Stochastic Passivity of Uncertain Neural Networks with Time-Varying Delays

Abstract

The passivity problem is investigated for a class of stochastic uncertain neural networks with time-varying delay as well as generalized activation functions. By constructing appropriate Lyapunov-Krasovskii functionals, and employing Newton-Leibniz formulation, the free-weighting matrix method, and stochastic analysis technique, a delay-dependent criterion for checking the passivity of the addressed neural networks is established in terms of linear matrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. An example with simulation is given to show the effectiveness and less conservatism of the proposed criterion. It is noteworthy that the traditional assumptions on the differentiability of the time-varying delays and the boundedness of its derivative are removed.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Jianting Zhou 1 and Qiankun Song 2 and Jianxi Yang 1

Recommended by Elena Litsyn

1, College of Civil Engineering and Architecture, Chongqing Jiaotong University, Chongqing 400074, China
2, Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, China

Received 22 July 2009; Accepted 18 October 2009

1. Introduction

During the last two decades, many artificial neural networks have been extensively investigated and successfully applied to various areas such as signal processing, pattern recognition, associative memory, and optimization problems [1]. In such applications, it is of prime importance to ensure that the designed neural networks are stable [2].

In hardware implementation, time delays are likely to be present due to the finite switching speed of amplifiers and communication time. It has also been shown that the processing of moving images requires the introduction of delay in the signal transmitted through the networks [3]. The time delays are usually variable with time, which will affect the stability of designed neural networks and may lead to some complex dynamic behavior such as oscillation, bifurcation, or chaos [4]. Therefore, the study of stability with consideration of time delays becomes extremely important to manufacture high quality neural networks [5]. Many important results on stability of delayed neural networks have been reported, see [1-10] and the references therein for some recent publications.

It is also well known that parameter uncertainties, which are inherent features of many physical systems, are great sources of instability and poor performance [11]. These uncertainties may arise due to the variations in system parameters, modelling errors, or some ignored factors [12]. It is not possible to perfectly characterize the evolution of an uncertain dynamical system as a deterministic set of state equations [13]. Recently, the problem on robust stability analysis of uncertain neural networks with delays has been extensively investigated, see [11-14] and the references therein for some recent publications.

Just as pointed out in [15], in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. In the implementation of artificial neural networks, noise is unavoidable and should be taken into consideration in modelling. Therefore, it is of significant importance to consider stochastic effects to the dynamical behavior of neural networks [16]. Some recent interest results on stability of stochastic neural networks can be found, see [15-26] and the references therein for some recent publications.

On the other hand, the passivity theory is another effective tool to the stability analysis of nonlinear system [27]. The main idea of passivity theory is that the passive properties of system can keep the system internal stability [27]. Thus, the passivity theory has received a lot of attention from the control community since 1970s [28-31]. Recently, the passivity theory for delayed neural networks was investigated, some criteria checking the passivity were provided for certain or uncertain neural networks, see [32-38] and references therein. In [32], the passivity-based approach is used to derive stability conditions for dynamic neural networks with different time scales. In [33-36], authors investigated the passivity of neural networks with time-varying delay. In [37, 38], stochastic neural networks with time-varying delays were considered, several sufficient conditions checking the passivity were obtained. It is worth pointing out that, the given criteria in [33-37] have been based on the following assumptions: (1 ) the time-varying delays are continuously differentiable; (2 ) the derivative of time-varying delay is bounded and is smaller than one; (3 ) the activation functions are bounded and monotonically nondecreasing. However, time delays can occur in an irregular fashion, and sometimes the time-varying delays are not differentiable. In such a case, the methods developed in [33-38] may be difficult to be applied, and it is therefore necessary to further investigate the passivity problem of neural networks with time-varying delays under milder assumptions. To the best of our knowledge, few authors have considered the passivity problem for stochastic uncertain neural networks with time-varying delays as well as generalized activation functions.

Motivated by the above discussions, the objective of this paper is to study the passivity of stochastic uncertain neural networks with time-varying delays as well as generalized activation functions by employing a combination of Lyapunov functional, the free-weighting matrix method and stochastic analysis technique. The obtained sufficient conditions require neither the differentiability of time-varying delays nor the monotony of the activation functions, and are expressed in terms of linear matrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. An example is given to show the effectiveness and less conservatism of the proposed criterion.

