Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 168375, 23 pages
doi:10.1155/2012/168375
Research Article
Global Exponential Stability of Antiperiodic
Solutions for Discrete-Time Neural Networks with
Mixed Delays and Impulses
Xiaofeng Chen and Qiankun Song
Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, China
Correspondence should be addressed to Qiankun Song, qiankunsong@163.com
Received 21 October 2011; Accepted 25 December 2011
Academic Editor: Taher S. Hassan
Copyright q 2012 X. Chen 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 problem on global exponential stability of antiperiodic solution is investigated for a class
of impulsive discrete-time neural networks with time-varying discrete delays and distributed
delays. By constructing an appropriate Lyapunov-Krasovskii functional, and using the contraction
mapping principle and the matrix inequality techniques, a new delay-dependent criterion for
checking the existence, uniqueness, and global exponential stability of anti-periodic solution is
derived in linear matrix inequalities LMIs. Two simulation examples are given to show the
effectiveness of the proposed result.
1. Introduction
Over the past decades, delayed neural networks have found successful applications in many
areas such as signal processing, pattern recognition, associative memories, and optimization
solvers. In such applications, the qualitative analysis of the dynamical behaviors is a
necessary step for the practical design of neural networks 1. Many important results
on the dynamical behaviors have been reported for delayed neural networks, see 1–5
and the references therein for some recent publications. Although neural networks are
mostly studied in the continuous-time setting, they are often discretized for experimental or
computational purposes. The dynamic characteristics of discrete-time neural networks have
been extensively investigated, for example, see 6–10 and the references cited therein.
Impulsive differential equations are mathematical apparatus for simulation of process
and phenomena observed in control theory, physics, chemistry, population dynamics,
biotechnologies, industrial robotics, economics, and so forth 11, 12. Consequently, many
neural networks with impulses have been studied extensively, and a great deal of the
2
Discrete Dynamics in Nature and Society
literature is focused on the existence and stability of an equilibrium point 13–17. In 18, 19,
the authors discussed the existence and global exponential stability of periodic solution of a
class of neural networks with impulse.
The study of antiperiodic solutions for nonlinear differential equations is closely
related to the study of periodic solutions, and it was initiated by Okochi in 1988 20.
Arising from problems in applied sciences, it is well-known that the existence and stability of
antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential
equations as a special periodic solution 21, 22. As pointed out in 23, antiperiodic
solutions arise naturally in the mathematical modeling of various physical processes. For
example, antiperiodic trigonometric polynomials are often studied in interpolation problems
24, 25, the signal transmission process of neural networks can often be described as an
antiperiodic process 26, and antiperiodic wavelets are discussed in 27. During the past
twenty years antiperiodic problems of nonlinear differential equations have been extensively
studied by many authors, for example, see 28–30 and references therein. Recently, the
problem of antiperiodic solutions for neural networks with or without time delays has
received considerable research interest, see for example 26, 31–38 and references therein.
In 31, 32, using some analysis skills and Lyapunov method, the authors studied the
existence and exponential stability of antiperiodic solutions for shunting inhibitory cellular
neural networks with time-varying discrete delays or distributed delays. In 33, some
sufficient conditions have been established for checking the existence and exponential
stability of antiperiodic solutions of high-order Hopfield neural networks with time-varying
delays. In 34, the recurrent neural networks with time-varying delays and continuously
distributed delays have been considered, and some sufficient conditions for the existence and
exponential stability of the antiperiodic solutions have been given. In 35, 36, the existence
and exponential stability of antiperiodic solutions have been studied for Cohen-Grossberg
neural networks with time-varying delays and continuously distributed delays. In 37, the
authors obtained some sufficient conditions to ensure existence and exponential stability of
the antiperiodic solutions for a class of Hopfield neural networks with periodic impulses.
In 38, several sufficient conditions are established for the existence and global exponential
stability of antiperiodic solutions to impulsive shunting inhibitory cellular neural networks
with distributed delays on time scale.
However, to the best of our knowledge, there are few papers published on the
existence and exponential stability of antiperiodic solutions for discrete-time neural networks
with mixed delays and impulses. Motivated by the above discussion, the objective of this
paper is to study the existence and exponential stability of antiperiodic solutions for
impulsive discrete-time recurrent neural networks with time-varying discrete delays and
distributed delays. Under more general description on the activation functions, we obtain
a new criterion for checking the existence, uniqueness, and global exponential stability of
antiperiodic solution, which can be checked numerically using the effective LMI toolbox in
MATLAB. Two simulation examples are given to show the effectiveness and less conservatism of the proposed criteria.
Notations
The notation used here is quite standard. Rn and Rm×n denote, respectively, the n-dimensional
Euclidean space and the set of all m × n real matrices. The superscript “T” denotes matrix
transposition. The notation X ≥ Y resp., X > Y means that X and Y are symmetric
matrices, and that X − Y is positive semidefinite resp., positive definite. · is the Euclidean
Discrete Dynamics in Nature and Society
3
norm in Rn . If A is a matrix, denote by λmax A resp., λmin A the largest resp., smallest
eigenvalue of A. For integers a, b, and a < b, Na, b denotes the discrete interval given by
Na, b {a, a 1, . . . , b}. CN−τ, 0, Rn denotes the set of all functions φ : N−τ, 0 → Rn .
In symmetric block matrices, the symbol ∗ is used as an ellipsis for terms induced by
symmetry. Sometimes, the arguments of a function or a matrix will be omitted in the analysis
when no confusion can arise.
