stability analysis for cellular neural networks with variable delays
TRANSCRIPT
Chaos, Solitons and Fractals 28 (2006) 331–336
www.elsevier.com/locate/chaos
Stability analysis for cellular neural networkswith variable delays
Qiang Zhang *, Xiaopeng Wei, Jin Xu
Liaoning Key Lab of Intelligent Information Processing, Dalian University, Dalian 116622, China
Accepted 6 May 2005
Abstract
Some sufficient conditions for the global exponential stability of cellular neural networks with variable delay areobtained by means of a method based on delay differential inequality. The method, which does not make use of Lyapu-nov functionals, is simple and effective for the stability analysis of neural networks with delay. Some previously estab-lished results in the literature are shown to be special cases of the presented result.� 2005 Elsevier Ltd. All rights reserved.
1. Introduction
Cellular neural networks with delay (DCNNs) first introduced in [1] have found many important applications in mo-tion-related areas such as classification of patterns, processing of moving images and recognition of moving objects. Inthese applications, the dynamics of networks such as the existence, uniqueness and global asymptotic stability or globalexponential stability of the equilibrium point of the networks plays a key role. For this reason, stability of DCNNs hasattracted many researchers� attention. In the studies of stability for DCNNs, a general assumption is that the delay isfixed. However, in some applications, delay is time-varying even drastic, see [2–15]. The aim of this brief is to providesome new results on global exponential stability for the DCNNs by utilizing a delay differential inequality. Some resultsin the aforementioned studies emerge as special cases of the main result presented here. An example is given to show thevalidity of the results.
2. Preliminaries
The dynamic behavior of a continuous time cellular neural networks with variable delays can be described by thefollowing state equations:
0960-0doi:10.
* CoE-m
x0iðtÞ ¼ �cixiðtÞ þXnj¼1
aijfjðxjðtÞÞ þXnj¼1
bijfjðxjðt � sjðtÞÞÞ þ I i; i ¼ 1; 2; . . . ; n ð1Þ
779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.1016/j.chaos.2005.05.026
rresponding author.ail address: [email protected] (Q. Zhang).
332 Q. Zhang et al. / Chaos, Solitons and Fractals 28 (2006) 331–336
or equivalently
x0ðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt � sðtÞÞÞ þ I; ð2Þ
where x(t) = [x1(t), . . .,xn(t)]T 2 Rn, f(x(t)) = [f1(x1(t)), . . ., fn(xn(t))]
T 2 Rn, f(x(t � s(t))) = [f1(x1(t � s1(t))), . . .,
fn(xn(t � sn(t)))]T 2 Rn. A = (aij) is referred to as the feedback matrix, B = (bij) represents the delayed feedback matrix,
while I = [I1, . . ., In]T is a constant input vector and time delays sj are bounded nonnegative functions satisfying
0 6 sj(t) 6 s for all j = 1,2, . . .,n. The activation function fi, i = 1,2, . . .,n satisfy the following condition:(H) Each fi is bounded continuous and satisfies
fiðn1Þ � fiðn2Þj j 6 Li n1 � n2j j
for each n1,n2 2 R.This type of activation functions is clearly more general than both the usual sigmoid activation functions in Hopfield
networks and the piecewise linear function (PWL): fiðxÞ ¼ 12ðjxþ 1j � jx� 1jÞ in standard cellular networks [16].
In the following discussion, we will use the notations: for any matrix A = (aij)n·n 2 Rn·n, jAj denotes absolute-valuematrix given by jAj = (jaijj)n·n. E denotes the identity matrix with appropriate dimension.Assume that the system (1) is supplemented with initial conditions of the form
xiðsÞ ¼ /iðsÞ; s 2 ½�s; 0; i ¼ 1; 2; . . . ; n;
in which /i(s) is continuous for s 2 [�s, 0].Due to the boundedness of the activation function fi, by employing the well-known Brouwer�s fixed point theorem,
we can easily obtain that there exists an equilibrium point of Eq. (1). Besides, the uniqueness of the equilibrium pointcan be derived from the global exponential stability established below.Suppose that (1) has a unique equilibrium x� ¼ ðx�1; x�2; . . . ; x�nÞ. Denote
k/ � x�k ¼ sup�s6s60
Xni¼1
j/iðsÞ � x�i jp
" #1=p.
