Convergence dynamics of CohenâGrossberg ... - Semantic Scholar
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Applied Mathematics and Computation 202 (2008) 188–199 www.elsevier.com/locate/amc
Convergence dynamics of Cohen–Grossberg neural networks with continuously distributed delays q Yimin Meng, Shangjiang Guo *, Lihong Huang College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, People’s Republic of China
Abstract In this paper, we consider the convergence dynamics of Cohen–Grossberg neural networks (CGNNs) with continuously distributed delays. Without assuming the differentiability and monotonicity of activation functions, the differentiability of amplification functions and the symmetry of synaptic interconnection weights, we construct suitable Lyapunov functionals and employ inequality technique to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, global exponential convergence, and even global exponential stability of equilibria. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria and can be applied to neural networks including Hopfield neural networks, bidirectional association memory neural networks and cellular neural networks. Ó 2008 Elsevier Inc. All rights reserved. Keywords: Cohen–Grossberg neural networks; Continuously distributed delays; Global exponential stability
1. Introduction It is well known that for neural networks with delays, it is rather difficult to analyze their stability properties due to introduction of delays. On the other hand, although the use of constant discrete delays in models with delayed feedback provides a good approximation to simple circuits consisting of a small number of neurons, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus, there will be a distribution of propagation delays. In this case, the signal propagation is no longer instantaneous and cannot be modelled with discrete time delay. A more appropriate way is to incorporate distributed delays. Tank and Hopfield [16] have proposed a neural circuit with distributed delays, which solves a general problem of recognizing patterns in a time-dependent signal. For the applications of neural networks with distributed delays as described in [16], the readers may also refer q This work was supported in part by the National Natural Science Foundation of PR China (Grant No. 10601016), by the Program for New Century Excellent Talents in University of Education Ministry of China (Grant No. [2007]70), and by the Hunan Provincial Natural Science Foundation (Grant No. 06JJ3001), and by the Hunan University Science Foundation. * Corresponding author. E-mail address: [email protected] (S. Guo).
0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.01.030
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
189
to [2,8–10,14,17,18]. Moreover, a neural network model with distributed delay is more general than one with discrete delay. This is because the distributed delay becomes a discrete delay when the delay kernel is a delta function at a certain time. Cohen–Grossberg neural networks (CGNN, in brief, see [3]), as an important recurrent neural networks model, have aroused a tremendous surge of investigation in these years. In this paper, we investigate the convergence dynamics of CGNNs with continuously distributed delays modelled by the following system of delayed differential equations: ! n Z t X x_ i ¼ ai ðxi Þ bi ðxi Þ tij ðt sÞfj ðxj ðsÞÞds þ I i ð1Þ j¼1
1
for i ¼ 1; 2; . . . ; n, where n P 2 is the number of neurons in the network, xi denotes the state variable associated to the ith neuron, ai is positive and continuous and represents an amplification function, and bi is an appropriately behaved continuous function, I i is the constant input from outside of the network. The activation function fj is continuous and shows how the jth neuron reacts to the input. The kernels tij for i; j ¼ 1; 2; . . . ; n, are real valued continuous functions defined on ½0; þ1Þ, and satisfy Z 1 Z 1 tij ðtÞdt ¼ mij and jtij ðtÞjdt ¼ k ij : 0
0
In particular, if tij ðsÞ ¼ mij dðs sij Þ, i; j ¼ 1; 2; . . . ; n, where sij P 0, and dðsÞ is the Dirac delta function. Then we can rewrite (1) as ! n X x_ i ¼ ai ðxi Þ bi ðxi Þ mij fj ðxj ðt sij ÞÞ þ I i ; i ¼ 1; . . . ; n; ð2Þ j¼1
which is a CGNNs with discrete delays and has been investigated by many researchers [5,12,19,20]. It is seen that (2) includes the Hopfield neural network as a special case, which is of the form n X xi þ mij fj ðxj ðt sij ÞÞ þ I i ; i ¼ 1; 2; . . . ; n; ð3Þ x_ i ¼ C i Ri j¼1 where the positive constant C i and Ri are the neuron amplifier input capacitances and resistances, respectively; xi , fj , I i , and T ¼ ðmij Þnn are the same as in (1). Our purpose in this paper is to discuss the convergence dynamics of (1), including the existence, uniqueness, global asymptotic stability (GAS), global exponential convergence (GEC), and even global exponential stability (GES) of equilibria. In fact, the GAS of a unique equilibrium for the model system is of great importance from a theoretical and an application point of view in several fields, and has been the major concern of many authors. Because in order to embed and solve many problems in applications of neural networks to parallel computations, signal processing and other problems involving the optimization, the dynamic neural networks have to be designed to have only a unique equilibrium point which is GAS (i.e., has the whole space as its domain of attraction) to avoid the risk of spurious responses or the problem of local minima. The initial conditions associated with system (1) are of the form xi ðtÞ ¼ ui ðtÞ;
t 2 ð1; 0; i ¼ 1; 2; . . . ; n;
ð4Þ
where each ui is given bounded continuous function on ð1; 0. It is well known that by the fundamental theory of functional differential equations [6], system (1) has a unique solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞ satisfying the initial conditions (4). The equilibrium or pattern x of (1) is said to be GAS if it is locally stable in sense of Lyapunov and globally attractive, i.e., every solution of (1) corresponding to an arbitrary given set of initial values satisfy limt!1 xi ðtÞ ¼ xi , i ¼ 1; 2; . . . ; n. To prove the GAS of the equilibrium, we will exploit the Lyapunov direct method. Moreover, if there exist constants M P 1 and k > 0 such that for every solution xðtÞ of (1) with any initial value x0 2 Cðð1; 0; Rn Þ, jxi ðtÞ xi j 6 M kx0 x kekt ;
i ¼ 1; 2; . . . ; n;
then x is called to be globally exponentially stable (GES) and k is called to be globally exponentially convergent rate.
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The following assumptions play an important role in analyzing the GAS and even GES of the equilibrium and convergent rate of solutions of system (1). (H1) For each i 2 f1; 2; . . . ; ng, fi is global Lipschitz with a Lipschitz constant di , bi is monotonically increasing and its inverse b1 is global Lipschitz with i R 1 a cLipschitz constant ai > 0. (H2) There exists a nonnegative constant c such that j 0 aixðxÞ dxj ¼ þ1 for all i ¼ 1; 2; . . . ; n. (H3) There exist positive real constants M i and mi , such that M i P ai ðxÞ P mi for all x 2 R and i 2 f1; 2; . . . ; ng. In literature such as [15,17,18], the activation functions fi ði ¼ 1; 2; . . . ; nÞ are usually assumed to be bounded and/or continuously differentiable or bounded continuously differentiable. Moreover, bi is assumed to be linear or differentiable with b0i > 0. Clearly, assumption (H1) is weaker than those given in the earlier references and is much more realistic because it makes the system include the important cellular neural networks. In addition, in most literature (to name a few, see [2,9–11,17,18]), assumption (H3) is employed to grantee the GAS or global attractivity of system (1). In this paper, however, we show that (H3) can be used to ensure the GES of system (1) and can be replaced by the much weaker assumption (H2) when considering the GAS of system (1). The paper is organized as follows. We first derive some new criteria to guarantee the existence and the GAS of an equilibrium in Section 2. Section 3 is devoted to GEC and even GES of an equilibrium. In Section 4, two examples are given to illustrate the results obtained in this paper. Finally, some conclusions are drawn in Section 5.
2. Global asymptotic stability In this section, we aim to find some new sufficient conditions ensuring the GAS of the equilibrium of (1). The following lemmas are needed in the proof of our main results. Lemma 2.1 [4]. If G : Rn ! Rn is continuous, and satisfies that G is injective on Rn and that limkxk!1 kGðxÞk ! 1, then GðxÞ is a homeomorphism of Rn . Lemma 2.2 [7]. cxy c1 6 xc þ ðc 1Þy c for all c P 0 and x; y > 0. Theorem 2.1. Under assumption (H1), system (1) has a GAS equilibrium if there exist positive real numbers d i ði ¼ 1; 2; . . . ; nÞ and c P 1 such that Z 1 c1 x dx ¼ þ1 ð5Þ ai ðxÞ 0 and one of the following inequalities is satisfied: (i)
Pn
j¼1 ½ðc
Pn
1Þd i ai k ij dj þ d j aj k ji di < d i c, i ¼ 1; 2; . . . ; n;
c j¼1 ½ðc 1Þd i ai dj þ d j aj k ji di < d i c, i ¼ 1; 2; . . . ; n; Pn (iii) j¼1 ½ðc 1Þd i ai k ij þ d j aj k ji dci < d i c, i ¼ 1; 2; . . . ; n; Pn (iv) j¼1 ½ðc 1Þd i ai k ij dc=ðc1Þ þ d j aj k ji < d i c, i ¼ 1; 2; . . . ; n; j Pn c=ðc1Þ (v) j¼1 ½ðc 1Þd i ai dj þ d j aj k cji < d i c, i ¼ 1; 2; . . . ; n; Pn (vi) j¼1 ½ðc 1Þd i k ij dj aj þ d j k ji di ai < d i c, i ¼ 1; 2; . . . ; n;
(ii)
(vii)
Pn
j¼1 ½ðc
Pn
1Þd i dj aj þ d j k cji di ai < d i c, i ¼ 1; 2; . . . ; n;
(viii) j¼1 ½ðc 1Þd i k ij aj þ d j k ji dci ai < d i c, i ¼ 1; 2; . . . ; n; P (ix) nj¼1 ½ðc 1Þd i k ij dj þ d j k ji di aci < d i c, i ¼ 1; 2; . . . ; n;
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
Pn
j¼1 ½ðc
(x) (xi)
(xii)
Pn
191
c
1Þd i k ij þ d j k ji ðdi ai Þ < d i c, i ¼ 1; 2; . . . ; n;
j¼1 ½ðc
Pn
1Þd i k ij aj djc=ðc1Þ þ d j k ji ai < d i c, i ¼ 1; 2; . . . ; n;
j¼1 ½ðc
1Þd i aj djc=ðc1Þ þ d j k cji ai < d i c, i ¼ 1; 2; . . . ; n.