2. Model Description and Preliminaries
In this paper, we consider the following impulsive discrete-time neural networks with timevarying discrete delays and distributed delays
xi k 1 ai xi k n
n
bij gj xj k cij gj xj k − τ1 k
j1
n
j1
dij
j1
kτ
2 −1
gj xj m − τ2 Ii k,
k
/ kr
2.1
mk
xi kr 1 xi kr eir xi kr ,
r 1, 2, . . .
or, in an equivalent vector form
xk 1 Axk Bgxk Cgxk − τ1 k
D
kτ
2 −1
gxm − τ2 Ik,
k
/ kr
2.2
mk
xkr 1 xkr er xkr ,
r 1, 2, . . .
for k 1, 2, . . ., where xk x1 k, x2 k, . . . , xn kT ∈ Rn , xi k is the state of the ith
neuron at time k; gxk g1 x1 k, g2 x2 k, . . . , gn xn kT ∈ Rn , gj xj k denotes
the activation function of the jth neuron at time k; Ik I1 k, I2 k, . . . , In kT ∈ Rn ,
Ii k represents the external input on the ith neuron at time k; the positive integer τ1 k
corresponds to the transmission delay and satisfies τ̌ ≤ τ1 k ≤ τ τ̌ and τ are known
integers such that τ > τ̌ ≥ 0; the positive integer τ2 describes the distributed time delay;
A diaga1 , a2 , . . . , cn , ai 0 ≤ ai < 1 describes the rate with which the ith neuron will
reset its potential to the resting state in isolation when disconnected from the networks and
external inputs; B bij n×n is the connection weight matrix, C cij n×n and D dij n×n are
the delayed connection weight matrices; kr are the impulse instants satisfying 0 k0 < k1 <
· · · < kr < kr1 < · · · and kr1 − kr > 1; er : Rn → Rn r 1, 2, . . . denotes a sequence of jump
operators.
2 −1
gxm−τ2 in the system 2.2 is the so-called distributed
Remark 2.1. The delay term kτ
mk
delay in the discrete-time setting, which can be regarded as the discretization of the integral
t
form t−τ2 gxsds for the continuous-time system.
4
Discrete Dynamics in Nature and Society
The initial condition associated with system 2.2 is given by
xs φs,
2.3
φ ∈ CN−τ, 0, Rn ,
where τ max{
τ , τ2 }.
Throughout this paper, we make the following assumptions.
H1 gu and er u are odd functions, τ1 k and Ik are ω-periodic function and ωantiperiodic function, respectively, that is,
g−u −gu,
er −u −er u,
τ1 k τ1 k ω,
r 1, 2, . . . ,
u ∈ Rn ,
Ik −Ik ω,
k 1, 2, . . . .
2.4
And there exists a positive integer p such that
krp kr ω,
erp u er u,
u ∈ Rn ,
r 1, 2, . . . .
2.5
H2 There exist constants ľj and lj j 1, 2, . . . , n such that
ľj ≤
gj α1 − gj α2 ≤ lj ,
α1 − α2
∀α1 , α2 ∈ R, α1 /
α2 .
2.6
i i 1, 2, . . . , n such that
H3 There exist constants ȟi and h
ȟi ≤
eir α1 − eir α2 ≤ hi ,
α1 − α2
∀α1 , α2 ∈ R, α1 /
α2 , r 1, 2, . . . .
2.7
Definition 2.2. The antiperiodic solution x∗ k of system 2.2 with 2.3 is said to be globally
exponentially stable if there exist two positive constants M > 0 and 0 < ε < 1 such that
xk − x∗ k ≤ Mεk sup φk − φ∗ k,
2.8
k∈N−τ,0
for all k 1, 2, . . ., where xk is any solution of system 2.2 with 2.3, φk and φ∗ k are
the initial functions of solutions xk and x∗ k, respectively.
Lemma 2.3. Let M ∈ Rn×n be positive definite matrix, αi ∈ Rn i 1, 2, . . . , m. Then the following
inequality holds:
m
αi
i1
T
m
m
M
αi ≤ m αTi Mαi .
i1
2.9
i1
The proof of Lemma 2.3 can be carried out by following a similar line as in 10, and
hence it is omitted.
Discrete Dynamics in Nature and Society
5
3. Main Result
The main objective of this section is to obtain sufficient conditions on the existence, uniqueness, and exponential stability of antiperiodic solution for system 2.2. For presentation
convenience, in the following, we denote
τ̌ τ
τ
2
L1 diag ľ1 , ľ2 , . . . , ľn ,
L2 diag l1 , l2 , . . . , ln ,
L3 diag ľ1 l1 , ľ2 l2 , . . . , ľn ln ,
ľn ln
ľ1 l1 ľ2 l2
L4 diag
,
,...,
2
2
2
2, . . . , h
n ,
1, h
H1 diag h
3.1
,
ȟ
,
max
,
.
.
.
,
max
h
H2 diag max ȟ1 , h
ȟn , hn .
