2
𝑗=1
6
4
𝑗=1
(37)
It follows from (𝐴 4 ) and (32) that
𝑥2
𝑛
󵄨𝑝
󵄨
󵄨𝑝
󵄨
× ( ∑ 𝑑𝑖𝑗 ) 󵄨󵄨󵄨𝑒𝑖 (𝑡)󵄨󵄨󵄨 + (𝑝 − 1) ∑𝑑𝑖𝑗 [󵄨󵄨󵄨󵄨𝑒𝑗 (𝑡)󵄨󵄨󵄨󵄨 ] .
𝜏
2
𝑛
𝑗=1
Phase plot of drive system
10
0
−2
−4
−6
−8
L𝑉𝑖 (𝑒 (𝑡)) ≤ 𝑑𝑖 𝑉𝑖 (𝑒 (𝑡))
−10
−4
𝑛
+ ∑ (𝑝𝑖𝑗 𝑉𝑗 (𝑒 (𝑡)) + 𝑞𝑖𝑗 [𝑉𝑗 (𝑒 (𝑡))]𝜏
𝑗=1
+∫
+∞
0
−3
−2
−1
0
𝑥1
(38)
1
2
3
4
Figure 1: Chaos behavior of drive system.
𝜐𝑖𝑗 (𝑠) 𝑉𝑗 (𝑒 (𝑡 − 𝑠)) 𝑑𝑠) .
By the continuity of E𝑉𝑖 (𝑒(𝑡)), we conclude that
Substituting (38) into (33) gives
𝐷+ E𝑉𝑖 (𝑒 (𝑡)) ≤ 𝑑𝑖 E𝑉𝑖 (𝑒 (𝑡))
𝑑𝑉𝑖 (𝑒 (𝑡)) ≤ [𝑑𝑖 𝑉𝑖 (𝑒 (𝑡))
𝑛
+ ∑ (𝑝𝑖𝑗 E𝑉𝑗 (𝑒 (𝑡)) + 𝑞𝑖𝑗 E[𝑉𝑗 (𝑒 (𝑡))]𝜏
𝑛
+ ∑ (𝑝𝑖𝑗 𝑉𝑗 (𝑒 (𝑡)) + 𝑞𝑖𝑗 [𝑉𝑗 (𝑒 (𝑡))]𝜏
𝑗=1
𝑗=1
+∫
+∞
0
+
+∫
𝜐𝑖𝑗 (𝑠) 𝑉𝑗 (𝑒 (𝑡 − 𝑠)) 𝑑𝑠)] 𝑑𝑡
𝜕𝑉𝑖 (𝑒)
𝜎𝑑𝑤 (𝑡) ,
𝜕𝑒
0
𝑡 ≥ 𝑡0 , 𝑡 ≠ 𝑡𝑘 , 𝑘 ∈ N.
(39)
𝑡
+ 𝑃E𝑉 (𝑒 (𝑡)) + 𝑄E[𝑉 (𝑒 (𝑡))]𝜏
+∫
+∞
0
(42)
Υ (𝑠) E𝑉 (𝑒 (𝑡 − 𝑠)) 𝑑𝑠.
Meanwhile, it follows from (𝐴 2 ) and (32) that
[𝑑𝑖 E𝑉𝑖 (𝑒 (𝑢))
E𝑉𝑖 (𝑒 (𝑡𝑘 ))
[
󵄨
󵄨𝑝
= E󵄨󵄨󵄨𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘− )) − 𝐼𝑖𝑘 (𝑥𝑖 (𝑡𝑘− )) + 𝑒𝑖 (𝑡𝑘− )󵄨󵄨󵄨
𝑛
+ ∑ (𝑝𝑖𝑗 E𝑉𝑗 (𝑒 (𝑢))
𝑗=1
(40)
+ 𝑞𝑖𝑗 E[𝑉𝑗 (𝑒 (𝑢))]𝜏
+∫
(41)
𝐷+ E𝑉 (𝑒 (𝑡)) ≤ 𝐷0 E𝑉 (𝑒 (𝑡))
E𝑉𝑖 (𝑒 (𝑡 + 𝛿)) − E𝑉𝑖 (𝑒 (𝑡))
𝑡+𝛿
𝜐𝑖𝑗 (𝑠) E𝑉𝑗 (𝑒 (𝑡 − 𝑠)) 𝑑𝑠) ,
which implies that for 𝑡 ≠ 𝑡𝑘 , 𝑘 ∈ N,
Integrating and taking the expectations on both sides of (39)
lead to
≤∫
+∞
+∞
0
𝜐𝑖𝑗 (𝑠)
×E𝑉𝑗 (𝑒 (𝑢 − 𝑠)) 𝑑𝑠) ] 𝑑𝑢,
]
where 𝛿 > 0 is small enough such that 𝑡, 𝑡 + 𝛿 ∈ [𝑡𝑘−1 , 𝑡𝑘 ) for
𝑘 ∈ N.