Proof. Since bi ði ¼ 1; 2; . . . ; nÞ are all monotonically increasing, sgnðxi xi Þ ¼ sgnðbðxi Þ bi ðxi ÞÞ, i ¼ 1; 2; . . . ; n. We first consider the existence of equilibria of system (1). Since ai is positive, a point t x ¼ ðx1 ; x2 ; . . . ; xn Þ in Rn is an equilibrium of system (1) if and only if this point x is a solution of the following equation: n X bi ðxi Þ mij fj ðxj Þ þ I i ¼ 0; i ¼ 1; 2; . . . ; n: ð6Þ j¼1
Generally, Eq. (6) may have more than one solution x , and hence system (1) may have more than one equi1 librium. It follows from that b1 is global Lipschitz that jx yj ¼ jb1 i i ðbi ðxÞÞ bi ðbi ðyÞÞj 6 ai jbi ðxÞ bi ðyÞj, t i.e., jbi ðxÞ bi ðyÞj P a1 i jx yj for all x; y 2 R, i ¼ 1; 2; . . . ; n. Let gðxÞ ¼ ðg 1 ðxÞ; g 2 ðxÞ; . . . ; g n ðxÞÞ , where n X gi ðxÞ ¼ bi ðxi Þ þ mij fj ðxj Þ I i ; i ¼ 1; 2; . . . ; n: j¼1
As we all know, if g is a homeomorphism on Rn , then there exists a unique equilibrium x 2 Rn such that gðx Þ ¼ 0. Moreover, it suffices to discuss the GAS of equilibrium x of the following system: " # n Z t X dxi ¼ ai ðxi Þ bi ðxi Þ bi ðxi Þ tij ðt sÞðfj ðxj ðsÞÞ fj ðxj ÞÞds : ð7Þ dt 1 j¼1 In what follows, we only prove that conditions (i) and (vi) both ensure that system (1) has a GAS equilibrium. The others conditions can be verified analogously. Case 1: We start to show that condition (i) ensures that g is a homeomorphism on Rn . In fact, under condition (i), in view of k ij P jmij j for all i; j, we have n X #1 :¼ min fd i c1 ½ðc 1Þd i ai jmij jdj þ d j aj jmji jdi g > 0: 16i6n
j¼1
We first claim that g is injective on Rn . By way of contrary, assume that exist x; y 2 Rn with x 6¼ y such that gðxÞ ¼ gðyÞ. Then, we have n X c1 c c1 d i ai jmij jdj jxj y j jjxi y i j : ð8Þ 0 ¼ d i ai ðgi ðxÞ gi ðyÞÞjxi y i j sgnðxi y i Þ 6 d i jxi y i j þ j¼1
Then (8) implies that n n X n X X c c c 06 d i jxi y i j þ c1 d i ai jmij jdj jxj y j j þ ðc 1Þjxi y i j i¼1
¼
n X i¼1
i¼1
d i þ c
1
n X
j¼1
ðc 1Þd i ai jmij jdj þ d j aj jmji jdi
! c
jxi y i j 6 #1
j¼1
n X
c
jxi y i j ;
i¼1 n
which is a contradiction. So the map g is injective on R . Using a similar argument as above, we can obtain n n n X n X X X c1 c1 c1 d i ai jgi ðxÞ gi ð0Þjjxi j P d i ai jbi ðxi Þ bi ð0Þjjxi j d i ai jmij jjfj ðxj Þ fj ð0Þjjxi j i¼1
i¼1
P
n X i¼1
i¼1 c
d i jxi j
n X n X i¼1
j¼1
j¼1 c1
d i ai jmij jdj jxj jjxi j
P #1
n X
c
jxi j :
i¼1
According to the classical results in functional analysis, for any two different vector norms k k1 and k k2 defined on Rn , they are equivalent in sense that there exist two positive constants c1 and c2 such that
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Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
x 2 Rn :
c1 kxk2 6 kxk1 6 c2 kxk2 ;
Therefore, there exists some suitable constant C 1 such that !1c n X c jxi j P C 1 kxk0 ; i¼1
Pn where k k0 is defined as kxk0 ¼ i¼1 jxi j. Thus, n X d i ai jgi ðxÞ gi ð0Þjjxi jc1 P C c1 #1 kxkc0 : i¼1
Noticing that n X c1 c1 d i ai jgi ðxÞ gi ð0Þjjxi j 6 akgðxÞ gð0Þk0 kxk ; i¼1
where a ¼ max16i6n fd i ai g, we have kgðxÞ gð0Þk0 P a1 C c1 #1 kxk0 : Let kxk ! 1, we get kgðxÞk ! 1. By Lemma 2.1, g is a homeomorphism on Rn . Therefore, (1) has a unique equilibrium, denoted by x . In what follows, we show that x is GAS. We consider the Lyapunov functional V 1 ðtÞ ¼ V 11 ðtÞ þ V 12 ðtÞ, where Z xi ðx x Þc1 n X i V 11 ðtÞ ¼ dx; d i ai a ðxÞ i xi i¼1 Z Z n n 1 t X X c V 12 ðtÞ ¼ c1 d i ai jtij ðsÞjdj jxj ðuÞ xj j du ds: i¼1
0
j¼1
ts
Obviously for any x except x , V 1 ðxÞðtÞ > 0. Calculating the upper right derivatives of V 11 and V 12 along the solution xðtÞ of (7), we have n X
c1
dxi sgn xi xi dt i¼1 # n n Z t X X ¼ d i ai bi ðxi Þ þ bi ðxi Þ þ tij ðt sÞðfj ðxj ðsÞÞ fj ðxj ÞÞds jxi xi jc1 sgn xi xi
Dþ V 11 ðtÞ ¼
i¼1
6
n X i¼1
6
n X
d i ai
jxi xi j ai ðxi Þ "
1
j¼1
" d i jxi
xi jc
n X
þ
j¼1
"
Z
ai jtij ðsÞjdj jxj ðt sÞ
n X
d i ai
i¼1
n X
Z
jxi
xi jc1
#
1
ai jtij ðsÞjdj jxj ðt sÞ
xj jc ds
0
j¼1
þ c1 ðc 1Þ
xj jds
0
n X c d i xi xi þ c1
i¼1
#
1
k ij dj jxi xi j
c
j¼1
and þ
D V 12 ðtÞ ¼ c
1
¼ c1
n X n X i¼1
j¼1
n X
n X
i¼1
j¼1
Z d i ai dj
1
h i c c jtij ðsÞj jxj ðtÞ xj j jxj ðt sÞ xj j ds
0 c
d i ai dj k ij jxj xj j c1
n X n X i¼1
j¼1
Z d i ai dj 0
1
c
jtij ðsÞjjxj ðt sÞ xj j ds:
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
193
Thus Dþ V 1 ðtÞ 6
n X
d i jxi xi jc þ c1
i¼1
n X n X i¼1
d i ai dj k ij jxj xj jc þ c1 ðc 1Þ
n X
j¼1
i¼1
d i ai
n X
k ij dj jxi xi jc
j¼1
n X xi x c < 0; ¼ #1 i i¼1
where
#1
¼ min16i6n fd i c1
Pn
j¼1 ½ðc
1Þd i ai k ij dj þ d j aj k ji di g > 0. Thus
c
Dþ V 1 ðtÞ 6 k1 kx x k < 0;
ð9Þ
where k1 is some suitable positive constant. Therefore, x is GAS. Case 2: We show that condition (vi) ensures that g is also a homeomorphism on Rn . In fact, under condition (vi), in view of k ij P jmij j for all i; j, we have ( #2 :¼ min
16i6n
di c
1
n X ½ðc 1Þd i jmij jdj aj þ d j jmji jdi ai
) > 0:
j¼1
We first claim that g is injective on Rn . By way of contrary, assume that exist x; y 2 Rn with x 6¼ y such that gðxÞ ¼ gðyÞ. Then, we have c1
0 ¼ d i ðgi ðxÞ gi ðyÞÞjbi ðxi Þ bi ðy i Þj sgnðbi ðxi Þ bi ðy i ÞÞ n X d i jmij jdj jxj y j jjbi ðxi Þ bi ðy i Þjc1 6 d i jbi ðxi Þ bi ðy i Þjc þ j¼1 c
6 d i jbi ðxi Þ bi ðy i Þj þ
n X
d i jmij jdj aj jbj ðxj Þ bj ðy j Þjjbi ðxi Þ bi ðy i Þj
c1
:
j¼1
Using a similar argument as above, we have n n X X c1 c d i ðgi ðxÞ gi ðyÞÞjbi ðxi Þ bi ðy i Þj sgnðbi ðxi Þ bi ðy i ÞÞ 6 #2 jbi ðxi Þ bi ðy i Þj ; 0¼ i¼1
i¼1 n
which is a contradiction. So the map g is injective on R . Similarly, we have n n X X c1 c d i jgi ðxÞ gi ð0Þjjbi ðxi Þ bi ð0Þj P #2 jbi ðxi Þ bi ð0Þj i¼1
i¼1
and hence kgðxÞ gð0Þk P C 2 kbðxÞ bð0Þk P C 2 bkxk; t