,
1
2
2
Theorem 3.1. Under assumptions (H1), (H2), and (H3), there exit exactly one ω-antiperiodic
solution of system 2.2 with 2.3 and all other solutions of system 2.2 with 2.3 converge
exponentially to it as k → ∞, if there exist ten n × n symmetric positive define matrices P,
Ri i 1, 2, 3, Qi , i 1, 2, 3, 4, 5, 6, and four n×n positive diagonal matrix Ti i 1, 2, Si i 1, 2
such that the following LMIs
Π Ωi < 0,
Ξ Ωi < 0,
i 1, 2
3.2
Π Ωi < 0,
Ξ Ωi < 0,
i 3, 4
3.3
hold or
hold, where
⎡
Π11 Π12 Π13
⎢
⎢ ∗ Π22 Π23
⎢
⎢
⎢ ∗
∗ Π33
⎢
⎢
⎢ ∗
∗
∗
⎢
Π⎢
⎢ ∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
⎣
∗
∗
∗
Π14
0
0
R1
Π24
0
0
0
Π34 Π35
0
0
Π44
0
0
0
∗
Π55
0
0
∗
∗
−Q1
0
∗
∗
∗
−Q2
∗
∗
∗
∗
0
⎤
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥,
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎦
−Q3
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Discrete Dynamics in Nature and Society
⎤
⎡
Ξ11 Ξ12 0
0
0
0
R1
0
⎥
⎢
⎢ ∗ Ξ22 0
0
0
0
0
0 ⎥
⎥
⎢
⎥
⎢
⎥
⎢∗
∗
Ξ
0
Ξ
0
0
0
33
35
⎥
⎢
⎥
⎢
⎢∗
∗
∗ Ξ44 0
0
0
0 ⎥
⎥
⎢
Ξ⎢
⎥,
⎥
⎢∗
∗
∗
∗
Ξ
0
0
0
55
⎥
⎢
⎥
⎢
⎥
⎢∗
∗
∗
∗
∗
−Q
0
0
1
⎥
⎢
⎥
⎢
⎢∗
∗
∗
∗
∗
∗ −Q2 0 ⎥
⎦
⎣
∗
∗
∗
∗
∗
∗
∗ −Q3
⎡
0 0 0 0
0
0
0
0 0 0
0
0
0
∗ 0 0
0
0
0
∗ ∗ 0
0
0
0
2R2
R2
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
Ω1 ⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎣
∗
⎡
∗ ∗ ∗ −3R2
∗ ∗ ∗
∗
−2R2 − R3
0
∗ ∗ ∗
∗
∗
−R1 − R2
∗ ∗ ∗
∗
∗
∗
0 0 0 0
0
0
0
0 0 0
0
0
0
∗ 0 0
0
0
0
∗ ∗ 0
0
0
0
R2
2R2
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
Ω2 ⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎣
∗ ∗ ∗ −3R2
∗ ∗ ∗
∗
−R2 − R3
0
∗ ∗ ∗
∗
∗
−R1 − 2R2
∗ ∗ ∗ ∗
∗
∗
∗
0 0 0 0
0
0
0
0 0 0
0
0
0
∗ 0 0
0
0
0
∗ ∗ 0
0
0
0
R3
0
⎡
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
Ω3 ⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎣
∗
∗ ∗ ∗ −3R3
∗ ∗ ∗
∗
−R2 − R3
R2
∗ ∗ ∗
∗
∗
−R1 − R2
∗ ∗ ∗
∗
∗
∗
0
⎤
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥,
0 ⎥
⎥
⎥
R3 ⎥
⎥
⎥
0 ⎥
⎦
−R3
0
⎤
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥,
0 ⎥
⎥
⎥
R3 ⎥
⎥
⎥
0 ⎥
⎦
−R3
0
⎤
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥,
2R3 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎦
−2R3
Discrete Dynamics in Nature and Society
⎡
7
0 0 0 0
0
0
0
0 0 0
0
0
0
∗ 0 0
0
0
0
∗ ∗ 0
0
0
0
2R3
0
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
Ω4 ⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎣
∗
∗ ∗ ∗ −3R3
∗ ∗ ∗
∗
−R2 − 2R3
R2
∗ ∗ ∗
∗
∗
−R1 − R2
∗ ∗ ∗
∗
∗
∗
0
⎤
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥,
R3 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎦
−R3
3.4
with
Π11 APA − P Q1 Q2 Q3
1 τ − τ̌Q4 − 2L1 S1 2L2 S2 A − ERA − E − R1 − L3 T1 ,
Π12 APB 1 τ − τ̌S1 − S2 A − ERB L4 T1 ,
Π13 APC A − ERC,
Π14 APD A − ERD,
Π22 BT PB BT RB 1 τ − τ̌Q5 τ2 Q6 − T1 ,
Π23 BT PC BT RC,
Π24 BT PD BT RD,
Π33 CT PC CT RC − Q5 − T2 ,
Π34 CT PD CT RD,
Π35 S2 − S1 L4 T2 ,
Π44 DT PD DT RD −
1
Q6 ,
τ2
Π55 −Q4 2L1 S1 − 2L2 S2 − L3 T2 ,
Ξ11 H1 P PT H1 H2 PH2 Q1 Q2 Q3
1 τ − τ̌Q4 − 2L1 S1 2L2 S2 H2 RH2 − R1 − L3 T1 ,
Ξ12 1 τ − τ̌S1 − S2 L4 T1 ,
Ξ22 1 τ − τ̌Q5 τ2 Q6 − T1 ,
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Discrete Dynamics in Nature and Society
Ξ33 −Q5 − T2 ,
Ξ35 Π35 ,
Ξ44 −
1
Q6 ,
τ2
Ξ55 Π55 ,
R τ̌ 2 R1 τ − τ̌2 R2 τ − τ2 R3 .