(43)
𝑝
≤ 𝜂𝑘 E𝑉𝑖 (𝑒 (𝑡𝑘− ))
for 𝑖 ∈ N and 𝑘 ∈ N, which means that
𝑝
E𝑉 (𝑒 (𝑡𝑘 )) ≤ 𝜂𝑘 E𝑉 (𝑒 (𝑡𝑘− )) .
(44)
Obviously, (42) and (44) indicate that E𝑉(𝑒(𝑡)) satisfies
inequality (14) in Lemma 6.
On the other hand, noting that 𝑒(𝑡0 + 𝑠) = 𝜓(𝑠) − 𝜙(𝑠) in
(8) and by a simple calculation, we get
󵄩𝑝
󵄨𝑝 󵄩
󵄨𝑝
󵄨
󵄨
E󵄨󵄨󵄨𝑒𝑖 (𝑡0 + 𝑠)󵄨󵄨󵄨 = E󵄨󵄨󵄨𝜓𝑖 (𝑠) − 𝜙𝑖 (𝑠)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩F
(45)
Abstract and Applied Analysis
9
𝑝
for any 𝑠 ∈ (−∞, 0], which means E|𝑒𝑖 (𝑡)|𝑝 ≤ ‖𝜓 − 𝜙‖F for
𝑡 ∈ (−∞, 𝑡0 ] and 𝑖 ∈ N. Recalling the definition of 𝑉(𝑒(𝑡)) in
(32) and 𝑧 > 0, we conclude that for 𝑡 ∈ (−∞, 𝑡0 ]
󵄩𝑝
󵄩
E𝑉 (𝑒 (𝑡)) ≤ 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩F (1, . . . , 1)𝑇
𝑧𝑛
𝑧1
󵄩𝑝
󵄩
≤ 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩F (
,...,
)
min𝑖∈N {𝑧𝑖 }
min𝑖∈N {𝑧𝑖 }
󵄩𝑝
󵄩󵄩
󵄩𝜓 − 𝜙󵄩󵄩󵄩F
≤ 󵄩
𝑧,
min𝑖∈N {𝑧𝑖 }
𝑇
which further indicates that
𝑡 ∈ (−∞, 𝑡0 ] ,
(47)
𝑝
where 𝑧 = (‖𝜓 − 𝜙‖F /min𝑖∈N {𝑧𝑖 })𝑧. This implies that
condition (16) in Lemma 6 holds.
Therefore, by Lemma 6, we derive that
𝑘−1
E𝑉 (𝑒 (𝑡)) ≤ (∏𝜁𝑗 ) 𝑧𝑒−𝜆(𝑡−𝑡0 ) ,
𝑗=0
𝑡𝑘−1 ≤ 𝑡 < 𝑡𝑘 , 𝑘 ∈ N,
(48)
with 𝜁0 = 1. Meanwhile, (30) implies that there is a small
enough constant 𝜖 (0 < 𝜖 < 𝜆) such that
𝜁𝑘 ≤ 𝑒(𝜆−𝜖)(𝑡𝑘 −𝑡𝑘−1 ) ,
𝑘 ∈ N.
