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Stability of periodic solution in fuzzy BAM neural networks with finite distributed delays
ARTICLE IN PRESS Neurocomputing 71 (2008) 3064– 3069
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Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Stability of periodic solution in fuzzy BAM neural networks with finite distributed delays Tingwen Huang a,, Yu Huang b, Chuandong Li c a b c
Texas A&M University at Qatar, P.O. Box 5825, Doha, Qatar Department of Mathematics, Zhongshan University, Guangzhou 510275, China Department of Computer Science and Engineering, Chongqing University, Chongqing 400030, PR China
a r t i c l e in f o
a b s t r a c t
Available online 20 June 2008
In this paper, stability is considered for a class of fuzzy cellular neural networks with distributed delay (FCNNDD). The condition for feedback kernel is minimal: the only condition required is that Rt 0 kðsÞ ds ¼ 1. Significant for applications of neural networks, the stability of FCNNDD is discussed when t is finite and infinite. Exponential stability is obtained with distributed delay in finite time, while asymptotic stability is obtained with distributed delay in infinite time. & 2008 Elsevier B.V. All rights reserved.
Keywords: Exponential stability Fuzzy cellular neural networks Distributed delay Equilibrium point
In this paper, we investigate the exponential periodicity for the following fuzzy BAM neural networks:
1. Introduction After Kosko [11,12] proposed bidirectional associative memory neural networks: n X dxi ðtÞ ¼ xi ðtÞ þ aij f j ðyj ðtÞÞ þ Ii ðtÞ; dt j¼1
i ¼ 1; . . . ; m;
m X dyj ðtÞ ¼ yj ðtÞ þ bji f nþi ðxi ðtÞÞ þ Imþj ðtÞ; dt i¼1
n Z t n ^ ^ dxi ðtÞ aji ðsÞf j ðyj ðt sÞÞ ds þ T ji mj ðtÞ ¼ di xi ðtÞ þ dt 0 j¼1 j¼1 n Z t n _ _ þ bji ðsÞf j ðyj ðt sÞÞ ds þ Hji mj ðtÞ þ Ii ðtÞ; i ¼ 1; . . . ; m; j¼1
0
j¼1
j ¼ 1; . . . ; n (1)
many researchers [1–6,14] have studied the dynamics of BAM neural networks with or without delays, including stability and periodic solutions. In this paper, we would like to integrate fuzzy operations into BAM neural networks and maintain local connectedness among cells. Speaking of fuzzy operations, Yang et al. [16–19] first combined those operations with cellular neural networks and investigated the stability of fuzzy cellular neural networks (FCNNs). So far researchers have found that FCNNs are useful in image processing, and some results have been reported on stability and periodicity of FCNNs [8–10,20]. However, to the best of our knowledge, few results reported on periodicity of FCNNs or fuzzy BAM neural networks with finite distributed delays. It is believed that the conclusions made from finite distributed delays could provide us insight into their counterparts with unbounded delays. Corresponding author.
E-mail addresses: [email protected] (T. Huang), [email protected] (Y. Huang), [email protected] (C. Li). 0925-2312/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2008.04.021
m Z t ^ dyj ðtÞ ¼ dj yj ðtÞ þ zij ðsÞf nþi ðxi ðt sÞÞ ds dt i¼1 0 m m Z t ^ _ þ T ij mnþi ðtÞ þ Zij ðsÞf nþi ðxi ðt sÞÞ ds
þ
i¼1 m _
i¼1
0
Hij mnþi ðtÞ þ Imþj ðtÞ;
j ¼ 1; . . . ; n,
(2)
i¼1
where ci 40, dj 40, for i ¼ 1; . . . ; m, j ¼ 1; 2; . . . ; n; aij ðsÞ and zij ðsÞ, bij ðsÞ and Zij ðsÞ which denote elements of fuzzy feedback MIN templates, fuzzy feedback MAX templates, are continuous in the interval ½0; t, where t is a constant; T ij and Hij are fuzzy feed forward MIN template and fuzzy feed-forward V W MAX template, respectively; and denote the fuzzy AND and fuzzy OR operation, respectively; xi ; yj are activations of the ith neuron and the jth neuron, respectively; for k ¼ 1; . . . ; m þ n, mk , Ik , denote inputs, bias of the ith neuron, respectively; function f k are activation functions; moreover, functions mk ðtÞ, Ik ðtÞ are continuously periodic functions with period o, i.e., mk ðtÞ ¼ mk ðt þ oÞ, Ik ðtÞ ¼ Ik ðt þ oÞ; t 2 R, for k ¼ 1; . . . ; m þ n.
ARTICLE IN PRESS T. Huang et al. / Neurocomputing 71 (2008) 3064–3069
The initial conditions associated with (2) are the following: tpsp0; i ¼ 1; . . . ; m;
xi ðsÞ ¼ ji ðsÞ;
tpsp0; j ¼ 1; 2; . . . ; n.
yj ðsÞ ¼ jmþj ðsÞ;
(3)
In this paper, we assume that H: f i is a bounded function defined on R and satisfies Lipschitz condition jf i ðxÞ f i ðyÞjpli jx yj;
i ¼ 1; . . . ; m þ n,
(4)
for any x; y 2 R. Let us define that for any o 2 Rmþn, kok ¼ max1pkpmþn jok j. Let C t ¼ Cð½t; 0; Rmþn Þ be the Banach space of all continuous functions mapping from ½t; 0 to Rmþn with norm defined as follows: for each j 2 C t , kjkt ¼ suptptp0 max1pipmþn jji ðtÞj. Now we define the exponentially periodic solution of fuzzy BAM neural networks. To be convenient, for given j 2 C t ; let zðt; jÞ ¼ ðx1 ðt; jÞ; . . . ; xmþn ðt; jÞÞ¼ ðx1 ðt; jÞ; . . . ; xm ðt; jÞ; y1 ðt; jÞ; . . . ; yn ðt; jÞÞT 2 Rmþn represent the solution of system (2) with initial condition: zðtÞ ¼ jðtÞ when tptp0. Definition 1. The solution zðt; j Þ is exponentially periodic if zðt; j Þ is periodic and there exist positive constants M, a such that any solution zðt; fÞ satisfies kzðt; fÞ zðt; j ÞkpMkf j kt eat ;
tX0.
(5)
Here, we present a lemma which we will use in the proof of the main theorem. Lemma 1 (Le et al. [14]). For any aij 2 R, xj ; yj 2 R, i; j ¼ 1; . . . ; n, we have the following estimations: ^ n ^ X n aij xj a y ðjaij j jxj yj jÞ (6) ij j p j¼1 1pjpn j¼1 and _ n _ X n aij xj aij yj p ðjaij j jxj yj jÞ. j¼1 1pjpn j¼1
(7)
constants M and e which depend on li , i ¼ 1; . . . ; m þ n and known coefficients of (2) such that jjzðt; fÞ zðt; jÞjjpMjjf jjjt eet ;
In this section, sufficient conditions to guarantee exponentially periodic solution of fuzzy BAM neural networks are obtained. First, we would like to present the main result. Theorem 1. Fuzzy BAM neural networks (2) is exponentially periodic if there exist l1 ; . . . ; lmþn such that the following two inequalities hold: 0
1 Z t Z t n n X X @ li 2di lj ðjaji ðsÞj þ jbji ðsÞjÞ dsA4lnþi lmþj ðjzij ðsÞj þ jZij ðsÞjÞ ds,
lmþj 2cj
i¼1
0
li
Z t 0
(9)
j¼1 n _
þ
j¼1
0
Hji mj ðtÞ þ Ii ðtÞ;
i ¼ 1; . . . ; m;
j¼1 n Z t ^ dxi ðt; fÞ ¼ di xi ðt; fÞ þ aji ðsÞf j ðyj ðt s; fÞÞ ds dt j¼1 0 n n Z t ^ _ þ T ji mj ðtÞ bji ðsÞf j ðyj ðt s; fÞÞ ds j¼1
þ
n _
j¼1
0
Hji mj ðtÞ þ Ii ðtÞ;
i ¼ 1; . . . ; m.
(10)
j¼1
Thus, for i ¼ 1; . . . ; m; we have dxi ðt; jÞ dxi ðt; fÞ dt dt ¼ di ðxi ðt; jÞ xi ðt; fÞÞ n Z t ^ aji ðsÞðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds þ þ
j¼1
0
j¼1
0
n Z t _
bji ðsÞðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds.