2. Problem Formulation and Preliminaries

In this paper, we consider the following stochastic uncertain neural networks with time-varying delay:

[figure omitted; refer to PDF] for t≥0 , where x(t)=(_x1 (t),_x2 (t),...,_xn (t)^)T ∈^Rn is the state vector of the network at time t , n corresponds to the number of neurons; C=diag (_c1 ,_c2 ,...,_cn ) is a positive diagonal matrix, A=(_aij)n×n , and B=(_bij)n×n are known constant matrices; ΔC(t) , ΔA(t) and ΔB(t) are time-varying parametric uncertainties; σ(t,x(t),x(t-τ(t)))∈^Rn×n is the diffusion coefficient matrix and ω(t)=(_ω1 (t),_ω2 (t),...,_ωn (t)^)T is an n -dimensional Brownian motion defined on a complete probability space (Ω,F,{_Ft}t≥0 ,P ) with a filtration {_Ft}t≥0 satisfying the usual conditions (i.e., it is right continuous and _F0 contains all P -null sets); f(x(t))=(_f1 (_x1 (t)),_f2 (_x2 (t)),...,_fn (_xn (t))^)T denotes the neuron activation at time t ; u(t)=(_u1 (t),_u2 (t),...,_un (t)^)T ∈^Rn is a varying external input vector; τ(t)>0 is the time-varying delay, and is assumed to satisfy 0≤τ(t)≤τ , where τ is constant.

The initial condition associated with model (2.1) is given by [figure omitted; refer to PDF]

Let x(t,[varphi]) denote the state trajectory of model (2.1) from the above initial condition and x(t,0) the corresponding trajectory with zero initial condition.

Throughout this paper, we make the following assumptions.

(H1) [33] The time-varying uncertainties ΔC(t) , ΔA(t) and ΔB(t) are of the form

[figure omitted; refer to PDF] where _H1 , _H2 , _H3 , _E1 , _E2 , and _E3 are known constant matrices of appropriate dimensions, _G1 (t) , _G2 (t), and _G3 (t) are known time-varying matrices with Lebesgue measurable elements bounded by

[figure omitted; refer to PDF]

(H2) [10] For any j∈{1,2,...,n} , _fj (0)=0 and there exist constants ^Fj- and ^Fj+ such that

[figure omitted; refer to PDF] for all _α1 ≠_α2 .

(H3) [15] There exist two scalars _ρ1 >0 , _ρ2 >0 such that the following inequality:

[figure omitted; refer to PDF] holds for all (t,u,v)∈R×^Rn ×^Rn .

Definition 2.1 (see [33]).

System (2.1) is called globally passive in the sense of expectation if there exists a scalar γ>0 such that [figure omitted; refer to PDF] for all _tp ≥0 and for all x(t,0) , where E{·} stands for the mathematical expectation operator with respect to the given probability measure P .

To prove our results, the following lemmas that can be found in [39] are necessary.

Lemma 2.2 (see [39]).

For given matrices H , E, and F with ^FT F≤I and a scalar [straight epsilon]>0 , the following holds: [figure omitted; refer to PDF]

Lemma 2.3 (see [39]).

For any constant matrix W∈^Rm×m , W>0 , scalar 0<h(t)<h , vector function ω:[0,h][arrow right]^Rm such that the integrations concerned are well defined, then [figure omitted; refer to PDF]

Lemma 2.4 (see [39]).

Given constant matrices P , Q, and R , where ^PT =P , ^QT =Q , then [figure omitted; refer to PDF] is equivalent to the following conditions: [figure omitted; refer to PDF]

3. Main Results

For presentation convenience, in the following, we denote

[figure omitted; refer to PDF]

Theorem 3.1.

Under assumptions (H1)-(H3), model (2.1) is passive in the sense of expectation if there exist two scalars γ>0 , λ>0 , three symmetric positive definite matrices _Pi (i=1,2,3 ), two positive diagonal matrices L and S , and matrices _Qi (i=1,2,3,4 ) such that the following two LMIs hold: [figure omitted; refer to PDF] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] in which _Ω11 =_P2 -_Q2 C-C^Q2T +(_{[straight epsilon]1} +_{[straight epsilon]4} )^E1T_E1 -_Q3 -^Q3T -_F1 L+(1+τ)λ_ρ1 I , _Ω12 =_P1 -C^Q1T -_Q2 , _Ω13 =_Q2 A+_F2 L , _Ω22 =-_Q1 -^Q1T +τ_P3 , _Ω33 =(_{[straight epsilon]2} +_{[straight epsilon]5} )^E2T_E2 -L , _Ω44 =(_{[straight epsilon]3} +_{[straight epsilon]6} )^E3T_E3 -S , _Ω55 =-_Q4 -^Q4T -_F1 S+(1+τ)λ_ρ2 I .