We say that an equilibrium point x� ¼ ðx�1; x�2; . . . ; x�nÞ is globally exponentially stable if there exist constants � > 0 andM P 1 such that
kxðtÞ � x�k 6 Mk/ � x�ke��t; t P 0.
Let y(t) = x(t) � x*, then Eq. (1) can be rewritten as
y0iðtÞ ¼ �ciyiðtÞ þXnj¼1
aijgjðyjðtÞÞ þXnj¼1
bijgjðyjðt � sjðtÞÞÞ; ð3Þ
where gjðyjÞ ¼ fjðyj þ x�j Þ � fjðx�j Þ; j ¼ 1; 2; . . . ; n. It is obvious that the function gj(Æ) also satisfies the hypothesis(H).To prove the stability of the equilibrium point x* of Eq. (1), it is sufficient to prove the stability of the trivial solution
of Eq. (3).
Definition 1 [17]. Let the n · n matrix A = (aij) have nonpositive off-diagonal elements and all principal minors of A
are positive, then A is said to be an M-matrix.
The following lemma will be used to study the global exponential convergence of (1).
Lemma 1 [18]. Let x(t) = (x1(t),x2(t), . . .,xn(t))T be a solution of the differential inequality (4)
x0ðtÞ 6 AxðtÞ þ B�xðtÞ; t P t0; ð4Þ
where
�xðtÞ ¼ supt�s6s6t
fx1ðsÞg; supt�s6s6t
fx2ðsÞg; . . . ; supt�s6s6t
fxnðsÞg� �T
A = (aij)n·n, B = (aij)n·n. If:
(H1) aij P 0 ði 6¼ jÞ; bij P 0; i; j ¼ 1; 2; . . . ; n;Pn
j¼1�xjðt0Þ > 0;(H2) The matrix �(A + B) is an M-matrix;
Q. Zhang et al. / Chaos, Solitons and Fractals 28 (2006) 331–336 333
then there always exist constants k > 0, ri > 0 (i = 1,2, . . ., n) such that
xiðtÞ 6 riXnj¼1
�xjðt0Þe�kðt�t0Þ. ð5Þ
3. Stability analysis
Theorem 1. If there exist real constants aij; a�ij; bij; b�ij; fij; f�ij; gij; g�ij ði; j ¼ 1; 2; . . . ; nÞ and positive constants p P 1
such that the matrix
Rij ¼ � �pcj þXni¼1
ðp � 1Þjajijpaji Lpbjii þ
Xni¼1
ðp � 1Þjbjijpfji Lpgjii
" #dij þ jaijjpa
�ij L
pb�ijj þ jbijjpf
�ij L
pg�ijj
( )n�n
is an M-matrix, where dij ¼1; i¼ j;0; i 6¼ j;
ðp�1Þaijþa�ij ¼ 1; ðp�1Þbijþb�
ij ¼ 1; ðp�1Þfijþ f�ij ¼ 1; ðp�1Þgijþg�ij ¼ 1, then
the equilibrium point x*of system (1) is globally exponentially stable.
Proof. Let ziðtÞ ¼ 1p jyiðtÞj
p, then the upper right derivative D+zi(t) along the solution of (3) as follows:
DþziðtÞ ¼ jyiðtÞjp�1DþjyiðtÞj 6 jyiðtÞj
p�1 �cijyiðtÞj þXnj¼1
jaijjLjjyjðtÞj þXnj¼1
jbijjLjjyjðt � sjðtÞÞj( )
6 jyiðtÞjp�1 �cijyiðtÞj þ
Xnj¼1
jaijjLjjyjðtÞj þXnj¼1
jbijjLjj�yjðtÞj( )
¼ �cijyiðtÞjp þ
Xnj¼1
jaijjLjjyiðtÞjp�1jyjðtÞj þ
Xnj¼1
jbijjLjjyiðtÞjp�1j�yjðtÞj
¼ �cijyiðtÞjp þ
Xnj¼1
jaijjpaij Lpbijj jyiðtÞj
p �p�1
p � jaijjpa�ij L
pb�ijj jyjðtÞj
p �1
p þXnj¼1
jbijjpfij Lpgijj jyiðtÞj
p �p�1
p
� jbijjpf�ij L
pg�ijj j�yjðtÞj
p �1
p. ð6Þ
