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Boundedness and stability for Cohen–Grossberg neural network with time-varying delays ✩ Jinde Cao ∗ , Jinling Liang Department of Mathematics, Southeast University, Nanjing 210096, China Received 9 February 2004 Available online 2 July 2004 Submitted by K. Gopalsamy

Abstract In this paper, a model is considered to describe the dynamics of Cohen–Grossberg neural network with variable coefficients and time-varying delays. Uniformly ultimate boundedness and uniform boundedness are studied for the model by utilizing the Hardy inequality. Combining with the Halanay inequality and the Lyapunov functional method, some new sufficient conditions are derived for the model to be globally exponentially stable. The activation functions are not assumed to be differentiable or strictly increasing. Moreover, no assumption on the symmetry of the connection matrices is necessary. These criteria are important in signal processing and the design of networks. 2004 Elsevier Inc. All rights reserved. Keywords: Ultimate boundedness; Lyapunov functional; Exponential stability; Hardy inequality; Halanay inequality; Cohen–Grossberg neural network

1. Introduction In the past few decades, neural networks such as Hopfield neural network [1], cellular neural network [2,3], and bi-directional associative memory neural network [4–6,10,11,33] have attracted the attention of many mathematicians, physicists, and computer scientists ✩ This work was jointly supported by the National Natural Science Foundation of China under Grant 60373067, the Natural Science Foundation of Jiangsu Province, China under Grants BK2003053 and BK2003001, Qing-Lan Engineering Project of Jiangsu Province, and the Foundation of Southeast University, Nanjing, China under Grant XJ030714. * Corresponding author. E-mail addresses: [email protected], [email protected] (J. Cao).

0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.04.039

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(see [7–17]) due to their wide range of applications in, for example, pattern recognition, associative memory, and combinatorial optimization. Among them, the Cohen–Grossberg neural network [18] is an important one, which can be described by the system of ordinary differential equations n dxi (t) cij gj xj (t) , i = 1, 2, . . . , n, (1) = −ai xi (t) bi xi (t) − dt j =1

in which n 2 is the number of neurons in the network; xi (t) denotes the state variable of the ith neuron at time t; gj (xj (t)) denotes the activation function of the j th neuron at time t; the feedback matrix C = (cij )n×n indicates the strength of the neuron interconnections within the network; ai (xi (t)) represents an amplification function; bi (xi (t)) is an appropriately behaved function such that the solutions of model (1) remain bounded. This model was firstly proposed and studied by Cohen and Grossberg (1983), it includes a lot of models from evolutionary theory, population biology and neurobiology. It should be pointed out that the Cohen–Grossberg neural network encompasses the Hopfield neural i network [1] as a special case (when ai (xi (t)) ≡ 1, bi (xi (t)) = C Ri xi (t) + Ii ), the latter could be described as n xi (t) dxi (t) =− Ci + cij gj xj (t) + Ii , i = 1, 2, . . . , n, (2) dt Ri j =1

where Ci and Ri are positive constants representing the neuron amplifier input capacitance and resistance, respectively; Ii is the constant input from outside of the network. In fact, due to the finite speeds of the switching and transmission of signals in a network, time delays do exist in a working network and thus should be incorporated into the model equations of the network. It was observed both experimentally and numerically in [19] that time delay could induce instability, causing sustained oscillations which may be harmful to a system. For the Cohen–Grossberg model (1), Ye et al. [20] also introduced delays by considering the following system of delayed differential equations: K n dxi (t) k = −ai xi (t) bi xi (t) − cij gj xj (t − τk ) , dt k=0 j =1

i = 1, 2, . . . , n,

(3)

where the time delays τk (k = 0, 1, . . . , K) are arranged such that 0 = τ0 < τ1 < · · · < τK . Further studies were taken by Wang and Zou [21,22], Lu and Chen [23], Chen and Rong [24] about the following model: n n dxi (t) = −ai xi (t) bi xi (t) − cij gj xj (t) − dij gj xj (t − τij ) + Ii , dt j =1

j =1

(4) in which D = (dij )n×n indicates the strength of the neuron interconnections within the network with time delay parameters τij . In [22,24], several sufficient conditions were obtained to ensure model (4) to be asymptotically stable. A set of conditions ensuring global

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667

exponential stability of system (4) were derived in [21] when cij ≡ 0 and dij ≡ 0, respectively. And, by the property of Lyapunov diagonal stable matrix, absolutely global stability was studied in [23] for model (4) when dij ≡ 0. A more generalized model was studied by Hwang in [25], n dxi (t) = −ai xi (t) bi xi (t) − cij gj xj (t) dt j =1 n − dij gj xj t − τij (t) + Ii (5) j =1

and exponential stability result was obtained. The purpose of this paper is to study the dynamic behavior of the generalized Cohen– Grossberg neural network with variable coefficients and time-varying delays. The organization of this paper is as follows. In Section 2, we give a model description and some prerequisite results. In Section 3, boundedness of the model will be discussed. And the conditions ensuring the exponential stability of the Cohen–Grossberg neural network are obtained in Section 4. In Section 5, some examples and their numerical simulations are given to confirm and illustrate the analysis. Finally, in Section 6, we give concluding remarks of the derived results. 2. Model description and preliminaries In this paper, we investigate the following delayed dynamical systems: n dxi (t) = −ai xi (t) bi xi (t) − cij (t)gj xj (t) dt j =1 n dij (t)gj xj t − τij (t) + Ii (t) , −