where C 2 is some suitable positive constant, b ¼ min16i6n fa1 i g, and bðxÞ ¼ ðb1 ðxÞ; b2 ðxÞ; . . . ; bn ðxÞÞ . Let kxk ! 1, we get kgðxÞk ! 1. By Lemma 2.1, g is a homeomorphism on Rn . Therefore, (1) has a unique equilibrium, denoted by x . In what follows, we show that x is GAS. We consider the Lyapunov functional V 2 ðtÞ ¼ V 21 ðtÞ þ V 22 ðtÞ, where Z xi ½b ðxÞ b ðx Þc1 n X i i i dx; V 21 ðtÞ ¼ d i a ðxÞ i xi i¼1 Z Z n n 1 t X X c V 22 ðtÞ ¼ c1 di jtij ðsÞjdj aj jbj ðxj ðuÞÞ bj ðxj Þj du ds: i¼1
j¼1
0
ts
Obviously for any x except x , V 2 ðxÞðtÞ > 0. Calculating the upper right derivatives of V 21 and V 22 along the solution xðtÞ of (7), we have
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Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199 n X
c1
dxi sgn xi xi dt i¼1 # n n Z t X X d i bi ðxi Þ þ bi ðxi Þ þ tij ðt sÞ fj ðxj ðsÞÞ fj ðxj Þ ds ¼
Dþ V 21 ðtÞ ¼
di
jbi ðxÞ bi ðxi Þj ai ðxi Þ "
i¼1
jbi ðxi Þ bi ðxi Þj þ
6
n X i¼1 n X
c1
sgn xi xi 6
1
0 c
d i ½jbi ðxi Þ bi ðxi Þj þ
n Z X
d i fjbi ðxi Þ
bi ðxi Þjc
þc
jtij ðsÞjdj aj jbj ðxj ðt sÞÞ bj ðxj Þj jbi ðxÞ bi ðxi Þj
1
n Z X
i¼1
1
n X n X
1
n X n X
i¼1
i¼1
c1
ds
1
jtij ðsÞjdj aj jbj ðxj ðt sÞÞ bj ðxj Þjc dsg
0
j¼1 n X n X
c
d i k ij dj aj jbi ðxi Þ bi ðxi Þj
j¼1
and
¼c
1 0
j¼1
þ c1 ðc 1Þ
D V 22 ðtÞ ¼ c
c
d i ½jbi ðxi Þ bi ðxi Þj
jtij ðsÞjdj jxj ðt sÞ xj jjbi ðxÞ bi ðxi Þjc1 ds
i¼1
þ
n X i¼1
n Z X j¼1
6
1
j¼1
Z
1
d i dj aj
h i c c jtij ðsÞj jbj ðxj Þ bj ðxj Þj jbj ðxj ðt sÞÞ bj ðxj Þj ds
0
j¼1
d i dj aj k ij jbj ðxj Þ
c bj ðxj Þj
j¼1
c
1
n X n X i¼1
Z
1
c
jtij ðsÞjjbj ðxj ðt sÞÞ bj ðxj Þj ds:
d i dj aj 0
j¼1
Thus Dþ V 2 ðtÞ 6
n X
c
d i fjbi ðxi Þ bi ðxi Þj þ c1
i¼1 n X n X i¼1 n X
d i k ij dj aj jbi ðxi Þ bi ðxi Þjc
j¼1
bi ðxi Þ bi ðx Þc < 0; a1 i i
i¼1
#1
c
k ij dj aj jbj ðxj Þ bj ðxj Þj g
j¼1
þ c1 ðc 1Þ ¼ #2
n X
where ¼ min16i6n fd i c1 completed. h
Pn
j¼1 ½ðc
1Þd i k ij dj aj þ d j k ji di ai g > 0. Therefore, x is GAS. The proof is
Remark 2.1. In the proof of Theorem 2.1, we can see the importance of assumption (H2), which ensures the radial unboundedness of the Lyapunov function. In most literatures, such as [2,10,11,17,18]), the upper bound of amplification functions ai ðÞ ði ¼ 1; 2; . . . ; nÞ is a prerequisite condition, which is obviously much stronger than (5). On the other hand, in literatures such as [1,13,14,21], the authors tried to drop this prerequisite condition by constructing a invariant set. However, their results are actually on the global attractivity, instead of GAS. Corollary 2.1. Under assumptions (H1) and (H2), if I BKD a non-singular M-matrix, then system (1) has a GAS equilibrium, where I is an identity matrix of size n, B ¼ diagða1 ; . . . ; an Þ, K ¼ ðk ij Þnn , and D ¼ diagðd1 ; . . . ; dn Þ. Proof. Since I BKD a non-singular M-matrix, then there exists a vector D ¼ ðd 1 ; . . . ; d n Þ > 0 such that Dð I BKDÞ > 0. Namely, n X d j aj k ji di : di > j¼1
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
195
Therefore, condition (1) of Theorem 2.1 holds. Hence, system (1) has a GAS equilibrium. The proof is completed. h 3. Global exponential stability If we put some more strict constraints on functions of system (1), for example, (H3), then we can obtain some further results about the GES and convergence rate of solutions of system (1). R1 Theorem 3.1. In addition to (H1) and (H3), assume that 0 jtij ðtÞjek0 t dt < 1 for some positive k0 , and that there exist positive real numbers d i ði ¼ 1; 2; . . . ; nÞ and c P 1 such that one of the following inequalities (i)–(v) in Theorem 2.1 is satisfied, then system (1) has exactly one equilibrium, which is GES. Proof. Similarly, we only prove that conditions (i) ensures that the GES of the equilibrium because the others conditions can be verified analogously. Under condition (i), in view of Theorem 1, system (1) has a GAS equilibrium x . In what follows, we show that x is GES. Obviously, there exists sufficiently small positive constants k < k0 =c such that n Z 1 X kai ½ðc 1Þd i ai jtij ðsÞjdj þ d j aj jtji ðsÞjdi ekcs ds di 1 > c1 mi 0 j¼1 for all i ¼ 1; 2; . . . ; n. Thus, we introduce the following positive constant,which will play a important role in our proof. ( ) n Z 1 X kai 1 kcs #3 ¼ min d i 1 ½ðc 1Þd i ai jtij ðsÞjdj þ d j aj jtji ðsÞjdi e ds : c 16i6n mi 0 j¼1 We consider the Lyapunov functional V 3 ðtÞ ¼ V 31 ðtÞ þ V 32 ðtÞ, where Z xi ðx x Þc1 n X i dxeckt ; V 31 ðtÞ ¼ d i ai a ðxÞ i x i¼1 i Z Z n n 1 t X X V 32 ðtÞ ¼ c1 d i ai jtij ðsÞjeckðuþsÞ dj jxj ðuÞ xj jc du ds: i¼1
j¼1
0
ts
Obviously for any x except x , V 3 ðxÞðtÞ > 0. Calculating the upper right derivatives of V 31 and V 32 along the solution xðtÞ of (7), we have " # n n Z t X X þ D V 31 ðtÞ ¼ d i ai bi ðxi Þ þ bi ðxi Þ þ tij ðt sÞ fj ðxj ðsÞÞ fj ðxj Þ ds i¼1
j¼1
1
Z xi ðx x Þc1 i dxeckt xi þ kc d i ai jxi a ðxÞ i xi i¼1 Z 1 n n X n X X c 6 ekct d i xi xi þ ekct d i ai jtij ðsÞjdj jxj ðt sÞ xj jds jxi xi jc1 xi jc1 eckt sgn xi
i¼1
þk
n X
i¼1
n X
j¼1
0
c
ckt d i ai m1 i jxi xi j e
i¼1
6 ekct
n X
Z n X n X xi x c þ c1 ekct d i ½kai m1 1 d a i i i i
i¼1
þ ð1 c1 Þekct
i¼1 n X
n X
i¼1
j¼1
d i ai k ij dj jxi xi jc
j¼1
0
1
c
jtij ðsÞjdj jxj ðt sÞ xj j ds
196
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
and Dþ V 12 ðtÞ ¼ c1
n X n X i¼1
¼ c1
Z
i¼1
h i jtij ðsÞj jxj ðtÞ xj jc jxj ðt sÞ xj jc ds
0
j¼1
n X n X
1
d i ai dj
c
d i ai dj k ij jxj xj j c1
j¼1
n X n X i¼1
Z
1
d i ai dj
c
jtij ðsÞjjxj ðt sÞ xj j ds:
0
j¼1
Thus, Dþ V 3 ðtÞ 6 ekct
n X
n X n X c c d i ½kai m1 þ ð1 c1 Þekct d i ai k ij dj jxi xi j i 1 xi xi
i¼1
þ c1 ekct
n X n X i¼1
i¼1
Z
1
j¼1
c
jtij ðsÞjecks dsjxj xj j ¼ #3
d i ai dj 0
j¼1
n X xi x c ekct < 0: i i¼1
Therefore, V 3 ðtÞ 6 V 3 ð0Þ. Recall that V 3 ðtÞ P ðM i cÞ1 d i ai jxi ðtÞ xi jc eckt ;
i ¼ 1; 2; . . . ; n
and V 3 ð0Þ 6
n X
1
d i ai ðmi cÞ jxi ð0Þ
i¼1
xi jc
þc
1
n X n X i¼1
j¼1
Z d i ai
1
jtij ðsÞjecks dsdj max jxj ðuÞ xj jc :
0
u60
We can see that there exist some suitable constant C 2 > 0 such that c
c
jxi ðtÞ xi j eckt 6 C c2 kx x k ; i ¼ 1; 2; . . . ; n; P where k k is defined by kxk ¼ ð ni¼1 maxu60 jxi ðuÞjc Þ1=c . That is, jxi ðtÞ xi j 6 C 2 kx x kekt ; This completes the proof.