3.5
Proof. Let X {φ | φ ∈ CN−τ, 0, Rn }. For φ ∈ X, define
φ sup φs,
3.6
s∈−τ,0
then X is a Banach space with the topology of uniform convergence.
For φ, ψ ∈ X, let xk, φ and xk, ψ be the solutions of system 2.2 with initial values
φ and ψ, respectively. Define
xk φ t x k t, φ ,
∀t ∈ N−τ, 0, k 1, 2, . . . ,
3.7
and then xk φ ∈ X for all k 1, 2, . . .. It follows from system 2.2 that
x k 1, φ − x k 1, ψ A x k, φ − x k, ψ B g x k, φ − g x k, ψ
C g x k − τ1 k, φ − g x k − τ1 k, ψ
D
τ2
g x k − m, φ − g x k − m, ψ ,
k
/ kr
m1
x kr 1, φ − x kr 1, ψ x kr , φ − x kr , ψ
e r x kr , φ − e r x kr , ψ ,
r 1, 2, . . . .
3.8
Letting
yk x k, φ − x k, ψ ,
fk g x k, φ − g x k, ψ ,
ur ykr er x kr , φ − er x kr , ψ ,
3.9
Discrete Dynamics in Nature and Society
9
system 3.8 can then be simplified as
yk 1 Ayk Bfk Cfk − τ1 k
D
τ2
fk − m,
k
/ kr
3.10
m1
ykr 1 ykr ur ykr ,
r 1, 2, . . . .
Defining ηk yk 1 − yk, we consider the following Lyapunov-Krasovskii
functional candidate for system 3.10 as
V k 8
3.11
Vi k,
i1
where
V1 k yT kPyk,
V2 k k−1
k−1
k−1
yT j Q1 y j yT j Q2 y j yT j Q3 y j ,
jk−τ̌
jk−τ
k−1
V3 k yT j Q4 y j jk−τ1 k
k−1
V4 k f T j Q5 f j τ2 k−1
k−1
k−τ̌ yT j Q4 y j ,
mk−
τ 1jm
jk−τ1 k
V5 k jk−
τ
k−1
k−τ̌ f T j Q5 f j ,
mk−
τ 1jm
f T j Q6 f j ,
m1jk−m
k−1
T
fj − L1 yj S1 y j
V6 k 2
jk−τ1 k
2
k−τ̌
k−1
T
fj − L1 yj S1 y j ,
mk−
τ 1 jm
k−1
T
S2 y j
L2 y j − f j
V7 k 2
jk−τ1 k
2
k−τ̌
k−1
mk−
τ 1 jm
T
S2 y j ,
L2 y j − f j
10
Discrete Dynamics in Nature and Society
V8 k τ̌
k−1
k−1 k−
τ̌−1 k−1
ηT j R1 η j τ − τ̌
η T j R2 η j
mk−τ̌ jm
τ − τ
mk−τ jm
k−1
k−τ−1
η T j R3 η j .
mk−
τ jm
3.12
We proceed by considering two possible cases of k / kr and k kr .
Case 1 k /
kr r 1, 2, . . .. Calculating the difference of Vi k i 1, 2, . . . , 8 along the
system 3.10, by Lemma 2.3 we have
ΔV1 k yT k 1Pyk 1 − yT kPyk
yT k AT PA − P yk 2yT kAT PBfk
τ2
2yT kAT PCfk − τ1 k 2yT kAT PD
fk − m
m1
f T kBT PBfk 2f T kBT PCfk − τ1 k
2f T kBT PD
τ2
fk − m f T k − τ1 kCT PCfk − τ1 k
m1
2f T k − τ1 kCT PD
τ2
fk − m
m1
τ2
T
fk − m
DT PD
m1
τ2
fk − m,
m1
3.13
ΔV2 k yT kQ1 Q2 Q3 yk − yT k − τQ1 yk − τ
− yT k − τ̌Q2 yk − τ̌ − yT k − τQ3 yk − τ,
k−τ̌
ΔV3 k k−1
yT j Q4 y j yT j Q4 y j yT kQ4 yk
jk1−τ1 k1
−
k−1
3.14
jk−τ̌1
yT j Q4 y j − yT k − τ1 kQ4 yk − τ1 k
jk−τ1 k1
τ − τ̌yT kQ4 yk −
k−τ̌
yT j Q4 y j
jk−
τ 1
≤ 1 τ − τ̌yT kQ4 yk − yT k − τ1 kQ4 yk − τ1 k,
3.15
Discrete Dynamics in Nature and Society
11
ΔV4 k ≤ 1 τ − τ̌f T kQ5 fk − f T k − τ1 kQ5 fk − τ1 k,
ΔV5 k 3.16
τ2 f T kQ6 fk − f T k − mQ6 fk − m ,
m1
T τ
τ
2
2
1 T
≤ τ2 f kQ6 fk −
fk − m Q6
fk − m ,
τ2 m1
m1
T
ΔV6 k ≤ 21 τ − τ̌ fk − L1 yk S1 yk
3.18
T
− 2 fk − τ1 k − L1 yk − τ1 k S1 yk − τ1 k,
T
ΔV7 k ≤ 21 τ − τ̌ L2 yk − fk S2 yk
3.19
T
− 2 L2 yk − τ1 k − fk − τ1 k S2 yk − τ1 k,
k−1
ΔV8 k ηT kRηk − τ̌
ηT j R1 η j
jk−τ̌
k−
τ̌−1
3.20
k−τ−1
η j R2 η j − τ − τ
η T j R3 η j .