(6)

j =1

where i = 1, 2, . . . , n; 0 τij (t) τ and supt ∈[−τ,+∞) τ˙ij (t) < 1 (where τ˙ij (t) represents the derivative of τij (t)); cij (t), dij (t) and Ii (t) are continuous and bounded functions defined on [−τ, +∞). Define xt (s) = x(t + s), s ∈ [−τ, 0], t 0. Let x(t) = (x1 (t), x2 (t), . . . , xn (t))T , its norm is defined as 1 n r r xi (t + s) xt r = sup x(t + s) , where x(t + s) = and r 1. −τ s0

r

r

i=1

(7) Assume that the nonlinear model (6) has initial values of the type xi (t) = ϕi (t),

t ∈ [−τ, 0],

in which ϕi (t) (i = 1, 2, . . . , n) are continuous functions. By the fundamental theory of functional differential equations [29], model (6) has a unique solution x(t) = (x1 (t), x2 (t), . . . , xn (t))T satisfying the initial condition in (7).

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To establish the main results of the model given in (6), some of the following assumptions will apply: (H1 ) Each function ai (u) is bounded, positive and locally Lipschitz continuous. Furthermore, 0 < α i ai (u) α i < +∞ for all u ∈ R and i = 1, 2, . . . , n. (H2 ) Each function bi (u) is locally Lipschitz continuous and there exists βi > 0 such that ubi (u) βi u2 for u ∈ R, i = 1, 2, . . . , n. (H2 ) Each function bi (u) ∈ C 1 (R, R) and b˙i (u) βi > 0; both bi (·) and bi−1 (·) are locally Lipschitz continuous. (H3 ) Each function gj : R → R satisfies the Lipschitz condition with a Lipschitz constant Lj > 0, i.e., |gj (u) − gj (v)| Lj |u − v| for all u, v ∈ R, j = 1, 2, . . . , n. (H3 ) Each function gj (·) is bounded and satisfies the Lipschitz condition with a Lipschitz constant Lj > 0. Definition 1. System (6) is uniformly bounded if, for any constant δ > 0, there is B = B(δ) > 0 such that, for all t0 ∈ [0, +∞), ϕ, and ϕ|r < δ, we have |x(t, t0 , ϕ)|r < B for all t t0 . Definition 2. System (6) is uniformly ultimately bounded if there is a B > 0 such that, for any δ > 0, there is a constant t˜ = t˜(δ) > 0 such that |x(t, t0 , ϕ)|r < B for t t0 + t˜ for all t0 ∈ [0, +∞), ϕr < δ. Under the assumptions (H1 ), (H2 ) and (H3 ), we know from [21] that system (5) has an equilibrium point x ∗ = (x1∗ , x2∗ , . . . , xn∗ )T . Let y(t) = x(t) − x ∗ , substitute x(t) = y(t) + x ∗ in (5) and we have dyi (t) ∗ = −ai yi (t) + xi bi yi (t) + xi∗ − bi xi∗ dt −

n j =1

−

n

cij gj yj (t) + xj∗ − gj xj∗ ∗ ∗ . dij gj yj t − τij (t) + xj − gj xj

(8)

j =1

Denote Ai (yi (t)) = ai (yi (t) + xi∗ ), Bi (yi (t)) = bi (yi (t) + xi∗ ) − bi (xi∗ ), fj (yj (t)) = gj (yj (t) + xj∗ ) − gj (xj∗ ); then system (8) becomes n n dyi (t) = −Ai yi (t) Bi yi (t) − cij fj yj (t) − dij fj yj t − τij (t) . dt j =1

j =1

(9) Definition 3. System (5) or (9) is globally exponentially stable if there exist constants ε > 0 and M > 0 such that

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669

yt r = xt − x ∗ r Mϕ − x ∗ r e−εt for all t 0. It is clear that x ∗ is globally exponentially stable for (5) if and only if the trivial solution of (8) or (9) is globally exponentially stable. Throughout this paper, the following Hardy inequality and Halanay inequality are used. Lemma 1 (Hardy inequality [26]). Assume there exist constants ak 0, pk > 0 (k = 1, 2, . . . , m + 1), then the following inequality holds: m+1 1 1 m+1 r p Sm+1 − 1r ak k pk akr Sm+1 , (10) k=1

where r > 0 and Sm+1 =

k=1

m+1 k=1

pk .

In (10), if we let pm+1 = 1, r = Sm+1 = m k=1 pk + 1, we will get m m p 1 r 1 ak k am+1 pk akr + am+1 . r r k=1

(11)

k=1

Lemma 2 (Halanay inequality [27,28]). Assume constant numbers k1 , k2 satisfy k1 > k2 > 0, V (t) is a nonnegative continuous function on [t0 − τ, t0 ], and as t t0 , satisfy the following inequality: D + V (t) −k1 V (t) + k2 V (t), where V (t) = supt −τ st {V (s)}, τ 0 is constant. Then as t t0 , we have V (t) V (t0 )e−λ(t −t0) , in which λ is the unique positive solution of the following equation: λ = k1 − k2 eλτ . 3. Boundedness results Consider the following equations: dxi (t) = fi (t, xt ), i = 1, 2, . . . , n, (12) dt where fi (t, ϕ) : [0, +∞) × C[−τ, 0] → R is continuous with respect to (t, ϕ) and satisfies the Lipschitz condition with respect to ϕ (i = 1, 2, . . . , n). Let Wj (s) : [0, +∞) → [0, +∞) (j = 1, 2, 3, 4) be continuous and increasing functions with Wj (0) = 0, Wj (s) → +∞ as s → +∞. Let the functional V (t, ϕ) : [0, +∞) × C[−τ, 0] → [0, +∞) be continuous with respect to (t, ϕ). Lemma 3 [29]. The solutions of system (12) are uniformly bounded and uniformly ultimately bounded if the functional V (t, ϕ) and functions Wj (s) (j = 1, 2, 3, 4) satisfy the following conditions:

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t (i) W1 (|x(t)|r ) V (t, xt ) W2 (|x(t)|r ) + W3 ( t −τ W4 (|x(s)|r ) ds); (ii) D + V (t, xt )|(12) −W4 (|x(t)|r ) + M for some constant M > 0. Lemma 4 [30]. For system (12), let fi (t + T , ϕ) = fi (t, ϕ) and the solutions be uniformly bounded and uniformly ultimately bounded. Then system (12) has a T -periodic solution if, for any constant δ > 0, there is a constant B = B(δ) > 0 such that, for all ϕ with ϕr < δ, we have |fi (t, ϕ)| < B for all t ∈ [−τ, 0] (i = 1, 2, . . . , n). For the boundedness of solutions for system (6), we have the following results. Theorem 1. Under assumptions (H1 )–(H3 ), all solutions of model (6) are uniformly bounded and uniformly ultimately bounded if there exist constants νk > 0 (k = 1, 2, . . . , K1 ), µk > 0 (k = 1, 2, . . . , K2 ), ωi > 0, σ > 0, pij , pij∗ , qij , qij∗ , ξij , ξij∗ , ηij , ηij∗ ∈ R such that rα i βi ωi −

K1 n

K2 n rξij νkij rpij µij νk α i ωi νk cij (t) Lj − α i ωi µk dij (t) µk Lj k rη

j =1 k=1

−

n

j =1 k=1

rξ ∗ rη∗ α j ωj cj i (t) ji Li ji −

j =1

rq

n

rq ∗ α j ωj Li ji

j =1

|dj i (ψj−1 i (t))|

∗ rpji

1 − τ˙j i (ψj−1 i (t))

> rσ

holds for all t 0; where ψij−1 (t) is the inverse function of ψij (t) = t − τij (t); r =

K 1

K 2 ∗ ∗ ∗ ∗ k=1 νk + 1 = k=1 µk + 1 and pij , pij , qij , qij , ξij , ξij , ηij , ηij are any real constant ∗ ∗ ∗ numbers with K1 ξij + ξij = 1, K1 ηij + ηij = 1, K2 pij + pij = 1, K2 qij + qij∗ = 1 (i, j = 1, 2, . . . , n). Proof. Let x(t) = (x1 (t), x2 (t), . . . , xn (t))T be any solution of model (6). Now consider the Lyapunov functional rp ∗ t n n r r |dij (ψij−1 (s))| ij rqij∗ 1 V (t, xt ) = xj (s) ds , ωi xi (t) + α i Lj r 1 − τ˙ij (ψij−1 (s)) j =1

i=1

t −τij (t )

obviously, we have r 1 r 1 V (t, xt ) ω xi (t) = ωx(t)r = W1 x(t)r , r r n

i=1

t n n r r rqij∗ 1 V (t, xt ) ω α i Dij Lj xj (s) ds xi (t) + r j =1

i=1

1 ω r

t −τ

t

n

n

i=1

i=1t −τ

xi (t)r + 1 ωLn r

xi (s)r ds

(13)

J. Cao, J. Liang / J. Math. Anal. Appl. 296 (2004) 665–685

= W2 x(t)r + W3

t

W4 x(s)r ds

671

(14)

t −τ

for all t 0, where ω = min ωi ,

ω = max ωi ,

1in

1in

∗

Dij =

|dij (ψij−1 (s))|rpij

sup s∈[−τ,+∞)

1 W1 (s) = ωs r , r

1 − τ˙ij (ψij−1 (s)) 1 W2 (s) = ωs r , r

Then D + V (t, xt )|(6) =

n

,

L = max

1i,j n

W3 (s) =

rq ∗ α i Dij Lj ij ,

2ωn Ls, rσ

1 W4 (s) = σ s r . 2

r−1 ωi xi (t) D + xi (t)

i=1 ∗

rp n r |dij (ψij−1 (t))| ij 1 rqij∗ + αi Lj xj (t) r 1 − τ˙ij (ψij−1 (t)) j =1 ∗

n r |dij (t)|rpij 1 rqij∗ − αi 1 − τ˙ij (t) Lj xj t − τij (t) r 1 − τ˙ij (t) j =1

n r−1 = ωi xi (t) sgn xi (t)

i=1

n × −ai xi (t) bi xi (t) − cij (t)gj xj (t) j =1

−

n

dij (t)gj xj t − τij (t) + Ii (t)

j =1 ∗

rp n r |dij (ψij−1 (t))| ij 1 rqij∗ Lj xj (t) + αi r 1 − τ˙ij (ψij−1 (t)) j =1

n rp∗ r 1 rqij∗ Lj xj t − τij (t) dij (t) ij − αi r j =1 n n r cij (t)xi (t)r−1 Lj xj (t) ωi −α i βi xi (t) + α i j =1

i=1

n dij (t)xi (t)r−1 Lj xj t − τij (t) + αi j =1

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n n r−1 + α i xi (t) cij (t) gj (0) + dij (t) gj (0) + Ii (t) j =1