Proof.

Let y(t)=-(C+ΔC(t))x(t)+(A+ΔA(t))f(x(t))+(B+ΔB(t))f(x(t-τ(t)))+u(t) , α(t)=σ(t,x(t),x(t-τ(t))) , then model (2.1) is rewritten as [figure omitted; refer to PDF] Consider the following Lyapunov-Krasovskii functional as [figure omitted; refer to PDF] By Ito... differential rule, the stochastic derivative of V(t) along the trajectory of model (3.5) can be obtained as [figure omitted; refer to PDF] From the definition of y(t) , we have [figure omitted; refer to PDF] By assumption (H1) and Lemma 2.2, we get [figure omitted; refer to PDF] It follows from (3.8) and (3.9) that [figure omitted; refer to PDF] Integrating both sides of (3.5) from t-τ(t) to t , we have [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] By Lemmas 2.2 and 2.3, and noting τ(t)≤τ , we get [figure omitted; refer to PDF] Integrating both sides of (3.5) from t-τ to t-τ(t) , we have [figure omitted; refer to PDF] Similarly, by using of the same way, and noting τ-τ(t)≤τ , we get [figure omitted; refer to PDF] From assumption (H2), we have [figure omitted; refer to PDF] which are equivalent to [figure omitted; refer to PDF] where _er denotes the unit column vector having 1 element on its r th row and zeros elsewhere. Let [figure omitted; refer to PDF] then [figure omitted; refer to PDF] that is [figure omitted; refer to PDF] Similarly, one has [figure omitted; refer to PDF] It follows from (3.7), (3.10), (3.13), (3.15), (3.20) and (3.21) that [figure omitted; refer to PDF] By assumption (H3) and inequality (3.2), we get [figure omitted; refer to PDF] From the proof of [19], we have [figure omitted; refer to PDF] Taking the mathematical expectation on both sides of (3.22), and noting (3.24), we get [figure omitted; refer to PDF] where ξ(t)=(^xT (t),^yT (t),^fT (x(t)),^fT (x(t-τ(t))),^xT (t-τ(t)),^xT (t-τ),^uT (t)^)T , and [figure omitted; refer to PDF] with _Π1 =_P2 -_Q2 C-C^Q2T +(_{[straight epsilon]1} +_{[straight epsilon]4} )^E1T_E1 +^{[straight epsilon]4-1}_Q2H1^H1TQ2T +^{[straight epsilon]5-1}_Q2H2^H2TQ2T +^{[straight epsilon]6-1}_Q2H3^H3TQ2T -_Q3 -^Q3T +τ_Q3^P3-1Q3T +_Q3^P1-1Q3T +_Q4^P1-1Q4T -_F1 L+(1+τ)λ_ρ1 I , _Π2 =-_Q1 -^Q1T +^{[straight epsilon]1-1}_Q1H1^H1TQ1T +^{[straight epsilon]2-1}_Q1H2^H2TQ1T +^{[straight epsilon]3-1}_Q1H3^H3TQ1T +τ_P3 , _Π3 =(_{[straight epsilon]2} +_{[straight epsilon]5} )^E2T_E2 -L , _Π4 =(_{[straight epsilon]3} +_{[straight epsilon]6} )^E3T_E3 -S , _Π5 =-_Q4 -^Q4T +τ_Q4^P3-1Q4T -_F1 S+(1+τ)λ_ρ2 I .

It is easy to verify the equivalence of Π<0 and Ω<0 by using Lemma 2.4. Thus, one can derive from (3.3) and (3.25) that

[figure omitted; refer to PDF] From (3.27) and the definition of V(t,x(t)) , we can get [figure omitted; refer to PDF] From Definition 2.1, we know that the stochastic neural networks (2.1) are globally passive in the sense of expectation, and the proof of Theorem 3.1 is then completed.

Remark 3.2.

Assumption (H2) was first proposed in [10]. The constants ^Fj- and ^Fj+ (i=1,2,...,n ) in assumption (H2) are allowed to be positive, negative or zero. Hence, Assumption (H2) is weaker than the assumption in [27-37]. In addition, the conditions in [32-37] that the time-varying delay is differentiable and the derivative is bounded or smaller than one have been removed in this paper.