− τ − τ̌
3.17
T
jk−
τ
jk−τ
From the definition of ηk and 3.10, we have
ηT kRηk yT kA − ET RA − Eyk 2yT kA − ET RBfk
2yT kA − ET RCfk − τ1 k 2yT kA − ET RD
τ2
fk − m
m1
f T kBT RBfk 2f T kBT RCfk − τ1 k
2f T kBT RD
τ2
fk − m f T k − τ1 kCT RCfk − τ1 k
m1
2f T k − τ1 kCT RD
τ2
fk − m
m1
T
fk − m
τ2
DT RD
m1
τ2
fk − m.
m1
3.21
From Lemma 2.3, it can be shown that the following inequality holds:
−τ̌
k−1
⎡
⎤T ⎡
⎤
k−1
k−1
η T j R1 η j ≤ − ⎣
η j ⎦ R1 ⎣
η j ⎦
jk−τ̌
jk−τ̌
yk
jk−τ̌
T yk − τ̌
−R1 R1
∗
−R1
yk
yk − τ̌
.
3.22
12
Discrete Dynamics in Nature and Society
When τ̌ ≤ τ1 k ≤ τ, let ak τ1 k − τ̌/τ − τ̌. Then 0 ≤ ak ≤ 1. It is easy to get
that
k−
τ̌−1
k−τ
k−
τ̌−1
1 k−1
ηT j R2 η j −τ − τ̌
ηT j R2 η j − τ − τ̌
η T j R2 η j
−τ − τ̌
jk−τ
jk−τ1 k
jk−τ
≤ −τ − τ1 k
k−τ
1 k−1
η T j R2 η j
jk−τ
k−τ
1 k−1
η T j R2 η j
− akτ − τ1 k
jk−τ
− 1 − akτ1 k − τ̌
k−
τ̌−1
η T j R2 η j
jk−τ1 k
− τ1 k − τ̌
k−τ
1 k−1
η T j R2 η j
jk−τ
≤−
k−τ
1 k−1
1 k−1
k−τ
η T j R2
η j
jk−τ
jk−τ
k−τ
1 k−1
1 k−1
k−τ
η T j R2
η j
− ak
jk−τ
− 1 − ak
jk−τ
k−
τ̌−1
k−
τ̌−1
η T j R2
η j
jk−τ1 k
−
k−
τ̌−1
jk−τ1 k
k−
τ̌−1
η T j R2
η j
jk−τ1 k
jk−τ1 k
⎡
⎤T ⎡
⎤
⎤⎡
yk − τ1 k
−2R2 R2 R2
yk − τ1 k
⎢
⎥ ⎢
⎥
⎥⎢
⎥ ⎢
⎥
⎢
⎢
−R2 0 ⎥
⎣ yk − τ ⎦ ⎣ ∗
⎦⎣ yk − τ ⎦
yk − τ̌
∗
∗ −R2
yk − τ̌
ak
T yk − τ1 k
−R2 R2
yk − τ1 k
yk − τ
1 − ak
∗
yk − τ
−R2
T yk − τ1 k
−R2 R2
yk − τ1 k
yk − τ̌
∗
−R2
yk − τ̌
,
3.23
T −R3 R3
yk − τ
yk − τ
−
τ − τ
.
η j R3 η j ≤
yk − τ
∗ −R3 yk − τ
jk−
τ
k−τ−1
T
3.24
Discrete Dynamics in Nature and Society
13
For positive diagonal matrices T1 > 0 and T2 > 0, we can get from assumption H2
that
T −L3 T1 L4 T1 yk
,
0≤
∗
−T1 fk
fk
0≤
yk
T −L3 T2 L4 T2 yk − τ1 k
yk − τ1 k
fk − τ1 k
∗
fk − τ1 k
−T2
3.25
.
3.26
T
3.27
Denoting
αk yT k, f T k, f T k − τ1 k,
τ2
f T k − m,
m1
yT k − τ1 k, yT k − τ, yT k − τ̌, yT k − τ
,
it follows from 3.13–3.26 that
ΔV k 8
ΔVi k
i1
≤ yT kAPA − P Q1 Q2 Q3
1 τ − τ̌Q4 − 2L1 S1 2L2 S2 A − ERA − E − R1 − L3 T1 yk
2yT kAPB 1 τ − τ̌S1 − S2 A − ERB L4 T1 fk
2yT kAPC A − ERCfk − τ1 k
2yT kAPD A − ERD
τ2
fk − m
m1
f T k BT PB BT RB 1 τ − τ̌Q5 τ2 Q6 − T1 fk
2f T k BT PC BT RC fk − τ1 k
τ2
2f T k BT PD BT RD
fk − m
m1
f T k − τ1 k CT PC CT RC − Q5 − T2 fk − τ1 k
τ2
2f T k − τ1 k CT PD CT RD
fk − m
m1
14
Discrete Dynamics in Nature and Society
τ2
T fk − m
m1
1
D PD D RD − Q6
τ2
T
T
τ2
fk − m
m1
2yT kR1 yT k − τ̌
2f T k − τ1 kS2 − S1 L4 T2 yk − τ1 k
yT k − τ1 k−Q4 2L1 S1 − 2L2 S2 − L3 T2 − 3R2 yk − τ1 k
2yT k − τ1 kR2 akR2 yk − τ
2yT k − τ1 kR2 1 − akR2 yk − τ̌
− yT k − τQ1 1 akR2 R3 yk − τ
2yT k − τR3 yk − τ
− yT k − τ̌Q2 R1 R2 1 − akR2 yk − τ̌
− yT k − τQ3 R3 yk − τ
αT kakΠ Ω1 1 − akΠ Ω2 αk.
