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Stability of Cohen–Grossberg neural networks with distributed delays
Applied Mathematics and Computation 160 (2005) 93–110 www.elsevier.com/locate/amc
Stability of Cohen–Grossberg neural networks with distributed delays Lin Wang Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Nfld, Canada A1C 5S7
Abstract Global asymptotic stability and global exponential stability of the Cohen–Grossberg neural networks and Hopfield neural networks with infinite and finite distributed delays are investigated in this paper. For Hopfield neural networks with distributed delays, local stability is also discussed. The existence and global stability of periodic solutions of Hopfield neural networks with distributed delays and periodic inputs are achieved by using the theory of dissipative systems. Ó 2003 Published by Elsevier Inc. Keywords: Cohen–Grossberg neural networks; Distributed delay; Equilibrium; Hopfield neural networks; Liapunov function; Periodic solution; Stability
1. Introduction Cohen and Grossberg [5] proposed a neural network model (CGNN) in 1983 described by the following system ! n X x_ i ðtÞ ¼ ai ðxi ðtÞÞ bi ðxi ðtÞÞ aij gj ðxj ðtÞÞ ; i ¼ 1; 2; . . . ; n; ð1:1Þ j¼1
where i ¼ 1; 2; . . . ; n and n P 2 is the number of neurons in the network; xi describes the activation of the ith neuron; ai represents an amplification Present address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1. E-mail address: [email protected] (L. Wang). 0096-3003/$ - see front matter Ó 2003 Published by Elsevier Inc. doi:10.1016/j.amc.2003.09.014
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function and the function bi can include a constant term indicating a fixed input to the network; the n n connection matrix A ¼ ðaij Þ tells how the neurons are connected in the network; the activation functions gj , j ¼ 1; 2; . . . ; n show how the neurons react to the input. As pointed out in [5] that the system (1.1) includes a number of models from neurobiology, population biology and evolution theory, among which is the Hopfield neural network (HNN) model n X aij gj ðxj ðtÞÞ þ Ii ; i ¼ 1; 2; . . . ; n; ð1:2Þ x_ i ðtÞ ¼ bi xi ðtÞ þ j¼1
where Ii (i ¼ 1; 2; . . . ; n) is a fixed input from outside of the network. Systems (1.1) and (1.2) have attracted great attention of the scientific community and have been extensively investigated, see, for example, [1–4,7,8,11,12,15– 19,21,23,24,26,27]. Among them, Gopalsamy and He [11], and van den Driessche and Zou [23] studied the case with multiple delays x_ i ðtÞ ¼ bi xi ðtÞ þ
n X
tij sj ðxj ðt sij ÞÞ þ Ii ;
i ¼ 1; 2; . . . ; n
ð1:3Þ
j¼1
Indeed, there has been lots of evidence [25] such as finite transmission speed showing that time delays always exist and should be taken into consideration in designing a neural network model. Instead of considering discrete time delays, we incorporate time delays which are continuously distributed over an infinite interval such that the distant past has less influence compared to the most recent neuronsÕ states into system (1.1) and obtain the following CGNN model with infinite distributed delays Z t ! n X x_ i ðtÞ ¼ ai ðxi ðtÞÞ bi ðxi ðtÞÞ aij gj kij ðt sÞxj ðsÞ ds ; ð1:4Þ 1
j¼1
where the delay kernel functions kij ðtÞ are assumed to be piecewise continuous and satisfy Z 1 Z 1 kij ðtÞ P 0; kij ðtÞ dt ¼ 1; tkij ðtÞ dt < 1: ð1:5Þ 0
0
Its a special case is the Hopfield neural network model with distributed delays Z t n X x_ i ðtÞ ¼ bi xi ðtÞ þ aij gj kij ðt sÞxj ðsÞ ds þ Ii ; i ¼ 1; 2; . . . ; n: j¼1
1
ð1:6Þ We will also study the case where the inputs Ii , i ¼ 1; 2; . . . ; n are functions of time t and are periodic with period x, that is, Ii ðtÞ ¼ Ii ðt þ xÞ for all t.
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
95
If delay kernel functions kij ðtÞ are of the form kij ðtÞ ¼ dðt sij Þ;
i; j 2 f1; 2; . . . ; ng;
ð1:7Þ
then system (1.6) reduces to (1.3). Therefore the discrete delays can be included in our models by choosing suitable kernel functions. System (1.6) has been briefly indicated by Gopalsamy and He [11] and Mohamad and Gopalsamy [21]. However, as we will see, their conclusions can be included in our results as special cases. An application of system (1.6) can be found in Tank and Hopfield [22]. The rest of the paper is organized as follows. In Section 2 we introduce our basic notation and assumptions. Section 3 is devoted to the global stability results and proofs for (1.4). Local stability and the applications of our main results to (HNNs) with distributed delays are given in Section 4. (HNNs) with periodic inputs is investigated in Section 5.
2. Preliminaries The initial conditions associated with (1.4) are of the form xi ðsÞ ¼ /i ðsÞ;
i ¼ 1; 2; . . . ; n;
ð2:1Þ
where /i ðsÞ 2 Cðð1; 0 ; RÞ are bounded. We first give some assumptions. (H1) For each i 2 f1; 2; . . . ; ng, ai is bounded, positive and continuous, furthermore we assume 0 < ai 6 ai ðuÞ 6 ai . (H2) For each i 2 f1; 2; . . . ; ng, bi is continuous increasing. For the activation functions gi ðxÞ, i ¼ 1; 2; . . . ; n, they are typically assumed to be sigmoid which implies that they are monotone and smooth, that is, they are required to satisfy the following (A1) gi 2 C 2 ðRÞ, gi0 ðxÞ > 0; supx2R gi0 ðuÞ ¼ gi0 ð0Þ ¼ 1, i ¼ 1; 2; . . . ; n; (A2) gi ð0Þ ¼ 0 and limx!1 gi ðxÞ ¼ 1. One commonly used function is gðxÞ ¼ tanhðxÞ. Without that assumptions on monotonicity and differentiability of the activation function, as pointed out in [23], we can assume (S1) For each i 2 f1; 2; . . . ; ng, gi : R ! R is globally Lipschitz continuous with a Lipschitz constant Li ; (S2) For each i 2 f1; 2; . . . ; ng, jgi ðxÞj 6 Mi , x 2 R for some constant Mi > 0.
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L. Wang / Appl. Math. Comput. 160 (2005) 93–110
3. CGNNs with distributed delays We first establish an existence result for the equilibrium of system (1.4). Theorem 3.1. If (H1 )–(H2 ) and (S1 )–(S2 ) (or (A1 )–(A2 )) hold, then there exists at least one equilibrium for system (1.4). Proof. By (H1 ), we know that x is an equilibrium of (1.4) if and only if T x ¼ ðx1 ; . . . ; xn Þ is a solution of equations Z t n X aij gj kij ðt sÞxj ðsÞ ds ¼ 0; i ¼ 1; . . . ; n ð3:1Þ bi ðxi Þ j¼1
1
From (S2 ), we have Z t X X n n aij gj kij ðt sÞxj ðsÞ ds 6 ja jM ¼: Pi : j¼1 j¼1 ij j 1 is increasing. Now consider Since (H2 ) holds, then b1 i Z t ! n X 1 xi ¼ hi ðx1 ; x2 ; . . . ; xn Þ ¼ bi aij gj kij ðt sÞxj ðsÞ ds j¼1
1
for i ¼ 1; 2; . . . ; n. We have
1 jhi ðx1 ; x2 ; . . . ; xn Þj 6 max jb1 i ðPi Þj; jbi ð Pi Þj ¼: Di ;
i ¼ 1; 2; . . . ; n:
T
It follows that ðh1 ; h2 ; . . . ; hn Þ maps a bounded set D :¼ ½D1 ; D1 ½D2 ; D2 ½Dn ; Dn to itself. Then the existence of the equilibrium follows from the BrouwerÕs fixed point theorem [6, Theorem 3.2] and the proof is thus complete. h Let x be an equilibrium of (1.4) and uðtÞ ¼ xðtÞ x . Substituting xðtÞ ¼ uðtÞ þ x into (1.4) leads to Z t ! n X u_ i ðtÞ ¼ ai ðui ðtÞÞ bi ðui ðtÞÞ aij fj kij ðt sÞuj ðsÞ ds ð3:2Þ j¼1
1
for i ¼ 1; 2; . . R. ; n, where ai ðui ðtÞÞ ¼ ai ðui ðtÞ þ xi Þ, bi ðui ðtÞÞ ¼ bi ðui ðtÞ þ xi Þ t bi ðxi Þ, fj ¼ gj ð 1 kij ðt sÞuj ðsÞ ds þ xj Þ gj ðxj Þ. Theorem 3.2. Assume that (H1 )–(H2 ) and (S1 )–(S2 ) hold. If there exist ci > 0, and qi > 0 such that bi ðuÞ bi ðvÞ P ci uv
for u 2 R;
i ¼ 1; 2; . . . ; n;
ð3:3Þ
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
97
and ( a i c i q i Li
l :¼ min
16i6n
n X
) aj jaji jqj
ð3:4Þ
> 0:
j¼1
Then the equilibrium x of (1.4) is globally asymptotically stable. Proof. Combining (3.2) with (3.3), we have djui ðtÞj ¼ sgnðui ðtÞÞu_ i ðtÞ dt 6 ai ci jui ðtÞj þ ai
n X
Lj jaij j
n X
t
kij ðt sÞjuj ðsÞj ds 1
j¼1
¼ ai ci jui ðtÞj þ ai
Z
Lj jaij j
Z
1
kij ðsÞjuj ðt sÞj ds;
0
j¼1
where 8 < 1; sgnðxÞ ¼ 0; : 1
x > 0; x ¼ 0; x < 0:
Let V ðtÞ ¼ V ðuÞðtÞ be defined as V ðtÞ ¼
n X
qi jui ðtÞj þ ai qi
i¼1
n X
Lj jaij j
Z
1
kij ðsÞ
0
j¼1
Z
!