+

j =1

rp ∗ r |dij (ψij−1 (t))| ij rqij∗ 1 αi Lj xj (t) r 1 − τ˙ij (ψij−1 (t)) j =1 n

n rp∗ r 1 rqij∗ − αi Lj dij (t) ij xj t − τij (t) . r j =1

From Lemma 1, we have K1 ξij ij ∗ ∗ cij (t)Lj xi (t)r−1 xj (t) = cij (t) νk L νk xi (t) νk cij (t)ξij Lηij xj (t) η

j

j

k=1 K1 rξij ν ij r 1 rξ ∗ rη∗ r 1 νk cij (t) νk Lj k xi (t) + cij (t) ij Lj ij xj (t) r r rη

k=1

and

dij (t)Lj xi (t)r−1 xj t − τij (t) =

q K2 pij ij ∗ q∗ dij (t) µk L µk xi (t) µk dij (t)pij L ij xj t − τij (t)

j

k=1 K2

1 r

k=1

rpij µij r 1 rp∗ rq ∗ r µk dij (t) µk Lj k xi (t) + dij (t) ij Lj ij xj t − τij (t) . r rq

So +

j

D V (t, xt )|(6)

n

n K1 rξij ν ij 1 νk cij (t) νk Lj k −α i βi + α i r i=1 j =1 k=1 rq n K2 rpij µkij 1 µk xi (t)r + αi µk dij (t) Lj r j =1 k=1 n rξ ∗ rη∗ 1 cij (t) ij Lj ij + αi r j =1 rp ∗ −1 n r 1 rqij∗ |dij (ψij (t))| ij xj (t) + αi Lj −1 r 1 − τ ˙ (ψ (t)) ij ij j =1 n n cij (t)gj (0) + ωi α i rη

ωi

i=1

j =1

n r−1 + dij (t) gj (0) + Ii (t) xi (t) j =1

J. Cao, J. Liang / J. Math. Anal. Appl. 296 (2004) 665–685

n

673

n K1 rξij ν ij 1 −α i βi ωi + α i ωi νk cij (t) νk Lj k r rη

j =1 k=1

i=1

2 rpij µij 1 + αi ωi µk dij (t) µk Lj k r

n

K

rq

j =1 k=1

+

1 r

n

rξ ∗ rη∗ α j ωj cj i (t) ji Li ji

j =1

rp ∗ n ∗ |dj i (ψ −1 (t))| ji rqji 1 ji xi (t)r + α j ωj Li −1 r 1 − τ˙j i (ψj i (t)) j =1

+M

n xi (t)r−1 i=1

n n r xi (t)r−1 xi (t) + M −σ i=1

i=1

r r−1 = − σ x(t)r + M x(t)r−1 in which M=

sup

max

t ∈[0,+∞) 1in

n n cij (t)gj (0) + dij (t)gj (0) + Ii (t) ωi α i . j =1

By the equivalence of the norms in θ |x(t)|r , so we obtain

Rn ,

j =1

there is a constant θ > 0 such that |x(t)|r−1

r r−1 1 1 D + V (t, xt )|(6) − σ x(t)r + x(t)r Mθ r−1 − σ x(t)r 2 2 r 1 − σ x(t)r + M ∗ , 2 where 1 ∗ r−1 r−1 Mθ − σs . M = sup s 2 s∈[0,+∞)

(15)

From (13)–(15) and Lemma 3 we know all solutions of model (6) are uniformly bounded and uniformly ultimately bounded. 2 In Theorem 1, when r = 1, define n n V (t, xt ) = ωi xi (t) + α i Lj i=1

j =1

t t −τij (t )

|dij (ψij−1 (s))| xj (s) ds , 1 − τ˙ij (ψij−1 (s))

not using Hardy inequality and by direct computation, we have the following corollary.

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Corollary 1. Under assumptions (H1 )–(H3), all solutions of model (6) are uniformly bounded and uniformly ultimately bounded if there exist constants ωi > 0 (i = 1, 2, . . . , n), σ > 0, such that α i βi ωi −

n |dj i (ψj−1 i (t))| >σ α j ωj cj i (t)Li − α j ωj Li 1 − τ˙j i (ψj−1 i (t)) j =1 j =1

n

holds for all t 0. From Theorem 1 and Lemma 4, we can easily derive the following results. Theorem 2. Under assumptions (H1 )–(H3), let cij (t), dij (t), τij (t), Ii (t) (i, j = 1, 2, . . . , n) be periodic functions with periodic T > 0, then model (6) has a unique T -periodic solution if there exist constants νk > 0 (k = 1, 2, . . . , K1 ), µk > 0 (k = 1, 2, . . . , K2 ), ωi > 0, σ > 0, pij , pij∗ , qij , qij∗ , ξij , ξij∗ , ηij , ηij∗ ∈ R such that rα i βi ωi −

K1 n

K2 n rξij ν ij rpij µij α i ωi νk cij (t) νk Lj k − α i ωi µk dij (t) µk Lj k rη

j =1 k=1

−

n

rq

j =1 k=1

rξ ∗ α j ωj cj i (t) ji Li

∗ rηji

−

j =1

n

rq ∗ α j ωj Li ji

j =1

∗

rpji |dj i (ψj−1 i (t))|

1 − τ˙j i (ψj−1 i (t))