3.28
τ − τ1 k/
τ − τ. Then 0 ≤ bk ≤ 1. In the similitude
When τ ≤ τ1 k ≤ τ, let bk of the proof of inequality 3.23, we have
⎡
⎤⎡
⎤T ⎡
⎤
yk − τ1 k
−2R3 R3 R3
yk − τ1 k
⎥⎢
⎥ ⎢
⎥
⎢
⎢
⎥ ⎢
⎥
−
τ − τ
η T j R3 η j ≤ ⎢
−R3 0 ⎥
⎣ yk − τ ⎦ ⎣ ∗
⎦⎣ yk − τ ⎦
jk−
τ
yk − τ
yk − τ
∗
∗ −R3
k−τ−1
bk
T yk − τ1 k
−R3 R3
yk − τ1 k
yk − τ
∗
−R3
yk − τ
T yk − τ1 k
−R3 R3
yk − τ1 k
,
1 − bk
yk − τ
yk − τ
∗ −R3
T −R2 R2
yk − τ̌
yk − τ̌
.
−τ − τ̌
η j R2 η j ≤
∗ −R2 yk − τ
yk − τ
jk−τ
k−
τ̌−1
T
3.29
By using similar method in 3.28, it follows from 3.13 to 3.22, and 3.25 to 3.26, and
3.29 that
ΔV k ≤ αT kbkΠ Ω3 1 − bkΠ Ω4 αk.
3.30
Discrete Dynamics in Nature and Society
15
Case 2 k kr r 1, 2, . . .. Note that the inequalities from 3.13 to 3.26 except 3.13
and 3.21 hold for k kr . Calculating ΔV1 kr and ηT kr Rηkr along the system 3.10, we
have
ΔV1 kr yT kr 1Pykr 1 − yT kr Pykr T ykr ur ykr P ykr ur ykr − yT kr Pykr 3.31
≤ y kr 2H1 P H2 PH2 ykr ηT kr Rηkr uTr ykr Rur ykr ≤ yT kr H2 RH2 ykr .
T
By using similar method in Case 1, we can obtain that
ΔV k αT kakΞ Ω1 1 − akΞ Ω2 αk
3.32
ΔV k αT kbkΞ Ω3 1 − bkΞ Ω4 αk.
3.33
or
Combining the above discussions in Case 1 and 2, we obtain from 3.2, 3.3, 3.28,
3.30, 3.32, and 3.33 that
ΔV k ≤ −γ1 αk2 ≤ −γ1 yk2 ,
3.34
where γ1 mini1,2,3,4 {λmin −Π − Ωi , λmin −Ξ − Ωi } > 0.
From the definition of V k, it is easy to verify that
k−1 2
y j 2 ,
V k ≤ λmax Pyk γ2
3.35
jk−τ
where
γ2 λmax Q1 λmax Q2 λmax Q3 1 τ − τ̌ λmax Q4 λmax Q5 λmax LT L 4λmax LS1 4λmax LS2 τ2 λmax Q6 4τ̌ 2 λmax R1 4τ − τ̌2 λmax R2 4
τ − τ2 λmax R3 ,
where L diagmax{|ľ1 |, |l1 |}, max{|ľ2 |, |l2 |}, . . . , max{|ľn |, |ln |}.
3.36
16
Discrete Dynamics in Nature and Society
For any scalar ρ ≥ 1, it follows from 3.34 and 3.35 that
ρi1 V i 1 − ρi V i ρi1 ΔV i ρi ρ − 1 V i
2
≤ ρi ρ − 1 λmax P − γ1 ρi1 yi
γ2 ρi ρ − 1
i−1
3.37
2
y j .
ji−τ
Summing up both sides of 3.37 from 0 to k − 1 with respect to i, we have
ρk V k − V 0 ≤
k−1 2
ρ − 1 λmax P − γ1 ρ
ρi yi
i0
k−1 i−1
2
γ2 ρ − 1
ρi y j .
3.38
i0 ji−τ
Note that
k−1 i−1
⎛
⎞
jτ
jτ
k−1
−1 k−1−τ
k−2 2
⎠ρi y j 2
ρ i y j ⎝
i0 ji−τ
j−τ i0
≤ τρτ
j0 ij1
jk−τ ij1
−1 k−1−τ
k−2
2
2
y j 2 τρτ
ρj y j τρτ
ρj y j j−τ
j0
3.39
jk−τ
k−2 2
2
≤ τ 2 ρτ sup ys τρτ ρj y j .
s∈N−τ,0
j0
From 3.35, we have
2
V 0 ≤ λmax P γ2 τ sup ys .
3.40
s∈N−τ,0
It follows from 3.38–3.40 that
k−1 2
2
ρk V k ≤ K1 ρ sup ys K2 ρ
ρj y j ,
s∈N−τ,0
3.41
j0
where
K1 ρ λmax P γ2 τ γ2 ρ − 1 τ 2 ρτ ,
K2 ρ ρ − 1 λmax P − γ1 ρ γ2 ρ − 1 τρτ .