t
juj ðxÞj dx ds :
ð3:5Þ
ts
Then the upper righthand derivative of V ðtÞ along the solution of (3.2) is given by ! Z 1 n n X X þ D V ðtÞ 6 qi ai ci jui ðtÞj þ ai Lj jaij j kij ðsÞjuj ðt sÞj ds i¼1
þ
n X i¼1
6
n X
ai qi
n X
Lj jaij j
j¼1
Z
¼
i¼1
kij ðsÞðjuj ðtÞj juj ðt sÞjÞ ds
0
ai ci qi jui ðtÞj þ ai qi
i¼1 n X
n X j¼1
a i c i q i Li
0
j¼1 1
n X j¼1
! jaij jLj juj ðtÞj !
aj jaij jqj jui ðtÞj 6 l
n X
jui ðtÞj:
i¼1
P The above inequality together with (1.5) shows that ni¼1 jui ðtÞj is bounded for all t P 0, thus the solutions of (3.2) exist globally. Moreover, we have
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L. Wang / Appl. Math. Comput. 160 (2005) 93–110
V ðtÞ þ l
Z
n X
t
0
! jui ðsÞj ds 6 V ð0Þ:
i¼1
On the other hand, V ð0Þ ¼
n X
qi jui ð0Þj þ ai qi
i¼1
6
n X
Lj jaij j
0
j¼1
n X
q i þ Li
i¼1
n X
aj qj jaji j
Z
Z
1
1
kij ðsÞ
!
0
juj ðxÞj dx ds
s
!
skji ðsÞ ds
sup j/i ðxÞ xi j < 1;
x2ð1;0
0
j¼1
Z
which implies that n X
jui ðtÞj 2 L1 ð0; 1Þ:
ð3:6Þ
i¼1
By Lemma 1.2.2 in [10], we can obtain n X
jui ðtÞj ! 0
as t ! 1;
ð3:7Þ
i¼1
i.e., xi ðtÞ ! xi
as t ! 1;
i ¼ 1; 2; . . . ; n:
ð3:8Þ
Corollary 3.1. Assume that (H1 ), (H2 ) and (S1 ), (S2 ) hold. In addition to (3.3), if ) ( n X l1 :¼ min ai ci Li aj jaji j > 0 ð3:9Þ 16i6n
j¼1
holds, then the equilibrium x of (1.4) is globally asymptotically stable. Proof. Condition (3.9) implies (3.4) holds for qi ¼ 1, i ¼ 1; 2; . . . ; n, and thus this corollary follows directly from Theorem 3.2. h By using a different Liapunov functional, we have Theorem 3.3. Assume that (H1 )–(H2 ), (S1 )–(S2 ) and (3.3) hold. If there exist positive real numbers pi > 0, and gi > 0 such that for i 2 N ð1; nÞ 2ai ci pi ai pi
n X j¼1
jaij jLj gj
n Li X jaji jaj pj > 0: gi j¼1
Then the equilibrium x of (1.4) is globally asymptotically stable.
ð3:10Þ
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
Proof. Define V ðtÞ ¼ V ðuÞðtÞ by V ðtÞ ¼
n X
pi u2i ðtÞ
þ ai p i
i¼1
n X j¼1
Lj jaij j gj
Z
1
kij ðsÞ
Z
0
99
!
t
u2j ðxÞ dx ds
:
ð3:11Þ
ts
Now we can estimate the upper righthand derivative of V ðtÞ along the solution of (3.2) as follows ! Z 1 n n X X Dþ V ðtÞ 6 pi 2ai ci u2i ðtÞ þ 2ai jui j Lj jaij j kij ðsÞjuj ðt sÞj ds i¼1
þ
j¼1
n X
ai p i
i¼1
6
n X
n X
jaij j
j¼1
Lj gj
Z
1
kij ðsÞðu2j ðtÞ u2j ðt sÞÞ ds
0
2ai ci pi u2i ðtÞ þ ai pi
i¼1
1 0
n X i¼1
¼
n X
jaij jLj
j¼1
Z
þ
0
n X
! ! 2 u ðt sÞ j kij ðsÞ gj u2i ðtÞ þ ds gj ai p i
n X
jaij j
j¼1
Lj gj
Z
2ai ci pi ai pi
i¼1
1
kij ðsÞðu2j ðtÞ u2j ðt sÞÞ ds
0 n X j¼1
! n Li X jaij Lj gj jaij jaj pj u2i ðtÞ: gi j¼1
The rest of the proof is similar to that of the above theorem. h By varying the parameters in Theorem 3.3, we immediately have Corollary 3.2. Assume that (H1 )–(H2 ), (S1 )–(S2 ) and (3.3) hold. If one of the following conditions holds for some positive real numbers gi , pi , i 2 N ð1; nÞ, then the equilibrium x of (1.4) is globally asymptotically stable. (1)
2ai ci ai
n X
jaij jLj gj
j¼1
(2)
2ai ci pi ai pi
n X j¼1
n Li X jaji jaj > 0; gi j¼1
jaij jLj Li
n X j¼1
jaji jaj pj > 0;
ð3:12Þ
ð3:13Þ
100
(3)
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
2ai ci ai
n X
jaij jLj Li
j¼1
(4)
2ai ci pi ai pi
n X
n X
n X
jaij j L2i
j¼1
(5)
2ai ci pi ai pi
2ai ci ai
n X
jaij jL2i
n X
jaij j L2i
j¼1
(7)
2ai ci ai
n X j¼1
ð3:14Þ
jaji jaj pj > 0;
ð3:15Þ
jaji jaj pj > 0;
ð3:16Þ
j¼1
n X
j¼1
(6)
jaji jaj > 0;
j¼1
j¼1
n X
jaji jaj > 0;
ð3:17Þ
jaji jaj > 0:
ð3:18Þ
j¼1
jaij jL2i
n X j¼1
Proof. The above conditions can be obtained by letting pi ¼ 1; gi ¼ 1; pi ¼ gi ¼ 1; gi ¼ 1=Li ; gi ¼ Li ; pi ¼ 1, gi ¼ 1=Li ; pi ¼ 1, gi ¼ Li in (3.10) respectively. h With respect to the global exponential stability of the equilibrium of (1.4), we have Theorem 3.4. Assume that all conditions in Theorem 3.2 are satisfied, in addition, if kij ðtÞ 6 edt ;
t > T0 ; i; j 2 f1; 2; . . . ng
ð3:19Þ
holds for some positive numbers d and T0 , then the equilibrium of (1.4) is globally exponentially asymptotically stable. Precisely, we have ! n n X X r1 t jxi ðtÞ xi j 6 C1 e sup j/j ðxÞ xj j ; t > 0; ð3:20Þ i¼1
j¼1
x2ð1;0
where C1 > 0 and r1 > 0 will be specified later. Proof. From Theorem 3.2, we have n X djui ðtÞj 6 ai ci jui ðtÞj þ ai jaij jLj dt j¼1
Z 0
1
kij ðsÞjuj ðt sÞj ds
ð3:21Þ
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
101
Since (3.4) holds, we can choose a positive real number r1 2 ð0; dÞ such that n X
qi ai ci qi r1 Li
m :¼ min
16i6n
Z
aj qj jaij j
!
1
kji ðsÞe
r1 s
ds
ð3:22Þ
> 0:
0
j¼1
Let yi ðtÞ ¼ er1 t jui ðtÞj, direct calculation shows that dyi ðtÞ 0 ¼ er1 t ðr1 jui ðtÞj þ jui ðtÞj Þ dt 6e
r1 t
r1 jui ðtÞj ai ci jui ðtÞj þ ai
n X
Lj jaij j
Z
¼ ðai ci r1 Þyi ðtÞ þ ai
Z
Lj jaij j
kij ðsÞjuj ðt sÞj ds
0
j¼1 n X
!