(11)
Similarly, for j ¼ 1; . . . ; n; we have dyj ðt; jÞ dyj ðt; fÞ dt dt ¼ dj ðyj ðt; jÞ yj ðt; fÞÞ m Z t ^ zij ðsÞðf nþi ðxi ðt s; jÞÞ f nþi ðxi ðt s; fÞÞÞ ds þ þ
i¼1 0 m Z t _ i¼1
0
Zij ðsÞðf nþi ðxi ðt s; jÞÞ f nþi ðxi ðt s; fÞÞÞ ds. (12)
By condition (8), there exists a positive number e such that 0 1 Z t n X @ li 2di e lj ðjaji ðsÞj þ jbji ðsÞjÞ dsA
2. Main results
m X
tX0:
Since zðt; fÞ, zðt; jÞ are the solutions to system (2), we have n Z t ^ dxi ðt; jÞ ¼ di xi ðt; jÞ þ aji ðsÞf j ðyj ðt s; jÞÞ ds dt j¼1 0 n n Z t ^ _ þ T ji mj ðtÞ bji ðsÞf j ðyj ðt s; jÞÞ ds
The rest of the paper is organized as follows. Section 2 presents some sufficient conditions for exponentially periodic solution to fuzzy BAM neural networks. Conclusion is drawn in Section 3.
j¼1
3065
! ðjzij ðsÞj þ jZij ðsÞjÞ ds 4lj
0
j¼1 m X i¼1
li
Z t 0
ðjaji ðsÞj þ jbji ðsÞjÞ ds,
(8)
0
j¼1
4lnþi
n X
lmþj
j¼1
lmþj 4lj
Z t 0
ees ðjzij ðsÞj þ jZij ðsÞjÞ ds,
! Z t m X 2cj e li ðjzij ðsÞj þ jZij ðsÞjÞ ds m X i¼1
li
Z t 0
i¼1
0
ees ðjaji ðsÞj þ jbji ðsÞjÞ ds.
Now let us define the Lyapunov functional Vðzðt; jÞ; zðt; fÞÞ ¼ V 1 ðzðt; jÞ; zðt; fÞÞ þ V 2 ðzðt; jÞ; zðt; fÞÞ, where 8 < m X li ðxi ðt; jÞ xi ðt; fÞÞ2 eet V 1 ðzðt; jÞ; zðt; fÞÞ ¼ : i¼1
where i ¼ 1; . . . ; m, j ¼ 1; 2; . . . ; n. Proof. First let us prove the following claim. Claim. Let zðt; jÞ ¼ ðx1 ðt; jÞ; . . . ; xm ðt; jÞ; y1 ðt; jÞ; . . . ; yn ðt; jÞÞT ; zðt; fÞ ¼ ðx1 ðt; fÞ; . . . ; xm ðt; fÞ; y1 ðt; fÞ; . . . ; yn ðt; fÞÞT be the solutions to system (2). Under the conditions of this theorem, there are positive
(13)
þ
Z t n X lj ðjaji ðsÞj þ jbji ðsÞjÞ j¼1
0
9 = ðyj ðr; jÞ yj ðr; fÞÞ2 eeðrþsÞ dr ds ; ts Z
t
ARTICLE IN PRESS 3066
T. Huang et al. / Neurocomputing 71 (2008) 3064–3069
V 2 ðzðt; jÞ; zðt; fÞÞ ¼
n X
(
þ
lmþj ðyj ðt; jÞ yj ðt; fÞÞ2 eet
0
)
t ts
ðxi ðr; jÞ xi ðr; fÞÞ
2
eeðrþsÞ dr ds
lj jbji ðsÞjððxi ðt; jÞ xi ðt; fÞÞ2
þ ðyj ðt s; jÞ yj ðt s; fÞÞ2 Þ ds Z t n X þ lj ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞ2 ees
Z t m X li ðjzij ðsÞj þ jZij ðsÞjÞ þ i¼1
0
j¼1
j¼1
Z
n Z t X
0
j¼1
. 2
(14) 2
Take the derivative of V 1 , using Lemma 1 and inequality 2abpa2 þ b in the following process, we have
ðyj ðt s; jÞ yj ðt s; fÞÞ ds
9 =
; 82 3 Z t m n < X X e t ¼e li 4e 2di þ lj ðjaji ðsÞj þ jbji ðsÞjÞ ds5 : 0 i¼1
j¼1
2
ðxi ðt; jÞ xi ðt; fÞÞ d V 1 ðzðt; jÞ; zðt; fÞÞ dt 8 < m X li e eet ðxi ðt; jÞ xi ðt; fÞÞ2 ¼ :
þ
n X
2
lj ðyj ðt; jÞ yj ðt; fÞÞ
Z t 0
j¼1
9 = ðjaji ðsÞj þ jbji ðsÞjÞ ees ds . ; (15)
i¼1
þ 2eet ðxi ðt; jÞ xi ðt; fÞÞðx_ i ðt; jÞ x_ i ðt; fÞÞ Z t n X lj eet ðjaji ðsÞj þ jbji ðsÞjÞ½ees ðyj ðt; jÞ yj ðt; fÞÞ2 þ 0
j¼1
2
ðyj ðt s; jÞ yj ðt s; fÞÞ ds
¼ eet
m X
8 <
9 = ;
Similarly, we can obtain that (" n m X X d e t V 2 ðzðt; jÞ; zðt; fÞÞpe lmþj e 2cj þ lnþi dt j¼1 i¼1 # Z t ðjzij ðsÞj þ jZij ðsÞjÞ ds ðyi ðt; jÞ yi ðt; fÞÞ2 0
li eðxi ðt; jÞ xi ðt; fÞÞ2 2di ðxi ðt; jÞ xi ðt; fÞÞ2
þ
:
i¼1
þ 2ðxi ðt; jÞ xi ðt; fÞÞ
n Z t ^ j¼1
0
i¼1
aji ðsÞ
0
0
9 = ðyj ðt s; jÞ yj ðt s; fÞÞ ds ; 8 m < X peet li ðe 2di Þðxi ðt; jÞ xi ðt; fÞÞ2 : 2
By inequalities (15), (16) and (13), we have d Vðzðt; jÞ; zðt; fÞÞ dt d d ¼ V 1 ðzðt; jÞ; zðt; fÞÞ þ V 2 ðzðt; jÞ; zðt; fÞÞ dt dt 82 m n < X X li 4e 2di þ lj peet : i¼1 j¼1 Z t ðjaji ðsÞj þ jbji ðsÞjÞ ds ðxi ðt; jÞ xi ðt; fÞÞ2 0
þ
n Z t X j¼1
0
lj jaji ðsÞj
jyj ðt s; jÞ yj ðt s; fÞj ds n Z t X lj jbji ðsÞj þ 2jxi ðt; jÞ xi ðt; fÞj j¼1
0
9 = 2 ðyj ðt s; jÞ yj ðt s; fÞÞ ds ; 8 m < X peet li ðe 2di Þðxi ðt; jÞ xi ðt; fÞÞ2 : i¼1
n Z t X j¼1
0
2
lj jaji ðsÞjððxi ðt; jÞ xi ðt; fÞÞ
þ ðyj ðt s; jÞ yj ðt s; fÞÞ Þ ds
lj ðyj ðt; jÞ yj ðt; fÞÞ n X
0
("
e 2cj þ
lmþj
Z t
ðjaji ðsÞj þ jbji
m X
9 =
ðsÞjÞ ees ds
;
lnþi
i¼1
# Z t ðjzij ðsÞj þ jZij ðsÞjÞ ds ðyi ðt; jÞ yi ðt; fÞÞ2 0
m X
2
lnþi ðxi ðt; jÞ xi ðt; fÞÞ
i¼1
¼ eet
m X
li 4e 2di þ
þlnþi
lmþj
j¼1
þ eet
n X
Z t 0
ees ðjz
ðjzij ðsÞj þ jZij
0
3
ij ðsÞj
"
lmþj e 2cj þ
j¼1
) ðsÞjÞ ees ds
Z t n X lj ðjaji ðsÞj þ jbji ðsÞjÞ ds j¼1
n X
Z t 0
2
i¼1
þ jZij ðsÞjÞ ds5ðxi ðt; jÞ xi ðt; fÞÞ2
m X i¼1
lnþi
Z t 0
ðjzij ðsÞj þ jZij ðsÞjÞ ds
3 Z t n X es þ lj ðjaji ðsÞj þ jbji ðsÞjÞ e ds5ðyi ðt; jÞ yi ðt; fÞÞ2 j¼1
2
2
j¼1
0
j¼1
þ
þ eet
þ
jyj ðt s; jÞ yj ðt s; fÞj ds Z t n X lj ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞ2 ees þ
n X j¼1
i¼1
þ 2jxi ðt; jÞ xi ðt; fÞj
(16)
0
ðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds Z t n X lj ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞ2 ees þ j¼1
lnþi ðxi ðt; jÞ xi ðt; fÞÞ2
) Z t ðjzij ðsÞj þ jZij ðsÞjÞ ees ds .
ðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds n Z t _ bji ðsÞ þ 2ðxi ðt; jÞ xi ðt; fÞÞ j¼1
m X
p0.
0
(17)
ARTICLE IN PRESS T. Huang et al. / Neurocomputing 71 (2008) 3064–3069
Thus, Vðzðt; jÞ; zðt; fÞÞpVðzð0; jÞ; zð0; fÞÞ. For k ¼ 1; . . . ; m þ n, we have eet jzk ðt; jÞ zk ðt; fÞj2 ( m X eet p l ðx ðt; jÞ xi ðt; fÞÞ2 min1plpmþn fll g i¼1 i i 9 n = X 2 lmþj ðyj ðt; jÞ yj ðt; fÞÞ þ ;
Theorem 2. Fuzzy BAM neural networks (1) is exponentially periodic if there exist l1 ; . . . ; lmþn such that the following two inequalities hold:
j¼1
0
pV 1 ðzð0; jÞ; zð0; fÞÞ þ V 2 ðzð0; jÞ; zð0; fÞÞ 8 Z t < m n X X ¼ li ðxi ð0; jÞ xi ð0; fÞÞ2 þ lj ðjaji ðsÞj þ jbji ðsÞjÞ : 0 i¼1 j¼1 ) Z 2
ðyj ðr; jÞ yj ðr; fÞÞ
to zðt; j Þ as t ! 1. So far, we have completed the proof of the theorem. & In the following, we will give a generalized result which includes Theorem 1 as a special case.
pV 1 ðzðt; jÞ; zðt; fÞÞ þ V 2 ðzðt; jÞ; zðt; fÞÞ
0
3067
1 Z t Z t n n X X p lj ðjaji ðsÞj þ jbji ðsÞjÞ dsA4 lnþi lmþj ðjzij ðsÞj þ jZij ðsÞjÞ ds, q 0 0 j¼1 j¼1 ! Z t Z t m m X p X pcj li ðjzij ðsÞj þ jZij ðsÞjÞ ds 4 lj li ðjaji ðsÞj þ jbji ðsÞjÞ ds, q i¼1 0 0 i¼1
li @pdi lmþj
eeðrþsÞ dr ds
(21)
s
þ
n X
lmþj fðyj ð0; jÞ yj ð0; fÞÞ2 þ
j¼1
Z
Z t m X li ðjzij ðsÞj þ jZij ðsÞjÞ i¼1
where i ¼ 1; . . . ; m, j ¼ 1; 2; . . . ; n, and ð1=pÞ þ ð1=qÞ ¼ 1, p; q are positives.
0
0
ðxi ðr; jÞ ðxi ðr; fÞÞ2 eeðrþsÞ dr dsg 8 Z t < m n X X ¼ li ðji ð0Þ fi ð0ÞÞ2 þ lj ðjaji ðsÞj þ jbji ðsÞjÞ : 0 i¼1 j¼1 ) Z
The proof is similar to that of Theorem 1. We will not present the details. Consider the Lyapunov functional
s
0
s
þ
where
ðjmþj ðrÞ fmþj ðrÞÞ2 eeðrþsÞ dr ds
n X
(
lmþj
Z t m X ðjmþj ð0Þ fmþj ð0ÞÞ2 þ li ðjzij ðsÞj þ jZij ðsÞjÞ
j¼1
Z
Vðzðt; jÞ; zðt; fÞÞ ¼ V 1 ðzðt; jÞ; zðt; fÞÞ þ V 2 ðzðt; jÞ; zðt; fÞÞ,
)
0
s
2
ðji ðrÞ fi ðrÞÞ
i¼1
i¼1
þ
eeðrþsÞ dr ds
8 9 2 Z 0 Z t m n < = X X li 1 þ lj ðjaji ðsÞj þ jbji ðsÞjÞ eeðrþsÞ dr ds pjjj fjj2t 4 : ; 0 s i¼1 j¼1 3 ( ) Z t Z 0 n m X X lmþj 1 þ li ðjzij ðsÞj þ jZij ðsÞjÞ eeðrþsÞ dr ds 5. þ j¼1
V 1 ðzðt; jÞ; zðt; fÞÞ ¼
0
i¼1
0
s
8 < li ðxi ðt; jÞ xi ðt; fÞÞp eet :
n pX l q j¼1 j
Z
V 2 ðzðt; jÞ; zðt; fÞÞ ¼
Z t 0
t ts
n X
ðjaji ðsÞj þ jbji ðsÞjÞ q
ðyj ðr; jÞ yj ðr; fÞÞ
9 = , ;
eeðrþsÞ dr ds
(
lmþj ðyj ðt; jÞ yj ðt; fÞÞp eet
j¼1
(18)
Z t m pX li ðjzij ðsÞj þ jZij ðsÞjÞ q i¼1 0 ) Z t ðxi ðr; jÞ ðxi ðr; fÞÞq eeðrþsÞ dr ds .
From the above inequalities, we have
þ
jzk ðt; fÞ zk ðt; j ÞjpMjjf j jjt eðe=2Þt , tX0;
k ¼ 1; . . . ; m þ n, (19) R 0 eðrþsÞ Rt Pn Pm dr dsgþ where M ¼ i¼1 li f1 þ j¼1 lj 0 ðjaji ðsÞj þ jbji ðsÞjÞ s e R 0 eðrþsÞ Pn Pm R t dr dsg. So far, we j¼1 lmþj f1 þ i¼1 li 0 ðjzij ðsÞj þ jZij ðsÞjÞ s e have completed the proof of the Claim. By the Claim, to complete this theorem, we just need to prove that there exists a o-periodic solution for model (2). Let zt ðjÞðWÞ ¼ zðt; jÞðWÞ ¼ zðt þ W; jÞ, for W 2 ½t; 0 and zðt þ W; jÞ is a solution of model (2) with initial solution of (3). Now, we define a Poincare mapping P : C ! C by P j ¼ zo ðjÞ. By Claim, there are positives M and e, jjzt ðfÞ zt ðj Þjjt pMjjf j jjt eðe=2ÞðttÞ ;
m X
tX0.
(20)
Choose a positive big integer l, such that Mee=2ðoltÞ p13. Thus, we have jjP l f P l jjjt p13jjf jjjt . It implies that Pl is a contraction mapping, so there exists a unique fixed point j 2 C such that Pl j ¼ j . Note that Pl ðP j Þ ¼ PðP l j Þ ¼ P j , so P j is a fixed point of mapping P l . By the uniqueness of the fixed point of the contraction map, P j ¼ j , i.e., j is a fixed point of P. Thus, ztþo ðj Þ ¼ zt ðzo ðj ÞÞ ¼ zt ðj Þ, i.e., zðt þ o; j Þ ¼ zðt; j Þ, for any t40. This proves that there exists one o-periodic solution, and from the Claim, it is clear that all solutions converge exponentially
ts
(22) Take the derivative of V 1 , using abpð1=pÞap þ ð1=qÞb following process, we have:
q
in the
d V 1 ðzðt; jÞ; zðt; fÞÞ dt 8 < m X ¼ li eeet ðxi ðt; jÞ xi ðt; fÞÞp : i¼1
þ peet ðxi ðt; jÞ xi ðt; fÞÞp1 ðx_ i ðt; jÞ x_ i ðt; fÞÞ Z t n pX þ l eet ðjaji ðsÞj þ jbji ðsÞjÞ½ees ðyj ðt; jÞ yj ðt; fÞÞq q j¼1 j 0 9 = q ðyj ðt s; jÞ yj ðt s; fÞÞ ds ; 8 < m X ¼ ee t li eðxi ðt; jÞ xi ðt; fÞÞp pdi ðxi ðt; jÞ xi ðt; fÞÞp : i¼1
þ pðxi ðt; jÞ xi ðt; fÞÞ
n Z t ^ j¼1
0
aji ðsÞðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds
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þ pðxi ðt; jÞ xi ðt; fÞÞ
n Z t _ j¼1
þ
n pX l q j¼1 j
Z t 0
0
bji ðsÞðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds
ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞq ees
ðyj ðt s; jÞ yj ðt s; fÞÞq ds
9 =
; 8 m < X peet li ðe pdi Þðxi ðt; jÞ xi ðt; fÞÞp :
(25) where i ¼ 1; . . . ; m, j ¼ 1; 2; . . . ; n, and 1=p þ 1=q ¼ 1, p; q are positives.