Remark 3.3.

In [36], authors considered the passivity of uncertain neural networks with both discrete and distributed time-varying delays. In [37], authors considered the passivity for stochastic neural networks with time-varying delays and random abrupt changes. It is worth pointing out that, the method in this paper can also analyze the passivity for models in [36, 37].

Remark 3.4.

It is known that the obtained criteria for checking passivity of neural networks depend on the constructed Lyapunov functionals or Lyapunov-Krasovskii functionals in varying degrees. Constructing proper Lyapunov functionals or Lyapunov-Krasovskii functionals can reduce conservatism. Recently, the delay fractioning approach has been used to investigate global synchronization of delayed complex networks with stochastic disturbances, which has shown the potential of reducing conservatism [22]. Using the delay fractioning approach, we can also investigate the passivity of delayed neural networks. The corresponding results will appear in the near future.

Remark 3.5.

When we do not consider the stochastic effect, model (2.1) turns into the following model: [figure omitted; refer to PDF] Furthermore, model (3.29) also comprises the following neural network model with neither stochastic effect nor uncertainty [figure omitted; refer to PDF] which have been considered in [33, 34]. For models (3.29) and (3.30), one can get the following results.

Corollary 3.6.

Under assumptions (H1)-(H2), model (3.29) is passive if there exist a scalar γ>0 , three symmetric positive definite matrices _Pi (i=1,2,3 ), two positive diagonal matrices L and S , and matrices _Qi (i=1,2,3,4 ) such that the following LMI holds: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] in which _Ω11 =_P2 -_Q2 C-C^Q2T +(_{[straight epsilon]1} +_{[straight epsilon]4} )^E1T_E1 -_Q3 -^Q3T -_F1 L , _Ω12 =_P1 -C^Q1T -_Q2 , _Ω13 =_Q2 A+_F2 L , _Ω22 =-_Q1 -^Q1T +τ_P3 , _Ω33 =(_{[straight epsilon]2} +_{[straight epsilon]5} )^E2T_E2 -L , _Ω44 =(_{[straight epsilon]3} +_{[straight epsilon]6} )^E3T_E3 -S , _Ω55 =-_Q4 -^Q4T -_F1 S .

Corollary 3.7.

Under assumption (H2), model (3.30) is passive if there exist a scalar γ>0 , three symmetric positive definite matrices _Pi (i=1,2,3 ), two positive diagonal matrices L and S , and matrices _Qi (i=1,2,3,4 ) such that the following LMI holds: [figure omitted; refer to PDF] where _Π1 =_P2 -_Q2 C-C^Q2T -_Q3 -^Q3T -_F1 L , _Π2 =_P1 -C^Q1T -_Q2 , _Π3 =_Q2 A+_F2 L , _Π4 =-_Q1 -^Q1T +τ_P3 , _Π5 =-_Q4 -^Q4T -_F1 S .

4. An Example

Consider a two-neuron neural network (3.30), where

[figure omitted; refer to PDF]

Figure 1 depicts the states of the considered network with initial conditions _x1 (t)=0.5 , _x2 (t)=0.45 , t∈[-2,0] .

Figure 1: State responses of _x1 (t) and _x2 (t) .

[figure omitted; refer to PDF]

It can be verified that assumption (H2) is satisfied, and _F1 =0 , _F2 =diag {0.5,0.5} , τ=2 . By the Matlab LMI Control Toolbox, we find a solution to the LMI in (3.33) as follows:

[figure omitted; refer to PDF]

Therefore, by Corollary 3.7, we know that model (3.30) is passive. It should be pointed out that the conditions in [33-36] cannot be applied to this example since that require the differentiability of the time-varying delay.

5. Conclusions

In this paper, the passivity has been investigated for a class of stochastic uncertain neural networks with time-varying delay as well as generalized activation functions. By employing a combination of Lyapunov-Krasovskii functionals, the free-weighting matrix method, Newton-Leibniz formulation, and stochastic analysis technique, a delay-independent criterion for checking the passivity of the addressed neural networks has been established in terms of linear matrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. The obtained results generalize and improve the earlier publications and remove the traditional assumptions on the differentiability of the time-varying delay and the boundedness of its derivative. An example has been provided to demonstrate the effectiveness and less conservatism of the proposed criterion.

