3_454_wu.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 5, 2010, 806-818 ISSN 1307-5543 – www.ejpam.com Global Asymptotical Stability of Delayed Impulsive Neural Networks without Lipschitz Neuron Activations Ailong Wu1, Jine Zhang2, and Chaojin Fu3,∗ 1 Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China 2 School of Basic Science, East China Jiaotong University, Nanchang 330013, China 3 College of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China Abstract. In this paper, based on the homeomorphism theory and Lyapunov functional method, we investigate global asymptotical stability for a novel class of delayed impulsive neural networks with- out Lipschitz neuron activations. Some sufficient conditions are derived which ensure the existence, uniqueness, and global asymptotical stability of the equilibrium point of neural networks. Finally, a numerical example is given to demonstrate the improvements of the paper. 2000 Mathematics Subject Classifications: 37B25; 92B20; 34D23 Key Words and Phrases: Neural networks, Global asymptotical stability, Impulses. 1. Introduction In the design of neural networks, the model of neural networks is descried by the system of nonlinear ordinary differential equations or the system of nonlinear functional differential equations. In generality, these nonlinear systems possibly show complex dynamic behaviors, such as, periodic oscillatory, bifurcation, chaos, etc. However, in practical applications, es- pecially for solving linear and quadratic programming problems in real time, it requires that networks have good convergent property. Under these good convergent property, the validity can be guaranteed during numeral solving. Due to these, stability analysis for neural net- works with or without time delays has received a great of attention (see [1− 10]). Recently, impulsive neural networks have been extensively studied in both theory and applications (see [5−10]). However, in the existing literatures, almost all results on the stability of neural net- works are obtained under Lipschitz neuron activations [1−6,9,10]. When neuron activation functions do not satisfy Lipschitz conditions, people want to know whether the neural net- works is stable. In practical engineering applications, people also need to present new neural ∗Corresponding author. Email addresses: alwu83�gmail. om (A. Wu), jezhang�126. om (J. Zhang), haojinfu�126. om (C. Fu) http://www.ejpam.com 806 c© 2010 EJPAM All rights reserved. A. Wu, J. Zhang, C. Fu / Eur. J. Pure Appl. Math, 3 (2010), 806-818 807 networks. Therefore, developing a new class of neural networks without Lipschitz neuron activation functions and giving the conditions of the stability of new neural networks are very interesting and valuable. In this paper, we investigate a general class of delayed neural networks with impulses where the neuron activations do not satisfy Lipschitz conditions. To the best of authors’ knowl- edge, this is the first time to study the existence, uniqueness, and global asymptotical stability of equilibrium point for the neural networks developed by us. Consider a general delayed neural