i¼1
þ pjxi ðt; jÞ xi ðt; fÞj
n Z t X j¼1
þ pjxi ðt; jÞ xi ðt; fÞj
n Z t X j¼1
n pX l þ q j¼1 j
Z t 0
0
0
Corollary 1. Fuzzy BAM neural networks (1) is exponentially periodic if the following two inequalities hold: Z t n n Z t X X p pdi lj ðjaji ðsÞj þ jbji ðsÞjÞ ds4 lnþi ðjzij ðsÞj þ jZij ðsÞjÞ ds, q 0 j¼1 j¼1 0 Z t m m Z t X p X pcj li ðjzij ðsÞj þ jZij ðsÞjÞ ds4 lj ðjaji ðsÞj þ jbji ðsÞjÞ ds, q i¼1 0 0 i¼1
lj jaji ðsÞjjyj ðt s; jÞ yj ðt s; fÞj ds
3. Conclusion lj jbji ðsÞjjyj ðt s; jÞ yj ðt s; fÞj ds
ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞ q
ðyj ðt s; jÞ yj ðt s; fÞÞ ds
q
In this paper, we discuss the existence and exponential stability of the equilibrium of the periodic solution of Fuzzy BAM neural networks with finite distributed delays. Sufficient condition set up here by using Lyapunov functional is easily verifiable.
ee s
9 =
; 8 m < X peet li ðe pdi Þðxi ðt; jÞ xi ðt; fÞÞp :
Acknowledgments
i¼1
þ
n Z t X j¼1
0
The first author is grateful for the support of Texas A and M University at Qatar. This project is partially support by NSFC (10771222), NSFC (60574024), NCET-06-0764 and NSF of Guangdong. Part of the above work is presented in ICONIP 2006.
lj jaji ðsÞjððxi ðt; jÞ xi ðt; fÞÞp
p þ ðyj ðt s; jÞ yj ðt s; fÞÞq Þ ds q n Z t X þ lj jbji ðsÞjððxi ðt; jÞ xi ðt; fÞÞp j¼1
0
p þ ðyj ðt s; jÞ yj ðt s; fÞÞq Þ ds q Z t n pX l ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞq ees þ q j¼1 j 0 9 = q ðyj ðt s; jÞ yj ðt s; fÞÞ ds ; 82 3 Z t < m n X X ¼ ee t li 4e pdi þ lj ðjaji ðsÞj þ jbji ðsÞjÞ ds5ðxi ðt; jÞ xi ðt; fÞÞp : 0 i¼1 j¼1 9 Z n = t pX q e s þ lj ðyj ðt; jÞ yj ðt; fÞÞ ðjaji ðsÞj þ jbji ðsÞjÞe ds . ; q 0 j¼1
(23)
Similarly, we can obtain that
d V 2 ðzðt; jÞ; zðt; fÞÞ dt (" n m X X e t pe lmþj e pcj þ lnþi j¼1
i¼1
# Z t ðjzij ðsÞj þ jZij ðsÞjÞ ds ðyi ðt; jÞ yi ðt; fÞÞq 0
þ
m pX l ðx ðt; jÞ xi ðt; fÞÞp q i¼1 nþi i
Z t 0
) ðjzij ðsÞj þ jZij ðsÞjÞees ds . (24)
Similar to that of proof of Theorem 1, we can obtain the ðd=dtÞVðzðt; jÞ; zðt; fÞÞp0, and follow the similar argument in Theorem 1, we can prove that fuzzy BAM neural networks (1) is exponentially periodic. Let l1 ¼ ¼ lmþn ¼ 0, we have the following corollary.
References [1] J. Cao, Exponential stability and periodic oscillation solution in BAM networks with delays, IEEE Trans. Neural Networks 13 (2002) 457–463. [2] J. Cao, Global asymptotic stability of delayed bi-directional associative memory neural networks, Appl. Math. Comput. 142 (2003) 333–339. [3] J. Cao, Q. Jiang, An analysis of periodic solutions of bi-directional associative memory networks with time-varying delays, Phys. Lett. A 330 (2004) 203–213. [4] J. Cao, J. Wang, Global exponential stability and periodicity of recurrent neural networks with time delays, IEEE Trans. Circuits Syst. Part I 52 (2005) 920–931. [5] A. Chen, L. Huang, J. Cao, Existence and stability of almost periodic solution for BAM neural networks with delays, Appl. Math. Comput. 137 (2003) 177–193. [6] A. Chen, L. Huang, Z. Liu, J. Cao, Periodic bidirectional associative memory neural networks with distributed delays, J. Math. Anal. Appl. 317 (2006) 80–102. [8] T. Huang, Exponential stability of fuzzy cellular neural networks with unbounded distributed delay, Phys. Lett. A 351 (2006) 48–52. [9] T. Huang, Exponential stability of delayed fuzzy cellular neural networks with diffusion, Chaos Solitons Fractals 31 (2007) 658–664. [10] T. Huang, L. Zhang, Exponential stability of fuzzy cellular neural networks, in: J. Wang, X. Liao, Z. Yi (Eds.), Advances in Neural Networks, Lecture Notes in Computer Science, vol. 3496, Springer, Berlin, Heidelberg, New York, 2005, pp. 168–173. [11] B. Kosko, Adaptive bidirectional associative memories, Appl. Opt. 26 (1987) 4947–4960. [12] B. Kosko, Bidirectional associative memories, IEEE. Trans. Syst. Man Cybern. SMC-18 (1988) 49–60. [14] C. Li, X. Liao, R. Zhang, Delay-dependent exponential stability analysis of Bidirectional associative memory neural networks with time delay: an LMI approach, Chaos Solitons Fractals 24 (2005) 1119–1134. [16] Q.K. Song, Z. Zhao, Y. Li, Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms, 335 (2005) 213–225. [17] T. Yang, L.B. Yang, The global stability of fuzzy cellular neural network, Circuits Syst. I Fundam. Theory Appl. 43 (1996) 880–883. [18] T. Yang, L.B. Yang, C.W. Wu, L.O. Chua, Fuzzy cellular neural networks: theory, in: Proceedings of IEEE International Workshop on Cellular Neural Networks and Applications, 1996, pp. 181–186. [19] T. Yang, L.B. Yang, C.W. Wu, L.O. Chua, Fuzzy cellular neural networks: applications, in: Proceedings of IEEE International Workshop on Cellular Neural Networks and Applications, 1996, pp. 225–230. [20] X. Yang, C. Li, X. Liao, D. Evans, G. Megson, Global exponential periodicity of a class of bidirectional associative memory networks with finite distributed delays, Appl. Math. Comput. 171 (2005) 108–121.
ARTICLE IN PRESS T. Huang et al. / Neurocomputing 71 (2008) 3064–3069
Tingwen Huang obtained his B.S. from Southwest Normal University in 1990, M.S. from Sichuan University in 1993 and Ph.D. from Texas A&M University in 2002. After he graduated at Texas A&M University, he has been working in Mathematics Department of Texas A&M University as Visiting Assistant Professor. In 2003, he started to work at Texas A&M University at Qatar until now. His research fields include neural networks, chaos and its applications, etc. He has published about 30 journal papers on neural networks and nonlinear dynamics.
3069
Chuandong Li received his B.S. degree in Applied Mathematics from Sichuan University, Chengdu, China in 1992, and his M.S. degree in operational research and control theory and Ph.D. degree in Computer Software and Theory from Chongqing University, Chongqing, China, in 2001 and 2005, respectively. He has been a Professor at the Chongqing University, Chongqing, China, since 2007. From November 2006 to November 2008, he serves as a research fellow in the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong. His current research interest covers iterative learning control of time-delay systems, neural networks, chaos control and synchronization, and impulsive dynamical systems.