i ¼ 1; 2; . . . ; n:
h
Corollary 3.1. Under assumptions (H1) and (H3), if I BKD a non-singular M-matrix, then system (1) has a GAS equilibrium, where matrices I, B, K, and D are defined as those in Corollary 2.1. Corollary 3.2. Under assumptions (H1) and (H3), if m BKDM a non-singular M-matrix, then system (1) has a GAS equilibrium, where m ¼ diagðm1 ; . . . ; mn Þ, M ¼ diagðM 1 ; . . . ; M n Þ, B, K, and D are defined as those in Corollary 2.1. Proof. As we know, if A is a M-matrix, then for any two positive diagonal matrices P and Q, PAQ is also Mmatrix. If m BKDM a non-singular M-matrix, then M BKD is a non-singular M-matrix, where M ¼ diagðm1 =M 1 ; . . . ; mn =M n Þ. Thus, there exists a vector D ¼ ðd 1 ; . . . ; d n Þ > 0 such that DðM BKDÞ > 0. Namely, d i mi > M i
n X
d j aj k ji di :
j¼1
Obviously, condition (i) in Theorem 2.1 holds. Hence, system (1) has a GES equilibrium. The proof is completed. h Corollary 3.2 is actually Theorem 4 of [2]. Therefore, Theorem 3.1 not only unifies but also improves the previous results. R1 Theorem 3.2. In addition to (H1) and (H3), assume that 0 jtij ðtÞjtek0 t dt < 1 for some positive k0 , and that there exist positive real numbers d i ði ¼ 1; 2; . . . ; nÞ and c P 1 such that one of the following inequalities (vi)–(xii) in
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
197
Theorem 2.1 is satisfied, then system (1) has exactly one equilibrium and all other solutions converge exponentially to it as t ! 1. Proof. We only prove that conditions (vi) ensures that the global exponential convergence of the equilibrium because the others conditions can be verified analogously. Under condition (vi), in view of Theorem 1, system (1) has a GAS equilibrium x . In what follows, we show that x is globally exponentially convergent. Obviously, there exists sufficiently small positive constants k < k0 =c such that n Z 1 X kai di 1 ½ðc 1Þd i aj jtij ðsÞjdj þ d j ai jtji ðsÞjdi ekcs ds > c1 mi 0 j¼1 for all i ¼ 1; 2; . . . ; n. Thus, we introduce the following positive constant,which will play a important role in our proof. ( ) n Z 1 X kai 1 kcs #4 ¼ min d i 1 ½ðc 1Þd i aj jtij ðsÞjdj þ d j ai jtji ðsÞjdi e ds : c 16i6n mi 0 j¼1 We consider the Lyapunov functional V 4 ðtÞ ¼ V 41 ðtÞ þ V 42 ðtÞ, where Z xi ½b ðxÞ b ðx Þc1 n X i i i V 41 ðtÞ ¼ dxeckt ; d i a ðxÞ i x i¼1 i Z Z n n 1 t X X V 42 ðtÞ ¼ c1 di jtij ðsÞjeckðuþsÞ dj aj jbj ðxj ðuÞÞ bj ðxj Þjc du ds: i¼1
j¼1
0
ts
Obviously for any x except x , V 4 ðxÞðtÞ > 0. Calculating the upper right derivatives of V 41 and V 42 along the solution xðtÞ of (7), we have " # n n Z t X X þ D V 41 ðtÞ ¼ d i bi ðxi Þ þ bi ðxi Þ þ tij ðt sÞ fj ðxj ðsÞÞ fj ðxj Þ ds j¼1
1
c1 bi ðxi Þj eckt sgn xi
xi
i¼1
jbi ðxi Þ
þ kc
n X i¼1
6 ekct
n n X n X X c d i bi ðxi Þ bi ðxi Þ þ ekct di i¼1
þk
n X i¼1
6 ekct
n X
i¼1
j¼1
Z
d i m1 i
Z
xi
½bi ðxÞ xi
c1 bi ðxi Þ dxeckt
1
jtij ðsÞjdj jxj ðt sÞ xj jds jbi ðxi Þ bi ðxi Þj 0
c ckt d i m1 i ai bi ðxi Þ bi ðxi Þ dxe Z n X n X bi ðxi Þ bi ðx Þc þ c1 ekct d i ½kai m1 1 d i i i
i¼1
i¼1 c
bj ðxj Þj ds þ ð1 c1 Þekct
n X n X i¼1
j¼1
d i k ij dj aj jbi ðxi Þ bi ðxi Þj
1
jtij ðsÞjdj aj jbj ðxj ðt sÞÞ
0 c
j¼1
and Dþ V 42 ðtÞ ¼ c1 ekct
Z 0
n X n X i¼1
1
j¼1
Z d i dj aj
1
c
jtij ðsÞjecks jbj ðxj Þ bj ðxj Þj ds c1 ekct
0
jtij ðsÞjjbj ðxj ðt sÞÞ bj ðxj Þjc ds:
n X n X i¼1
j¼1
d i dj aj
c1
198
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
Thus Dþ V 4 ðtÞ 6 ekct
n X
n X n X bi ðxi Þ bi ðx Þc þ ð1 c1 Þekct d i ½kai m1 1 d i k ij dj aj jbi ðxi Þ bi ðxi Þjc i i
i¼1
þ c1 ekct
n X n X i¼1
Z
i¼1 1
d i dj aj
j¼1
c
jtij ðsÞjecks jbj ðxj Þ bj ðxj Þj ds ¼ #4
0
j¼1
n X bi ðxi Þ bi ðx Þc ekct < 0: i i¼1
Therefore, V 4 ðtÞ 6 V 4 ð0Þ. Recall that 1
c
V 4 ðtÞ P ðM i cÞ ai1c d i jxi ðtÞ xi j eckt ;
i ¼ 1; 2; . . . ; n
and V 4 ð0Þ 6
n X
1
c
d i ai ðmi cÞ jbi ðxi ð0ÞÞ bi ðxi Þj þ c1
i¼1
n X n X i¼1
j¼1
Z di
1
c
jtij ðsÞjecks dsdj max jbj ðxj ðuÞÞ bj ðxj Þj : u60
0
We can see that there exist some suitable constant C 3 > 0 such that jxi ðtÞ xi jc eckt 6 C c3 kbðxÞ bðx Þkc ;
i ¼ 1; 2; . . . ; n;
where bðxÞ ¼ ðb1 ðx1 Þ; b2 ðx2 Þ; . . . ; bn ðxn ÞÞ and k k is defined by kxk ¼ ð jxi ðtÞ
xi j
6 C 3 kbðxÞ bðx Þke
This completes the proof.
kt
;
Pn
c 1=c . i¼1 maxu60 jxi ðuÞj Þ
i ¼ 1; 2; . . . ; n;
That is, ð10Þ
h
Remark 3.1. As a result of Theorem 3.2, all solutions of system (1) converge exponentially to the equilibrium x as t ! 1. However, the equilibrium x maybe not GES. Of course, if we put a further restriction on bi , i ¼ 1; 2; . . . ; n, for example, bi ði ¼ 1; 2; . . . ; nÞ are all globally Lipshitz, then it follows from (10) that the equilibrium x is GES. 4. Examples In this section, we illustrate our main results by two examples. We will see that it is not hard to verify the conditions stated in our main theorems. Consider the following CGNNs with distributed delays: " # 2 Z t X x_ i ¼ ai ðxi Þ bðxi Þ tij ðt sÞf ðxj ðsÞÞds ; i ¼ 1; 2: ð11Þ j¼1
1
Example 1. Consider system (11) with a1 ðxÞ ¼ 1 þ x, a2 ðxÞ ¼ 2 þ cos x, bðxÞ ¼ 2x þ jxj, f ðxÞ ¼ 12 ðx þ sin xÞ. Obviously, assumptions (H1) and (H2) hold with d1 ¼ d2 ¼ 1 and a1 ¼ a2 ¼ 1. If the kernels tij ði; j ¼ 1; 2Þ are given by t11 ðsÞ ¼ 0:20es ;
t12 ðsÞ ¼ 1:8e2s ;
t21 ðsÞ ¼ 1:4e2s ;
t22 ðsÞ ¼ 0:15es ;
then k 11 ¼ 0:2, k 12 ¼ 0:9, k 21 ¼ 0:7, k 22 ¼ 0:15. Thus, condition (i) of Theorem 2.1 holds with c ¼ 1, d 1 ¼ 8, and d 2 ¼ 9. Therefore, system (11) has a globally asymptotically stable equilibrium. Example 2. Consider system (11) with bðxÞ ¼ 2x þ jxj and 8 if jxj 6 1; > : 1 if jxj P 2; 3 jxj if jxj 6 1; a2 ðxÞ ¼ 2 if jxj > 1;
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
f ðxÞ ¼
0
199
if x > 0;
x if x 6 0: Obviously, assumptions (H1) and (H3) hold with d1 ¼ d2 ¼ 1, a1 ¼ a2 ¼ 1, m1 ¼ 1, m2 ¼ 2, M 1 ¼ 2, and M 2 ¼ 3. If the kernels tij ði; j ¼ 1; 2Þ are given by t11 ðsÞ ¼ 0:25es ;
t12 ðsÞ ¼ 0:30e2s ;
t21 ðsÞ ¼ 0:50e2s ;
t22 ðsÞ ¼ 0:30es ;
R1 then k 11 ¼ 0:25, k 12 ¼ 0:15, k 21 ¼ 0:25, and k 22 ¼ 0:30. Moreover, it is easy to see that 0 jtij ðsÞjsek0 s < 1 for some k0 2 ð0; 1Þ, and that condition (1) of Theorem 3.1 holds with c ¼ 1 and d 1 < d 2 . Thus, system (11) has a globally exponentially stable equilibrium. 5. Conclusion This Letter is concerned with the convergence dynamics of Cohen–Grossberg neural networks with continuously distributed delays. Without assuming the differentiability and monotonicity of activation functions, the differentiability of amplification functions and the symmetry of synaptic interconnection weights, we construct suitable Lyapunov functionals and employ inequality technique to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, global exponential convergence, and even global exponential stability of equilibria. Our new criteria will definitely be significant for the designs and applications of delayed neural networks. References [1] J. Cao, Q. Song, Stability in Cohen–Grossberg-type bidirectional associative memory neural networks with time-varying delays, Nonlinearity 19 (2006) 1601–1617. [2] Y. Chen, Global asymptotic stability of delayed Cohen–Grossberg neural networks, IEEE Transactions on Circuits and Systems: Regular Papers 53 (2006) 351–357. [3] M.A. Cohen, S. Grossberg, Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybernet. 13 (1983) 815–826. [4] M. Forti, A. Tesi, New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE Trans. Circ. Syst. I Fund. Theory Appl. 42 (1995) 354–366. [5] S. Guo, L. Huang, Stability analysis of Cohen–Grossberg neural networks, IEEE Trans. Neural Networks 17 (2006) 106–117. [6] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993. [7] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, second ed., Cambridge University Press, London, 1952. [8] T. Huang, J. Cao, C. Li, Necessary and sufficient condition for the absolute exponential stability of a class of neural networks with finite delay, Phys. Lett. A 352 (2006) 94–98. [9] T. Huang, A. Chan, Y. Huang, J. Cao, Stability of Cohen–Grossberg neural networks with time-varying delays, Neural Networks 20 (2007) 868–873. [10] T. Huang, C. Li, G. Chen, Stability of Cohen–Grossberg neural networks with unbounded distributed delays, Chaos Solitons Fract. 34 (2007) 992–996. [11] X. Liao, C. Li, Global attractivity of Cohen–Grossberg model with finite and infinite delays, J. Math. Anal. Appl. 315 (2006) 244–262. [12] X. Liao, C. Li, K. Wong, Criteria for exponential stability of Cohen–Grossberg neural networks, Neural Networks 17 (2004) 1401– 1414. [13] K. Lu, D. Xu, Z. Yang, Global attraction and stability for Cohen–Grossberg neural networks with delays, Neural Networks 19 (2006) 1538–1549. [14] Z. Mao, H. Zhao, Dynamical analysis of Cohen–Grossberg neural networks with distributed delays, Phys. Lett. A 364 (2007) 38–47. [15] J. Sun, L. Wan, Global exponential stability and periodic solutions of Cohen–Grossberg neural networks with continuously distributed delays, Phys. D: Nonlinear Phenom. 208 (2005) 1–20. [16] D.W. Tank, J.J. Hopfield, Neural computation by concentrating information in time, Proc. Natl. Acad. Sci. USA 84 (1987) 1896– 1900. [17] L. Wan, J. Sun, Global asymptotic stability of Cohen–Grossberg neural network with continuously distributed delays, Phys. Lett. A 342 (2005) 331–340. [18] L. Wang, Stability of Cohen–Grossberg neural networks with distributed delays, Appl. Math. Comput. 160 (2005) 93–110. [19] L. Wang, X. Zou, Harmless delays in Cohen–Grossberg neural networks, Physica D 170 (2002) 162–173. [20] H. Ye, A.N. Michel, K. Wang, Qualitative analysis of Cohen–Grossberg neural networks with multiple delays, Phys. Rev. E 51 (1995) 2611–2618. [21] J. Zhang, Y. Suda, H. Komine, Global exponential stability of Cohen–Grossberg neural networks with variable delays, Phys. Lett. A 338 (2005) 44–50.