(49)
Thus, inequality (48) together with (49) shows that for 𝑘 = 1
E𝑉 (𝑒 (𝑡)) ≤ 𝜁0 𝑧𝑒−𝜆(𝑡−𝑡0 ) ≤ 𝑧𝑒−𝜖(𝑡−𝑡0 ) ,
𝑡0 ≤ 𝑡 < 𝑡1 ,
If the random noise has not been considered, which
means 𝜎 ≡ 0 in (5), then the response system reduces to
𝑛
𝑑𝑦𝑖 (𝑡)
= −𝑐𝑖 𝑦𝑖 + ∑𝑎𝑖𝑗 𝑓𝑗 (𝑦𝑗 )
𝑑𝑡
𝑗=1
(46)
E𝑉 (𝑒 (𝑡)) ≤ 𝑧𝑒−𝜆(𝑡−𝑡0 ) ,
Remark 9. In synchronization scheme, we take both impulsive perturbations and random noise into account. Comparing with the results in [25–32], Theorem 7 can reflect a more
realistic dynamical behavior and synchronization procedure.
(50)
𝑛
𝑛
𝑗=1
𝑗=1
+ ∑𝑏𝑖𝑗 𝜈𝑗 + 𝐽𝑖 + ⋀𝑇𝑖𝑗 𝜇𝑗
𝑛
+ ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑗=1
𝑛
+∞
+ ⋀𝛾𝑖𝑗 ∫
𝑗=1
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠
0
(53)
𝑛
𝑛
𝑗=1
𝑗=1
+ ⋁𝑆𝑖𝑗 𝜇𝑗 + ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)))
𝑛
+∞
+ ⋁𝜃𝑖𝑗 ∫
𝑗=1
0
+ 𝑈𝑖 (𝑡) ,
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠
𝑡 ≥ 𝑡0 , 𝑡 ≠ 𝑡𝑘 ,
Δ𝑦𝑖 (𝑡𝑘 ) = 𝑦𝑖 (𝑡𝑘+ ) − 𝑦𝑖 (𝑡𝑘− ) = 𝐼𝑖𝑘 (𝑦𝑖 (𝑡𝑘− )) ,
𝑦𝑖 (𝑡0 + 𝑠) = 𝜓𝑖 (𝑠) ,
𝑘 ∈ N,
−∞ < 𝑠 ≤ 0.
In this case, the following globally exponential synchronization scheme for drive-response IFCNNs (2) and (53) can be
derived.
and for any 𝑘 ≥ 2,
E𝑉 (𝑒 (𝑡)) ≤ 𝑧𝜁0 𝜁1 ⋅ ⋅ ⋅ 𝜁𝑘−1 𝑒−𝜆(𝑡−𝑡0 )
Theorem 10. Assume that (𝐴 1 ) and (𝐴 2 ) hold and
≤ 𝑧𝑒(𝜆−𝜖)(𝑡1 −𝑡0 ) ⋅ ⋅ ⋅ 𝑒(𝜆−𝜖)(𝑡𝑘−1 −𝑡𝑘−2 ) 𝑒−𝜆(𝑡−𝑡0 )
+∞
≤ 𝑧𝑒(𝜆−𝜖)(𝑡−𝑡0 ) 𝑒−𝜆(𝑡−𝑡0 )
̂ = −(𝐷1 +𝑃1 +𝑄1 +∫ Υ(𝑠)𝑑𝑠) is a nonsingular M(𝐴6 ) 𝐷
0
matrix, where 𝑃1 = [𝐴]+ 𝐿 + 𝑀, 𝑄1 = ([𝛼]+ + [𝛽]+ )𝐿 +
[𝑁]+ ,
≤ 𝑧𝑒−𝜖(𝑡−𝑡0 ) ,
(𝐴7 ) the impulsive perturbations satisfy
= 𝑧𝑒(𝜆−𝜖)(𝑡𝑘−1 −𝑡0 ) 𝑒−𝜆(𝑡−𝑡0 )
(51)
𝑡𝑘−1 ≤ 𝑡 < 𝑡𝑘 .