networks with impulses:    ẋ i(t) = −di x i(t) + n ∑ j=1 ai j g j(x j(t)) + n ∑ j=1 bi j g j(x j(t −τi j)) + Ii , t 6= tk, △x i(tk) = x i(t + k )− x i(t − k ) = Jik(x i(tk)), k = 1,2, · · · , i = 1, · · · , n, (1) where n denotes the number of the neurons; x i(t) is the state of the ith neuron at time t; di > 0 is the neural self-inhibitions of the ith neuron; g j(·) represents the input-output activation of the jth neuron, g j(·) is continuous and monotone nondecreasing; ai j and bi j denote the connection of the jth neuron to the ith neuron at time t and t −τi j, respectively; Ii is the external bias on the ith neuron; 0 ≤ τi j ≤ τ, τ is a positive constant; Jik shows impulsive perturbation of the ith neuron at time tk; △x i(tk) = x i(t + k )− x i(t − k ), k = 1,2, · · · , are the impulses at moments tk, and 0 < t1 < t2 < · · · is a strictly increasing sequence such that lim k→∞ tk =∞. The system (1) is supplemented with the initial conditions of the type x(t) = φ(t) = (φ1, · · · ,φn) T ,−τ ≤ t ≤ 0 in which φ(t) ∈ C([−τ, 0]; Rn) is a continuous function. C([−τ, 0]; Rn) is a Banach space of continuous mapping which maps [−τ, 0] into Rn with a topology of uniform convergence. For convenience, we introduce the following notations: Let matrix Q = (qi j)n×n, R = (ri j)m×n, Q−1 denotes the inverse of Q, ‖R‖1 and ‖R‖∞ represent the first norm and infin- ity norm of matrix R, respectively. That is, ‖R‖1 = max 1≤ j≤n m ∑ i=1 � �ri j � �, ‖R‖∞ = max 1≤i≤m n ∑ j=1 � �ri j � �. Symmetric matrix S = (si j)n×n, S > 0 (S ≥ 0,S < 0,S ≤ 0) means that S is positive definite (positive semi-definite, negative definite, negative semi-definite). Given the vector ψ = (ψ1, · · · ,ψn) T ∈ Rn, ψ = max 1≤i≤n � �ψi � �. I denotes identical matrix. We will sometimes write x(t) as x , f (x(t)) as f (x). Definition 1. A function x(t) : [−τ,+∞]→ Rn is said to be a solution of system (1) with initial conditions x(t) = φ(t), t ∈ [−τ, 0], if the following conditions are satisfied: (1) x(t) is piecewise continuous with first kind discontinuity at points tk, k = 1,2, · · · . Moreover, x(t) is right continuous at each discontinuity points; (2) x(t) satisfies system (1) for t ≥ 0, and x(s) = φ(s) for s ∈ [−τ, 0]. Definition 2. A constant vector x∗ = (x∗1, · · · , x∗n) T ∈ Rn is an equilibrium point of system (1.1) A. Wu, J. Zhang, C. Fu / Eur. J. Pure Appl. Math, 3 (2010), 806-818 808 if and only if x∗ is a solution of the following equations: −di x i + n ∑ j=1 (ai j + bi j)g j(x j) + Ii = 0, i = 1, · · · , n, (2) and the impulsive jumps Jik(·) are assumed to satisfy Jik(x ∗ i ) = 0, k = 1,2, · · · , i = 1, · · · , n. Lemma 1. Continuous map H(x) : Rn → Rn is homeomorphic, if: (1) H(x) is injective; (2) lim ‖x‖p→∞ ‖H(x)‖p =∞. 2. Existence and Uniqueness of the Equilibrium Point First, we definite the map H(x) associated with (2) as follows: H(x) = (H1(x), · · · , Hn(x)) T (3) where Hi(x) = −di x i + n ∑ j=1 (ai j + bi j)g j(x j) + Ii , i = 1, · · · , n. Theorem 1. If there exist positive constants pi > 0, i = 1, · · · , n, such that pi(−aii − � �bii � �)− n ∑ j=1, j 6=i p j( � �a ji � �+ � �b ji � �)≥ 0, i = 1, · · · , n, then Eq.