1
1
kij ðsÞer1 s yj ðt sÞj ds:
0
j¼1
Define V ðtÞ ¼ V ðyÞðtÞ ¼
n X
qi yi ðtÞ þ ai qi
i¼1
n X
Lj jaij j
Z
Z
1
kij ðsÞer1 s
0
j¼1
!
t
yj ðxÞ dx ds :
ts
ð3:23Þ It is easy to show that þ
D V ðtÞ ¼
n X
qi y_ i ðtÞ þ ai qi
i¼1
6
n X
Lj jaij j
Z
qi ðai ci r1 Þyi ðtÞ þ ai qi
i¼1
¼
qi ðai ci r1 Þ Li
i¼1 n X i¼1
which indicates that V ðtÞ 6 V ð0Þ:
kij ðsÞe ðyj ðtÞ yj ðt sÞÞ ds n X
Lj jaij j
n X j¼1
yi ðtÞ 6 0;
aj qj jaji j
Z
!
1 r1 s
kij ðsÞe yi ðtÞ ds
0
j¼1
n X
6 m
r1 s
0
j¼1
n X
!
1
Z 0
!
1
kji ðsÞe
r1 s
yi ðtÞ
102
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
Hence, we have n X
qi yi ðtÞ 6 V ðtÞ 6 V ð0Þ
i¼1
6
n X
qi yi ð0Þ þ ai qi
i¼1
6
n X
Lj jaij j
q i þ Li
i¼1
n X
aj qj jaji j
!
1
kij ðsÞse
0
j¼1
n X
Z
Z
!
1
kij ðsÞser1 s ds
ds sup yj ðxÞ x2ð1;0
sup yj ðxÞ: x2ð1;0
0
j¼1
r1 s
By virtue of (3.19) and (1.5), it follows that Z 1 kji :¼ kji ðsÞser1 s ds 0 Z T0 Z 1 ¼ þ kji ðsÞser1 s ds 0 T0 Z 1 r1 T0 6 T0 e þ seðdr1 Þs ds 0
¼ T0 e
r1 T0
þ
1 ðd r1 Þ
2
< 1:
Putting Pn max1 6 i 6 n fqi þ Li j¼1 aj qj jaji jkji g C1 :¼ min qi 16i6n
and noting that sup yi ðxÞ ¼ x2ð1;0
sup er1 x j/i ðxÞ xi j 6
x2ð1;0
sup j/i ðxÞ xi j;
x2ð1;0
we have n X
yi ðtÞ 6 C1
n X
sup j/j ðxÞ xj j;
j¼1 x2ð1;0
i¼1
which implies that n X
jxi ðtÞ xi j 6 C1 er1 t
i¼1
Thus the proof is complete. Similarly, we have
n X
sup j/j ðxÞ xj j:
j¼1 x2ð1;0
h
ð3:24Þ
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
103
Theorem 3.5. Assume that all conditions in Theorem 3.3 and (3.19) are satisfied, Then the equilibrium of (1.4) is globally exponentially stable in the sense that the following inequality holds. ! n n X X 2 r2 t 2 ðxi ðtÞ xi Þ 6 C2 e sup ð/j ðxÞ xj Þ ; t > 0; ð3:25Þ i¼1
j¼1
x2ð1;0
where C2 > 0 and r2 > 0 are given by n o Pn max1 6 i 6 n pi þ Lgii j¼1 aj pj jaji jkji C2 :¼ min1 6 i 6 n fpi g and
( r2 :¼ max
16i6n
r 2 ð0; dÞ : 2ai pi ci pi r ai pi
n X
) Z 1 n Li X rs ai pj jaji j kji ðsÞe ds > 0; : gi j¼1 0
jaij jLj gj
j¼1
Corollary 3.1 together with Theorem 3.4 immediately gives Corollary 3.3. If all conditions of Corollary 3.1 and (3.19) are satisfied, then the equilibrium of (1.4) is globally exponentially stable. Combining Corollary 3.2 and Theorem 3.5, we have Corollary 3.4. If all conditions of Corollary 3.2 and (3.19) are satisfied, then the equilibrium of (1.4) is globally exponentially stable. If in (1.4), the kernel functions kij ðtÞ are assumed to have special forms, such as kij ðtÞ ¼
lij ðtÞ; 0;
t 2 ½0; sij ; otherwise:
Then the duration intervals for time delays are finite, and thus the corresponding Cohen–Grossberg neural network model can be described by !! Z t n X x_ i ðtÞ ¼ ai ðxi ðtÞÞ bi ðxi ðtÞÞ aij gj lij ðt sÞxj ðsÞ ds ; ð3:26Þ j¼1
tsij
where i ¼ 1; 2; . . . ; n, and the delay kernel functions lij ðtÞ are subject to Z sij lij ðtÞ dt ¼ 1: ð3:27Þ lij ðtÞ P 0; 0
104
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
Using a similar argument, we have Theorem 3.6. Suppose that (H1 ), (H2 ), (S1 ), (S2 ), (3.3) and (3.4) hold. Then the equilibrium x of (3.26) is globally exponentially stable in the sense that n n X X jxi ðtÞ xi j 6 C3 er3 t max j/j ðxÞ xj j ; t > 0 ð3:28Þ i¼1
j¼1
x2½s;0
with s ¼ maxðsij ; i; j 2 f1; 2; . . . ; ngÞ, r3 > 0 such that ) ( Z sji n X r3 s min ai ci qi qi r3 Li aj qj jaji j lji ðsÞe ds > 0 16i6n
0
j¼1
and Pn max1 6 i 6 n fqi þ Li j¼1 aj qj jaji jsji er3 sji g C3 :¼ : minfqi ; i ¼ 1; 2; . . . ; ng Theorem 3.7. Assume that (H1 )–(H2 ), (S1 )–(S2 ) and (3.3) hold. If there exist positive real numbers p1 > 0, and gi > 0 such that (3.10) holds, then the equilibrium x of (3.26) is globally exponential stable with ! n n X X 2 r4 t 2 ðxi ðtÞ xi Þ 6 C4 e sup ð/j ðxÞ xj Þ ; t > 0; ð3:29Þ i¼1
j¼1
x2ðs;0
where ( r4 :¼ max r > 0 : 2ai pi ci pi r ai pi
n X j¼1
Z s n Li X ai pj jaji j lji ðsÞers ds > 0; gi j¼1 0
jaij jLj gj ) i ¼ 1; 2; . . . ; n :
4. HNNs with distributed delays Clearly the Hopfield neural networks with distributed delays (1.6) has the same equilibria as the system (1.4) does. Applying Theorems 3.2 and 3.4 to system (1.6), we have Theorem 4.1. Suppose there exist qi > 0, i ¼ 1; 2; . . . ; n such that ( ) n X m :¼ min bi qi Li jaji jqj > 0: 16i6n
j¼1
ð4:1Þ
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
105
Then the equilibrium x is globally asymptotically stable. In addition, if (3.19) holds, the x is globally exponentially stable. Letting qi ¼ 1, i ¼ 1; 2; . . . ; n in (4.1), we have Corollary 4.1. Instead of (4.1), if ) ( n X mi :¼ min bi Li jaji j > 0 16i6n
ð4:2Þ
j¼1
holds, then we have the same results as Theorem 4.1. Remark 4.1. Corollary 4.1 coincides with Theorem 3.3 in [21]. Applying Theorems 3.3 and 3.5 to system (1.6), we have Theorem 4.2. Suppose there exist pi > 0, i ¼ 1; 2; . . . ; n such that ( ) n n X Li X min 2bi pi pi jaij jLj gj jaji jpj > 0: 16i6n gi j¼1 j¼1
ð4:3Þ
Then the equilibrium x is globally asymptotically stable. In addition, if (3.19) holds, the x is globally exponentially stable. For the HNNs with finite distributed delays ! Z t n X x_ i ðtÞ ¼ bi xi ðtÞ aij gj lij ðt sÞxj ðsÞds þ Ii ;
i ¼ 1; 2; . . . ; n;
tsij
j¼1
ð4:4Þ applying Theorems 3.6 and 3.7, we have Theorem 4.3. Assume that ðS1 Þ and ðS2 Þ hold. If (4.1) or (4.3) holds, then the equilibrium x of (4.4) is globally exponentially stable. Instead of ðS1 Þ and ðS2 Þ; if we assume that ðA1 Þ and ðA2 Þ hold and bi > 0 for T i ¼ 1; 2; . . . ; n. Linearizing (1.6) at an equilibrium point x ¼ ðx1 ; x2 ; . . . ; xn Þ gives Z 1 n X x_ i ¼ bi xi þ aij gj0 ðxj Þ kij ðsÞxj ðt sÞ ds: ð4:5Þ j¼1
0
Then the characteristic equation would be F ðkÞ ¼ 0;
ð4:6Þ
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L. Wang / Appl. Math. Comput. 160 (2005) 93–110
where
0
1 c1n k1n ðkÞ k b1 þ c11 k11 ðkÞ B C c12 k21 ðkÞ c2n k2n ðkÞ B C F ðkÞ ¼ det B C; .. .. .. @ A . . . cn1 kn1 ðkÞ k bn þ cnn knn ðkÞ R1 0 in which cij ¼ aij gj ðxj Þ and kij ðkÞ ¼ 0 kij ðsÞeks ds.