> rσ

1

K 2 ∗ holds for all t 0; where r = K k=1 νk + 1 = k=1 µk + 1 and K1 ξij + ξij = 1, K1 ηij + ∗ ∗ ∗ ηij = 1, K2 pij + pij = 1, K2 qij + qij = 1 (i, j = 1, 2, . . . , n). Corollary 2. Under assumptions (H1 )–(H3), let cij (t), dij (t), τij (t), Ii (t) (i, j = 1, 2, . . . , n) be periodic functions with periodic T > 0, then model (6) has a unique T -periodic solution if there exist constants ωi > 0 (i = 1, 2, . . . , n), σ > 0, such that n |dj i (ψj−1 i (t))| >σ α i βi ωi − α j ωj cj i (t) Li − α j ωj Li 1 − τ˙j i (ψj−1 i (t)) j =1 j =1 n

holds for all t 0. Corollary 3. Under assumptions (H1 )–(H3), let cij (t), dij (t), τij (t), Ii (t) (i, j = 1, 2, . . . , n) be periodic functions with periodic T > 0, then model (6) has a unique T -periodic solution if there exist a constant σ > 0, such that α i βi −

n j =1

n α i cij (t)Lj − α i Lj dij (t) > σ j =1

holds for all t 0 and i = 1, 2, . . . , n. Proof. In Theorem 2, if we take K1 = K2 = 1, νk = µk = r − 1, ωi = 1, ξij = ηij = pij = 1 ∗ ∗ ∗ ∗ qij = r−1 r and ξij = ηij = pij = qij = r (i, j = 1, 2, . . . , n), a condition to ensure system (6) has a T -periodic solution is obtained as

J. Cao, J. Liang / J. Math. Anal. Appl. 296 (2004) 665–685

α i βi − −

1 r

675

r − 1 1 r − 1 α i cij (t)Lj − α j cj i (t)Li − α i dij (t)Lj r r r n

n

n

j =1

j =1

j =1

|dj i (ψj−1 i (t))| α j Li 1 − τ˙j i (ψj−1 i (t)) j =1

n

> σ.

Let r tends to +∞, and the result follows. 2 Remark 1. For system (6), when ai (xi (t)) ≡ 1, bi (xi (t)) = bi (t)xi (t) (in which bi (t) is not only differentiable but also bounded on interval [−τ, +∞), and its maximal lower bound is denoted as βi > 0) then system (6) turns out to be a recurrent neural network model with variable coefficients and time-varying delays. In this case, Theorems 1 and 2 turn out to be a generalized result for those in [31,32], that is, the results in [31,32] are special cases of ours.

4. Stability results In this section, we will obtain some criteria for global exponential stability of (5) or (9). Moreover, the uniqueness of the equilibrium point follows directly from its global exponential stability. Theorem 3. Under assumptions (H1 ), (H2 ) and (H3 ), model (9) is globally exponentially stable if there exist constants νk > 0 (k = 1, 2, . . . , K1 ), µk > 0 (k = 1, 2, . . . , K2 ), pij , pij∗ , qij , qij∗ , ξij , ξij∗ , ηij , ηij∗ ∈ R such that σ1 > σ2 > 0,

1

K 2 ∗ ∗ ∗ where r = K k=1 νk +1 = k=1 µk +1; K1 ξij +ξij = 1, K1 ηij +ηij = 1, K2 pij +pij = 1, K2 qij + qij∗ = 1 and

rηij rqij K1 K2 n n rξij rpij νk µ νk µk νk |cij | Lj α i µk |dij | Lj k σ1 = min rα i βi − α i 1in

j =1 k=1

−

n

α j |cj i |

∗ rξji

∗ rηji

Li

j =1 k=1

,

j =1

σ2 = max

n

1in

∗

∗ rqji

α j |dj i |rpji Li

.

j =1

Proof. Define r 1 yi (t) , r n

V (t, yt ) =

i=1

it can easily be verified that V (t, yt ) is a nonnegative function over [−τ, +∞) and that it is radically unbounded, that is, V (t, yt ) → +∞ as |y(t)|r → +∞.

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Next, evaluating the Dini derivative of V along the trajectory of (9) gives n r−1 yi (t) D + V (t, yt ) = sgn yi (t) −Ai yi (t) Bi yi (t) i=1

−

n

n cij fj yj (t) − dij fj yj t − τij (t)

j =1

n

j =1

n r r−1 −α i βi yi (t) + α i |cij |Lj yi (t) yj (t) j =1

i=1 n

+ αi

n

r−1 |dij |Lj yi (t) yj t − τij (t)

j =1

r −α i βi yi (t)

i=1

K n 1 1

+ αi

j =1

r

j =1

r

rηij νk

Lj

k=1

K n 2 1

+ αi

νk |cij |

rξij νk

µk |dij |

rpij µk

∗ rη∗ yi (t)r + 1 |cij |rξij L ij yj (t)r j r

rqij µk

Lj

yi (t)r

k=1

r ∗ rqij∗ 1 rpij + |dij | Lj yj t − τij (t) r rηij K1 n n rξij 1 ν νk rα i βi − α i νk |cij | Lj k =− r j =1 k=1

i=1

−

n

α j |cj i |

∗ rξji

rη∗ Li ji

− αi

j =1

K2 n

µk |dij |

rpij µk

rqij µk

Lj

yi (t)r

j =1 k=1

n n ∗ r ∗ rqji 1 rpji + α j |dj i | Li yi t − τj i (t) r i=1

j =1

−σ1 V (t, yt ) + σ2 V (t), and from Lemma 2, it can be drawn that if σ1 > σ2 > 0, then sup V (s) e−λs , V (t, yt ) −τ s0

where λ is the unique positive solution of equation: λ = σ1 − σ2 eλτ . Therefore, V (t, yt ) converges to zero exponentially, which in turn implies that y(t) also converges globally and exponentially to zero with a convergence rate λr , and this completes the proof. 2