By Lemma 5, we get
󵄩𝑝
󵄩
E|𝑒 (𝑡)|𝑝 ≤ 𝐾󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩F 𝑒−𝜖(𝑡−𝑡0 ) ,
ln 𝜉𝑘
< 𝜆,
𝑡
𝑘∈𝑁 𝑘 − 𝑡𝑘−1
sup
𝑡 ≥ 𝑡0 ,
(52)
where 𝐾 = (∑𝑛𝑖=1 𝑧𝑖 )/(𝑒𝑝 (𝑛) min𝑖∈N {𝑧𝑖 }). The proof is com-
where 𝜉𝑘 = max{1, 𝜂𝑘 }, and 𝜆 ∈ (0, 𝜎0 ] is determined by
plete.
Remark 8. In [25–32], the authors established some useful criteria for ensuring synchronization of FCNNs with
delays, respectively. However, once the unbounded distributed delays are involved, all results in [25–32] will be
invalid. Hence, in this sense, the proposed Theorem 7 has a
wider range of applications than those in previous papers.
(54)
(𝜆𝐸 + 𝐷1 + 𝑃1 + 𝑄1 𝑒𝜆𝜏 + ∫
+∞
0
Υ (𝑠) 𝑒𝜆𝑠 𝑑𝑠) 𝑧 < 0,
(55)
for a given 𝑧 ∈ M𝐷̂.
Then, the drive-response systems (2) and (53) are globally
exponential synchronization.
10
Abstract and Applied Analysis
Synchronization of 𝑥1 and 𝑦1
8
Synchronization of 𝑥2 and 𝑦2
15
6
10
𝑥2 /𝑦2
𝑥1 /𝑦1
4
2
0
5
0
−2
−5
−4
0
10
20
30
0
10
𝑡
Error trajectory 𝑒1
2
Error trajectory 𝑒2
2
1
1
0
0
𝑒2
𝑒1
30
𝑥2
𝑦2
𝑥1
𝑦1
−1
−1
−2
20
𝑡
0
10
20
𝑡
30
𝐷+ 𝑉 (𝑒 (𝑡)) ≤ 𝐷1 𝑉 (𝑒 (𝑡))
+ 𝑃1 𝑉 (𝑒 (𝑡)) + 𝑄1 [𝑉 (𝑒 (𝑡))]𝜏
+∞
0
0
10
Figure 2: Synchronization of (57) and (59).
Proof. Let 𝑉(𝑒(𝑡)) = [𝑒(𝑡)]+ = (|𝑒1 (𝑡)|, . . . , |𝑒𝑛 (𝑡)|)𝑇 . Calculating the time derivative of 𝑉(𝑒(𝑡)) along with the trajectory of
error system can give
+∫
−2
20
30
𝑡
for 𝑖 ∈ N, 𝑘 ∈ N have been imposed on the impulsive
perturbations. However, Theorem 10 drops these restrictions.
4. Illustrative Example
(56)
Υ (𝑠) 𝑉 (𝑒 (𝑡 − 𝑠)) 𝑑𝑠.
The rest proof is similar to Theorem 7 and omitted here. We
complete the proof.
Remark 11. In [31], Feng et al. derived some criteria on
the globally exponential synchronization for a special case
of (2) and (53) with 𝛾𝑖𝑗 = 𝜃𝑖𝑗 = 0, 𝜏𝑖𝑗 (𝑡) = 𝜏𝑖𝑗 and
𝐼𝑖𝑘 (𝑥𝑖 (𝑡𝑘− )) = −𝜂𝑖𝑘 𝑥𝑖 (𝑡𝑘− ) for 𝑖, 𝑗 ∈ N, 𝑘 ∈ N. In order to
achieve synchronous control, the conditions as 0 ≤ 𝜂𝑖𝑘 ≤ 2
In this section, a numerical example and its simulations are
given to illustrate the effectiveness of our results.
Example 1. Consider the following 2-dimensional IFCNNs
with mixed delays as the drive system
2
𝑑𝑥𝑖 (𝑡)
= −𝑐𝑖 𝑥𝑖 + ∑𝑎𝑖𝑗 𝑓𝑗 (𝑥𝑗 )
𝑑𝑡
𝑗=1
2
+ ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡 − 1))
𝑗=1
Abstract and Applied Analysis
11
2
The control gain matrices 𝑀 and 𝑁 can be chosen as
+ ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡 − 1))
𝑗=1
2
+ ⋀𝛾𝑖𝑗 ∫
+∞
0
𝑗=1
2
+ ⋁𝜃𝑖𝑗 ∫
+∞
0
𝑗=1
−10.6 0
𝑀= (
),
0 −20.6
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝑠)) 𝑑𝑠
𝑒−20 𝑒−20
).