ð4:7Þ
Theorem 4.4. Suppose F ð0Þ 6¼ 0 and there exist qi > 0, i ¼ 1; 2; . . . ; n such that X ðbi þ jcii jÞqi þ jcij jqj 6 0; for i ¼ 1; 2; . . . ; n: ð4:8Þ j6¼i
Then the equilibrium of (1.6) is asymptotically stable. Proof. Let k be a root of (4.6). Then k is an eigenvalue of the matrix D ¼ ðdij Þ ^ d^ij Þ with with dii ¼ bi þ cii kii ðkÞ dij ¼ cij kii ðkÞ for i, j 2 f1; 2; . . . ; ng. Let Dð 1 1 ^ ^ ^ dij ¼ qi dij qj . Then D ¼ Q DQ, where Q ¼ diagðq1 ; q2 ; . . . ; qn Þ. So D and D are similar and thus have the same eigenvalues. Let k^ be an eigenvalue of D. Ap^ we have for some i 2 f1; 2; . . . ; ng plying the GershgorinÕs theorem [9] to D, X jk^ d^ii j 6 jd^ij j; j6¼i
that is, jk^ dii j 6
X
q1 dij qj :
j6¼i
Therefore, we have ^ 6 Reðdii Þ þ ReðkÞ ¼ ReðkÞ
X
q1 jdij jqj :
j6¼i
If ReðkÞ P 0, then Reðdii Þ ¼ Reðbi þ cii kii ðkÞÞ 6 bi þ jcii jReðjkii ðkÞjÞ 6 bi þ jcii j; and jdij j ¼ jcij kij ðkÞj 6 jcij j: Hence, we have ReðkÞ 6 bi þ jcii j þ
X j6¼i
q1 jcij jqj :
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
107
Multiplying both sides of the above inequalities by qi , we get X qi ReðkÞ 6 ðbi þ jcii jÞqi þ jcij jqj ; j6¼i
which implies that ReðkÞ < 0. From the analysis, we know the equality occurs only when k is real and F ð0Þ 6¼ 0 implies k could not be 0. So we must have ReðkÞ < 0. Which is a contradiction with our assumption ReðkÞ > 0. Thus we show that all the roots of F ðkÞ have negative real parts, which implies that the equilibrium x is locally asymptotically stable. h Remark 4.2. Our proof of Theorem 4.4 is similar to that of Lemma 2.3 in [2]. Furthermore, if we denote K ¼ B þ jCj with B ¼ diagðb1 ; b2 ; . . . ; bn Þ and jCj ¼ ðjcij jÞ and use a similar argument as in [2], then Theorem 4.4 can be modified to: If K is weakly diagonally dominant [14] and F ð0Þ 6¼ 0, then the conclusion in Theorem 4.4 still holds.
5. HNNs with periodic inputs In this section, we consider the HNNs with periodic inputs ! Z t n X x_ i ðtÞ ¼ bi xi ðtÞ aij gj lij ðt sÞxj ðsÞ ds þ Ii ðtÞ; j¼1
tsij
t P 0; i ¼ 1; 2; . . . ; n;
ð5:1Þ
where Ii ðtÞ ¼ Ii ðt þ xÞ for t P 0 and the activation functions gj , j ¼ 1; 2; . . . ; n satisfy ðS1 Þ and ðS2 Þ. The initial conditions associated with (5.1) are given by xi ðsÞ ¼ /i ðsÞ;
s 2 ½s; 0
with jj/jj ¼ max1 6 i 6 n jj/i jj, jj/i jj ¼ maxs2½s;0 j/i ðsÞj for i ¼ 1; 2; . . . ; n. Letting Qi ¼ Pi þ max jIi ðsÞj; s2½0;x
from (5.1), we have x_ i ðtÞ 6 bi xi ðtÞ þ Qi ; which gives
Qi bi t Qi xi ðtÞ 6 xi ð0Þ þ e : bi bi
ð5:2Þ
Therefore, we have jxi ðtÞ 6 2
Qi Qi þ jxi ð0Þjebi t 6 2 þ jxi ð0Þj; bi bi
ð5:3Þ
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which shows that the solution of (5.1) is defined on ½s; 1Þ. Denote xðtÞ ¼ T T ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞ , / ¼ ð/1 ; /2 ; . . . ; /n g . Let ðT ðtÞ/ÞðhÞ ¼ xðtþ h; /Þ for h 2 ½s; 0 . Then T ðtÞ/ is defined for all t P 0 and is an x-periodic process. From (5.3), we can see T ðtÞÞ is also a bounded map (orbits bounded sets to bounded sets) and point dissipative. Hence from Theorem 4.1.11 in [13], we have Theorem 5.1. Assume that the activation functions gi ; i ¼ 1; 2; . . . ; n in (5.1) are continuous and satisfy ðS2 Þ, then (5.1) has an x-periodic solution, we may denote T it as xðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞÞ . We can even prove that the x-periodic solution xðtÞ may be globally exponentially stable. Indeed, we have Theorem 5.2. Assume that ðS1 Þ and ðS2 Þ hold. If (4.1) or (4.3) holds, then the xperiodic solution xðtÞ (5.1) is globally exponentially stable. Proof. Let xðtÞ be a solution of (5.1) other than xðtÞ. Then we have ! Z t n X d ½xi ðtÞ xi ðtÞ ¼ bi ½xi ðtÞ xi ðtÞ þ aij gj lij ðt sÞxj ðsÞ ds dt tsij j¼1 ! Z t n X aij gj lij ðt sÞxj ðsÞ ds ð5:4Þ j¼1
tsij
Letting ui ðtÞ ¼ xi ðtÞ xi ðtÞ, we then have Z t n X djui ðtÞj 6 bi jui ðtÞj þ jaij jLj lij ðt sÞjuj ðsÞj ds: dt tsij j¼1 Using the similar arguments as in Section 3, we can complete the proof.
h
For the HNNs with infinite distributed delays and periodic inputs Z t n X x_ i ðtÞ ¼ bi xi ðtÞ aij gj kij ðt sÞxj ðsÞds þ Ii ðtÞ; t P 0;
ð5:5Þ
j¼1
1
with Ii ðtÞ ¼ Ii ðt þ xÞ, i ¼ 1; 2; . . . ; n,. Using the result in [20], we can show Theorem 5.3. Assume that ðS1 Þ and ðS2 Þ hold. Then system (5.5) admits an xperiodic solution. Moreover, if (3.19) and (4.1) (or (4.3)) hold, then x-periodic solution is globally (exponentially) stable.
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109
References [1] J. Belair, Stability in a model of a delayed neural network, J. Dynam. Differ. Equat. 5 (1993) 607–623. [2] S. Campbell, Delay independent stability for additive neural networks, Differ. Equat. Dynam. Syst. 3&4 (9) (2001) 115–138. [3] Y. Cao, Q. Wu, A note on stability of analog neural networks with time delays, IEEE Trans. Neural Networks 7 (1996) 1533–1535. [4] G. Carpenter, M. Cohen, S. Grossberg, Computing with neural networks, Science 235 (1987) 1226–1227. [5] M. Cohen, S. Grossberg, Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern. SMC 13 (5) (1983) 815–825. [6] K. Deimling, Nonlinear Functional Analysis, Springer, New York, 1985. [7] M. Forti, On global asymptotic stability of a class of nonlinear systems arising in neural network theory, J. Differ. Equat. 113 (1994) 246–264. [8] M. Forti, S. Manetti, M. Marini, A condition for global convergence of a class of symmetric neural networks, IEEE Trans. Circ. Syst. 39 (1992) 480–483. [9] J. Franklin, Matrix Theory, Prentice-Hall, London, 1968. [10] K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Kluwer, Dordrecht, 1992. [11] K. Gopalsamy, X. He, Stability in asymmetric Hopfield nets with transmission delays, Physica D 76 (1994) 344–358. [12] Z. Guan, G. Chen, Y. Qin, On equilibria, stability, and instability of Hopfield neural networks, IEEE Trans. Neural Networks 2 (2000) 534–540. [13] J.K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, Amer. Math. Soc., RI, 1988. [14] J. Hofbauer, J. So, Diagonal dominance and harmless off-diagonal delays, Proc. Amer. Math. Soc. 128 (2000) 2675–2682. [15] J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci. USA 81 (1984) 3088–3092. [16] D. Kelly, Stability in contractive, nonlinear neural networks, IEEE Trans. Biomed. Eng. 37 (1990) 231–242. [17] H. Lu, On stability of nonlinear continuous-time neural networks with delays, Neural Networks 13 (2000) 1135–1143. [18] C. Marcus, R. Westervelt, Stability of analog neural networks with delay, Phys. Rev. A 39 (1989) 347–359. [19] K. Martsuoka, Stability conditions for nonlinear continuous neural networks with asymmetric connection weights, Neural Networks 5 (1992) 495–500. [20] P. Massatt, Stability and fixed points of point dissipative systems, J. Differ. Equat. 40 (1981) 217–231. [21] S. Mohamad, K. Gopalsamy, Dynamics of a class of discrete-time neural networks and their continuous-time counter parts, Math. Comput. Simulation 53 (2000) 1–39. [22] D. Tank, J. Hopfield, Simple neural optimization networks: an A/D converter, signal decision circuit and a linear programming circuit, IEEE Trans. Circ. Syst. 33 (1986) 533–541. [23] P. van den Driessche, X. Zou, Global attractivity in delayed Hopfield neural network models, SIAM J. Appl. Math. 58 (1998) 1878–1890. [24] L. Wang, X. Zou, Harmless delays in Cohen–Grossberg neural network, Physica D 170 (2002) 162–173. [25] J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Walter de Gruyter, New York, 2001.