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677

In Theorem 3, if we take K1 = K2 = 1, νk = µk = r − 1, ξij = ηij = pij = qij = and ξij∗ = ηij∗ = pij∗ = qij∗ = 1r (i, j = 1, 2, . . . , n), we have the following result.

r−1 r

Corollary 4. Under assumptions (H1 ), (H2 ) and (H3 ), model (9) is globally exponentially stable if σ1 > σ2 > 0, where

σ1 = min

1in

rα i βi − (r − 1)α i −

n

n

|cij |Lj − (r − 1)α i

j =1

n

|dij |Lj

j =1

α j |cj i |Li ,

j =1

σ2 = max

n

1in

α j |dj i |Li

j =1

and r > 1 is a constant number. Remark 2. In [25] the authors gave a condition L(C + D)η < 1 to ensure system (5) to be globally exponentially stable. To obtain the result, firstly system (5) was written into a matrix form dx(t) = −A x(t) B x(t) − Cg x(t) − Dg x t − τ (t) + I ; dt secondly norm of matrix was utilized. However, in the term gj (xj (t − τij (t))), τij (t) is not only related to index j but also to index i, it is thus impossible to write system (5) into the required matrix form as suggested in [25]. Theorem 4. Under assumptions (H1 ), (H2 ) and (H3 ), model (9) is globally exponentially stable if there exist constants νk > 0 (k = 1, 2, . . . , K1 ), µk > 0 (k = 1, 2, . . . , K2 ), ωi > 0, pij , pij∗ , qij , qij∗ , ξij , ξij∗ , ηij , ηij∗ ∈ R such that rα i βi ωi −

K1 n

α i ωi νk |cij |

rξij νk

Lj

j =1 k=1

−

n

α j ωj |cj i |

j =1

rηij νk

−

K2 n

α i ωi µk |dij |

rpij µk

rqij µk

Lj

j =1 k=1 ∗ rξji

rη∗ Li ji

−

n j =1

rq ∗ α j ωj Li ji

∗

|dj i |rpji 1 − τ˙j i (ψj−1 i (t))

>0

1

K 2 ∗ holds for all t 0; where r = K k=1 νk + 1 = k=1 µk + 1 and K1 ξij + ξij = 1, K1 ηij + ηij∗ = 1, K2 pij + pij∗ = 1, K2 qij + qij∗ = 1 (i, j = 1, 2, . . . , n). Proof. Suppose y(t) = (y1 (t), y2 (t), . . . , yn (t))T is a solution of model (9) with ϕ − x ∗ as its initial function, that is, yi (t) = ϕi (t) − xi∗ ,

t ∈ [−τ, 0], i = 1, 2, . . . , n.

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Since rα i βi ωi −

K1 n

α i ωi νk |cij |

rξij νk

rηij νk

Lj

−

j =1 k=1

−

n

K2 n

α i ωi µk |dij |

rpij µk

rqij µk

Lj

j =1 k=1

α j ωj |cj i |

∗ rξji

rη∗ Li ji

−

j =1

n

rq ∗ α j ωj Li ji

j =1

∗

|dj i |rpji 1 − τ˙j i (ψj−1 i (t))

> 0,

we can choose a small ε > 0 such that ωi (ε − rα i βi ) +

K1 n

α i ωi νk |cij |

rξij νk

rηij νk

Lj

j =1 k=1

+

n

α j ωj |cj i |

∗ rξji

+

K2 n

α i ωi µk |dij |

rpij µk

rqij µk

Lj

j =1 k=1

rη∗ Li ji

+e

ετ

j =1

n

rq ∗ α j ωj Li ji

j =1

∗

|dj i |rpji 1 − τ˙j i (ψj−1 i (t))

< 0.

Now we consider the Lyapunov functional n r 1 V (t, yt ) = ωi yi (t) eεt r i=1

+ αi

n

t

rq ∗ Lj ij

j =1

yj (s)r

t −τij (t )

|dij |

∗ rpij

1 − τ˙ij (ψij−1 (s))

e

ε(s+τij (ψij−1 (s)))

calculating the upper right Dini derivative of V (t, yt ), we obtain n r r−1 1 + ωi εeεt yi (t) + reεt yi (t) D + yi (t) D V (t, yt )|(9) = r i=1