Applied Mathematics and Computation 202 (2008) 188–199 www.elsevier.com/locate/amc
Convergence dynamics of Cohen–Grossberg neural networks with continuously distributed delays q Yimin Meng, Shangjiang Guo *, Lihong Huang College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, People’s Republic of China
Abstract In this paper, we consider the convergence dynamics of Cohen–Grossberg neural networks (CGNNs) with continuously distributed delays. Without assuming the differentiability and monotonicity of activation functions, the differentiability of amplification functions and the symmetry of synaptic interconnection weights, we construct suitable Lyapunov functionals and employ inequality technique to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, global exponential convergence, and even global exponential stability of equilibria. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria and can be applied to neural networks including Hopfield neural networks, bidirectional association memory neural networks and cellular neural networks. Ó 2008 Elsevier Inc. All rights reserved. Keywords: Cohen–Grossberg neural networks; Continuously distributed delays; Global exponential stability
1. Introduction It is well known that for neural networks with delays, it is rather difficult to analyze their stability properties due to introduction of delays. On the other hand, although the use of constant discrete delays in models with delayed feedback provides a good approximation to simple circuits consisting of a small number of neurons, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus, there will be a distribution of propagation delays. In this case, the signal propagation is no longer instantaneous and cannot be modelled with discrete time delay. A more appropriate way is to incorporate distributed delays. Tank and Hopfield [16] have proposed a neural circuit with distributed delays, which solves a general problem of recognizing patterns in a time-dependent signal. For the applications of neural networks with distributed delays as described in [16], the readers may also refer q This work was supported in part by the National Natural Science Foundation of PR China (Grant No. 10601016), by the Program for New Century Excellent Talents in University of Education Ministry of China (Grant No. [2007]70), and by the Hunan Provincial Natural Science Foundation (Grant No. 06JJ3001), and by the Hunan University Science Foundation. * Corresponding author. E-mail address: [email protected] (S. Guo).
0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.01.030
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
189
to [2,8–10,14,17,18]. Moreover, a neural network model with distributed delay is more general than one with discrete delay. This is because the distributed delay becomes a discrete delay when the delay kernel is a delta function at a certain time. Cohen–Grossberg neural networks (CGNN, in brief, see [3]), as an important recurrent neural networks model, have aroused a tremendous surge of investigation in these years. In this paper, we investigate the convergence dynamics of CGNNs with continuously distributed delays modelled by the following system of delayed differential equations: ! n Z t X x_ i ¼ ai ðxi Þ bi ðxi Þ tij ðt sÞfj ðxj ðsÞÞds þ I i ð1Þ j¼1
1
for i ¼ 1; 2; . . . ; n, where n P 2 is the number of neurons in the network, xi denotes the state variable associated to the ith neuron, ai is positive and continuous and represents an amplification function, and bi is an appropriately behaved continuous function, I i is the constant input from outside of the network. The activation function fj is continuous and shows how the jth neuron reacts to the input. The kernels tij for i; j ¼ 1; 2; . . . ; n, are real valued continuous functions defined on ½0; þ1Þ, and satisfy Z 1 Z 1 tij ðtÞdt ¼ mij and jtij ðtÞjdt ¼ k ij : 0
0
In particular, if tij ðsÞ ¼ mij dðs sij Þ, i; j ¼ 1; 2; . . . ; n, where sij P 0, and dðsÞ is the Dirac delta function. Then we can rewrite (1) as ! n X x_ i ¼ ai ðxi Þ bi ðxi Þ mij fj ðxj ðt sij ÞÞ þ I i ; i ¼ 1; . . . ; n; ð2Þ j¼1
which is a CGNNs with discrete delays and has been investigated by many researchers [5,12,19,20]. It is seen that (2) includes the Hopfield neural network as a special case, which is of the form n X xi þ mij fj ðxj ðt sij ÞÞ þ I i ; i ¼ 1; 2; . . . ; n; ð3Þ x_ i ¼ C i Ri j¼1 where the positive constant C i and Ri are the neuron amplifier input capacitances and resistances, respectively; xi , fj , I i , and T ¼ ðmij Þnn are the same as in (1). Our purpose in this paper is to discuss the convergence dynamics of (1), including the existence, uniqueness, global asymptotic stability (GAS), global exponential convergence (GEC), and even global exponential stability (GES) of equilibria. In fact, the GAS of a unique equilibrium for the model system is of great importance from a theoretical and an application point of view in several fields, and has been the major concern of many authors. Because in order to embed and solve many problems in applications of neural networks to parallel computations, signal processing and other problems involving the optimization, the dynamic neural networks have to be designed to have only a unique equilibrium point which is GAS (i.e., has the whole space as its domain of attraction) to avoid the risk of spurious responses or the problem of local minima. The initial conditions associated with system (1) are of the form xi ðtÞ ¼ ui ðtÞ;
t 2 ð1; 0; i ¼ 1; 2; . . . ; n;
ð4Þ
where each ui is given bounded continuous function on ð1; 0. It is well known that by the fundamental theory of functional differential equations [6], system (1) has a unique solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞ satisfying the initial conditions (4). The equilibrium or pattern x of (1) is said to be GAS if it is locally stable in sense of Lyapunov and globally attractive, i.e., every solution of (1) corresponding to an arbitrary given set of initial values satisfy limt!1 xi ðtÞ ¼ xi , i ¼ 1; 2; . . . ; n. To prove the GAS of the equilibrium, we will exploit the Lyapunov direct method. Moreover, if there exist constants M P 1 and k > 0 such that for every solution xðtÞ of (1) with any initial value x0 2 Cðð1; 0; Rn Þ, jxi ðtÞ xi j 6 M kx0 x kekt ;
i ¼ 1; 2; . . . ; n;
then x is called to be globally exponentially stable (GES) and k is called to be globally exponentially convergent rate.
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Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
The following assumptions play an important role in analyzing the GAS and even GES of the equilibrium and convergent rate of solutions of system (1). (H1) For each i 2 f1; 2; . . . ; ng, fi is global Lipschitz with a Lipschitz constant di , bi is monotonically increasing and its inverse b1 is global Lipschitz with i R 1 a cLipschitz constant ai > 0. (H2) There exists a nonnegative constant c such that j 0 aixðxÞ dxj ¼ þ1 for all i ¼ 1; 2; . . . ; n. (H3) There exist positive real constants M i and mi , such that M i P ai ðxÞ P mi for all x 2 R and i 2 f1; 2; . . . ; ng. In literature such as [15,17,18], the activation functions fi ði ¼ 1; 2; . . . ; nÞ are usually assumed to be bounded and/or continuously differentiable or bounded continuously differentiable. Moreover, bi is assumed to be linear or differentiable with b0i > 0. Clearly, assumption (H1) is weaker than those given in the earlier references and is much more realistic because it makes the system include the important cellular neural networks. In addition, in most literature (to name a few, see [2,9–11,17,18]), assumption (H3) is employed to grantee the GAS or global attractivity of system (1). In this paper, however, we show that (H3) can be used to ensure the GES of system (1) and can be replaced by the much weaker assumption (H2) when considering the GAS of system (1). The paper is organized as follows. We first derive some new criteria to guarantee the existence and the GAS of an equilibrium in Section 2. Section 3 is devoted to GEC and even GES of an equilibrium. In Section 4, two examples are given to illustrate the results obtained in this paper. Finally, some conclusions are drawn in Section 5.
2. Global asymptotic stability In this section, we aim to find some new sufficient conditions ensuring the GAS of the equilibrium of (1). The following lemmas are needed in the proof of our main results. Lemma 2.1 [4]. If G : Rn ! Rn is continuous, and satisfies that G is injective on Rn and that limkxk!1 kGðxÞk ! 1, then GðxÞ is a homeomorphism of Rn . Lemma 2.2 [7]. cxy c1 6 xc þ ðc 1Þy c for all c P 0 and x; y > 0. Theorem 2.1. Under assumption (H1), system (1) has a GAS equilibrium if there exist positive real numbers d i ði ¼ 1; 2; . . . ; nÞ and c P 1 such that Z 1 c1 x dx ¼ þ1 ð5Þ ai ðxÞ 0 and one of the following inequalities is satisfied: (i)
Pn
j¼1 ½ðc
Pn
1Þd i ai k ij dj þ d j aj k ji di < d i c, i ¼ 1; 2; . . . ; n;
c j¼1 ½ðc 1Þd i ai dj þ d j aj k ji di < d i c, i ¼ 1; 2; . . . ; n; Pn (iii) j¼1 ½ðc 1Þd i ai k ij þ d j aj k ji dci < d i c, i ¼ 1; 2; . . . ; n; Pn (iv) j¼1 ½ðc 1Þd i ai k ij dc=ðc1Þ þ d j aj k ji < d i c, i ¼ 1; 2; . . . ; n; j Pn c=ðc1Þ (v) j¼1 ½ðc 1Þd i ai dj þ d j aj k cji < d i c, i ¼ 1; 2; . . . ; n; Pn (vi) j¼1 ½ðc 1Þd i k ij dj aj þ d j k ji di ai < d i c, i ¼ 1; 2; . . . ; n;
(ii)
(vii)
Pn
j¼1 ½ðc
Pn
1Þd i dj aj þ d j k cji di ai < d i c, i ¼ 1; 2; . . . ; n;
(viii) j¼1 ½ðc 1Þd i k ij aj þ d j k ji dci ai < d i c, i ¼ 1; 2; . . . ; n; P (ix) nj¼1 ½ðc 1Þd i k ij dj þ d j k ji di aci < d i c, i ¼ 1; 2; . . . ; n;
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
Pn
j¼1 ½ðc
(x) (xi)
(xii)
Pn
191
c
1Þd i k ij þ d j k ji ðdi ai Þ < d i c, i ¼ 1; 2; . . . ; n;
j¼1 ½ðc
Pn
1Þd i k ij aj djc=ðc1Þ þ d j k ji ai < d i c, i ¼ 1; 2; . . . ; n;
j¼1 ½ðc
1Þd i aj djc=ðc1Þ þ d j k cji ai < d i c, i ¼ 1; 2; . . . ; n.