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[26] J. Wu, Symmetric functional-differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1999) 4799–4838. [27] H. Ye, A. Michel, K. Wang, Qualitative analysis of Cohen–Grossberg neural networks with multiple delays, Phys. Rev. E 51 (1995) 2611–2618.
Stability of Cohen–Grossberg neural networks with distributed delays Lin Wang Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Nfld, Canada A1C 5S7
Abstract Global asymptotic stability and global exponential stability of the Cohen–Grossberg neural networks and Hopfield neural networks with infinite and finite distributed delays are investigated in this paper. For Hopfield neural networks with distributed delays, local stability is also discussed. The existence and global stability of periodic solutions of Hopfield neural networks with distributed delays and periodic inputs are achieved by using the theory of dissipative systems. Ó 2003 Published by Elsevier Inc. Keywords: Cohen–Grossberg neural networks; Distributed delay; Equilibrium; Hopfield neural networks; Liapunov function; Periodic solution; Stability
1. Introduction Cohen and Grossberg [5] proposed a neural network model (CGNN) in 1983 described by the following system ! n X x_ i ðtÞ ¼ ai ðxi ðtÞÞ bi ðxi ðtÞÞ aij gj ðxj ðtÞÞ ; i ¼ 1; 2; . . . ; n; ð1:1Þ j¼1
where i ¼ 1; 2; . . . ; n and n P 2 is the number of neurons in the network; xi describes the activation of the ith neuron; ai represents an amplification Present address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1. E-mail address: [email protected] (L. Wang). 0096-3003/$ - see front matter Ó 2003 Published by Elsevier Inc. doi:10.1016/j.amc.2003.09.014
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function and the function bi can include a constant term indicating a fixed input to the network; the n n connection matrix A ¼ ðaij Þ tells how the neurons are connected in the network; the activation functions gj , j ¼ 1; 2; . . . ; n show how the neurons react to the input. As pointed out in [5] that the system (1.1) includes a number of models from neurobiology, population biology and evolution theory, among which is the Hopfield neural network (HNN) model n X aij gj ðxj ðtÞÞ þ Ii ; i ¼ 1; 2; . . . ; n; ð1:2Þ x_ i ðtÞ ¼ bi xi ðtÞ þ j¼1
where Ii (i ¼ 1; 2; . . . ; n) is a fixed input from outside of the network. Systems (1.1) and (1.2) have attracted great attention of the scientific community and have been extensively investigated, see, for example, [1–4,7,8,11,12,15– 19,21,23,24,26,27]. Among them, Gopalsamy and He [11], and van den Driessche and Zou [23] studied the case with multiple delays x_ i ðtÞ ¼ bi xi ðtÞ þ
n X
tij sj ðxj ðt sij ÞÞ þ Ii ;
i ¼ 1; 2; . . . ; n
ð1:3Þ
j¼1
Indeed, there has been lots of evidence [25] such as finite transmission speed showing that time delays always exist and should be taken into consideration in designing a neural network model. Instead of considering discrete time delays, we incorporate time delays which are continuously distributed over an infinite interval such that the distant past has less influence compared to the most recent neuronsÕ states into system (1.1) and obtain the following CGNN model with infinite distributed delays Z t ! n X x_ i ðtÞ ¼ ai ðxi ðtÞÞ bi ðxi ðtÞÞ aij gj kij ðt sÞxj ðsÞ ds ; ð1:4Þ 1
j¼1
where the delay kernel functions kij ðtÞ are assumed to be piecewise continuous and satisfy Z 1 Z 1 kij ðtÞ P 0; kij ðtÞ dt ¼ 1; tkij ðtÞ dt < 1: ð1:5Þ 0
0
Its a special case is the Hopfield neural network model with distributed delays Z t n X x_ i ðtÞ ¼ bi xi ðtÞ þ aij gj kij ðt sÞxj ðsÞ ds þ Ii ; i ¼ 1; 2; . . . ; n: j¼1
1
ð1:6Þ We will also study the case where the inputs Ii , i ¼ 1; 2; . . . ; n are functions of time t and are periodic with period x, that is, Ii ðtÞ ¼ Ii ðt þ xÞ for all t.
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
95
If delay kernel functions kij ðtÞ are of the form kij ðtÞ ¼ dðt sij Þ;
i; j 2 f1; 2; . . . ; ng;
ð1:7Þ
then system (1.6) reduces to (1.3). Therefore the discrete delays can be included in our models by choosing suitable kernel functions. System (1.6) has been briefly indicated by Gopalsamy and He [11] and Mohamad and Gopalsamy [21]. However, as we will see, their conclusions can be included in our results as special cases. An application of system (1.6) can be found in Tank and Hopfield [22]. The rest of the paper is organized as follows. In Section 2 we introduce our basic notation and assumptions. Section 3 is devoted to the global stability results and proofs for (1.4). Local stability and the applications of our main results to (HNNs) with distributed delays are given in Section 4. (HNNs) with periodic inputs is investigated in Section 5.
2. Preliminaries The initial conditions associated with (1.4) are of the form xi ðsÞ ¼ /i ðsÞ;
i ¼ 1; 2; . . . ; n;
ð2:1Þ
where /i ðsÞ 2 Cðð1; 0 ; RÞ are bounded. We first give some assumptions. (H1) For each i 2 f1; 2; . . . ; ng, ai is bounded, positive and continuous, furthermore we assume 0 < ai 6 ai ðuÞ 6 ai . (H2) For each i 2 f1; 2; . . . ; ng, bi is continuous increasing. For the activation functions gi ðxÞ, i ¼ 1; 2; . . . ; n, they are typically assumed to be sigmoid which implies that they are monotone and smooth, that is, they are required to satisfy the following (A1) gi 2 C 2 ðRÞ, gi0 ðxÞ > 0; supx2R gi0 ðuÞ ¼ gi0 ð0Þ ¼ 1, i ¼ 1; 2; . . . ; n; (A2) gi ð0Þ ¼ 0 and limx!1 gi ðxÞ ¼ 1. One commonly used function is gðxÞ ¼ tanhðxÞ. Without that assumptions on monotonicity and differentiability of the activation function, as pointed out in [23], we can assume (S1) For each i 2 f1; 2; . . . ; ng, gi : R ! R is globally Lipschitz continuous with a Lipschitz constant Li ; (S2) For each i 2 f1; 2; . . . ; ng, jgi ðxÞj 6 Mi , x 2 R for some constant Mi > 0.