Recall that the inequality apbq6 pa + qb holds for any a > 0, b > 0 and 0 6 p 6 1 with p + q = 1, see, for instance,
[19]. Let a ¼ ðjaijjpaij Lpbijj jyiðtÞj
pÞp�1p , b ¼ ðjaijjpa
�ij L
pb�ijj jyjðtÞj
pÞ1p, then we have
jaijjpaij Lpbijj jyiðtÞj
p �p�1
p jaijjpa�ij L
pb�ijj jyjðtÞj
p �1
p6
p � 1p
jaijjpaij Lpbijj jyiðtÞj
p þ 1pjaijjpa
�ij L
pb�ijj jyjðtÞj
p. ð7Þ
Similarly, let a ¼ ðjbijjpfij Lpgijj jyiðtÞj
pÞp�1p , b ¼ ðjbijjpf
�ij L
pg�ijj j�yjðtÞj
pÞ1p, then we can get
jbijjpfij Lpgijj jyiðtÞj
p �p�1
p jbijjpf�ij L
pg�ijj j�yjðtÞj
p �1
p6
p � 1p
jbijjpfij Lpgijj jyiðtÞj
p þ 1pjbijjpf
�ij L
pg�ijj j�yjðtÞj
p. ð8Þ
Substituting the inequalities (7) and (8) into (6), we obtain
DþziðtÞ 6 �pciziðtÞ þXnj¼1
ðp � 1Þjaijjpaij Lpbijj ziðtÞ þ
Xnj¼1
jaijjpa�ij L
pb�ijj zjðtÞ þ
Xnj¼1
ðp � 1Þjbijjpfij Lpgijj ziðtÞ
þXnj¼1
jbijjpf�ij L
pg�ijj �zjðtÞ
¼ �pci þXnj¼1
ðp � 1Þjaijjpaij Lpbijj þ
Xnj¼1
ðp � 1Þjbijjpfij Lpgijj
( )ziðtÞ þ
Xnj¼1
jaijjpa�ij L
pb�ijj zjðtÞ þ
Xnj¼1
jbijjpf�ij L
pg�ijj �zjðtÞ
¼Xnj¼1
�pcj þXni¼1
ðp � 1Þjajijpaji Lpbjii þ
Xni¼1
ðp � 1Þjbjijpfji Lpgjii
" #dij þ jaijjpa
�ij L
pb�ijj
( )zjðtÞ þ
Xnj¼1
jbijjpf�ij L
pg�ijj �zjðtÞ.
ð9Þ
334 Q. Zhang et al. / Chaos, Solitons and Fractals 28 (2006) 331–336
Let R1 ¼ f½�pcj þPn
i¼1ðp � 1Þjajijpaji L
pbjii þ
Pni¼1ðp � 1Þjbjij
pfji Lpgjii dij þ jaijjpa
�ij L
pb�ijj g, R2 ¼ fjbijjpf
�ij L
pg�ijj g, then the
above inequality can be written as
DþzðtÞ 6 R1zðtÞ þ R2�zðtÞ.
According to Lemma 1, if the matrix R = �(R1 + R2) is an M-matrix, then there must exist constants k > 0, ri > 0(i = 1,2, . . .,n) such that
ziðtÞ ¼1
pjxiðtÞ � x�i j
p6 ri
Xnj¼1
�zjðt0Þe�kðt�t0Þ ¼ riXnj¼1
1
pj�xjðt0Þ � x�i j
pe�kðt�t0Þ;
that is, " #
jxiðtÞ � x�i j 6 r1=piXnj¼1
j�xjðt0Þ � x�i jp
1=p
e�kðt�t0Þ=p.
This implies that the unique equilibrium point of Eq. (1) is globally exponentially stable. h
By using Theorem 1 above, we can easily obtain the following corollaries.
Corollary 1. If the matrix R = C � (jAj + jBj)L is an M-matrix, where L = diag(L1,L2, . . .,Ln), then the equilibrium point
x*of system (1) is globally exponentially stable.
Proof. Let p = 1 in Theorem 1, then we can easily get Corollary 1. h
Remark 1. Due to R being an M-matrix, applying property of M-matrix, we can get CL�1 � (jAj + jBj) is an M-matrix, which is the result of [9, Theorem 3].
Remark 2. For the case C = E, from E � (jAj + jBj)L is an M-matrix, we can conclude that the spectral radiusq((jAj + jBj)L) < 1, which is the result of [10, Theorem 1].