Proof. Since bi ði ¼ 1; 2; . . . ; nÞ are all monotonically increasing, sgnðxi xi Þ ¼ sgnðbðxi Þ bi ðxi ÞÞ, i ¼ 1; 2; . . . ; n. We first consider the existence of equilibria of system (1). Since ai is positive, a point t x ¼ ðx1 ; x2 ; . . . ; xn Þ in Rn is an equilibrium of system (1) if and only if this point x is a solution of the following equation: n X bi ðxi Þ mij fj ðxj Þ þ I i ¼ 0; i ¼ 1; 2; . . . ; n: ð6Þ j¼1
Generally, Eq. (6) may have more than one solution x , and hence system (1) may have more than one equi1 librium. It follows from that b1 is global Lipschitz that jx yj ¼ jb1 i i ðbi ðxÞÞ bi ðbi ðyÞÞj 6 ai jbi ðxÞ bi ðyÞj, t i.e., jbi ðxÞ bi ðyÞj P a1 i jx yj for all x; y 2 R, i ¼ 1; 2; . . . ; n. Let gðxÞ ¼ ðg 1 ðxÞ; g 2 ðxÞ; . . . ; g n ðxÞÞ , where n X gi ðxÞ ¼ bi ðxi Þ þ mij fj ðxj Þ I i ; i ¼ 1; 2; . . . ; n: j¼1
As we all know, if g is a homeomorphism on Rn , then there exists a unique equilibrium x 2 Rn such that gðx Þ ¼ 0. Moreover, it suffices to discuss the GAS of equilibrium x of the following system: " # n Z t X dxi ¼ ai ðxi Þ bi ðxi Þ bi ðxi Þ tij ðt sÞðfj ðxj ðsÞÞ fj ðxj ÞÞds : ð7Þ dt 1 j¼1 In what follows, we only prove that conditions (i) and (vi) both ensure that system (1) has a GAS equilibrium. The others conditions can be verified analogously. Case 1: We start to show that condition (i) ensures that g is a homeomorphism on Rn . In fact, under condition (i), in view of k ij P jmij j for all i; j, we have n X #1 :¼ min fd i c1 ½ðc 1Þd i ai jmij jdj þ d j aj jmji jdi g > 0: 16i6n
j¼1
We first claim that g is injective on Rn . By way of contrary, assume that exist x; y 2 Rn with x 6¼ y such that gðxÞ ¼ gðyÞ. Then, we have n X c1 c c1 d i ai jmij jdj jxj y j jjxi y i j : ð8Þ 0 ¼ d i ai ðgi ðxÞ gi ðyÞÞjxi y i j sgnðxi y i Þ 6 d i jxi y i j þ j¼1
Then (8) implies that n n X n X X c c c 06 d i jxi y i j þ c1 d i ai jmij jdj jxj y j j þ ðc 1Þjxi y i j i¼1
¼
n X i¼1
i¼1
d i þ c
1
n X
j¼1
ðc 1Þd i ai jmij jdj þ d j aj jmji jdi
! c
jxi y i j 6 #1
j¼1
n X
c
jxi y i j ;
i¼1 n
which is a contradiction. So the map g is injective on R . Using a similar argument as above, we can obtain n n n X n X X X c1 c1 c1 d i ai jgi ðxÞ gi ð0Þjjxi j P d i ai jbi ðxi Þ bi ð0Þjjxi j d i ai jmij jjfj ðxj Þ fj ð0Þjjxi j i¼1
i¼1
P
n X i¼1
i¼1 c
d i jxi j
n X n X i¼1
j¼1
j¼1 c1
d i ai jmij jdj jxj jjxi j
P #1
n X
c
jxi j :
i¼1
According to the classical results in functional analysis, for any two different vector norms k k1 and k k2 defined on Rn , they are equivalent in sense that there exist two positive constants c1 and c2 such that
192
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
x 2 Rn :
c1 kxk2 6 kxk1 6 c2 kxk2 ;
Therefore, there exists some suitable constant C 1 such that !1c n X c jxi j P C 1 kxk0 ; i¼1
Pn where k k0 is defined as kxk0 ¼ i¼1 jxi j. Thus, n X d i ai jgi ðxÞ gi ð0Þjjxi jc1 P C c1 #1 kxkc0 : i¼1
Noticing that n X c1 c1 d i ai jgi ðxÞ gi ð0Þjjxi j 6 akgðxÞ gð0Þk0 kxk ; i¼1
where a ¼ max16i6n fd i ai g, we have kgðxÞ gð0Þk0 P a1 C c1 #1 kxk0 : Let kxk ! 1, we get kgðxÞk ! 1. By Lemma 2.1, g is a homeomorphism on Rn . Therefore, (1) has a unique equilibrium, denoted by x . In what follows, we show that x is GAS. We consider the Lyapunov functional V 1 ðtÞ ¼ V 11 ðtÞ þ V 12 ðtÞ, where Z xi ðx x Þc1 n X i V 11 ðtÞ ¼ dx; d i ai a ðxÞ i xi i¼1 Z Z n n 1 t X X c V 12 ðtÞ ¼ c1 d i ai jtij ðsÞjdj jxj ðuÞ xj j du ds: i¼1
0
j¼1
ts
Obviously for any x except x , V 1 ðxÞðtÞ > 0. Calculating the upper right derivatives of V 11 and V 12 along the solution xðtÞ of (7), we have n X
c1
dxi sgn xi xi dt i¼1 # n n Z t X X ¼ d i ai bi ðxi Þ þ bi ðxi Þ þ tij ðt sÞðfj ðxj ðsÞÞ fj ðxj ÞÞds jxi xi jc1 sgn xi xi
Dþ V 11 ðtÞ ¼
i¼1
6
n X i¼1
6
n X
d i ai
jxi xi j ai ðxi Þ "
1
j¼1
" d i jxi
xi jc
n X
þ
j¼1
"
Z
ai jtij ðsÞjdj jxj ðt sÞ
n X
d i ai
i¼1
n X
Z
jxi
xi jc1
#
1
ai jtij ðsÞjdj jxj ðt sÞ
xj jc ds
0
j¼1
þ c1 ðc 1Þ
xj jds
0
n X c d i xi xi þ c1
i¼1
#
1
k ij dj jxi xi j
c
j¼1
and þ
D V 12 ðtÞ ¼ c
1
¼ c1
n X n X i¼1
j¼1
n X
n X
i¼1
j¼1
Z d i ai dj
1
h i c c jtij ðsÞj jxj ðtÞ xj j jxj ðt sÞ xj j ds
0 c
d i ai dj k ij jxj xj j c1
n X n X i¼1
j¼1
Z d i ai dj 0
1
c
jtij ðsÞjjxj ðt sÞ xj j ds:
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
193
Thus Dþ V 1 ðtÞ 6
n X
d i jxi xi jc þ c1
i¼1
n X n X i¼1
d i ai dj k ij jxj xj jc þ c1 ðc 1Þ
n X
j¼1
i¼1
d i ai
n X
k ij dj jxi xi jc
j¼1
n X xi x c < 0; ¼ #1 i i¼1
where
#1
¼ min16i6n fd i c1
Pn
j¼1 ½ðc
1Þd i ai k ij dj þ d j aj k ji di g > 0. Thus
c
Dþ V 1 ðtÞ 6 k1 kx x k < 0;
ð9Þ
where k1 is some suitable positive constant. Therefore, x is GAS. Case 2: We show that condition (vi) ensures that g is also a homeomorphism on Rn . In fact, under condition (vi), in view of k ij P jmij j for all i; j, we have ( #2 :¼ min
16i6n
di c
1
n X ½ðc 1Þd i jmij jdj aj þ d j jmji jdi ai
) > 0:
j¼1
We first claim that g is injective on Rn . By way of contrary, assume that exist x; y 2 Rn with x 6¼ y such that gðxÞ ¼ gðyÞ. Then, we have c1
0 ¼ d i ðgi ðxÞ gi ðyÞÞjbi ðxi Þ bi ðy i Þj sgnðbi ðxi Þ bi ðy i ÞÞ n X d i jmij jdj jxj y j jjbi ðxi Þ bi ðy i Þjc1 6 d i jbi ðxi Þ bi ðy i Þjc þ j¼1 c
6 d i jbi ðxi Þ bi ðy i Þj þ
n X
d i jmij jdj aj jbj ðxj Þ bj ðy j Þjjbi ðxi Þ bi ðy i Þj
c1
:
j¼1
Using a similar argument as above, we have n n X X c1 c d i ðgi ðxÞ gi ðyÞÞjbi ðxi Þ bi ðy i Þj sgnðbi ðxi Þ bi ðy i ÞÞ 6 #2 jbi ðxi Þ bi ðy i Þj ; 0¼ i¼1
i¼1 n
which is a contradiction. So the map g is injective on R . Similarly, we have n n X X c1 c d i jgi ðxÞ gi ð0Þjjbi ðxi Þ bi ð0Þj P #2 jbi ðxi Þ bi ð0Þj i¼1
i¼1
and hence kgðxÞ gð0Þk P C 2 kbðxÞ bð0Þk P C 2 bkxk; t
where C 2 is some suitable positive constant, b ¼ min16i6n fa1 i g, and bðxÞ ¼ ðb1 ðxÞ; b2 ðxÞ; . . . ; bn ðxÞÞ . Let kxk ! 1, we get kgðxÞk ! 1. By Lemma 2.1, g is a homeomorphism on Rn . Therefore, (1) has a unique equilibrium, denoted by x . In what follows, we show that x is GAS. We consider the Lyapunov functional V 2 ðtÞ ¼ V 21 ðtÞ þ V 22 ðtÞ, where Z xi ½b ðxÞ b ðx Þc1 n X i i i dx; V 21 ðtÞ ¼ d i a ðxÞ i xi i¼1 Z Z n n 1 t X X c V 22 ðtÞ ¼ c1 di jtij ðsÞjdj aj jbj ðxj ðuÞÞ bj ðxj Þj du ds: i¼1