+ αi

n

r −1 rq ∗ Lj ij eε(t +τij (ψij (t )))yj (t)

j =1

− αi

n j =1

e

εt

n

rq ∗

Lj ij |dij |

ωi

i=1

+ αi

n

∗ rpij

|dij |

1 − τ˙ij (ψij−1 (t))

r yj t − τij (t) eεt

r r 1 ε yi (t) − α i βi yi (t) r

r−1 |cij |yi (t) Lj yj (t)

j =1

+ αi

n

r−1 |dij |yi (t) Lj yj t − τij (t)

j =1 ∗

n rp r |dij | ij 1 rqij∗ ετ + αi Lj e yj (t) r 1 − τ˙ij (ψij−1 (t)) j =1

∗ rpij

ds ,

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679

n r ∗ 1 rqij∗ rpij . − αi Lj |dij | yj t − τij (t) r j =1

Estimating the right of inequality above by the Hardy inequality, we have D + V (t, yt )|(9) eεt

n

ωi

i=1

r r 1 ε yi (t) − α i βi yi (t) r

K rηij n 1 rξij ∗ ∗ rηij 1 1 ν r r rξ + αi νk |cij | νk Lj k yi (t) + |cij | ij Lj yj (t) r r j =1

k=1

rqij K2 n rpij r 1 µ µk + αi µk |dij | Lj k yi (t) r

j =1

k=1

r ∗ rq ∗ 1 + |dij |rpij Lj ij yj t − τij (t) r

∗

n rp r |dij | ij 1 rqij∗ ετ + αi Lj e yj (t) r 1 − τ˙ij (ψij−1 (t)) j =1

n r 1 rqij∗ rp ∗ − αi Lj |dij | ij yj t − τij (t) r

j =1

rηij K1 n n rξij 1 εt ν = e α i ωi νk |cij | νk Lj k ωi (ε − rα i βi ) + r j =1 k=1

i=1

+

K2 n

α i ωi µk |dij |

rpij µk

rqij µk

Lj

+

j =1 k=1

+e

ετ

n j =1

j =1 rq ∗ α j ωj Li ji

∗

|dj i |rpji 1 − τ˙j i (ψj−1 i (t))

0, and so V (t) V (0),

t 0,

since n 1 yi (t)r = 1 ωeεt y(t)r , V (t) eεt ω r r r i=1

n

t 0,

∗

∗ rηji

α j ωj |cj i |rξji Li yi (t)r

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n r 1 V (0) = ωi ϕi (0) − xi∗ r i=1

+ αi

n

0

rq ∗ Lj ij

j =1

yj (s)r

−τij (0)

∗

|dj i |rpij 1 − τ˙j i (ψj−1 i (s))

e

ε(s+τij (ψij−1 (s)))

ds

n n 0 r 1 ∗ r ετ ϕi (0) − xi + Lne yi (s) ds ω r i=1 i=1 −τ n r 1 ετ ∗ ϕi (s) − x ω(1 + Lnτ e ) sup i r s∈[−τ,0] i=1

1 = ω(1 + Lnτ eετ )ϕ − x ∗ rr , r then it easily follows that yt rr = xt − x ∗ rr

ω (1 + Lnτ eετ )ϕ − x ∗ rr e−εt , ω

and this means yt r = xt − x ∗ r Mϕ − x ∗ r e−ε

∗t

for all t 0, where M 1 is a constant, ε∗ = (5) or (9) is globally exponentially stable. 2

ε r

> 0. This implies the solutions of model

Remark 3. Notice that (5) becomes (4) when τij (t) = τij . For this model, it has been reported in [24] that if (i) (ii)

gj (·) (j and

= 1, 2, . . . , n) are nondecreasing n n 1 1 + |dij | min βi − aii Li − |cij |Lj + |cj i |Li − 1in 2 2 j =i, j =1 j =1 n 1 2 |dj i |Li > 0, − 2 j =1

then (4) has a unique and globally asymptotically stable equilibrium point. In Theorem 4 above, by taking K1 = K2 = νk = µk = 1, ξij = ηij = pij = ξij∗ = ηij∗ = pij∗ = 12 , ωi = 1, qij∗ = 1 and qij = 0, a similar result can be derived (we obtained a stronger result of exponential stability). In other words, [24, Theorem 1] is a special case of ours. Remark 4. In the theoretical development in [24, Theorem 3], when y(t) = x(t) − x ∗ = 0, f (y(t)) = g(y(t) + x ∗ ) − g(x ∗ ) = 0 and f (y(t − τ )) = g(y(t − τ ) + x ∗ ) − g(x ∗ ) = 0, system (4) becomes dyi (t) = −Ai yi (t) Bi yi (t) dt

J. Cao, J. Liang / J. Math. Anal. Appl. 296 (2004) 665–685

681

(see [24, Eq. (38)]). The authors then concluded that dyi (t) −Ai yi (t) βi yi (t) dt which is incorrect and should be modified to dyi (t) = −Ai yi (t) sgn yi (t) Bi yi (t) D + yi (t) = sgn yi (t) dt = −Ai yi (t) Bi yi (t) −Ai yi (t) βi yi (t). Nevertheless, the conclusion that |yi (t)| → 0 (as t → +∞) remains valid. When r = 1, in the proof of Theorem 4, if one defines n ωi eεt yi (t) V (t, yt ) = i=1

+ αi

n j =1

t Lj

yj (s)

t −τij (t )

|dij | 1 − τ˙ij (ψij−1 (s))

e

ε(s+τij (ψij−1 (s)))

ds ,

and not employing the Hardy inequality, direct computation leads to the following results. Corollary 5. Under assumptions (H1 ), (H2 ) and (H3 ), all solutions of model (9) are globally exponentially stable if there exist constants ωi > 0 (i = 1, 2, . . . , n), such that α i βi ωi −

n |dj i (ψj−1 i (t))| α j ωj cj i (t)Li − α j ωj Li >0 1 − τ˙j i (ψj−1 i (t)) j =1 j =1

n

holds for all t 0. Corollary 6. Under assumptions (H1 ), (H2 ) and (H3 ), all solutions of model (9) are globally exponentially stable if n n α i βi − α i |cij |Lj − α i Lj |dij | > 0 j =1

j =1

holds for all t 0 and i = 1, 2, . . . , n. 5. Illustrative examples Example 1. Consider the following system: dx1 (t) = − 2 + cos x1 (t) 8x1 (t) − sin t × f x1 (t − 0.5 sin t − 1) dt − cos t × f x2 (t − 0.5 sin t − 1) + sin t , dx2 (t) = − 2 + sin x2 (t) 8x2 (t) − cos t × f x1 (t − 0.5 sin t − 1) dt − sin t × f x2 (t − 0.5 sin t − 1) + cos t ,