(2) has a unique solution. Proof. In order to complete the proof, we divide the proof into two steps. Step 1. Let x and y be two different vectors in Rn, then we have n ∑ i=1 pisi gn(x i − yi)(Hi(x)−Hi(y)) ≤ − n ∑ i=1 pidi � �x i − yi � �+ n ∑ i=1 pi(aii + bii) � �gi(x i)− gi(yi) � � + n ∑ i=1 n ∑ j=1, j 6=i pi( � �ai j � �+ � �bi j � �) · � �g j(x j)− g j(y j) � � ≤ − n ∑ i=1 pidi � �x i − yi � �+ n ∑ i=1 pi(aii + � �bii � �) � �gi(x i)− gi(yi) � � + n ∑ i=1 n ∑ j=1, j 6=i p j( � �a ji � �+ � �b ji � �) · � �gi(x i)− gi(yi) � � ≤ − n ∑ i=1 pidi � �x i − yi � � < 0 (4) A. Wu, J. Zhang, C. Fu / Eur. J. Pure Appl. Math, 3 (2010), 806-818 809 Moreover, there exists k0 ∈ {1, · · · , n} such that Hk0 (x) 6= Hk0 (y). That is, H(x) 6= H(y) for all x 6= y. Step 2. In (4), let y = 0, we get n ∑ i=1 pi(Hi(x)−Hi(0))si gn(x i − 0)≤ − n ∑ i=1 pidi � �x i � � ≤ −pmin n ∑ i=1 � �x i � � (5) where pmin = min 1≤i≤n � pidi . From (5), it follows that pmin‖x‖1 ≤ � � � � � n ∑ i=1 pi(Hi(x)−Hi(0)) � � � � � ≤ pmax n ∑ i=1 � �(Hi(x)−Hi(0)) � � ≤ pmax‖H(x)−H(0)‖1 ≤ pmax(‖H(x)‖1 + ‖H(0)‖1) where pmax =max � p1, · · · , pn . We obtain ‖H(x)‖1 ≥ pmin‖x‖1 − pmax‖H(0)‖1 pmax from which it can be easily concluded that ‖H(x)‖1 → ∞ as ‖x‖1 → ∞. Hence, we have proved that H(x) is a homeomorphism on Rn. That is, Eq.(2) has a unique solution. Theorem 2. If � �gi(x i) � � → ∞ as � �x i � � → ∞, i = 1, · · · , n, and there exists a positive constant r such that A+AT + ( 1 r ‖B‖∞ + r‖B‖1)I ≤ 0 where A= (ai j)n×n and B = (bi j)n×n, then Eq.(2) has a unique solution. Proof. In order to complete the proof, we divide the proof into two steps. Step 1. Let x and y be two different vectors in Rn, by gi(·) is monotone nondecreasing, x 6= y will imply two cases: (i) x 6= y and g(x)− g(y) 6= 0; (ii) x 6= y and g(x)− g(y) = 0. First, consider the case (i) where x 6= y and g(x)− g(y) 6= 0. In this case, there exists h ∈ {1, · · · , n} such that (xh− yh)(gh(xh)− gh(yh)) > 0 and (x i − yi)(gi(x i)− gi(yi)) ≥ 0 for i 6= h. Moreover, we have 2(g(x)− g(y))T · (H(x)−H(y)) ≤ −2 n ∑ i=1 di(x i − yi)(gi(x i)− gi(yi))+ (g(x)− g(y))T (A+AT )(g(x)− g(y)) + n ∑ i=1 n ∑ j=1 1 r � �bi j � � (gi(x i)− gi(yi)) 2+ n ∑ i=1 n ∑ j=1 r � �bi j � � (g j(x j)− g j(y j)) 2 A. Wu, J. Zhang, C. Fu / Eur. J. Pure Appl. Math, 3 (2010), 806-818 810 ≤ −2 n ∑ i=1 di(x i − yi)(gi(x i)− gi(yi)) + (g(x)− g(y))T [A+ AT + ( 1 r ‖B‖∞ + r‖B‖1)I] · (g(x)− g(y)) ≤ −2 n ∑ i=1 di(x i − yi)(gi(x i)− gi(yi)) ≤ −2dh(xh− yh)(gh(xh)− gh(yh))< 0 (6) Hence, Hh(x) 6= Hh(y). That is, H(x) 6= H(y) when x 6= y and g(x) 6= g(y). Now consider the case (ii) where x 6= y and g(x)− g(y) = 0. In the case, we have H(x)−H(y) = −D(x − y) 6= 0 where D = diag(d1, · · · , dn). Thus, H(x) 6= H(y) for all x 6= y and g(x) = g(y). Hence, we have proved that H(x) 6= H(y) when x 6= y. Step 2. In (2.4), let y = 0, we get 2(g(x)− g(0))T · (H(x)−H(0))≤ −2 n ∑ i=1 di(x i − 0)(gi(x i)− gi(0)) ≤ −2d x T(g(x)− g(0)) (7) where d =min � d1, · · · , dn . From (7) and gi is monotone nondecreasing, it follows that 0≤ d n ∑ i=1 x i(gi(x i)− gi(0))≤ n ∑ i=1 � �(gi(x i)− gi(0)) · (Hi(x)−Hi(0)) � � . (8) If lim ‖x‖→∞ ‖H(x)‖ 6=∞, then there exists a sequence {x p} such that lim p→∞ ‖x p‖ =∞ and for all p, ‖H(x p)‖ ≤ M1, where M1 is a positive constant. Therefore, there exists a subsequence (for convenience, we also denote it as {x p}) and a nonempty set W ⊂ {1, · · · , n}, such that the follows hold: (1) lim p→∞ � �x p i � � =∞ for all i ∈W ; (2) there exists a positive constant M2 such that � �x p i � � ≤ M2 for all p and i ∈ {1, · · · , n} \W ; (3) Hi(x p) ≤ M1 for all i and p. Since gi(s) is continuous on [−M2, M2], there exists a positive constant M3 such that � �gi(s) � � ≤ M3 for all s ∈ [−M2, M2] and i ∈ {1, · · · , n} \W . Thus, � �gi(x p i ) � � ≤ M3 for all p and i ∈ {1, · · · , n}\W . Moreover, we have n ∑ i=1 � �(gi(x p i )− gi(0)) · (Hi(x p)−Hi(0)) � � = ∑ i∈W � �(gi(x p i )− gi(0)) · (Hi(x p)−Hi(0)) � � A. Wu, J. Zhang, C. Fu / Eur. J. Pure Appl. Math, 3 (2010), 806-818 811 + ∑ i /∈W � �(gi(x p i )− gi(0)) · (Hi(x p)−Hi(0)) � � ≤ ∑ i∈W � �gi(x p i )− gi(0) � � · (M1 + � �Hi(0) � �) + ∑ i /∈W (M3 + � �gi(0) � �) · (M1 + � �Hi(0) � �) ≤ M ∑ i∈W � �gi(x p i )− gi(0) � �+M (9) where M =max{max 1≤ j≤n ¦ M1 + � �H j(0) � � © , ∑ i /∈W (M3 + � �gi(0) � �) · (M1 + � �Hi(0) � �)}. By gi(·) is mono- tone nondecreasing, then we have n ∑ i=1 x p i (gi(x p i )− gi(0)) = ∑ i∈W x p i (gi(x p i )− gi(0))+ ∑ i /∈W x p i (gi(x p i )− gi(0)) ≥ ∑ i∈W � �x p i � � · � �(gi(x p i )− gi(0) � � (10) Substituting (9) and (10) into (8), we can obtain for all p d ∑ i∈W � �x p i � � · � �(gi(x p i )− gi(0) � �≤ d n ∑ i=1 x p i (gi(x p i )− gi(0)) ≤ n ∑ i=1 � �(gi(x p i )− gi(0)) · (Hi(x p i )−Hi(0)) � � ≤ M ∑ i∈W � �gi(x p i )− gi(0) � �+M So we have ∑ i∈W (d � �x p i � �−M) · � �gi(x p i )− gi(0) � �≤ M (11) Since lim p→∞ � �x p i � � =∞ for all i ∈ W and lim |xi|→∞ � �gi(x i) � � =∞, there exists a positive constant P such that for all p > P, � �(gi(x p i )− gi(0) � �≥ 1 and � �x p i � �> 2M d . Hence, ∑ i∈W (d � �x p i � �−M) · � �gi(x p i )− gi(0) � �> M (12) which is contradict to (11). So H(x) is a homeomorphism on Rn. That is, Eq.(2) has a unique solution. Theorem 3. Under assumptions of Theorem 1 (or Theorem 2), system (1) has a unique equilib- rium point x∗ = (x∗1, · · · , x∗n) T . Proof. By Definition 2 and Theorem 1 (or Theorem 2), it is obvious that the constant vector x∗ is the unique equilibrium point of system (1). A. Wu, J. Zhang, C. Fu / Eur. J. Pure Appl. Math, 3 (2010), 806-818 812 3. Global Asymptotic Stability of the Equilibrium Point In this section, we aim to find some sufficient conditions ensuring the global asymptotic stability of the equilibrium point of system (1). The equilibrium point of system (1) is said to be globally asymptotically stale if it is locally stable in sense of Lyapunov and globally attractive, i.e., every solution of system (1) corresponding to an arbitrary given set of initial conditions satisfy lim t→∞ x i(t) = x∗ i , i = 1, · · · , n. To prove the global asymptotic stability of the equilibrium point, we will employ the Lyapunov direct method. Namely, the equilibrium point x∗ is stable and every solution is bounded if there exists a continuously differentiable Lyapunov