𝑁 = ( −20
10𝑒
5𝑒−20
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑥𝑗 (𝑡 − 𝑠)) 𝑑𝑠,
By simple calculation, we get that
̂ = ( −6 1.5)
𝐷
14.4 −5
𝑡 ≠ 𝑡𝑘 ,
Δ𝑥𝑖 (𝑡𝑘 ) = 𝑥𝑖 (𝑡𝑘+ ) − 𝑥𝑖 (𝑡𝑘− ) = −𝐽𝑘 𝑥𝑖 (𝑡𝑘− ) ,
(57)
where 𝑖, 𝑗 = 1, 2, 𝑓𝑗 (𝑢) = tanh(𝑢). 𝐽𝑘 = 𝑒0.15 + 1, 𝑡𝑘 = 𝑡𝑘−1 + 4
for 𝑘 ∈ N. For the simplicity of computer simulations, we
choose 𝑘𝑖𝑗 (𝑠) = 𝑒−𝑠 for 𝑠 ∈ [0, 20], 𝑘𝑖𝑗 (𝑠) = 0 for 𝑠 ∈ [20, +∞).
The system parameters are as follows:
−1 0
𝐷1 = (
),
0 −1
2 −0.1
𝐴=(
),
−4 3.2
−1.3 −0.2
𝛼=𝛽=(
),
−0.2 −4.2
−0.5 −0.5
Γ=Θ=(
).
5 −2.5
(60)
(58)
(61)
is a nonsingular M-matrix, which implies that (𝐴 6 ) holds.
Moreover, we can choose 𝜆 = 0.2 and 𝑧 = (1, 3.5)𝑇 ∈ M𝐷̂
such that (𝐴 7 ) holds. Therefore, by Theorem 10, the driveresponse systems (57) and (59) are globally exponentially
synchronized. The simulation result with 𝜓(𝑠) = (1.3, 1.8)𝑇
is shown in Figure 2.
Remark 12. It is worth noting that the impulsive perturbations 𝐽𝑘 > 2, which are not a satisfied condition (𝐻2) in
[31]. That is to say, even in the absence of the unbounded
distributed delays, the results in [31] still cannot be applied
to the synchronization problem of (57) and (59).
Case 2. Consider the response system with random noise as
follows:
We can choose 𝜎0 = 0.8 and 𝐿 1 = 𝐿 2 = 1 such that 𝑘𝑖𝑗 ∈ L𝑒
and (𝐴 1 ) holds, respectively. Obviously, (𝐴 2 ) holds with 𝜂𝑘 =
𝑒0.15 , 𝑘 ∈ N.
Choosing the initial value 𝜙(𝑠) = (3, −6)𝑇 for 𝑠 ∈ (−∞, 0],
the drive system (57) possesses a chaotic behavior as shown
in Figure 1.