We would like to point out that it is possible to generalize our main results to more complex neural networks, such as neural networks with discrete and distributed delays [10, 26], and neural networks of neutral-type [7, 20], neural networks with Markovian jumping [24, 25]. The corresponding results will appear in the near future.

Acknowledgments

The authors would like to thank the reviewers and the editor for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the National Natural Science Foundation of China under Grants 60974132 and 50608072, and in part by Natural Science Foundation Project of CQ CSTC2008BA6038 and 2008BB2351, and Scientific & Technological Research Projects of CQ KJ090406, and the Ministry of Education for New Century Excellent Talent Support Program.

References

[1] Ju H. Park, O. M. Kwon, "Delay-dependent stability criterion for bidirectional associative memory neural networks with interval time-varying delays," Modern Physics Letters B , vol. 23, no. 1, pp. 35-46, 2009., [email protected]; [email protected]

[2] K. Gopalsamy, "Stability of artificial neural networks with impulses," Applied Mathematics and Computation , vol. 154, no. 3, pp. 783-813, 2004.

[3] S. Arik, "An analysis of exponential stability of delayed neural networks with time varying delays," Neural Networks , vol. 17, no. 7, pp. 1027-1031, 2004., [email protected]

[4] W. Lu, T. Chen, "Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions," Neural Networks , vol. 18, no. 3, pp. 231-242, 2005., [email protected]; [email protected]

[5] T. Chen, L. Wang, "Global μ -stability of delayed neural networks with unbounded time-varying delays," IEEE Transactions on Neural Networks , vol. 18, no. 6, pp. 1836-1840, 2007., [email protected]

[6] J. Cao, X. Li, "Stability in delayed Cohen-Grossberg neural networks: LMI optimization approach," Physica D , vol. 212, no. 1-2, pp. 54-65, 2005.

[7] Ju H. Park, O. M. Kwon, "Further results on state estimation for neural networks of neutral-type with time-varying delay," Applied Mathematics and Computation , vol. 208, no. 1, pp. 69-75, 2009.

[8] Ju H. Park, "Global exponential stability of cellular neural networks with variable delays," Applied Mathematics and Computation , vol. 183, no. 2, pp. 1214-1219, 2006.

[9] Ju H. Park, "On global stability criterion for neural networks with discrete and distributed delays," Chaos, Solitons & Fractals , vol. 30, no. 4, pp. 897-902, 2006.

[10] Y. Liu, Z. Wang, X. Liu, "Global exponential stability of generalized recurrent neural networks with discrete and distributed delays," Neural Networks , vol. 19, no. 5, pp. 667-675, 2006., [email protected]

[11] N. Ozcan, S. Arik, "A new sufficient condition for global robust stability of bidirectional associative memory neural networks with multiple time delays," Nonlinear Analysis: Real World Applications , vol. 10, no. 5, pp. 3312-3320, 2009.

[12] V. Singh, "Novel global robust stability criterion for neural networks with delay," Chaos, Solitons & Fractals , vol. 41, no. 1, pp. 348-353, 2009.

[13] Ju H. Park, "Further note on global exponential stability of uncertain cellular neural networks with variable delays," Applied Mathematics and Computation , vol. 188, no. 1, pp. 850-854, 2007.

[14] O. M. Kwon, Ju H. Park, "Exponential stability analysis for uncertain neural networks with interval time-varying delays," Applied Mathematics and Computation , vol. 212, no. 2, pp. 530-541, 2009.

[15] Z. Wang, Y. Liu, K. Fraser, X. Liu, "Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays," Physics Letters A , vol. 354, no. 4, pp. 288-297, 2006., [email protected]

[16] X. Wang, Q. Guo, D. Xu, "Exponential p -stability of impulsive stochastic Cohen-Grossberg neural networks with mixed delays," Mathematics and Computers in Simulation , vol. 79, no. 5, pp. 1698-1710, 2009.

[17] P. Balasubramaniam, R. Rakkiyappan, "Global asymptotic stability of stochastic recurrent neural networks with multiple discrete delays and unbounded distributed delays," Applied Mathematics and Computation , vol. 204, no. 2, pp. 680-686, 2008.

[18] C. Huang, Y. He, H. Wang, "Mean square exponential stability of stochastic recurrent neural networks with time-varying delays," Computers & Mathematics with Applications , vol. 56, no. 7, pp. 1773-1778, 2008.