(16)

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J. Cao, J. Liang / J. Math. Anal. Appl. 296 (2004) 665–685

where f (x) = 0.5(|x + 1| − |x − 1|). The system satisfies all assumptions in this paper with α 1 = α 2 = 1, α 1 = α 2 = 3, L1 = L2 = 1, β1 = β2 = 8, 0 τij (t) = 0.5 sin t + 1 1.5 and supt ∈[−1.5,+∞) τ˙ij (t) = 0.5 < 1 (i, j = 1, 2). In Corollary 3, if we take σ = 1, then α 1 β1 − α 1 d11(t)L1 − α 1 d12 (t)L2 = 8 − 3| sin t| − 3| cos t| > σ = 1, α β2 − α 2 d21(t)L1 − α 2 d22 (t)L2 = 8 − 3| cos t| − 3| sin t| > σ = 1, 2

therefore we can deduce that system (16) has a 2π -periodic solution and it is uniformly bounded, uniformly ultimately bounded.

Fig. 1. Transient response of state variables x1 (t) and x2 (t) for Example 1.

Fig. 2. Phase plots of state variables x1 (t) and x2 (t) for Example 1.

J. Cao, J. Liang / J. Math. Anal. Appl. 296 (2004) 665–685

683

For numerical simulation, the following four cases are given: Case 1 with the initial state (ϕ1 (t), ϕ2 (t)) = (0.09, −0.09) for t ∈ [−1.5, 0]; Case 2 with the initial state (ϕ1 (t), ϕ2 (t)) = (0.115, −0.115) for t ∈ [−1.5, 0]; Case 3 with the initial state (ϕ1 (t), ϕ2 (t)) = (0.08, −0.08) for t ∈ [−1.5, 0]; Case 4 with the initial state (ϕ1 (t), ϕ2 (t)) = (0.125, −0.125) for t ∈ [−1.5, 0]. Figure 1 depicts the time responses of state variables of x1 (t) and x2 (t) with step h = 0.01, and Fig. 2 depicts the phase plots of state variables x1 (t) and x2 (t). It confirms that the proposed condition leads to the unique 2π -periodic solution for the model. Example 2. Consider dx1 (t) = − 7 + sin x1 (t) 3x1 (t) − tanh x1 (t − 1) − tanh x2 (t − 1) + 2 , dt dx2 (t) = − 4 + cos x2 (t) 4x2(t) − tanh x1 (t − 1) − tanh x2 (t − 1) + 3 . (17) dt This system satisfies all assumptions in this paper with α 1 = 6, α 2 = 3, α 1 = 8, α 2 = 5, L1 = L2 = 1, β1 = 3, β2 = 4, τij (t) ≡ 1 (i, j = 1, 2), then α 1 β1 − α 1 |d11|L1 − α 1 |d12 |L2 = 18 − 8 − 8 = 2 > 0, α 2 β2 − α 2 |d21|L1 − α 2 |d22 |L2 = 12 − 5 − 5 = 2 > 0, from Corollary 6 we know the solutions of system (17) are globally exponentially stable. For numerical simulation, the following two cases are given: Case 1 with the initial state (ϕ1 (t), ϕ2 (t)) = (−0.1 − | sin t|, −0.1 − | cos t|) for t ∈ [−2, 0]; Case 2 with the initial state (ϕ1 (t), ϕ2 (t)) = (−0.1 − | sin 2t|, −0.1 − | cos 2t|) for t ∈ [−2, 0].

Fig. 3. Transient response of state variables x1 (t) and x2 (t) for Example 2.

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Figure 3 depicts the time responses of state variables of x1 (t) and x2 (t) with step h = 0.01. It confirms that the proposed condition leads to the unique and globally exponentially stable solution for the model.

6. Conclusions The dynamics of the Cohen–Grossberg neural network is studied in this paper with variable coefficients and time-varying delays. By employing the Hardy inequality, several sufficient conditions have been obtained which guarantee the model to be uniformly bounded and ultimately uniformly bounded under appropriate assumptions. The Halanay inequality and Lyapunov functional method are also used in this paper to derive some new sufficient conditions ensuring the model to be globally exponentially stable. It is noted that the criteria derived in this paper are less restrictive than those reported in [21,22,24]. Several examples and their numerical simulations are also given to illustrate the effectiveness. The results obtained in this paper are delay-independent, which implies the strong self-regulation is dominant in the networks, and moreover they are useful in the design and applications of Cohen–Grossberg neural network. In addition, the methods given in this paper may be extended for more complex systems.

Acknowledgments The authors thank the reviewers and the editor for their helpful comments and constructive suggestions, and also thank Dr. James Lam for his help in improving our English writing, which have been very useful for improving the presentation of this paper.

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