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3. CGNNs with distributed delays We first establish an existence result for the equilibrium of system (1.4). Theorem 3.1. If (H1 )–(H2 ) and (S1 )–(S2 ) (or (A1 )–(A2 )) hold, then there exists at least one equilibrium for system (1.4). Proof. By (H1 ), we know that x is an equilibrium of (1.4) if and only if T x ¼ ðx1 ; . . . ; xn Þ is a solution of equations Z t n X aij gj kij ðt sÞxj ðsÞ ds ¼ 0; i ¼ 1; . . . ; n ð3:1Þ bi ðxi Þ j¼1
1
From (S2 ), we have Z t X X n n aij gj kij ðt sÞxj ðsÞ ds 6 ja jM ¼: Pi : j¼1 j¼1 ij j 1 is increasing. Now consider Since (H2 ) holds, then b1 i Z t ! n X 1 xi ¼ hi ðx1 ; x2 ; . . . ; xn Þ ¼ bi aij gj kij ðt sÞxj ðsÞ ds j¼1
1
for i ¼ 1; 2; . . . ; n. We have
1 jhi ðx1 ; x2 ; . . . ; xn Þj 6 max jb1 i ðPi Þj; jbi ð Pi Þj ¼: Di ;
i ¼ 1; 2; . . . ; n:
T
It follows that ðh1 ; h2 ; . . . ; hn Þ maps a bounded set D :¼ ½D1 ; D1 ½D2 ; D2 ½Dn ; Dn to itself. Then the existence of the equilibrium follows from the BrouwerÕs fixed point theorem [6, Theorem 3.2] and the proof is thus complete. h Let x be an equilibrium of (1.4) and uðtÞ ¼ xðtÞ x . Substituting xðtÞ ¼ uðtÞ þ x into (1.4) leads to Z t ! n X u_ i ðtÞ ¼ ai ðui ðtÞÞ bi ðui ðtÞÞ aij fj kij ðt sÞuj ðsÞ ds ð3:2Þ j¼1
1
for i ¼ 1; 2; . . R. ; n, where ai ðui ðtÞÞ ¼ ai ðui ðtÞ þ xi Þ, bi ðui ðtÞÞ ¼ bi ðui ðtÞ þ xi Þ t bi ðxi Þ, fj ¼ gj ð 1 kij ðt sÞuj ðsÞ ds þ xj Þ gj ðxj Þ. Theorem 3.2. Assume that (H1 )–(H2 ) and (S1 )–(S2 ) hold. If there exist ci > 0, and qi > 0 such that bi ðuÞ bi ðvÞ P ci uv
for u 2 R;
i ¼ 1; 2; . . . ; n;
ð3:3Þ
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
97
and ( a i c i q i Li
l :¼ min
16i6n
n X
) aj jaji jqj
ð3:4Þ
> 0:
j¼1
Then the equilibrium x of (1.4) is globally asymptotically stable. Proof. Combining (3.2) with (3.3), we have djui ðtÞj ¼ sgnðui ðtÞÞu_ i ðtÞ dt 6 ai ci jui ðtÞj þ ai
n X
Lj jaij j
n X
t
kij ðt sÞjuj ðsÞj ds 1
j¼1
¼ ai ci jui ðtÞj þ ai
Z
Lj jaij j
Z
1
kij ðsÞjuj ðt sÞj ds;
0
j¼1
where 8 < 1; sgnðxÞ ¼ 0; : 1
x > 0; x ¼ 0; x < 0:
Let V ðtÞ ¼ V ðuÞðtÞ be defined as V ðtÞ ¼
n X
qi jui ðtÞj þ ai qi
i¼1
n X
Lj jaij j
Z
1
kij ðsÞ
0
j¼1
Z
!
t
juj ðxÞj dx ds :
ð3:5Þ
ts
Then the upper righthand derivative of V ðtÞ along the solution of (3.2) is given by ! Z 1 n n X X þ D V ðtÞ 6 qi ai ci jui ðtÞj þ ai Lj jaij j kij ðsÞjuj ðt sÞj ds i¼1
þ
n X i¼1
6
n X
ai qi
n X
Lj jaij j
j¼1
Z
¼
i¼1
kij ðsÞðjuj ðtÞj juj ðt sÞjÞ ds
0
ai ci qi jui ðtÞj þ ai qi
i¼1 n X
n X j¼1
a i c i q i Li
0
j¼1 1
n X j¼1
! jaij jLj juj ðtÞj !
aj jaij jqj jui ðtÞj 6 l
n X
jui ðtÞj:
i¼1
P The above inequality together with (1.5) shows that ni¼1 jui ðtÞj is bounded for all t P 0, thus the solutions of (3.2) exist globally. Moreover, we have
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L. Wang / Appl. Math. Comput. 160 (2005) 93–110
V ðtÞ þ l
Z
n X
t
0
! jui ðsÞj ds 6 V ð0Þ:
i¼1
On the other hand, V ð0Þ ¼
n X
qi jui ð0Þj þ ai qi
i¼1
6
n X
Lj jaij j
0
j¼1
n X
q i þ Li
i¼1
n X
aj qj jaji j
Z
Z
1
1
kij ðsÞ
!
0
juj ðxÞj dx ds
s
!
skji ðsÞ ds
sup j/i ðxÞ xi j < 1;
x2ð1;0
0
j¼1
Z
which implies that n X
jui ðtÞj 2 L1 ð0; 1Þ:
ð3:6Þ
i¼1
By Lemma 1.2.2 in [10], we can obtain n X
jui ðtÞj ! 0
as t ! 1;
ð3:7Þ
i¼1
i.e., xi ðtÞ ! xi
as t ! 1;
i ¼ 1; 2; . . . ; n:
ð3:8Þ
Corollary 3.1. Assume that (H1 ), (H2 ) and (S1 ), (S2 ) hold. In addition to (3.3), if ) ( n X l1 :¼ min ai ci Li aj jaji j > 0 ð3:9Þ 16i6n
j¼1
holds, then the equilibrium x of (1.4) is globally asymptotically stable. Proof. Condition (3.9) implies (3.4) holds for qi ¼ 1, i ¼ 1; 2; . . . ; n, and thus this corollary follows directly from Theorem 3.2. h By using a different Liapunov functional, we have Theorem 3.3. Assume that (H1 )–(H2 ), (S1 )–(S2 ) and (3.3) hold. If there exist positive real numbers pi > 0, and gi > 0 such that for i 2 N ð1; nÞ 2ai ci pi ai pi
n X j¼1
jaij jLj gj
n Li X jaji jaj pj > 0: gi j¼1
Then the equilibrium x of (1.4) is globally asymptotically stable.
ð3:10Þ
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
Proof. Define V ðtÞ ¼ V ðuÞðtÞ by V ðtÞ ¼
n X
pi u2i ðtÞ
þ ai p i
i¼1
n X j¼1
Lj jaij j gj
Z
1
kij ðsÞ
Z
0
99
!
t
u2j ðxÞ dx ds
:
ð3:11Þ
ts
Now we can estimate the upper righthand derivative of V ðtÞ along the solution of (3.2) as follows ! Z 1 n n X X Dþ V ðtÞ 6 pi 2ai ci u2i ðtÞ þ 2ai jui j Lj jaij j kij ðsÞjuj ðt sÞj ds i¼1
þ
j¼1
n X
ai p i
i¼1
6
n X
n X
jaij j
j¼1
Lj gj
Z
1
kij ðsÞðu2j ðtÞ u2j ðt sÞÞ ds
0
2ai ci pi u2i ðtÞ þ ai pi
i¼1
1 0
n X i¼1
¼
n X
jaij jLj
j¼1
Z
þ
0
n X
! ! 2 u ðt sÞ j kij ðsÞ gj u2i ðtÞ þ ds gj ai p i
n X
jaij j
j¼1
Lj gj
Z
2ai ci pi ai pi
i¼1
1
kij ðsÞðu2j ðtÞ u2j ðt sÞÞ ds
0 n X j¼1
! n Li X jaij Lj gj jaij jaj pj u2i ðtÞ: gi j¼1
The rest of the proof is similar to that of the above theorem. h By varying the parameters in Theorem 3.3, we immediately have Corollary 3.2. Assume that (H1 )–(H2 ), (S1 )–(S2 ) and (3.3) hold. If one of the following conditions holds for some positive real numbers gi , pi , i 2 N ð1; nÞ, then the equilibrium x of (1.4) is globally asymptotically stable. (1)
2ai ci ai
n X
jaij jLj gj
j¼1
(2)
2ai ci pi ai pi
n X j¼1
n Li X jaji jaj > 0; gi j¼1
jaij jLj Li
n X j¼1
jaji jaj pj > 0;
ð3:12Þ
ð3:13Þ
100
(3)
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
2ai ci ai
n X
jaij jLj Li
j¼1
(4)
2ai ci pi ai pi
n X
n X
n X
jaij j L2i
j¼1
(5)
2ai ci pi ai pi
2ai ci ai
n X
jaij jL2i
n X
jaij j L2i
j¼1
(7)
2ai ci ai
n X j¼1
ð3:14Þ
jaji jaj pj > 0;
ð3:15Þ
jaji jaj pj > 0;
ð3:16Þ
j¼1
n X
j¼1
(6)
jaji jaj > 0;
j¼1
j¼1
n X
jaji jaj > 0;
ð3:17Þ
jaji jaj > 0:
ð3:18Þ
j¼1
jaij jL2i
n X j¼1
Proof. The above conditions can be obtained by letting pi ¼ 1; gi ¼ 1; pi ¼ gi ¼ 1; gi ¼ 1=Li ; gi ¼ Li ; pi ¼ 1, gi ¼ 1=Li ; pi ¼ 1, gi ¼ Li in (3.10) respectively. h With respect to the global exponential stability of the equilibrium of (1.4), we have Theorem 3.4. Assume that all conditions in Theorem 3.2 are satisfied, in addition, if kij ðtÞ 6 edt ;
t > T0 ; i; j 2 f1; 2; . . . ng
ð3:19Þ
holds for some positive numbers d and T0 , then the equilibrium of (1.4) is globally exponentially asymptotically stable. Precisely, we have ! n n X X r1 t jxi ðtÞ xi j 6 C1 e sup j/j ðxÞ xj j ; t > 0; ð3:20Þ i¼1
j¼1
x2ð1;0
where C1 > 0 and r1 > 0 will be specified later. Proof. From Theorem 3.2, we have n X djui ðtÞj 6 ai ci jui ðtÞj þ ai jaij jLj dt j¼1
Z 0
1
kij ðsÞjuj ðt sÞj ds
ð3:21Þ
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
101
Since (3.4) holds, we can choose a positive real number r1 2 ð0; dÞ such that n X
qi ai ci qi r1 Li
m :¼ min
16i6n
Z
aj qj jaij j
!