[19] J. Yu, K. Zhang, S. Fei, "Further results on mean square exponential stability of uncertain stochastic delayed neural networks," Communications in Nonlinear Science and Numerical Simulation , vol. 14, no. 4, pp. 1582-1589, 2009.

[20] Ju H. Park, O. M. Kwon, "Synchronization of neural networks of neutral type with stochastic perturbation," Modern Physics Letters B , vol. 23, no. 14, pp. 1743-1751, 2009., [email protected]

[21] Ju H. Park, O. M. Kwon, "Analysis on global stability of stochastic neural networks of neutral type," Modern Physics Letters B , vol. 22, no. 32, pp. 3159-3170, 2008.

[22] Y. Wang, Z. Wang, J. Liang, "A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances," Physics Letters A , vol. 372, no. 39, pp. 6066-6073, 2008., [email protected]

[23] Z. Wang, H. Shu, J. Fang, X. Liu, "Robust stability for stochastic Hopfield neural networks with time delays," Nonlinear Analysis: Real World Applications , vol. 7, no. 5, pp. 1119-1128, 2006.

[24] Z. Wang, Y. Liu, L. Yu, X. Liu, "Exponential stability of delayed recurrent neural networks with Markovian jumping parameters," Physics Letters A , vol. 356, no. 4-5, pp. 346-352, 2006., [email protected]

[25] Y. Liu, Z. Wang, J. Liang, X. Liu, "Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays," IEEE Transactions on Neural Networks , vol. 20, no. 7, pp. 1102-1116, 2009., [email protected]; [email protected]; [email protected]

[26] Z. Wang, Y. Liu, X. Liu, "State estimation for jumping recurrent neural networks with discrete and distributed delays," Neural Networks , vol. 22, no. 1, pp. 41-48, 2009., [email protected]

[27] W. Yu, "Passivity analysis for dynamic multilayer neuro identifier," IEEE Transactions on Circuits and Systems , vol. 50, no. 1, pp. 173-178, 2003.

[28] C. Li, H. Zhang, X. Liao, "Passivity and passification of fuzzy systems with time delays," Computers & Mathematics with Applications , vol. 52, no. 6-7, pp. 1067-1078, 2006.

[29] H. Gao, T. Chen, T. Chai, "Passivity and passification for networked control systems," SIAM Journal on Control and Optimization , vol. 46, no. 4, pp. 1299-1322, 2007.

[30] J. Zhao, D. J. Hill, "A notion of passivity for switched systems with state-dependent switching," Journal of Control Theory and Applications , vol. 4, no. 1, pp. 70-75, 2006.

[31] W. Gui, B. Liu, Z. Tang, "A delay-dependent passivity criterion of linear neutral delay systems," Journal of Control Theory and Applications , vol. 4, no. 2, pp. 201-206, 2006.

[32] W. Yu, X. Li, "Passivity analysis of dynamic neural networks with different time-scales," Neural Processing Letters , vol. 25, no. 2, pp. 143-155, 2007., [email protected]

[33] C. Li, X. Liao, "Passivity analysis of neural networks with time delay," IEEE Transactions on Circuits and Systems II , vol. 52, no. 8, pp. 471-475, 2005., [email protected]; [email protected]

[34] Ju H. Park, "Further results on passivity analysis of delayed cellular neural networks," Chaos, Solitons & Fractals , vol. 34, no. 5, pp. 1546-1551, 2007.

[35] X. Lou, B. Cui, "Passivity analysis of integro-differential neural networks with time-varying delays," Neurocomputing , vol. 70, no. 4-6, pp. 1071-1078, 2007., [email protected]; [email protected]

[36] B. Chen, H. Li, C. Lin, Q. Zhou, "Passivity analysis for uncertain neural networks with discrete and distributed time-varying delays," Physics Letters A , vol. 373, no. 14, pp. 1242-1248, 2009.

[37] X. Lou, B. Cui, "On passivity analysis of stochastic delayed neural networks with random abrupt changes," New Mathematics and Natural Computation , vol. 3, no. 3, pp. 321-330, 2007.

[38] Q. Song, J. Liang, Z. Wang, "Passivity analysis of discrete-time stochastic neural networks with time-varying delays," Neurocomputing , vol. 72, no. 7-9, pp. 1782-1788, 2009., [email protected]; [email protected]; [email protected]

[39] Q. Song, J. Cao, "Global dissipativity analysis on uncertain neural networks with mixed time-varying delays," Chaos , vol. 18, no. 4, 2008.