function V : Rn→ R which is positive definite and radially unbounded, i.e., V (x∗) = 0, V (x) > 0 for x 6= 0, lim ‖x−x∗‖→∞ V (x) = ∞, such that the time-derivative of V along the solution of system (1) is negative semi-definite. If, in addition, V̇ (x) is negative definite, then the equilibrium point of system (1) is globally asymptotically stable. Theorem 4. Under assumptions of Theorem 1, further if the following conditions are satisfied Jik(x i(tk)) = −γik(x i(tk)− x∗i ), k = 1,2, · · · , i = 1, · · · , n, where x∗ = (x∗1, · · · , x∗n) T is the equilibrium point of system (1), 0 < γik < 2, then system (1) has a unique equilibrium point which is globally asymptotically stable. Proof. In order to complete the proof, we divide the proof into four steps. Step 1. Consider the following system:    ẋ i(t) = −di x i(t) + n ∑ j=1 ai j g j(x j(t)) + n ∑ j=1 bi j g j(x j(t −τi j)) + Ii , t ∈ [0, t1], x i(t) = φi(t), t ∈ [−τ, 0], i = 1, · · · , n. (13) By g j is a continuous function, Ui(t) = −di x i(t)+ n ∑ j=1 ai j g j(x j(t))+ n ∑ j=1 bi j g j(x j(t−τi j))+ Ii is continuous and local bounded. It is easy to obtain the existence of a solution of system (3.1) on [0, t∗(φ)), where t∗(φ) ∈ (0, t1) or t∗(φ) = t1, and [0, t∗(φ)) is the maximal right- side existence interval of the solution of system (13). We denote this solution by x(t,φ), x(t,φ) = (x1(t,φ1), · · · , xn(t,φn)) T . Make a transformation z(t) = x(t)− x∗, system (13) is transformed into żi(t) = −dizi(t) + n ∑ j=1 ai j f j(z j(t)) + n ∑ j=1 bi j f j(z j(t −τi j)), t ∈ [0, t1], i = 1, · · · , n, (14) where fi(zi(t)) = gi(zi(t)+ x∗i )− gi(x ∗ i ). Hence, z(t, φ̃) = x(t,φ)− x∗ is a solution of system (14) with initial conditions z(t) = φ(t)− x∗, t ∈ [−τ, 0] on [0, t∗(φ)). Step 2. Consider the following Lyapunov functional V (z(t)) = n ∑ i=1 pi{ � �zi(t) � �+ n ∑ j=1 ∫ t t−τi j � �bi j � � · � � f j(z j(θ)) � �dθ}. (15) A. Wu, J. Zhang, C. Fu / Eur. J. Pure Appl. Math, 3 (2010), 806-818 813 Obviously, V (z) is positive definite and lim ‖z‖→∞ V (z) = ∞. Calculating the derivative of V (z) along the solution z(t, φ̃) of system (14) on [0, t∗(φ)), then we can get V̇ (z(t, φ̃))≤ − n ∑ i=1 {pidi � �zi(t, φ̃) � �− pi(aii + � �bii � �) � � fi(zi(t, φ̃)) � � − n ∑ j=1, j 6=i pi( � �ai j � �+ � �bi j � �) · � � f j(z j(t, φ̃)) � �} = − n ∑ i=1 {pidi � �zi(t, φ̃) � �− pi(aii + � �bii � �) � � fi(zi(t, φ̃)) � � − n ∑ j=1, j 6=i p j( � �a ji � �+ � �b ji � �) · � � fi(zi(t, φ̃)) � �} ≤ −pmin n ∑ i=1 � �zi(t, φ̃) � �< 0 where pmin = min 1≤i≤n � pidi . This implies V (z(t, φ̃))< V (z(0)), t ∈ [0, t∗(φ)). By (15), we can get n ∑ i=1 pi � �zi(t, φ̃) � � < V (z(0)) (16) According to (16), it is easy to derive that zi(t, φ̃), i = 1, · · · , n, are bounded on [0, t∗(φ)). By virtue of the continuous theorem of differential equations, we can conclude that system (14) has a solution on [0, t1], i.e., system (13) has a solution on [0, t1]. We denote this solution of system (13) by x0(t). Step 3. Consider the following system:    ẋ i(t) = −di x i(t) + n ∑ j=1 ai j g j(x j(t)) + n ∑ j=1 bi j g j(x j(t −τi j))+ Ii , t ∈ [t1, t2], x i(t1) = x0 i (t1) + Ji1(x 0 i (t1)), i = 