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Stability of periodic solution in fuzzy BAM neural networks with finite distributed delays Tingwen Huang a,, Yu Huang b, Chuandong Li c a b c
Texas A&M University at Qatar, P.O. Box 5825, Doha, Qatar Department of Mathematics, Zhongshan University, Guangzhou 510275, China Department of Computer Science and Engineering, Chongqing University, Chongqing 400030, PR China
a r t i c l e in f o
a b s t r a c t
Available online 20 June 2008
In this paper, stability is considered for a class of fuzzy cellular neural networks with distributed delay (FCNNDD). The condition for feedback kernel is minimal: the only condition required is that Rt 0 kðsÞ ds ¼ 1. Significant for applications of neural networks, the stability of FCNNDD is discussed when t is finite and infinite. Exponential stability is obtained with distributed delay in finite time, while asymptotic stability is obtained with distributed delay in infinite time. & 2008 Elsevier B.V. All rights reserved.
Keywords: Exponential stability Fuzzy cellular neural networks Distributed delay Equilibrium point
In this paper, we investigate the exponential periodicity for the following fuzzy BAM neural networks:
1. Introduction After Kosko [11,12] proposed bidirectional associative memory neural networks: n X dxi ðtÞ ¼ xi ðtÞ þ aij f j ðyj ðtÞÞ þ Ii ðtÞ; dt j¼1
i ¼ 1; . . . ; m;
m X dyj ðtÞ ¼ yj ðtÞ þ bji f nþi ðxi ðtÞÞ þ Imþj ðtÞ; dt i¼1
n Z t n ^ ^ dxi ðtÞ aji ðsÞf j ðyj ðt sÞÞ ds þ T ji mj ðtÞ ¼ di xi ðtÞ þ dt 0 j¼1 j¼1 n Z t n _ _ þ bji ðsÞf j ðyj ðt sÞÞ ds þ Hji mj ðtÞ þ Ii ðtÞ; i ¼ 1; . . . ; m; j¼1
0
j¼1
j ¼ 1; . . . ; n (1)
many researchers [1–6,14] have studied the dynamics of BAM neural networks with or without delays, including stability and periodic solutions. In this paper, we would like to integrate fuzzy operations into BAM neural networks and maintain local connectedness among cells. Speaking of fuzzy operations, Yang et al. [16–19] first combined those operations with cellular neural networks and investigated the stability of fuzzy cellular neural networks (FCNNs). So far researchers have found that FCNNs are useful in image processing, and some results have been reported on stability and periodicity of FCNNs [8–10,20]. However, to the best of our knowledge, few results reported on periodicity of FCNNs or fuzzy BAM neural networks with finite distributed delays. It is believed that the conclusions made from finite distributed delays could provide us insight into their counterparts with unbounded delays. Corresponding author.
E-mail addresses: [email protected] (T. Huang), [email protected] (Y. Huang), [email protected] (C. Li). 0925-2312/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2008.04.021
m Z t ^ dyj ðtÞ ¼ dj yj ðtÞ þ zij ðsÞf nþi ðxi ðt sÞÞ ds dt i¼1 0 m m Z t ^ _ þ T ij mnþi ðtÞ þ Zij ðsÞf nþi ðxi ðt sÞÞ ds
þ
i¼1 m _
i¼1
0
Hij mnþi ðtÞ þ Imþj ðtÞ;
j ¼ 1; . . . ; n,
(2)
i¼1
where ci 40, dj 40, for i ¼ 1; . . . ; m, j ¼ 1; 2; . . . ; n; aij ðsÞ and zij ðsÞ, bij ðsÞ and Zij ðsÞ which denote elements of fuzzy feedback MIN templates, fuzzy feedback MAX templates, are continuous in the interval ½0; t, where t is a constant; T ij and Hij are fuzzy feed forward MIN template and fuzzy feed-forward V W MAX template, respectively; and denote the fuzzy AND and fuzzy OR operation, respectively; xi ; yj are activations of the ith neuron and the jth neuron, respectively; for k ¼ 1; . . . ; m þ n, mk , Ik , denote inputs, bias of the ith neuron, respectively; function f k are activation functions; moreover, functions mk ðtÞ, Ik ðtÞ are continuously periodic functions with period o, i.e., mk ðtÞ ¼ mk ðt þ oÞ, Ik ðtÞ ¼ Ik ðt þ oÞ; t 2 R, for k ¼ 1; . . . ; m þ n.
ARTICLE IN PRESS T. Huang et al. / Neurocomputing 71 (2008) 3064–3069
The initial conditions associated with (2) are the following: tpsp0; i ¼ 1; . . . ; m;
xi ðsÞ ¼ ji ðsÞ;
tpsp0; j ¼ 1; 2; . . . ; n.
yj ðsÞ ¼ jmþj ðsÞ;
(3)
In this paper, we assume that H: f i is a bounded function defined on R and satisfies Lipschitz condition jf i ðxÞ f i ðyÞjpli jx yj;
i ¼ 1; . . . ; m þ n,
(4)
for any x; y 2 R. Let us define that for any o 2 Rmþn, kok ¼ max1pkpmþn jok j. Let C t ¼ Cð½t; 0; Rmþn Þ be the Banach space of all continuous functions mapping from ½t; 0 to Rmþn with norm defined as follows: for each j 2 C t , kjkt ¼ suptptp0 max1pipmþn jji ðtÞj. Now we define the exponentially periodic solution of fuzzy BAM neural networks. To be convenient, for given j 2 C t ; let zðt; jÞ ¼ ðx1 ðt; jÞ; . . . ; xmþn ðt; jÞÞ¼ ðx1 ðt; jÞ; . . . ; xm ðt; jÞ; y1 ðt; jÞ; . . . ; yn ðt; jÞÞT 2 Rmþn represent the solution of system (2) with initial condition: zðtÞ ¼ jðtÞ when tptp0. Definition 1. The solution zðt; j Þ is exponentially periodic if zðt; j Þ is periodic and there exist positive constants M, a such that any solution zðt; fÞ satisfies kzðt; fÞ zðt; j ÞkpMkf j kt eat ;
tX0.
(5)
Here, we present a lemma which we will use in the proof of the main theorem. Lemma 1 (Le et al. [14]). For any aij 2 R, xj ; yj 2 R, i; j ¼ 1; . . . ; n, we have the following estimations: ^ n ^ X n aij xj a y ðjaij j jxj yj jÞ (6) ij j p j¼1 1pjpn j¼1 and _ n _ X n aij xj aij yj p ðjaij j jxj yj jÞ. j¼1 1pjpn j¼1
(7)
constants M and e which depend on li , i ¼ 1; . . . ; m þ n and known coefficients of (2) such that jjzðt; fÞ zðt; jÞjjpMjjf jjjt eet ;
In this section, sufficient conditions to guarantee exponentially periodic solution of fuzzy BAM neural networks are obtained. First, we would like to present the main result. Theorem 1. Fuzzy BAM neural networks (2) is exponentially periodic if there exist l1 ; . . . ; lmþn such that the following two inequalities hold: 0
1 Z t Z t n n X X @ li 2di lj ðjaji ðsÞj þ jbji ðsÞjÞ dsA4lnþi lmþj ðjzij ðsÞj þ jZij ðsÞjÞ ds,
lmþj 2cj
i¼1
0
li
Z t 0
(9)
j¼1 n _
þ
j¼1
0
Hji mj ðtÞ þ Ii ðtÞ;
i ¼ 1; . . . ; m;
j¼1 n Z t ^ dxi ðt; fÞ ¼ di xi ðt; fÞ þ aji ðsÞf j ðyj ðt s; fÞÞ ds dt j¼1 0 n n Z t ^ _ þ T ji mj ðtÞ bji ðsÞf j ðyj ðt s; fÞÞ ds j¼1
þ
n _
j¼1
0
Hji mj ðtÞ þ Ii ðtÞ;
i ¼ 1; . . . ; m.
(10)
j¼1
Thus, for i ¼ 1; . . . ; m; we have dxi ðt; jÞ dxi ðt; fÞ dt dt ¼ di ðxi ðt; jÞ xi ðt; fÞÞ n Z t ^ aji ðsÞðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds þ þ
j¼1
0
j¼1
0
n Z t _
bji ðsÞðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds.