Recall that for a given matrix A, its spectral radius q(A) equals to the infimum of its all matrix norms of A, i.e., forany matrix norm k Æk, q(A) 6 kAk, so we can derive some more restrictive but practical conditions for globally expo-nentially stable.
Corollary 2. For any matrix norm k Æk, if k(jAj + jBj)Lk < 1, then the equilibrium point x* of system (1) is globally
exponentially stable.
Taking p = 2, aij ¼ a�ij ¼ bij ¼ b�
ij ¼ fij ¼ f�ij ¼ gij ¼ g�ij ¼ 1
2in Theorem 1, then we have
Corollary 3. If the matrix
R ¼ � �2cj þXni¼1
Li jajij þ jbjij� " #
dij � Lj jaijj þ jbijj� ( )
n�n
is an M-matrix, then the equilibrium point x* of system (1) is globally exponentially stable.
Remark 3. In [12–14], some results on the global asymptotic stability of Eq. (1) are presented by constructing Lyapu-nov functional. Different from our results, all of their results require that the delay function s(t) be differentiable. Thus,compared with the results presented here, their conditions are more restrictive and conservative.
4. An example
In this section, we will give an example to show the conditions given in the brief are milder than those given in someearlier literature. First, we restate the results in [15].
Theorem 2. The equilibrium point of Eq. (1) is globally exponentially stable if
min16i6n
ci � Li
Xnj¼1
jajij !
> max16i6n
Li
Xnj¼1
jbjij !
.
Q. Zhang et al. / Chaos, Solitons and Fractals 28 (2006) 331–336 335
Theorem 3. The equilibrium point of Eq. (1) is globally exponentially stable if
min16i6n
2ci �Xnj¼1
ðLjðjaijj þ jbijjÞ þ LijajijÞ !
> max16i6n
Li
Xnj¼1
jbjij !
.
Example 1. Consider cellular neural networks with variable delays
x01ðtÞ ¼ �c1x1ðtÞ þ a11f ðx1ðtÞÞ þ a12f ðx2ðtÞÞ þ b11f ðx1ðt � s1ðtÞÞÞ þ b12f ðx2ðt � s2ðtÞÞÞ þ I1;
x02ðtÞ ¼ �c2x2ðtÞ þ a21f ðx1ðtÞÞ þ a22f ðx2ðtÞÞ þ b21f ðx1ðt � s1ðtÞÞÞ þ b22f ðx2ðt � s2ðtÞÞÞ þ I2;ð10Þ
where the activation function is described by PWL function: fiðxÞ ¼ 12ðjxþ 1j � jx� 1jÞ. Obviously, this function satis-
fies (H) with L1 = L2 = 1.
In (10), taking a11 = 0.5, a12 = �0.1, a21 = 0.3, a22 = �0.2; b11 = �0.1, b12 = 0.1, b21 = �0.1, b22 = 0.4; c1 = c2 = 1;
s1(t) = s2(t) = jt + 1j � jt � 1j, i.e., C ¼ 1 00 1
� �, A ¼ 0.5 �0.1
0.3 �0.2
� �, B ¼ �0.1 0.1
�0.1 0.4
� �. We can easily check that the
matrix in Corollary 3
R ¼0.6 �0.2�0.4 0.4
� �
is an M-matrix. Hence, the equilibrium point of Eq. (10) is globally exponentially stable.On the other hand, by a simple computation, it is easy to verify that
min16i6n
ci � Li
Xnj¼1
jajij !
¼ 0.2 < max16i6n
Li
Xnj¼1
jbjij !
¼ 0.5
and
min16i6n
2ci �Xnj¼1
ðLjðjaijj þ jbijjÞ þ LijajijÞ !
¼ 0.4 < max16i6n
Li
Xnj¼1
jbjij !
¼ 0.5.
This implies that the main results in [15] do not hold for this example. Since the delay function is not differentiable,the results in [12–14] cannot be applied to this example.
5. Conclusions
Some criteria have been given ensuring the global exponential stability of cellular neural networks with variable de-lays by using an approach based on delay differential inequality. The results established here extend those earlier givenin the literature. Compared with the method of Lyapunov functionals as in most previous studies, our method is simplerand more effective for stability analysis.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No: 60403001) and ChinaPostdoctoral Foundation.
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