3.42
Discrete Dynamics in Nature and Society
17
Since K2 1 < 0, by the continuity of function K2 ρ, we can choose a scalar ξ > 1 such that
K2 ξ < 0. Obviously, K1 ξ > 0. From 3.41, we obtain
2
ξk V k ≤ K1 ξ sup ys .
3.43
s∈N−τ,0
From the definition of V k, we have
2
V k ≥ λmin Pyk .
Let M that
3.44
"
"
K1 ξ/λmin P, ε 1/ξ, then M > 0, 0 < ε < 1. It follows form 3.43 and 3.44
yk ≤ Mεk sup ys.
3.45
s∈N−τ,0
That is
x k, φ − x k, ψ ≤ Mk φ − ψ .
3.46
We can choose a positive integer N such that
MεNω ≤
1
.
4
3.47
Define a Poincaré mapping Γ : X → X by
Γ φ −xω φ .
3.48
Then, we can derive from 3.46 and 3.47 that
1
N φ − ΓN ψ ≤ φ − ψ ,
Γ
4
3.49
which shows that ΓN is a contraction mapping and therefore there exits a unique fixed point
φ∗ ∈ X of ΓN , which is also the unique fixed point of Γ such that
Γ φ∗ φ∗ .
3.50
Therefore,
−x w k, φ∗ φ∗ k,
∀k ∈ N−τ, 0.
3.51
18
Discrete Dynamics in Nature and Society
Let xk, φ∗ be the solution of system 2.2 through 0, φ∗ . From assumption H1
we know that −xk ω, φ∗ is also a solution of system 2.2. It follows from 3.51 that
−xk ω, φ∗ is also through 0, φ∗ . By the uniqueness of solution we can know
x k, φ∗ −x k ω, φ∗ ,
3.52
for k 1, 2, . . ., which indicates that xk, φ∗ is exactly one ω-antiperiodic solution of system
2.2. To this end, it is easy to see that all other solutions converge exponentially to it as
k → ∞. The proof is completed.
Remark 3.2. The conditions are dependent on both the lower bound and upper bound of
delays. It has been shown that the delay-dependent stability conditions are generally less conservative than the delay-independent ones, especially when the size of the delay is small.
Remark 3.3. In this paper, the model includes both discrete and distributed delays simultaneously, and can be used to describe some well-known neural networks owing to its generality.
In 38, only one kind of delay has been considered, which is a special case of neural networks
with mixed delays. Furthermore, in 38, the activations were assumed to be bounded
functions, while the boundedness condition is removed in this paper.
4. Examples
In this section, some examples and numerical simulations are provided to illustrate our results.
Example 4.1. Consider a discrete-time neural networks 2.2 with two neurons, where
0.2 0
A
,
0 0.3
B
0.06 0.01
−0.06 0.1
,
C
−0.05
0.1
−0.05 −0.02
,
−0.02 0.02
D
,
−0.05 0.07
#
g1 u arctan2u,
g2 u − arctan4u,
#
I2 k 4 cos
kr 8r − 3,
$
kπ
,
8
I1 k −3 sin
#
τ 1 k 3 sin
$
kπ
,
4
e1r u e2r u 0.08 sinu − 0.68u,
$
kπ
,
8
τ 2 2,
r 1, 2, . . . .
4.1
It can be verified that assumptions H1, H2, and H3 are satisfied with ľ1 0, l1 2,
1 h
2 −0.6. Thus,
ľ2 −4, l2 0, τ̌ 2, τ 4, ω 8, ȟ1 ȟ2 −0.76, h
L1 diag0, −4, L2 diag2, 0, L3 diag0, 0, L4 diag1, −2, H1 diag−0.6, −0.6,
Discrete Dynamics in Nature and Society
19
H2 diag0.76, 76. By the MATLAB Control Toolbox, we find a solution to the LMI in 3.2
as follows:
3.5412 −0.8274
0.1931 −0.0932
,
R2 ,
−0.0149 0.0106
−0.0932 0.1155
0.1941 −0.0937
0.1396 −0.0553
0.1762 −0.0693
,
Q1 ,
Q2 ,
R3 −0.0937 0.1158
−0.0553 0.0807
−0.0693 0.0917
0.1877 −0.0771
0.0230 −0.0077
0.1197 0.0554
,
Q4 ,
Q5 ,
Q3 −0.0771 0.0986
−0.0077 0.0222
0.0554 0.0274
0.1370 −0.0709
2.5687
0
1.2461
0
,
T1 ,
T2 ,
Q6 −0.0709 0.1201
0
0.8572
0
0.0252
0.0056
0
0.0756
0
,
S2 .
S1 0
0.0102
0
0.0034
4.2
P
−0.8274 8.5025
,
R1 0.0254 −0.0149
Therefore, by Theorem 3.1, we know that system 2.2 with above given parameters
has exactly one 8-antiperiodic solution and all other solutions of the system converge
exponentially to it as k → ∞, which is further verified by the simulation given in Figure 1.