Case 1. The response system without random noise is given
by
2
𝑑𝑦𝑖 (𝑡) = [−𝑐𝑖 𝑦𝑖 + ∑𝑎𝑖𝑗 𝑓𝑗 (𝑦𝑗 )
𝑗=1
[
2
+ ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 1))
𝑗=1
2
+ ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 1))
𝑗=1
2
𝑑𝑦𝑖 (𝑡)
= −𝑐𝑖 𝑦𝑖 + ∑ 𝑎𝑖𝑗 𝑓𝑗 (𝑦𝑗 )
𝑑𝑡
𝑗=1
2
+∞
+ ⋀𝛾𝑖𝑗 ∫
𝑗=1
2
2
+ ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 1))
0
+∞
+ ⋁𝜃𝑖𝑗 ∫
𝑗=1
𝑗=1
0
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠
𝑘𝑖𝑗 (𝑠)
2
+ ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑦𝑗 (𝑡 − 1))
𝑗=1
2
+∞
+ ⋀𝛾𝑖𝑗 ∫
𝑗=1
2
0
+∞
+ ⋁𝜃𝑖𝑗 ∫
𝑗=1
0
(59)
𝑘𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠
×𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠 + 𝑈𝑖 (𝑡) ] 𝑑𝑡
]
2
+ ∑𝜎𝑖𝑗 (𝑡, 𝑥𝑗 (𝑡) − 𝑦𝑗 (𝑡) ,
𝑗=1
𝑥𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡)) − 𝑦𝑗 (𝑡 − 𝜏𝑖𝑗 (𝑡))) 𝑑𝑤𝑗 (𝑡) ,
𝑘𝑖𝑗 (𝑠) 𝑓𝑗
× (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠 + 𝑈𝑖 (𝑡) ,
𝑡 ≠ 𝑡𝑘 ,
Δ𝑦𝑖 (𝑡𝑘 ) = 𝑦𝑖 (𝑡𝑘+ ) − 𝑦𝑖 (𝑡𝑘− ) = −𝐽𝑘 𝑦𝑖 (𝑡𝑘− ) .
𝑡 ≠ 𝑡𝑘 ,
Δ𝑦𝑖 (𝑡𝑘 ) = 𝑦𝑖 (𝑡𝑘+ ) − 𝑦𝑖 (𝑡𝑘− ) = −𝐽𝑘 𝑦𝑖 (𝑡𝑘− ) ,
(62)
12
Abstract and Applied Analysis
Synchronization of 𝑥1 and 𝑦1
8
Synchronization of 𝑥2 and 𝑦2
15
6
10
𝑥2 /𝑦2
𝑥1 /𝑦1
4
2
0
5
0
−2
−5
−4
10
0
20
0
30
10
𝑡
Error trajectory 𝑒1
2
Error trajectory 𝑒2
2
1
1
0
0
𝑒2
𝑒1
30
𝑥2
𝑦2
𝑥1
𝑦1
−1
−1
−2
20
𝑡
0
10
20
𝑡
30
−𝑥1 (𝑡) 0.5𝑥1 (𝑡 − 1)
).
𝑥2 (𝑡) −0.5𝑥2 (𝑡 − 1)
0.25 0
𝐷=(
)
0 0.25
(63)
(64)
such that (𝐴 3 ) holds.
For 𝑝 = 2, let control gain matrices be
−8.475
0
𝑀= (
),
0
−25.925
𝑒−20 𝑒−20
).
𝑁 = ( −20
10𝑒
5𝑒−20
10
20
30
𝑡
It is easy to deduce that
Clearly, we can choose
1 0
𝐶= (
),
0 1
0
Figure 3: Synchronization of (57) and (62).
and the noise intensity matrix is
𝜎=(
−2
(65)
̃ = ( −5 1.5)
𝐷
14.4 −5
(66)
is a nonsingular M-matrix, which implies that (𝐴 4 ) holds.
Meanwhile, we can choose 𝜆 = 0.1 and 𝑧 = (1, 3)𝑇 ∈
M𝐷̃ such that (𝐴 5 ) holds. Hence, by Theorem 7, the driveresponse systems (57) and (62) are globally exponentially
synchronized in mean square. The simulation result based on
Euler-Maruyama method is illustrated in Figure 3.
Remark 13. The schemes proposed in [25–29] cannot solve
the synchronization problem of (57) and (62) due to the
impulsive perturbations and random noise. Besides this, the
distributed delays make those methods in [30–32] cannot be
applied to synchronization problem of (57) and (62).
Abstract and Applied Analysis
5. Conclusion
In this paper, we investigate the synchronization problem of IFCNNs with mixed delays. Based on the properties of nonsingularM-matrix and some stochastic analysis
approaches, some useful synchronization criteria under both
impulse and random noise are obtained. The methods used
in this paper are novel and can be extended to many other
types of neural networks. These problems will be considered
in near future.
Acknowledgments
The authors thank the editor and reviewers for their useful
comments. This work was supported by the National Natural
Science Foundation of China under Grants 11271270 and
61273021 and the Natural Science Foundation Project of CQ
CSTC 2011jjA00012.
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