(11)
Similarly, for j ¼ 1; . . . ; n; we have dyj ðt; jÞ dyj ðt; fÞ dt dt ¼ dj ðyj ðt; jÞ yj ðt; fÞÞ m Z t ^ zij ðsÞðf nþi ðxi ðt s; jÞÞ f nþi ðxi ðt s; fÞÞÞ ds þ þ
i¼1 0 m Z t _ i¼1
0
Zij ðsÞðf nþi ðxi ðt s; jÞÞ f nþi ðxi ðt s; fÞÞÞ ds. (12)
By condition (8), there exists a positive number e such that 0 1 Z t n X @ li 2di e lj ðjaji ðsÞj þ jbji ðsÞjÞ dsA
2. Main results
m X
tX0:
Since zðt; fÞ, zðt; jÞ are the solutions to system (2), we have n Z t ^ dxi ðt; jÞ ¼ di xi ðt; jÞ þ aji ðsÞf j ðyj ðt s; jÞÞ ds dt j¼1 0 n n Z t ^ _ þ T ji mj ðtÞ bji ðsÞf j ðyj ðt s; jÞÞ ds
The rest of the paper is organized as follows. Section 2 presents some sufficient conditions for exponentially periodic solution to fuzzy BAM neural networks. Conclusion is drawn in Section 3.
j¼1
3065
! ðjzij ðsÞj þ jZij ðsÞjÞ ds 4lj
0
j¼1 m X i¼1
li
Z t 0
ðjaji ðsÞj þ jbji ðsÞjÞ ds,
(8)
0
j¼1
4lnþi
n X
lmþj
j¼1
lmþj 4lj
Z t 0
ees ðjzij ðsÞj þ jZij ðsÞjÞ ds,
! Z t m X 2cj e li ðjzij ðsÞj þ jZij ðsÞjÞ ds m X i¼1
li
Z t 0
i¼1
0
ees ðjaji ðsÞj þ jbji ðsÞjÞ ds.
Now let us define the Lyapunov functional Vðzðt; jÞ; zðt; fÞÞ ¼ V 1 ðzðt; jÞ; zðt; fÞÞ þ V 2 ðzðt; jÞ; zðt; fÞÞ, where 8 < m X li ðxi ðt; jÞ xi ðt; fÞÞ2 eet V 1 ðzðt; jÞ; zðt; fÞÞ ¼ : i¼1
where i ¼ 1; . . . ; m, j ¼ 1; 2; . . . ; n. Proof. First let us prove the following claim. Claim. Let zðt; jÞ ¼ ðx1 ðt; jÞ; . . . ; xm ðt; jÞ; y1 ðt; jÞ; . . . ; yn ðt; jÞÞT ; zðt; fÞ ¼ ðx1 ðt; fÞ; . . . ; xm ðt; fÞ; y1 ðt; fÞ; . . . ; yn ðt; fÞÞT be the solutions to system (2). Under the conditions of this theorem, there are positive
(13)
þ
Z t n X lj ðjaji ðsÞj þ jbji ðsÞjÞ j¼1
0
9 = ðyj ðr; jÞ yj ðr; fÞÞ2 eeðrþsÞ dr ds ; ts Z
t
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V 2 ðzðt; jÞ; zðt; fÞÞ ¼
n X
(
þ
lmþj ðyj ðt; jÞ yj ðt; fÞÞ2 eet
0
)
t ts
ðxi ðr; jÞ xi ðr; fÞÞ
2
eeðrþsÞ dr ds
lj jbji ðsÞjððxi ðt; jÞ xi ðt; fÞÞ2
þ ðyj ðt s; jÞ yj ðt s; fÞÞ2 Þ ds Z t n X þ lj ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞ2 ees
Z t m X li ðjzij ðsÞj þ jZij ðsÞjÞ þ i¼1
0
j¼1
j¼1
Z
n Z t X
0
j¼1
. 2
(14) 2
Take the derivative of V 1 , using Lemma 1 and inequality 2abpa2 þ b in the following process, we have
ðyj ðt s; jÞ yj ðt s; fÞÞ ds
9 =
; 82 3 Z t m n < X X e t ¼e li 4e 2di þ lj ðjaji ðsÞj þ jbji ðsÞjÞ ds5 : 0 i¼1
j¼1
2
ðxi ðt; jÞ xi ðt; fÞÞ d V 1 ðzðt; jÞ; zðt; fÞÞ dt 8 < m X li e eet ðxi ðt; jÞ xi ðt; fÞÞ2 ¼ :
þ
n X
2
lj ðyj ðt; jÞ yj ðt; fÞÞ
Z t 0
j¼1
9 = ðjaji ðsÞj þ jbji ðsÞjÞ ees ds . ; (15)
i¼1
þ 2eet ðxi ðt; jÞ xi ðt; fÞÞðx_ i ðt; jÞ x_ i ðt; fÞÞ Z t n X lj eet ðjaji ðsÞj þ jbji ðsÞjÞ½ees ðyj ðt; jÞ yj ðt; fÞÞ2 þ 0
j¼1
2
ðyj ðt s; jÞ yj ðt s; fÞÞ ds
¼ eet
m X
8 <
9 = ;
Similarly, we can obtain that (" n m X X d e t V 2 ðzðt; jÞ; zðt; fÞÞpe lmþj e 2cj þ lnþi dt j¼1 i¼1 # Z t ðjzij ðsÞj þ jZij ðsÞjÞ ds ðyi ðt; jÞ yi ðt; fÞÞ2 0
li eðxi ðt; jÞ xi ðt; fÞÞ2 2di ðxi ðt; jÞ xi ðt; fÞÞ2
þ
:
i¼1
þ 2ðxi ðt; jÞ xi ðt; fÞÞ
n Z t ^ j¼1
0
i¼1
aji ðsÞ
0
0
9 = ðyj ðt s; jÞ yj ðt s; fÞÞ ds ; 8 m < X peet li ðe 2di Þðxi ðt; jÞ xi ðt; fÞÞ2 : 2
By inequalities (15), (16) and (13), we have d Vðzðt; jÞ; zðt; fÞÞ dt d d ¼ V 1 ðzðt; jÞ; zðt; fÞÞ þ V 2 ðzðt; jÞ; zðt; fÞÞ dt dt 82 m n < X X li 4e 2di þ lj peet : i¼1 j¼1 Z t ðjaji ðsÞj þ jbji ðsÞjÞ ds ðxi ðt; jÞ xi ðt; fÞÞ2 0
þ
n Z t X j¼1
0
lj jaji ðsÞj
jyj ðt s; jÞ yj ðt s; fÞj ds n Z t X lj jbji ðsÞj þ 2jxi ðt; jÞ xi ðt; fÞj j¼1
0
9 = 2 ðyj ðt s; jÞ yj ðt s; fÞÞ ds ; 8 m < X peet li ðe 2di Þðxi ðt; jÞ xi ðt; fÞÞ2 : i¼1
n Z t X j¼1
0
2
lj jaji ðsÞjððxi ðt; jÞ xi ðt; fÞÞ
þ ðyj ðt s; jÞ yj ðt s; fÞÞ Þ ds
lj ðyj ðt; jÞ yj ðt; fÞÞ n X
0
("
e 2cj þ
lmþj
Z t
ðjaji ðsÞj þ jbji
m X
9 =
ðsÞjÞ ees ds
;
lnþi
i¼1
# Z t ðjzij ðsÞj þ jZij ðsÞjÞ ds ðyi ðt; jÞ yi ðt; fÞÞ2 0
m X
2
lnþi ðxi ðt; jÞ xi ðt; fÞÞ
i¼1
¼ eet
m X
li 4e 2di þ
þlnþi
lmþj
j¼1
þ eet
n X
Z t 0
ees ðjz
ðjzij ðsÞj þ jZij
0
3
ij ðsÞj
"
lmþj e 2cj þ
j¼1
) ðsÞjÞ ees ds
Z t n X lj ðjaji ðsÞj þ jbji ðsÞjÞ ds j¼1
n X
Z t 0
2
i¼1
þ jZij ðsÞjÞ ds5ðxi ðt; jÞ xi ðt; fÞÞ2
m X i¼1
lnþi
Z t 0
ðjzij ðsÞj þ jZij ðsÞjÞ ds
3 Z t n X es þ lj ðjaji ðsÞj þ jbji ðsÞjÞ e ds5ðyi ðt; jÞ yi ðt; fÞÞ2 j¼1
2
2
j¼1
0
j¼1
þ
þ eet
þ
jyj ðt s; jÞ yj ðt s; fÞj ds Z t n X lj ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞ2 ees þ
n X j¼1
i¼1
þ 2jxi ðt; jÞ xi ðt; fÞj
(16)
0
ðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds Z t n X lj ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞ2 ees þ j¼1
lnþi ðxi ðt; jÞ xi ðt; fÞÞ2
) Z t ðjzij ðsÞj þ jZij ðsÞjÞ ees ds .
ðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds n Z t _ bji ðsÞ þ 2ðxi ðt; jÞ xi ðt; fÞÞ j¼1
m X
p0.
0
(17)
ARTICLE IN PRESS T. Huang et al. / Neurocomputing 71 (2008) 3064–3069
Thus, Vðzðt; jÞ; zðt; fÞÞpVðzð0; jÞ; zð0; fÞÞ. For k ¼ 1; . . . ; m þ n, we have eet jzk ðt; jÞ zk ðt; fÞj2 ( m X eet p l ðx ðt; jÞ xi ðt; fÞÞ2 min1plpmþn fll g i¼1 i i 9 n = X 2 lmþj ðyj ðt; jÞ yj ðt; fÞÞ þ ;
Theorem 2. Fuzzy BAM neural networks (1) is exponentially periodic if there exist l1 ; . . . ; lmþn such that the following two inequalities hold:
j¼1
0
pV 1 ðzð0; jÞ; zð0; fÞÞ þ V 2 ðzð0; jÞ; zð0; fÞÞ 8 Z t < m n X X ¼ li ðxi ð0; jÞ xi ð0; fÞÞ2 þ lj ðjaji ðsÞj þ jbji ðsÞjÞ : 0 i¼1 j¼1 ) Z 2
ðyj ðr; jÞ yj ðr; fÞÞ
to zðt; j Þ as t ! 1. So far, we have completed the proof of the theorem. & In the following, we will give a generalized result which includes Theorem 1 as a special case.
pV 1 ðzðt; jÞ; zðt; fÞÞ þ V 2 ðzðt; jÞ; zðt; fÞÞ
0
3067
1 Z t Z t n n X X p lj ðjaji ðsÞj þ jbji ðsÞjÞ dsA4 lnþi lmþj ðjzij ðsÞj þ jZij ðsÞjÞ ds, q 0 0 j¼1 j¼1 ! Z t Z t m m X p X pcj li ðjzij ðsÞj þ jZij ðsÞjÞ ds 4 lj li ðjaji ðsÞj þ jbji ðsÞjÞ ds, q i¼1 0 0 i¼1
li @pdi lmþj
eeðrþsÞ dr ds
(21)
s
þ
n X
lmþj fðyj ð0; jÞ yj ð0; fÞÞ2 þ
j¼1
Z
Z t m X li ðjzij ðsÞj þ jZij ðsÞjÞ i¼1
where i ¼ 1; . . . ; m, j ¼ 1; 2; . . . ; n, and ð1=pÞ þ ð1=qÞ ¼ 1, p; q are positives.
0
0
ðxi ðr; jÞ ðxi ðr; fÞÞ2 eeðrþsÞ dr dsg 8 Z t < m n X X ¼ li ðji ð0Þ fi ð0ÞÞ2 þ lj ðjaji ðsÞj þ jbji ðsÞjÞ : 0 i¼1 j¼1 ) Z
The proof is similar to that of Theorem 1. We will not present the details. Consider the Lyapunov functional
s
0
s
þ
where
ðjmþj ðrÞ fmþj ðrÞÞ2 eeðrþsÞ dr ds
n X
(
lmþj
Z t m X ðjmþj ð0Þ fmþj ð0ÞÞ2 þ li ðjzij ðsÞj þ jZij ðsÞjÞ
j¼1
Z
Vðzðt; jÞ; zðt; fÞÞ ¼ V 1 ðzðt; jÞ; zðt; fÞÞ þ V 2 ðzðt; jÞ; zðt; fÞÞ,
)
0
s
2
ðji ðrÞ fi ðrÞÞ
i¼1
i¼1
þ
eeðrþsÞ dr ds
8 9 2 Z 0 Z t m n < = X X li 1 þ lj ðjaji ðsÞj þ jbji ðsÞjÞ eeðrþsÞ dr ds pjjj fjj2t 4 : ; 0 s i¼1 j¼1 3 ( ) Z t Z 0 n m X X lmþj 1 þ li ðjzij ðsÞj þ jZij ðsÞjÞ eeðrþsÞ dr ds 5. þ j¼1
V 1 ðzðt; jÞ; zðt; fÞÞ ¼
0
i¼1
0
s
8 < li ðxi ðt; jÞ xi ðt; fÞÞp eet :
n pX l q j¼1 j
Z
V 2 ðzðt; jÞ; zðt; fÞÞ ¼
Z t 0
t ts
n X
ðjaji ðsÞj þ jbji ðsÞjÞ q
ðyj ðr; jÞ yj ðr; fÞÞ
9 = , ;
eeðrþsÞ dr ds
(
lmþj ðyj ðt; jÞ yj ðt; fÞÞp eet
j¼1
(18)
Z t m pX li ðjzij ðsÞj þ jZij ðsÞjÞ q i¼1 0 ) Z t ðxi ðr; jÞ ðxi ðr; fÞÞq eeðrþsÞ dr ds .
From the above inequalities, we have
þ
jzk ðt; fÞ zk ðt; j ÞjpMjjf j jjt eðe=2Þt , tX0;
k ¼ 1; . . . ; m þ n, (19) R 0 eðrþsÞ Rt Pn Pm dr dsgþ where M ¼ i¼1 li f1 þ j¼1 lj 0 ðjaji ðsÞj þ jbji ðsÞjÞ s e R 0 eðrþsÞ Pn Pm R t dr dsg. So far, we j¼1 lmþj f1 þ i¼1 li 0 ðjzij ðsÞj þ jZij ðsÞjÞ s e have completed the proof of the Claim. By the Claim, to complete this theorem, we just need to prove that there exists a o-periodic solution for model (2). Let zt ðjÞðWÞ ¼ zðt; jÞðWÞ ¼ zðt þ W; jÞ, for W 2 ½t; 0 and zðt þ W; jÞ is a solution of model (2) with initial solution of (3). Now, we define a Poincare mapping P : C ! C by P j ¼ zo ðjÞ. By Claim, there are positives M and e, jjzt ðfÞ zt ðj Þjjt pMjjf j jjt eðe=2ÞðttÞ ;
m X
tX0.
(20)
Choose a positive big integer l, such that Mee=2ðoltÞ p13. Thus, we have jjP l f P l jjjt p13jjf jjjt . It implies that Pl is a contraction mapping, so there exists a unique fixed point j 2 C such that Pl j ¼ j . Note that Pl ðP j Þ ¼ PðP l j Þ ¼ P j , so P j is a fixed point of mapping P l . By the uniqueness of the fixed point of the contraction map, P j ¼ j , i.e., j is a fixed point of P. Thus, ztþo ðj Þ ¼ zt ðzo ðj ÞÞ ¼ zt ðj Þ, i.e., zðt þ o; j Þ ¼ zðt; j Þ, for any t40. This proves that there exists one o-periodic solution, and from the Claim, it is clear that all solutions converge exponentially
ts
(22) Take the derivative of V 1 , using abpð1=pÞap þ ð1=qÞb following process, we have:
q
in the
d V 1 ðzðt; jÞ; zðt; fÞÞ dt 8 < m X ¼ li eeet ðxi ðt; jÞ xi ðt; fÞÞp : i¼1
þ peet ðxi ðt; jÞ xi ðt; fÞÞp1 ðx_ i ðt; jÞ x_ i ðt; fÞÞ Z t n pX þ l eet ðjaji ðsÞj þ jbji ðsÞjÞ½ees ðyj ðt; jÞ yj ðt; fÞÞq q j¼1 j 0 9 = q ðyj ðt s; jÞ yj ðt s; fÞÞ ds ; 8 < m X ¼ ee t li eðxi ðt; jÞ xi ðt; fÞÞp pdi ðxi ðt; jÞ xi ðt; fÞÞp : i¼1
þ pðxi ðt; jÞ xi ðt; fÞÞ
n Z t ^ j¼1
0
aji ðsÞðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds
ARTICLE IN PRESS 3068
T. Huang et al. / Neurocomputing 71 (2008) 3064–3069
þ pðxi ðt; jÞ xi ðt; fÞÞ
n Z t _ j¼1
þ
n pX l q j¼1 j
Z t 0
0
bji ðsÞðf j ðyj ðt s; jÞÞ f j ðyj ðt s; fÞÞÞ ds
ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞq ees
ðyj ðt s; jÞ yj ðt s; fÞÞq ds
9 =
; 8 m < X peet li ðe pdi Þðxi ðt; jÞ xi ðt; fÞÞp :
(25) where i ¼ 1; . . . ; m, j ¼ 1; 2; . . . ; n, and 1=p þ 1=q ¼ 1, p; q are positives.