1
kji ðsÞe
r1 s
ds
ð3:22Þ
> 0:
0
j¼1
Let yi ðtÞ ¼ er1 t jui ðtÞj, direct calculation shows that dyi ðtÞ 0 ¼ er1 t ðr1 jui ðtÞj þ jui ðtÞj Þ dt 6e
r1 t
r1 jui ðtÞj ai ci jui ðtÞj þ ai
n X
Lj jaij j
Z
¼ ðai ci r1 Þyi ðtÞ þ ai
Z
Lj jaij j
kij ðsÞjuj ðt sÞj ds
0
j¼1 n X
!
1
1
kij ðsÞer1 s yj ðt sÞj ds:
0
j¼1
Define V ðtÞ ¼ V ðyÞðtÞ ¼
n X
qi yi ðtÞ þ ai qi
i¼1
n X
Lj jaij j
Z
Z
1
kij ðsÞer1 s
0
j¼1
!
t
yj ðxÞ dx ds :
ts
ð3:23Þ It is easy to show that þ
D V ðtÞ ¼
n X
qi y_ i ðtÞ þ ai qi
i¼1
6
n X
Lj jaij j
Z
qi ðai ci r1 Þyi ðtÞ þ ai qi
i¼1
¼
qi ðai ci r1 Þ Li
i¼1 n X i¼1
which indicates that V ðtÞ 6 V ð0Þ:
kij ðsÞe ðyj ðtÞ yj ðt sÞÞ ds n X
Lj jaij j
n X j¼1
yi ðtÞ 6 0;
aj qj jaji j
Z
!
1 r1 s
kij ðsÞe yi ðtÞ ds
0
j¼1
n X
6 m
r1 s
0
j¼1
n X
!
1
Z 0
!
1
kji ðsÞe
r1 s
yi ðtÞ
102
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
Hence, we have n X
qi yi ðtÞ 6 V ðtÞ 6 V ð0Þ
i¼1
6
n X
qi yi ð0Þ þ ai qi
i¼1
6
n X
Lj jaij j
q i þ Li
i¼1
n X
aj qj jaji j
!
1
kij ðsÞse
0
j¼1
n X
Z
Z
!
1
kij ðsÞser1 s ds
ds sup yj ðxÞ x2ð1;0
sup yj ðxÞ: x2ð1;0
0
j¼1
r1 s
By virtue of (3.19) and (1.5), it follows that Z 1 kji :¼ kji ðsÞser1 s ds 0 Z T0 Z 1 ¼ þ kji ðsÞser1 s ds 0 T0 Z 1 r1 T0 6 T0 e þ seðdr1 Þs ds 0
¼ T0 e
r1 T0
þ
1 ðd r1 Þ
2
< 1:
Putting Pn max1 6 i 6 n fqi þ Li j¼1 aj qj jaji jkji g C1 :¼ min qi 16i6n
and noting that sup yi ðxÞ ¼ x2ð1;0
sup er1 x j/i ðxÞ xi j 6
x2ð1;0
sup j/i ðxÞ xi j;
x2ð1;0
we have n X
yi ðtÞ 6 C1
n X
sup j/j ðxÞ xj j;
j¼1 x2ð1;0
i¼1
which implies that n X
jxi ðtÞ xi j 6 C1 er1 t
i¼1
Thus the proof is complete. Similarly, we have
n X
sup j/j ðxÞ xj j:
j¼1 x2ð1;0
h
ð3:24Þ
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
103
Theorem 3.5. Assume that all conditions in Theorem 3.3 and (3.19) are satisfied, Then the equilibrium of (1.4) is globally exponentially stable in the sense that the following inequality holds. ! n n X X 2 r2 t 2 ðxi ðtÞ xi Þ 6 C2 e sup ð/j ðxÞ xj Þ ; t > 0; ð3:25Þ i¼1
j¼1
x2ð1;0
where C2 > 0 and r2 > 0 are given by n o Pn max1 6 i 6 n pi þ Lgii j¼1 aj pj jaji jkji C2 :¼ min1 6 i 6 n fpi g and
( r2 :¼ max
16i6n
r 2 ð0; dÞ : 2ai pi ci pi r ai pi
n X
) Z 1 n Li X rs ai pj jaji j kji ðsÞe ds > 0; : gi j¼1 0
jaij jLj gj
j¼1
Corollary 3.1 together with Theorem 3.4 immediately gives Corollary 3.3. If all conditions of Corollary 3.1 and (3.19) are satisfied, then the equilibrium of (1.4) is globally exponentially stable. Combining Corollary 3.2 and Theorem 3.5, we have Corollary 3.4. If all conditions of Corollary 3.2 and (3.19) are satisfied, then the equilibrium of (1.4) is globally exponentially stable. If in (1.4), the kernel functions kij ðtÞ are assumed to have special forms, such as kij ðtÞ ¼
lij ðtÞ; 0;
t 2 ½0; sij ; otherwise:
Then the duration intervals for time delays are finite, and thus the corresponding Cohen–Grossberg neural network model can be described by !! Z t n X x_ i ðtÞ ¼ ai ðxi ðtÞÞ bi ðxi ðtÞÞ aij gj lij ðt sÞxj ðsÞ ds ; ð3:26Þ j¼1
tsij
where i ¼ 1; 2; . . . ; n, and the delay kernel functions lij ðtÞ are subject to Z sij lij ðtÞ dt ¼ 1: ð3:27Þ lij ðtÞ P 0; 0
104
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
Using a similar argument, we have Theorem 3.6. Suppose that (H1 ), (H2 ), (S1 ), (S2 ), (3.3) and (3.4) hold. Then the equilibrium x of (3.26) is globally exponentially stable in the sense that n n X X jxi ðtÞ xi j 6 C3 er3 t max j/j ðxÞ xj j ; t > 0 ð3:28Þ i¼1
j¼1
x2½s;0
with s ¼ maxðsij ; i; j 2 f1; 2; . . . ; ngÞ, r3 > 0 such that ) ( Z sji n X r3 s min ai ci qi qi r3 Li aj qj jaji j lji ðsÞe ds > 0 16i6n
0
j¼1
and Pn max1 6 i 6 n fqi þ Li j¼1 aj qj jaji jsji er3 sji g C3 :¼ : minfqi ; i ¼ 1; 2; . . . ; ng Theorem 3.7. Assume that (H1 )–(H2 ), (S1 )–(S2 ) and (3.3) hold. If there exist positive real numbers p1 > 0, and gi > 0 such that (3.10) holds, then the equilibrium x of (3.26) is globally exponential stable with ! n n X X 2 r4 t 2 ðxi ðtÞ xi Þ 6 C4 e sup ð/j ðxÞ xj Þ ; t > 0; ð3:29Þ i¼1
j¼1
x2ðs;0
where ( r4 :¼ max r > 0 : 2ai pi ci pi r ai pi
n X j¼1
Z s n Li X ai pj jaji j lji ðsÞers ds > 0; gi j¼1 0
jaij jLj gj ) i ¼ 1; 2; . . . ; n :
4. HNNs with distributed delays Clearly the Hopfield neural networks with distributed delays (1.6) has the same equilibria as the system (1.4) does. Applying Theorems 3.2 and 3.4 to system (1.6), we have Theorem 4.1. Suppose there exist qi > 0, i ¼ 1; 2; . . . ; n such that ( ) n X m :¼ min bi qi Li jaji jqj > 0: 16i6n
j¼1
ð4:1Þ
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
105
Then the equilibrium x is globally asymptotically stable. In addition, if (3.19) holds, the x is globally exponentially stable. Letting qi ¼ 1, i ¼ 1; 2; . . . ; n in (4.1), we have Corollary 4.1. Instead of (4.1), if ) ( n X mi :¼ min bi Li jaji j > 0 16i6n
ð4:2Þ
j¼1
holds, then we have the same results as Theorem 4.1. Remark 4.1. Corollary 4.1 coincides with Theorem 3.3 in [21]. Applying Theorems 3.3 and 3.5 to system (1.6), we have Theorem 4.2. Suppose there exist pi > 0, i ¼ 1; 2; . . . ; n such that ( ) n n X Li X min 2bi pi pi jaij jLj gj jaji jpj > 0: 16i6n gi j¼1 j¼1
ð4:3Þ
Then the equilibrium x is globally asymptotically stable. In addition, if (3.19) holds, the x is globally exponentially stable. For the HNNs with finite distributed delays ! Z t n X x_ i ðtÞ ¼ bi xi ðtÞ aij gj lij ðt sÞxj ðsÞds þ Ii ;
i ¼ 1; 2; . . . ; n;
tsij
j¼1
ð4:4Þ applying Theorems 3.6 and 3.7, we have Theorem 4.3. Assume that ðS1 Þ and ðS2 Þ hold. If (4.1) or (4.3) holds, then the equilibrium x of (4.4) is globally exponentially stable. Instead of ðS1 Þ and ðS2 Þ; if we assume that ðA1 Þ and ðA2 Þ hold and bi > 0 for T i ¼ 1; 2; . . . ; n. Linearizing (1.6) at an equilibrium point x ¼ ðx1 ; x2 ; . . . ; xn Þ gives Z 1 n X x_ i ¼ bi xi þ aij gj0 ðxj Þ kij ðsÞxj ðt sÞ ds: ð4:5Þ j¼1
0
Then the characteristic equation would be F ðkÞ ¼ 0;
ð4:6Þ
106
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
where
0
1 c1n k1n ðkÞ k b1 þ c11 k11 ðkÞ B C c12 k21 ðkÞ c2n k2n ðkÞ B C F ðkÞ ¼ det B C; .. .. .. @ A . . . cn1 kn1 ðkÞ k bn þ cnn knn ðkÞ R1 0 in which cij ¼ aij gj ðxj Þ and kij ðkÞ ¼ 0 kij ðsÞeks ds.