1, · · · , n. (17) Arguing as in step 1 and step 2, system (17) has a solution x1(t) on [t1, t2]. As inductive step, we can derive that the following system:    ẋ i(t) = −di x i(t) + n ∑ j=1 ai j g j(x j(t)) + n ∑ j=1 bi j g j(x j(t −τi j)) + Ii , t ∈ [tm, tm+1], x i(tm) = xm−1 i (tm) + Jim(x m−1 i (tm)), i = 1, · · · , n, also has a solution xm(t) on [tm, tm+1], m= 2,3, · · · . A. Wu, J. Zhang, C. Fu / Eur. J. Pure Appl. Math, 3 (2010), 806-818 814 Define x(t,φ) =        x0(t), t ∈ [0, t1], x1(t), t ∈ (t1, t2], · · · , xm(t), t ∈ (tm, tm+1], · · · , then x(t,φ) is the solution of system (1) with initial conditions x(s) = φ(s), s ∈ [−τ, 0]. This completes the proof of the existence of solutions of system (1). Step 4. Assume that x(t) is a solution of system (1), and x∗ is the unique equilibrium point of system (1). Make a transformation z(t) = x(t)− x∗, then system (1) is transformed into the following system:    żi(t) = −dizi(t) + n ∑ j=1 ai j f j(z j(t)) + n ∑ j=1 bi j f j(z j(t −τi j)), t 6= tk, △zi(tk) = Jik(zi(tk)) = −γikzi(tk), k = 1,2, · · · , i = 1, · · · , n, (18) where fi(zi(t)) = gi(zi(t) + x∗ i )− gi(x ∗ i ). Consider Lyapunov functional V (z(t)), the V (z(t)) is the same as (15). Calculating the derivative of V (z(t)) along the solution z(t) of system (18) for any t, t 6= tk, k = 1,2, · · · . Arguing as in step 2, we have V̇ (z(t)) < 0, t 6= tk, k = 1,2, · · · . Also, V (z(tk + 0)) = n ∑ i=1 pi{ � �zi(tk + 0) � �+ n ∑ j=1 ∫ (tk+0) (tk+0)−τi j � �bi j � � · � � f j(z j(θ)) � �dθ} = n ∑ i=1 pi{ � �(1− γik)zi(tk) � �+ n ∑ j=1 ∫ tk tk−τi j � �bi j � � · � � f j(z j(θ)) � �dθ} < V (z(tk)), k = 1,2, · · · . Then we can easily follow that V̇ (z(t)) < 0 for t > 0. Therefore, the equilibrium point x∗ of system (1) is globally asymptotically stable. This proof is completed. Theorem 5. Under assumptions of Theorem 2, further if the following conditions are satisfied Jik(x i(tk)) = −γik(x i(tk)− x∗i ), k = 1,2, · · · , i = 1, · · · , n, where x∗ = (x∗1, · · · , x∗n) T is the equilibrium point of system (1), 0 < γik < 1, then system (1) has a unique equilibrium point which is stable and every solution is bounded. If, in addition, gi(s), i = 1, · · · , n, are strictly increasing functions, then the unique equilibrium point of system (1) is globally asymptotically stable. Proof. In order to complete the proof, we divide the proof into four steps: step A, step B, step C, step D. A. Wu, J. Zhang, C. Fu / Eur. J. Pure Appl. Math, 3 (2010), 806-818 815 Step A. It is the same as step 1 of Theorem 4, so we do not repeat it here. Step B. Consider the following Lyapunov functional V (z(t)) = 2 n ∑ i=1 ∫ zi(t) 0 fi(s)ds+ r n ∑ i=1 n ∑ j=1 ∫ t t−τ j i � �b ji � � · f 2 i (zi(θ))dθ . (19) Obviously, V (z) is positive definite and lim ‖z‖→∞ V (z) = ∞. Calculating the derivative of V (z) along the solution z(t, φ̃) of system (14) on [0, t∗(φ)), then we can get V̇ (z(t, φ̃)) = −2 n ∑ i=1 di fi(zi(t, φ̃))zi(t, φ̃) + 2 n ∑ i=1 n ∑ j=1 ai j fi(zi(t, φ̃)) · f j(z j(t, φ̃)) + 2 n ∑ i=1 n ∑ j=1 bi j fi(zi(t, φ̃)) · f j(z j(t −τi j, φ̃)) + r n ∑ i=1 n ∑ j=1 � �b ji � � f 2 i (zi(t, φ̃)) − r n ∑ i=1 n ∑ j=1 � �b ji � � f 2 i (zi(t −τ ji, φ̃)) ≤ −2 n ∑ i=1 di fi(zi(t, φ̃))zi(t, φ̃) + 2 n ∑ i=1 n ∑ j=1 ai j fi(zi(t, φ̃)) · f j(z j(t, φ̃)) + 2 n ∑ i=1 n ∑ j=1 � �bi j � � � � fi(zi(t, φ̃)) � � · � � f j(z j(t −τi j, φ̃)) � �+ r n ∑ i=1 n ∑ j=1 � �b ji � � f 2 i (zi(t, φ̃)) − r n ∑ i=1 n ∑ j=1 � �b ji � � f 2 i (zi(t −τ ji, φ̃)). (20) Since 2 � � fi(zi(t, φ̃)) � � · � � f j(z j(t −τi j, φ̃)) � �≤ 1 r f 2 i (zi(t, φ̃))+ r f 2 j (z j(t −τi j, φ̃)) then we get V̇ (z(t, φ̃))≤ −2 n ∑ i=1 di fi(zi(t, φ̃))zi(t, φ̃) + 2 n ∑ i=1 n ∑ j=1 ai j fi(zi(t, φ̃)) · f j(z j(t, φ̃)) + n ∑ i=1 n ∑ j=1 1 r � �bi j � � f 2 i (zi(t, φ̃)) + n ∑ i=1 n ∑ j=1 r � �bi j � � f 2 j (z j(t −τi j, φ̃)) + r n ∑ i=1 n ∑ j=1 � �b ji � � f 2 i (zi(t, φ̃))− r n ∑ i=1 n ∑ j=1 � �b ji � � f 2 i (zi(t −τ ji, φ̃)) = −2 n ∑ i=1 di fi(zi(t, φ̃))zi(t, φ̃) + 2 n ∑ i=1 n ∑ j=1 ai j fi(zi(t, φ̃)) · f j(z j(t, φ̃)) A. Wu, J. Zhang, C. Fu / Eur. J. Pure Appl. Math, 3 (2010), 806-818 816 + n ∑ i=1 n ∑ j=1 1 r � �bi j � � f 2 i (zi(t, φ̃)) + r n ∑ i=1 n ∑ j=1 � �b ji � � f 2 i (zi(t, φ̃)) ≤ −2 n ∑ i=1 di fi(zi(t, φ̃))zi(t, φ̃) + f T (z(t, φ̃)) · (A+AT ) · f (z(t, φ̃)) + 1 r ‖B‖∞ f T (z(t, φ̃)) f (z(t, φ̃)) + r‖B‖1 f T (z(t, φ̃)) f (z(t, φ̃)) = −2 n ∑ i=1 di fi(zi(t, φ̃))zi(t, φ̃) + f T (z(t, φ̃)) · [A+ AT + ( 1 r ‖B‖∞ + r‖B‖1)I] · f (z(t, φ̃)) ≤ −2 n ∑ i=1 di fi(zi(t, φ̃))zi(t, φ̃)≤ 0. The rest of step B and step C are similar as step 2 and step 3 of Theorem 4, respectively, so we also do not repeat them, then we can easily obtain the existence of solutions of system (1). Step D. Consider Lyapunov functional V (z(t)), the V (z(t)) is the same as (19). Calculating the derivative of V (z(t)) along the solution z(t) of system (3.6) for any t, t 6= tk, k = 1,2, · · · . Arguing as in step B, we have V̇ (z(t)) ≤ 0, t 6= tk, k = 1,2, · · · . Also, V (z(tk + 0)) = 2 n ∑ i=1 ∫ zi(tk+0) 0 fi(s)ds+ r n ∑ i=1 n ∑ j=1 ∫ (tk+0) (tk+0)−τ j i � �b ji � � · f 2 i (zi(θ))dθ = V (z(tk))+ 2 n ∑ i=1 ∫ (1−γik)zi(tk) zi(tk) fi(s)ds ≤ V (z(tk)), k = 1,2, · · · . Then we can easily follow that V̇ (z(t)) ≤ 0 for t > 0. Therefore, the equilibrium point x∗ of system (1) is stable and every solution is bounded. If, in addition, gi(s), i = 1, · · · , n, are strictly increasing functions, then V̇ (z(t)) < 0 is negative definite. Thus, the unique equilibrium point of system (1) is globally asymptotically stable. Remark 1. Compared to the existing literatures, our results have improved those results. The results of [5, 6, 9, 10] are obtained under Lipschitz neuron activations. At the same time, our results are easy to be checked. In order to show that the conditions we have obtained in this paper provide new sufficient criteria for system (1), we consider the following example. Example 1. Consider the following delayed impulsive two-neuron network model (1) descried by: ẋ1(t) = − 2 21 x1(t)− 10 21 g1(x1(t)) + 2 21 g2(x2(t −τ2))+ I1, REFERENCES 817 ẋ2(t) = − 5 21 x2(t) + 2 21 g1(x1(t −τ1))− 6 21 g2(x2(t)) + I2, △x1(tk) = −0.2(x1(tk)− x∗1), △x2(tk) = −0.3(x2(tk)− x∗2), tk = kT, k = 1,2, · · · . (21) where T > 0 is a positive constant, g1(θ) = arctan(θ), g2(θ) = θ 3, x∗ = (x∗1, x∗2) T is the