i¼1
þ pjxi ðt; jÞ xi ðt; fÞj
n Z t X j¼1
þ pjxi ðt; jÞ xi ðt; fÞj
n Z t X j¼1
n pX l þ q j¼1 j
Z t 0
0
0
Corollary 1. Fuzzy BAM neural networks (1) is exponentially periodic if the following two inequalities hold: Z t n n Z t X X p pdi lj ðjaji ðsÞj þ jbji ðsÞjÞ ds4 lnþi ðjzij ðsÞj þ jZij ðsÞjÞ ds, q 0 j¼1 j¼1 0 Z t m m Z t X p X pcj li ðjzij ðsÞj þ jZij ðsÞjÞ ds4 lj ðjaji ðsÞj þ jbji ðsÞjÞ ds, q i¼1 0 0 i¼1
lj jaji ðsÞjjyj ðt s; jÞ yj ðt s; fÞj ds
3. Conclusion lj jbji ðsÞjjyj ðt s; jÞ yj ðt s; fÞj ds
ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞ q
ðyj ðt s; jÞ yj ðt s; fÞÞ ds
q
In this paper, we discuss the existence and exponential stability of the equilibrium of the periodic solution of Fuzzy BAM neural networks with finite distributed delays. Sufficient condition set up here by using Lyapunov functional is easily verifiable.
ee s
9 =
; 8 m < X peet li ðe pdi Þðxi ðt; jÞ xi ðt; fÞÞp :
Acknowledgments
i¼1
þ
n Z t X j¼1
0
The first author is grateful for the support of Texas A and M University at Qatar. This project is partially support by NSFC (10771222), NSFC (60574024), NCET-06-0764 and NSF of Guangdong. Part of the above work is presented in ICONIP 2006.
lj jaji ðsÞjððxi ðt; jÞ xi ðt; fÞÞp
p þ ðyj ðt s; jÞ yj ðt s; fÞÞq Þ ds q n Z t X þ lj jbji ðsÞjððxi ðt; jÞ xi ðt; fÞÞp j¼1
0
p þ ðyj ðt s; jÞ yj ðt s; fÞÞq Þ ds q Z t n pX l ðjaji ðsÞj þ jbji ðsÞjÞ½ðyj ðt; jÞ yj ðt; fÞÞq ees þ q j¼1 j 0 9 = q ðyj ðt s; jÞ yj ðt s; fÞÞ ds ; 82 3 Z t < m n X X ¼ ee t li 4e pdi þ lj ðjaji ðsÞj þ jbji ðsÞjÞ ds5ðxi ðt; jÞ xi ðt; fÞÞp : 0 i¼1 j¼1 9 Z n = t pX q e s þ lj ðyj ðt; jÞ yj ðt; fÞÞ ðjaji ðsÞj þ jbji ðsÞjÞe ds . ; q 0 j¼1
(23)
Similarly, we can obtain that
d V 2 ðzðt; jÞ; zðt; fÞÞ dt (" n m X X e t pe lmþj e pcj þ lnþi j¼1
i¼1
# Z t ðjzij ðsÞj þ jZij ðsÞjÞ ds ðyi ðt; jÞ yi ðt; fÞÞq 0
þ
m pX l ðx ðt; jÞ xi ðt; fÞÞp q i¼1 nþi i
Z t 0
) ðjzij ðsÞj þ jZij ðsÞjÞees ds . (24)
Similar to that of proof of Theorem 1, we can obtain the ðd=dtÞVðzðt; jÞ; zðt; fÞÞp0, and follow the similar argument in Theorem 1, we can prove that fuzzy BAM neural networks (1) is exponentially periodic. Let l1 ¼ ¼ lmþn ¼ 0, we have the following corollary.
References [1] J. Cao, Exponential stability and periodic oscillation solution in BAM networks with delays, IEEE Trans. Neural Networks 13 (2002) 457–463. [2] J. Cao, Global asymptotic stability of delayed bi-directional associative memory neural networks, Appl. Math. Comput. 142 (2003) 333–339. [3] J. Cao, Q. Jiang, An analysis of periodic solutions of bi-directional associative memory networks with time-varying delays, Phys. Lett. A 330 (2004) 203–213. [4] J. Cao, J. Wang, Global exponential stability and periodicity of recurrent neural networks with time delays, IEEE Trans. Circuits Syst. Part I 52 (2005) 920–931. [5] A. Chen, L. Huang, J. Cao, Existence and stability of almost periodic solution for BAM neural networks with delays, Appl. Math. Comput. 137 (2003) 177–193. [6] A. Chen, L. Huang, Z. Liu, J. Cao, Periodic bidirectional associative memory neural networks with distributed delays, J. Math. Anal. Appl. 317 (2006) 80–102. [8] T. Huang, Exponential stability of fuzzy cellular neural networks with unbounded distributed delay, Phys. Lett. A 351 (2006) 48–52. [9] T. Huang, Exponential stability of delayed fuzzy cellular neural networks with diffusion, Chaos Solitons Fractals 31 (2007) 658–664. [10] T. Huang, L. Zhang, Exponential stability of fuzzy cellular neural networks, in: J. Wang, X. Liao, Z. Yi (Eds.), Advances in Neural Networks, Lecture Notes in Computer Science, vol. 3496, Springer, Berlin, Heidelberg, New York, 2005, pp. 168–173. [11] B. Kosko, Adaptive bidirectional associative memories, Appl. Opt. 26 (1987) 4947–4960. [12] B. Kosko, Bidirectional associative memories, IEEE. Trans. Syst. Man Cybern. SMC-18 (1988) 49–60. [14] C. Li, X. Liao, R. Zhang, Delay-dependent exponential stability analysis of Bidirectional associative memory neural networks with time delay: an LMI approach, Chaos Solitons Fractals 24 (2005) 1119–1134. [16] Q.K. Song, Z. Zhao, Y. Li, Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms, 335 (2005) 213–225. [17] T. Yang, L.B. Yang, The global stability of fuzzy cellular neural network, Circuits Syst. I Fundam. Theory Appl. 43 (1996) 880–883. [18] T. Yang, L.B. Yang, C.W. Wu, L.O. Chua, Fuzzy cellular neural networks: theory, in: Proceedings of IEEE International Workshop on Cellular Neural Networks and Applications, 1996, pp. 181–186. [19] T. Yang, L.B. Yang, C.W. Wu, L.O. Chua, Fuzzy cellular neural networks: applications, in: Proceedings of IEEE International Workshop on Cellular Neural Networks and Applications, 1996, pp. 225–230. [20] X. Yang, C. Li, X. Liao, D. Evans, G. Megson, Global exponential periodicity of a class of bidirectional associative memory networks with finite distributed delays, Appl. Math. Comput. 171 (2005) 108–121.
ARTICLE IN PRESS T. Huang et al. / Neurocomputing 71 (2008) 3064–3069
Tingwen Huang obtained his B.S. from Southwest Normal University in 1990, M.S. from Sichuan University in 1993 and Ph.D. from Texas A&M University in 2002. After he graduated at Texas A&M University, he has been working in Mathematics Department of Texas A&M University as Visiting Assistant Professor. In 2003, he started to work at Texas A&M University at Qatar until now. His research fields include neural networks, chaos and its applications, etc. He has published about 30 journal papers on neural networks and nonlinear dynamics.
3069
Chuandong Li received his B.S. degree in Applied Mathematics from Sichuan University, Chengdu, China in 1992, and his M.S. degree in operational research and control theory and Ph.D. degree in Computer Software and Theory from Chongqing University, Chongqing, China, in 2001 and 2005, respectively. He has been a Professor at the Chongqing University, Chongqing, China, since 2007. From November 2006 to November 2008, he serves as a research fellow in the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong. His current research interest covers iterative learning control of time-delay systems, neural networks, chaos control and synchronization, and impulsive dynamical systems.