Example 4.2. Consider a discrete-time neural networks 2.2 with three neurons, where
⎤
⎡
0.15 0 0
⎥
⎢
⎥
A⎢
⎣ 0 0.2 0 ⎦,
0
0 0.3
⎡
⎤
−0.04 −0.02 0.06
⎢
⎥
⎥
C⎢
⎣ 0.09 −0.02 0.02⎦,
0.06 0.01 0.01
⎡
0.05
0.04 −0.04
⎤
⎥
⎢
⎥
B⎢
⎣ 0.08 0.04 −0.04⎦,
−0.04 −0.08 0.04;
⎤
⎡
−0.02 0.05 0.02
⎥
⎢
⎥
D⎢
⎣−0.05 0.02 −0.01⎦,
0.02 −0.05 −0.02
#
$
kπ
g1 u g2 u g3 u sin2u − u,
I1 k −2 sin
,
12
$
$
#
#
kπ
kπ
,
τ1 k 2 cos
,
τ2 2,
I2 k I3 k cos
12
6
r−1
3−1r 7,
kr 12
e1r u e2r u e3r u −u, r 1, 2, . . . .
2
4.3
It can be verified that assumptions H1, H2, and H3 are satisfied with ľ1 ľ2 ľ3 −3,
1 h
2 h
3 −1.
l1 l2 l3 1, τ̌ 1, τ 3, ω 12, ȟ1 ȟ2 ȟ3 −1, h
Thus, L1 diag−3, −3, −3, L2 diag1, 1, 1, L3 diag−3, −3, −3, L4 diag−1, −1, −1,
20
Discrete Dynamics in Nature and Society
4
x1
2
0
−2
−4
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
5
x2
0
−5
Figure 1: State responses of the discrete-time neural networks with initial conditions x1 s, x2 sT −1, 0T , s ∈ N−4, 0.
H1 diag−1, −1, −1, H2 diag1, 1, 1. By the MATLAB Control Toolbox, we find a solution
to the LMI in 3.3 as follows:
⎤
⎤
⎡
⎡
41.8020 −4.3473 4.5690
0.4005 −0.2958 0.1172
⎥
⎥
⎢
⎢
⎥
⎥
R1 ⎢
P⎢
⎣−4.3473 30.0757 0.2682 ⎦,
⎣−0.2958 0.8118 0.2942⎦,
4.5690 0.2682 25.8887
0.1172 0.2942 0.6825
⎤
⎤
⎡
⎡
0.5178 −0.2326 −0.1006
0.4881 −0.2381 −0.1038
⎥
⎥
⎢
⎢
⎥
⎥
R2 ⎢
R3 ⎢
⎣−0.2326 0.6843 0.2146 ⎦,
⎣−0.2381 0.7032 0.2240 ⎦,
−0.1006 0.2146 0.6101
−0.1038 0.2240 0.6280
⎤
⎤
⎡
⎡
1.0470 −0.1515 0.4772
1.0644 −0.1626 0.4402
⎥
⎥
⎢
⎢
⎥
⎥
Q1 ⎢
Q2 ⎢
⎣−0.1515 0.6361 −0.1655⎦,
⎣−0.1626 0.6310 −0.1597⎦,
0.4772 −0.1655 1.0423
0.4402 −0.1597 0.9820
⎤
⎤
⎡
⎡
1.2250 −0.1486 0.6129
1.7504 −0.3124 0.7392
⎥
⎥
⎢
⎢
⎥
⎥
Q3 ⎢
Q4 ⎢
4.4
⎣−0.1486 0.8285 −0.1864⎦,
⎣−0.3124 0.3251 −0.4540⎦,
0.6129 −0.1864 1.2586
0.7392 −0.4540 0.9975
⎤
⎤
⎡
⎡
0.2637 0.0305 −0.0300
0.5803 −0.1963 0.1481
⎥
⎥
⎢
⎢
⎥
⎥
Q5 ⎢
Q6 ⎢
⎣ 0.0305 0.0846 −0.1291⎦,
⎣−0.1963 0.8232 0.3296⎦,
−0.0300 −0.1291 0.2397
0.1481 0.3296 0.3495
⎤
⎤
⎡
⎡
3.4923
0
0
0.9595
0
0
⎥
⎥
⎢
⎢
T1 ⎢
3.4632
0 ⎥
T2 ⎢
0.4567
0 ⎥
⎦,
⎦,
⎣ 0
⎣ 0
0
0
2.3878
0
0
0.3671
⎤
⎤
⎡
⎡
0.6575
0
0
0.3709
0
0
⎥
⎥
⎢
⎢
S1 ⎢
S2 ⎢
0.4853
0 ⎥
0.0163
0 ⎥
⎦,
⎦.
⎣ 0
⎣ 0
0
0
0.2337
0
0
0.0507
Discrete Dynamics in Nature and Society
21
4
2
0
x1
−2
−4
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0
20
40
60
80
100
120
2
x2
0
−2
2
x3
0
−2
Figure 2: State responses of the discrete-time neural networks with initial conditions x1 s, x2 s, x3 sT −1, 0, 1T , s ∈ N−3, 0.
Therefore, by Theorem 3.1, we know that system 2.2 with the above-given parameters has exactly one 12-antiperiodic solution and all other solutions of the system converge
exponentially to it as k → ∞, which is further verified by the simulation given in Figure 2.
5. Conclusions
In this paper, the discrete-time neural networks with mixed delays and impulses have been
studied. A delay-dependent LMI criterion for the existence and global exponential stability
of antiperiodic solutions has been established by constructing an appropriate LyapunovKrasovskii functional, and using the contraction mapping principle and the matrix inequality
techniques. Moreover, two examples are given to illustrate the effectiveness of the results.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant
60974132 and the Natural Science Foundation Project of CQ CSTC2011jjA00012.
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