j¼1
0
ts
Obviously for any x except x , V 2 ðxÞðtÞ > 0. Calculating the upper right derivatives of V 21 and V 22 along the solution xðtÞ of (7), we have
194
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199 n X
c1
dxi sgn xi xi dt i¼1 # n n Z t X X d i bi ðxi Þ þ bi ðxi Þ þ tij ðt sÞ fj ðxj ðsÞÞ fj ðxj Þ ds ¼
Dþ V 21 ðtÞ ¼
di
jbi ðxÞ bi ðxi Þj ai ðxi Þ "
i¼1
jbi ðxi Þ bi ðxi Þj þ
6
n X i¼1 n X
c1
sgn xi xi 6
1
0 c
d i ½jbi ðxi Þ bi ðxi Þj þ
n Z X
d i fjbi ðxi Þ
bi ðxi Þjc
þc
jtij ðsÞjdj aj jbj ðxj ðt sÞÞ bj ðxj Þj jbi ðxÞ bi ðxi Þj
1
n Z X
i¼1
1
n X n X
1
n X n X
i¼1
i¼1
c1
ds
1
jtij ðsÞjdj aj jbj ðxj ðt sÞÞ bj ðxj Þjc dsg
0
j¼1 n X n X
c
d i k ij dj aj jbi ðxi Þ bi ðxi Þj
j¼1
and
¼c
1 0
j¼1
þ c1 ðc 1Þ
D V 22 ðtÞ ¼ c
c
d i ½jbi ðxi Þ bi ðxi Þj
jtij ðsÞjdj jxj ðt sÞ xj jjbi ðxÞ bi ðxi Þjc1 ds
i¼1
þ
n X i¼1
n Z X j¼1
6
1
j¼1
Z
1
d i dj aj
h i c c jtij ðsÞj jbj ðxj Þ bj ðxj Þj jbj ðxj ðt sÞÞ bj ðxj Þj ds
0
j¼1
d i dj aj k ij jbj ðxj Þ
c bj ðxj Þj
j¼1
c
1
n X n X i¼1
Z
1
c
jtij ðsÞjjbj ðxj ðt sÞÞ bj ðxj Þj ds:
d i dj aj 0
j¼1
Thus Dþ V 2 ðtÞ 6
n X
c
d i fjbi ðxi Þ bi ðxi Þj þ c1
i¼1 n X n X i¼1 n X
d i k ij dj aj jbi ðxi Þ bi ðxi Þjc
j¼1
bi ðxi Þ bi ðx Þc < 0; a1 i i
i¼1
#1
c
k ij dj aj jbj ðxj Þ bj ðxj Þj g
j¼1
þ c1 ðc 1Þ ¼ #2
n X
where ¼ min16i6n fd i c1 completed. h
Pn
j¼1 ½ðc
1Þd i k ij dj aj þ d j k ji di ai g > 0. Therefore, x is GAS. The proof is
Remark 2.1. In the proof of Theorem 2.1, we can see the importance of assumption (H2), which ensures the radial unboundedness of the Lyapunov function. In most literatures, such as [2,10,11,17,18]), the upper bound of amplification functions ai ðÞ ði ¼ 1; 2; . . . ; nÞ is a prerequisite condition, which is obviously much stronger than (5). On the other hand, in literatures such as [1,13,14,21], the authors tried to drop this prerequisite condition by constructing a invariant set. However, their results are actually on the global attractivity, instead of GAS. Corollary 2.1. Under assumptions (H1) and (H2), if I BKD a non-singular M-matrix, then system (1) has a GAS equilibrium, where I is an identity matrix of size n, B ¼ diagða1 ; . . . ; an Þ, K ¼ ðk ij Þnn , and D ¼ diagðd1 ; . . . ; dn Þ. Proof. Since I BKD a non-singular M-matrix, then there exists a vector D ¼ ðd 1 ; . . . ; d n Þ > 0 such that Dð I BKDÞ > 0. Namely, n X d j aj k ji di : di > j¼1
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
195
Therefore, condition (1) of Theorem 2.1 holds. Hence, system (1) has a GAS equilibrium. The proof is completed. h 3. Global exponential stability If we put some more strict constraints on functions of system (1), for example, (H3), then we can obtain some further results about the GES and convergence rate of solutions of system (1). R1 Theorem 3.1. In addition to (H1) and (H3), assume that 0 jtij ðtÞjek0 t dt < 1 for some positive k0 , and that there exist positive real numbers d i ði ¼ 1; 2; . . . ; nÞ and c P 1 such that one of the following inequalities (i)–(v) in Theorem 2.1 is satisfied, then system (1) has exactly one equilibrium, which is GES. Proof. Similarly, we only prove that conditions (i) ensures that the GES of the equilibrium because the others conditions can be verified analogously. Under condition (i), in view of Theorem 1, system (1) has a GAS equilibrium x . In what follows, we show that x is GES. Obviously, there exists sufficiently small positive constants k < k0 =c such that n Z 1 X kai ½ðc 1Þd i ai jtij ðsÞjdj þ d j aj jtji ðsÞjdi ekcs ds di 1 > c1 mi 0 j¼1 for all i ¼ 1; 2; . . . ; n. Thus, we introduce the following positive constant,which will play a important role in our proof. ( ) n Z 1 X kai 1 kcs #3 ¼ min d i 1 ½ðc 1Þd i ai jtij ðsÞjdj þ d j aj jtji ðsÞjdi e ds : c 16i6n mi 0 j¼1 We consider the Lyapunov functional V 3 ðtÞ ¼ V 31 ðtÞ þ V 32 ðtÞ, where Z xi ðx x Þc1 n X i dxeckt ; V 31 ðtÞ ¼ d i ai a ðxÞ i x i¼1 i Z Z n n 1 t X X V 32 ðtÞ ¼ c1 d i ai jtij ðsÞjeckðuþsÞ dj jxj ðuÞ xj jc du ds: i¼1
j¼1
0
ts
Obviously for any x except x , V 3 ðxÞðtÞ > 0. Calculating the upper right derivatives of V 31 and V 32 along the solution xðtÞ of (7), we have " # n n Z t X X þ D V 31 ðtÞ ¼ d i ai bi ðxi Þ þ bi ðxi Þ þ tij ðt sÞ fj ðxj ðsÞÞ fj ðxj Þ ds i¼1
j¼1
1
Z xi ðx x Þc1 i dxeckt xi þ kc d i ai jxi a ðxÞ i xi i¼1 Z 1 n n X n X X c 6 ekct d i xi xi þ ekct d i ai jtij ðsÞjdj jxj ðt sÞ xj jds jxi xi jc1 xi jc1 eckt sgn xi
i¼1
þk
n X
i¼1
n X
j¼1
0
c
ckt d i ai m1 i jxi xi j e
i¼1
6 ekct
n X
Z n X n X xi x c þ c1 ekct d i ½kai m1 1 d a i i i i
i¼1
þ ð1 c1 Þekct
i¼1 n X
n X
i¼1
j¼1
d i ai k ij dj jxi xi jc
j¼1
0
1
c
jtij ðsÞjdj jxj ðt sÞ xj j ds
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Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
and Dþ V 12 ðtÞ ¼ c1
n X n X i¼1
¼ c1
Z
i¼1
h i jtij ðsÞj jxj ðtÞ xj jc jxj ðt sÞ xj jc ds
0
j¼1
n X n X
1
d i ai dj
c
d i ai dj k ij jxj xj j c1
j¼1
n X n X i¼1
Z
1
d i ai dj
c
jtij ðsÞjjxj ðt sÞ xj j ds:
0
j¼1
Thus, Dþ V 3 ðtÞ 6 ekct
n X
n X n X c c d i ½kai m1 þ ð1 c1 Þekct d i ai k ij dj jxi xi j i 1 xi xi
i¼1
þ c1 ekct
n X n X i¼1
i¼1
Z
1
j¼1
c
jtij ðsÞjecks dsjxj xj j ¼ #3
d i ai dj 0
j¼1
n X xi x c ekct < 0: i i¼1
Therefore, V 3 ðtÞ 6 V 3 ð0Þ. Recall that V 3 ðtÞ P ðM i cÞ1 d i ai jxi ðtÞ xi jc eckt ;
i ¼ 1; 2; . . . ; n
and V 3 ð0Þ 6
n X
1
d i ai ðmi cÞ jxi ð0Þ
i¼1
xi jc
þc
1
n X n X i¼1
j¼1
Z d i ai
1
jtij ðsÞjecks dsdj max jxj ðuÞ xj jc :
0
u60
We can see that there exist some suitable constant C 2 > 0 such that c
c
jxi ðtÞ xi j eckt 6 C c2 kx x k ; i ¼ 1; 2; . . . ; n; P where k k is defined by kxk ¼ ð ni¼1 maxu60 jxi ðuÞjc Þ1=c . That is, jxi ðtÞ xi j 6 C 2 kx x kekt ; This completes the proof.