ð4:7Þ
Theorem 4.4. Suppose F ð0Þ 6¼ 0 and there exist qi > 0, i ¼ 1; 2; . . . ; n such that X ðbi þ jcii jÞqi þ jcij jqj 6 0; for i ¼ 1; 2; . . . ; n: ð4:8Þ j6¼i
Then the equilibrium of (1.6) is asymptotically stable. Proof. Let k be a root of (4.6). Then k is an eigenvalue of the matrix D ¼ ðdij Þ ^ d^ij Þ with with dii ¼ bi þ cii kii ðkÞ dij ¼ cij kii ðkÞ for i, j 2 f1; 2; . . . ; ng. Let Dð 1 1 ^ ^ ^ dij ¼ qi dij qj . Then D ¼ Q DQ, where Q ¼ diagðq1 ; q2 ; . . . ; qn Þ. So D and D are similar and thus have the same eigenvalues. Let k^ be an eigenvalue of D. Ap^ we have for some i 2 f1; 2; . . . ; ng plying the GershgorinÕs theorem [9] to D, X jk^ d^ii j 6 jd^ij j; j6¼i
that is, jk^ dii j 6
X
q1 dij qj :
j6¼i
Therefore, we have ^ 6 Reðdii Þ þ ReðkÞ ¼ ReðkÞ
X
q1 jdij jqj :
j6¼i
If ReðkÞ P 0, then Reðdii Þ ¼ Reðbi þ cii kii ðkÞÞ 6 bi þ jcii jReðjkii ðkÞjÞ 6 bi þ jcii j; and jdij j ¼ jcij kij ðkÞj 6 jcij j: Hence, we have ReðkÞ 6 bi þ jcii j þ
X j6¼i
q1 jcij jqj :
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
107
Multiplying both sides of the above inequalities by qi , we get X qi ReðkÞ 6 ðbi þ jcii jÞqi þ jcij jqj ; j6¼i
which implies that ReðkÞ < 0. From the analysis, we know the equality occurs only when k is real and F ð0Þ 6¼ 0 implies k could not be 0. So we must have ReðkÞ < 0. Which is a contradiction with our assumption ReðkÞ > 0. Thus we show that all the roots of F ðkÞ have negative real parts, which implies that the equilibrium x is locally asymptotically stable. h Remark 4.2. Our proof of Theorem 4.4 is similar to that of Lemma 2.3 in [2]. Furthermore, if we denote K ¼ B þ jCj with B ¼ diagðb1 ; b2 ; . . . ; bn Þ and jCj ¼ ðjcij jÞ and use a similar argument as in [2], then Theorem 4.4 can be modified to: If K is weakly diagonally dominant [14] and F ð0Þ 6¼ 0, then the conclusion in Theorem 4.4 still holds.
5. HNNs with periodic inputs In this section, we consider the HNNs with periodic inputs ! Z t n X x_ i ðtÞ ¼ bi xi ðtÞ aij gj lij ðt sÞxj ðsÞ ds þ Ii ðtÞ; j¼1
tsij
t P 0; i ¼ 1; 2; . . . ; n;
ð5:1Þ
where Ii ðtÞ ¼ Ii ðt þ xÞ for t P 0 and the activation functions gj , j ¼ 1; 2; . . . ; n satisfy ðS1 Þ and ðS2 Þ. The initial conditions associated with (5.1) are given by xi ðsÞ ¼ /i ðsÞ;
s 2 ½s; 0
with jj/jj ¼ max1 6 i 6 n jj/i jj, jj/i jj ¼ maxs2½s;0 j/i ðsÞj for i ¼ 1; 2; . . . ; n. Letting Qi ¼ Pi þ max jIi ðsÞj; s2½0;x
from (5.1), we have x_ i ðtÞ 6 bi xi ðtÞ þ Qi ; which gives
Qi bi t Qi xi ðtÞ 6 xi ð0Þ þ e : bi bi
ð5:2Þ
Therefore, we have jxi ðtÞ 6 2
Qi Qi þ jxi ð0Þjebi t 6 2 þ jxi ð0Þj; bi bi
ð5:3Þ
108
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
which shows that the solution of (5.1) is defined on ½s; 1Þ. Denote xðtÞ ¼ T T ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞ , / ¼ ð/1 ; /2 ; . . . ; /n g . Let ðT ðtÞ/ÞðhÞ ¼ xðtþ h; /Þ for h 2 ½s; 0 . Then T ðtÞ/ is defined for all t P 0 and is an x-periodic process. From (5.3), we can see T ðtÞÞ is also a bounded map (orbits bounded sets to bounded sets) and point dissipative. Hence from Theorem 4.1.11 in [13], we have Theorem 5.1. Assume that the activation functions gi ; i ¼ 1; 2; . . . ; n in (5.1) are continuous and satisfy ðS2 Þ, then (5.1) has an x-periodic solution, we may denote T it as xðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞÞ . We can even prove that the x-periodic solution xðtÞ may be globally exponentially stable. Indeed, we have Theorem 5.2. Assume that ðS1 Þ and ðS2 Þ hold. If (4.1) or (4.3) holds, then the xperiodic solution xðtÞ (5.1) is globally exponentially stable. Proof. Let xðtÞ be a solution of (5.1) other than xðtÞ. Then we have ! Z t n X d ½xi ðtÞ xi ðtÞ ¼ bi ½xi ðtÞ xi ðtÞ þ aij gj lij ðt sÞxj ðsÞ ds dt tsij j¼1 ! Z t n X aij gj lij ðt sÞxj ðsÞ ds ð5:4Þ j¼1
tsij
Letting ui ðtÞ ¼ xi ðtÞ xi ðtÞ, we then have Z t n X djui ðtÞj 6 bi jui ðtÞj þ jaij jLj lij ðt sÞjuj ðsÞj ds: dt tsij j¼1 Using the similar arguments as in Section 3, we can complete the proof.
h
For the HNNs with infinite distributed delays and periodic inputs Z t n X x_ i ðtÞ ¼ bi xi ðtÞ aij gj kij ðt sÞxj ðsÞds þ Ii ðtÞ; t P 0;
ð5:5Þ
j¼1
1
with Ii ðtÞ ¼ Ii ðt þ xÞ, i ¼ 1; 2; . . . ; n,. Using the result in [20], we can show Theorem 5.3. Assume that ðS1 Þ and ðS2 Þ hold. Then system (5.5) admits an xperiodic solution. Moreover, if (3.19) and (4.1) (or (4.3)) hold, then x-periodic solution is globally (exponentially) stable.
L. Wang / Appl. Math. Comput. 160 (2005) 93–110
109
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