equilibrium point of system (21). Obviously, those results in [5, 6, 8, 9, 10] would fail when applying to this example. However, we select p1 = p2 = 1 (or by A+ AT + (‖B‖∞ + ‖B‖1)I = −diag(16 21 , 8 21 ) < 0), from Theorem 4 (or Theorem 5), system (21) has a unique equilibrium point x∗ which is globally asymptotically stable. Therefore, our results establish new criteria for the global asymptotic stability of delayed neural networks with impulses and improve those results in the existing literatures. 4. Conclusion This paper has developed a class of delayed neural networks with impulses, where the neuron activations do not satisfy Lipschitz conditions. Some general sufficient conditions are derived for the global asymptotic stability of delayed neural networks with impulses. A comparison between our results and the previous results is also given, which shows that our results establish a new criteria for global asymptotic stability of delayed neural networks with impulses. At the same time, the criteria in this paper is also valuable in the design of neural networks which is used to solve efficiently classes of optimization problems arising in practical engineering applications. ACKNOWLEDGEMENTS The work is supported by Key Science Foundation of Educational Department of Hubei Province under Grant D20082201 and Innovation Teams of Hubei Nor- mal University. References [1] L. Zhou and G. Hu, Global exponential periodicity and stability of cellular neural net- works with variable and distributed delays, Applied Mathematics and Computation, 195, 402-411. 2008. [2] T. Li and S. Fei, Stability analysis of Cohen-Grossberg neural networks with time-varying and distributed delays, Neurocomputing, 71, 1069-1081. 2008. [3] Q. Zhang, X. Wei, and J. Xu, Delay-dependent exponential stability criteria for non- autonomous cellular neural networks with time-varying delays, Chaos, Solitons and Fractals, 36, 985-990. 2008. REFERENCES 818 [4] W. Zhao, Dynamics of Cohen-Grossberg neural network with variable coefficients and time-varying delays, Nonlinear Analysis: Real World Applications, 9, 1024-1037. 2008. [5] W. Xiong, Q. Zhou, B. Xiao, and Y. Yu, Global exponential stability of cellular neural networks with mixed delays and impulses, Chaos, Solitons and Fractals, 34, 896-902. 2007. [6] Z. Huang, Q. Yang, and X. Luo, Exponential stability of impulsive neural networks with time-varying delays, Chaos, Solitons and Fractals, 35, 770-780. 2008. [7] S. Long and D. Xu, Delay-dependent stability analysis for impulsive neural networks with time-varying delays, Neurocomputing, 71, 1705-1713. 2008. [8] H. Wu and X. Xue, Stability analysis for neural networks with inverse Lipschitzian neu- ron activations and impulses, Applied Mathematical Modelling, 32, 2347-2359. 2008. [9] Q. Song and J. Zhang, Global exponential stability of impulsive Cohen-Grossberg neural network with time-varying delays, Nonlinear Analysis: Real World Applications, 9, 500- 510. 2008. [10] S. Ahmada and I. Stamov, Global exponential stability for impulsive cellular neural net- works with time-varying delays, Nonlinear Analysis, 69, 786-795. 2008.