i ¼ 1; 2; . . . ; n:
h
Corollary 3.1. Under assumptions (H1) and (H3), if I BKD a non-singular M-matrix, then system (1) has a GAS equilibrium, where matrices I, B, K, and D are defined as those in Corollary 2.1. Corollary 3.2. Under assumptions (H1) and (H3), if m BKDM a non-singular M-matrix, then system (1) has a GAS equilibrium, where m ¼ diagðm1 ; . . . ; mn Þ, M ¼ diagðM 1 ; . . . ; M n Þ, B, K, and D are defined as those in Corollary 2.1. Proof. As we know, if A is a M-matrix, then for any two positive diagonal matrices P and Q, PAQ is also Mmatrix. If m BKDM a non-singular M-matrix, then M BKD is a non-singular M-matrix, where M ¼ diagðm1 =M 1 ; . . . ; mn =M n Þ. Thus, there exists a vector D ¼ ðd 1 ; . . . ; d n Þ > 0 such that DðM BKDÞ > 0. Namely, d i mi > M i
n X
d j aj k ji di :
j¼1
Obviously, condition (i) in Theorem 2.1 holds. Hence, system (1) has a GES equilibrium. The proof is completed. h Corollary 3.2 is actually Theorem 4 of [2]. Therefore, Theorem 3.1 not only unifies but also improves the previous results. R1 Theorem 3.2. In addition to (H1) and (H3), assume that 0 jtij ðtÞjtek0 t dt < 1 for some positive k0 , and that there exist positive real numbers d i ði ¼ 1; 2; . . . ; nÞ and c P 1 such that one of the following inequalities (vi)–(xii) in
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
197
Theorem 2.1 is satisfied, then system (1) has exactly one equilibrium and all other solutions converge exponentially to it as t ! 1. Proof. We only prove that conditions (vi) ensures that the global exponential convergence of the equilibrium because the others conditions can be verified analogously. Under condition (vi), in view of Theorem 1, system (1) has a GAS equilibrium x . In what follows, we show that x is globally exponentially convergent. Obviously, there exists sufficiently small positive constants k < k0 =c such that n Z 1 X kai di 1 ½ðc 1Þd i aj jtij ðsÞjdj þ d j ai jtji ðsÞjdi ekcs ds > c1 mi 0 j¼1 for all i ¼ 1; 2; . . . ; n. Thus, we introduce the following positive constant,which will play a important role in our proof. ( ) n Z 1 X kai 1 kcs #4 ¼ min d i 1 ½ðc 1Þd i aj jtij ðsÞjdj þ d j ai jtji ðsÞjdi e ds : c 16i6n mi 0 j¼1 We consider the Lyapunov functional V 4 ðtÞ ¼ V 41 ðtÞ þ V 42 ðtÞ, where Z xi ½b ðxÞ b ðx Þc1 n X i i i V 41 ðtÞ ¼ dxeckt ; d i a ðxÞ i x i¼1 i Z Z n n 1 t X X V 42 ðtÞ ¼ c1 di jtij ðsÞjeckðuþsÞ dj aj jbj ðxj ðuÞÞ bj ðxj Þjc du ds: i¼1
j¼1
0
ts
Obviously for any x except x , V 4 ðxÞðtÞ > 0. Calculating the upper right derivatives of V 41 and V 42 along the solution xðtÞ of (7), we have " # n n Z t X X þ D V 41 ðtÞ ¼ d i bi ðxi Þ þ bi ðxi Þ þ tij ðt sÞ fj ðxj ðsÞÞ fj ðxj Þ ds j¼1
1
c1 bi ðxi Þj eckt sgn xi
xi
i¼1
jbi ðxi Þ
þ kc
n X i¼1
6 ekct
n n X n X X c d i bi ðxi Þ bi ðxi Þ þ ekct di i¼1
þk
n X i¼1
6 ekct
n X
i¼1
j¼1
Z
d i m1 i
Z
xi
½bi ðxÞ xi
c1 bi ðxi Þ dxeckt
1
jtij ðsÞjdj jxj ðt sÞ xj jds jbi ðxi Þ bi ðxi Þj 0
c ckt d i m1 i ai bi ðxi Þ bi ðxi Þ dxe Z n X n X bi ðxi Þ bi ðx Þc þ c1 ekct d i ½kai m1 1 d i i i
i¼1
i¼1 c
bj ðxj Þj ds þ ð1 c1 Þekct
n X n X i¼1
j¼1
d i k ij dj aj jbi ðxi Þ bi ðxi Þj
1
jtij ðsÞjdj aj jbj ðxj ðt sÞÞ
0 c
j¼1
and Dþ V 42 ðtÞ ¼ c1 ekct
Z 0
n X n X i¼1
1
j¼1
Z d i dj aj
1
c
jtij ðsÞjecks jbj ðxj Þ bj ðxj Þj ds c1 ekct
0
jtij ðsÞjjbj ðxj ðt sÞÞ bj ðxj Þjc ds:
n X n X i¼1
j¼1
d i dj aj
c1
198
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
Thus Dþ V 4 ðtÞ 6 ekct
n X
n X n X bi ðxi Þ bi ðx Þc þ ð1 c1 Þekct d i ½kai m1 1 d i k ij dj aj jbi ðxi Þ bi ðxi Þjc i i
i¼1
þ c1 ekct
n X n X i¼1
Z
i¼1 1
d i dj aj
j¼1
c
jtij ðsÞjecks jbj ðxj Þ bj ðxj Þj ds ¼ #4
0
j¼1
n X bi ðxi Þ bi ðx Þc ekct < 0: i i¼1
Therefore, V 4 ðtÞ 6 V 4 ð0Þ. Recall that 1
c
V 4 ðtÞ P ðM i cÞ ai1c d i jxi ðtÞ xi j eckt ;
i ¼ 1; 2; . . . ; n
and V 4 ð0Þ 6
n X
1
c
d i ai ðmi cÞ jbi ðxi ð0ÞÞ bi ðxi Þj þ c1
i¼1
n X n X i¼1
j¼1
Z di
1
c
jtij ðsÞjecks dsdj max jbj ðxj ðuÞÞ bj ðxj Þj : u60
0
We can see that there exist some suitable constant C 3 > 0 such that jxi ðtÞ xi jc eckt 6 C c3 kbðxÞ bðx Þkc ;
i ¼ 1; 2; . . . ; n;
where bðxÞ ¼ ðb1 ðx1 Þ; b2 ðx2 Þ; . . . ; bn ðxn ÞÞ and k k is defined by kxk ¼ ð jxi ðtÞ
xi j
6 C 3 kbðxÞ bðx Þke
This completes the proof.
kt
;
Pn
c 1=c . i¼1 maxu60 jxi ðuÞj Þ
i ¼ 1; 2; . . . ; n;
That is, ð10Þ
h
Remark 3.1. As a result of Theorem 3.2, all solutions of system (1) converge exponentially to the equilibrium x as t ! 1. However, the equilibrium x maybe not GES. Of course, if we put a further restriction on bi , i ¼ 1; 2; . . . ; n, for example, bi ði ¼ 1; 2; . . . ; nÞ are all globally Lipshitz, then it follows from (10) that the equilibrium x is GES. 4. Examples In this section, we illustrate our main results by two examples. We will see that it is not hard to verify the conditions stated in our main theorems. Consider the following CGNNs with distributed delays: " # 2 Z t X x_ i ¼ ai ðxi Þ bðxi Þ tij ðt sÞf ðxj ðsÞÞds ; i ¼ 1; 2: ð11Þ j¼1
1
Example 1. Consider system (11) with a1 ðxÞ ¼ 1 þ x, a2 ðxÞ ¼ 2 þ cos x, bðxÞ ¼ 2x þ jxj, f ðxÞ ¼ 12 ðx þ sin xÞ. Obviously, assumptions (H1) and (H2) hold with d1 ¼ d2 ¼ 1 and a1 ¼ a2 ¼ 1. If the kernels tij ði; j ¼ 1; 2Þ are given by t11 ðsÞ ¼ 0:20es ;
t12 ðsÞ ¼ 1:8e2s ;
t21 ðsÞ ¼ 1:4e2s ;
t22 ðsÞ ¼ 0:15es ;
then k 11 ¼ 0:2, k 12 ¼ 0:9, k 21 ¼ 0:7, k 22 ¼ 0:15. Thus, condition (i) of Theorem 2.1 holds with c ¼ 1, d 1 ¼ 8, and d 2 ¼ 9. Therefore, system (11) has a globally asymptotically stable equilibrium. Example 2. Consider system (11) with bðxÞ ¼ 2x þ jxj and 8 if jxj 6 1; > : 1 if jxj P 2; 3 jxj if jxj 6 1; a2 ðxÞ ¼ 2 if jxj > 1;
Y. Meng et al. / Applied Mathematics and Computation 202 (2008) 188–199
f ðxÞ ¼
0
199
if x > 0;
x if x 6 0: Obviously, assumptions (H1) and (H3) hold with d1 ¼ d2 ¼ 1, a1 ¼ a2 ¼ 1, m1 ¼ 1, m2 ¼ 2, M 1 ¼ 2, and M 2 ¼ 3. If the kernels tij ði; j ¼ 1; 2Þ are given by t11 ðsÞ ¼ 0:25es ;
t12 ðsÞ ¼ 0:30e2s ;
t21 ðsÞ ¼ 0:50e2s ;
t22 ðsÞ ¼ 0:30es ;
R1 then k 11 ¼ 0:25, k 12 ¼ 0:15, k 21 ¼ 0:25, and k 22 ¼ 0:30. Moreover, it is easy to see that 0 jtij ðsÞjsek0 s < 1 for some k0 2 ð0; 1Þ, and that condition (1) of Theorem 3.1 holds with c ¼ 1 and d 1 < d 2 . Thus, system (11) has a globally exponentially stable equilibrium. 5. Conclusion This Letter is concerned with the convergence dynamics of Cohen–Grossberg neural networks with continuously distributed delays. Without assuming the differentiability and monotonicity of activation functions, the differentiability of amplification functions and the symmetry of synaptic interconnection weights, we construct suitable Lyapunov functionals and employ inequality technique to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, global exponential convergence, and even global exponential stability of equilibria. Our new criteria will definitely be significant for the designs and applications of delayed neural networks. References [1] J. Cao, Q. Song, Stability in Cohen–Grossberg-type bidirectional associative memory neural networks with time-varying delays, Nonlinearity 19 (2006) 1601–1617. [2] Y. Chen, Global asymptotic stability of delayed Cohen–Grossberg neural networks, IEEE Transactions on Circuits and Systems: Regular Papers 53 (2006) 351–357. [3] M.A. Cohen, S. Grossberg, Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybernet. 13 (1983) 815–826. [4] M. Forti, A. Tesi, New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE Trans. Circ. Syst. I Fund. Theory Appl. 42 (1995) 354–366. [5] S. Guo, L. Huang, Stability analysis of Cohen–Grossberg neural networks, IEEE Trans. Neural Networks 17 (2006) 106–117. [6] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993. [7] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, second ed., Cambridge University Press, London, 1952. [8] T. Huang, J. Cao, C. Li, Necessary and sufficient condition for the absolute exponential stability of a class of neural networks with finite delay, Phys. Lett. A 352 (2006) 94–98. [9] T. Huang, A. Chan, Y. Huang, J. Cao, Stability of Cohen–Grossberg neural networks with time-varying delays, Neural Networks 20 (2007) 868–873. [10] T. Huang, C. Li, G. Chen, Stability of Cohen–Grossberg neural networks with unbounded distributed delays, Chaos Solitons Fract. 34 (2007) 992–996. [11] X. Liao, C. Li, Global attractivity of Cohen–Grossberg model with finite and infinite delays, J. Math. Anal. Appl. 315 (2006) 244–262. [12] X. Liao, C. Li, K. Wong, Criteria for exponential stability of Cohen–Grossberg neural networks, Neural Networks 17 (2004) 1401– 1414. [13] K. Lu, D. Xu, Z. Yang, Global attraction and stability for Cohen–Grossberg neural networks with delays, Neural Networks 19 (2006) 1538–1549. [14] Z. Mao, H. Zhao, Dynamical analysis of Cohen–Grossberg neural networks with distributed delays, Phys. Lett. A 364 (2007) 38–47. [15] J. Sun, L. Wan, Global exponential stability and periodic solutions of Cohen–Grossberg neural networks with continuously distributed delays, Phys. D: Nonlinear Phenom. 208 (2005) 1–20. [16] D.W. Tank, J.J. Hopfield, Neural computation by concentrating information in time, Proc. Natl. Acad. Sci. USA 84 (1987) 1896– 1900. [17] L. Wan, J. Sun, Global asymptotic stability of Cohen–Grossberg neural network with continuously distributed delays, Phys. Lett. A 342 (2005) 331–340. [18] L. Wang, Stability of Cohen–Grossberg neural networks with distributed delays, Appl. Math. Comput. 160 (2005) 93–110. [19] L. Wang, X. Zou, Harmless delays in Cohen–Grossberg neural networks, Physica D 170 (2002) 162–173. [20] H. Ye, A.N. Michel, K. Wang, Qualitative analysis of Cohen–Grossberg neural networks with multiple delays, Phys. Rev. E 51 (1995) 2611–2618. [21] J. Zhang, Y. Suda, H. Komine, Global exponential stability of Cohen–Grossberg neural networks with variable delays, Phys. Lett. A 338 (2005) 44–50.
