Dynamics of Uncertain Discrete-Time Neural Network with Delay and Impulses

Abstract

The stability of discrete-time impulsive delay neural networks with and without uncertainty is investigated. First, by using Razumikhin-type theorem, a new less conservative condition for the exponential stability of discrete-time neural network with delay and impulse is proposed. Moreover, some new sufficient conditions are derived to guarantee the stability of uncertain discrete-time neural network with delay and impulse by using Lyapunov function and linear matrix inequality (LMI). Finally, several examples with numerical simulation are presented to demonstrate the effectiveness of the obtained results.

Xuehui Mei 1,2 and Liwei Zhang 1 and Haijun Jiang 2 and Zhiyong Yu 2

Academic Editor:Luca Gori

1, College of Mathematics Sciences, Dalian University of Technology, Dalian 116024, China
2, College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China

Received 14 September 2015; Accepted 27 October 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In the last three decades, much attention has been paid to the study of neural networks due to their potential applications in a variety of fields such as signal processing, pattern recognition, associative memory, combinational optimization, and other areas [1-5]. As we all know, it is very important to know the dynamical behaviors of neural networks. In particular, stability properties of dynamical neural networks are very essential in their design for solving practical problems.

It is well known that time delay is always encountered in the implementation of a signal transmission and even small time delay may lead to significantly deteriorated performance or instability for the system. So stability of delayed neural networks has been widely investigated. Impulse is also a common phenomenon in control schemes and evolution processes of dynamic system. For the dynamics of system, impulse can bring about many complex influences. For some stable systems, impulse may play a role as perturbation and may destabilize the system. However, for others, impulse may serve as control power and makes unstable system stabilization. Consequently, it is essential to study how impulse affects the stability of system. Meanwhile, over the past few decades, many results about the stability of neural networks with time delay and impulse were obtained [6-12].

In general, most neural networks were assumed to be continuous time and many considerable results were obtained. However, in the fields of engineering especially in the numerical simulations, many phenomena are often described by discrete-time systems. Generally speaking, there always exist different dynamic properties between continuous-time neural networks and discrete-time neural networks. And results about the discrete-time neural networks are still few [13-32].

Furthermore, uncertainty is unavoidable in many evolution processes, which may destroy the stability of the neural networks. Therefore, it is necessary to analyze the robust stability of neural networks with uncertainty. But for the discrete-time impulsive neural networks with delay and uncertainty, there are very few results. In previous articles, several approaches were proposed to study the stability of impulsive delayed systems, most of which were based on Lyapunov functional, inequality technique, and Razumikhin technique. For instance, in [18], exponential stability for discrete-time impulsive delayed neural networks with and without uncertainty was investigated by using Lyapunov functionals. In [24], the global exponential stability of the discrete-time systems of Cohen-Grossberg neural networks with and without delays was studied by using Lyapunov methods. Global dissipativity criteria of uncertain discrete-time stochastic neural networks with time-varying delays were obtained in [25] by using linear matrix inequalities. The authors in [26] studied the stability effects of impulses in discrete-time delayed neural networks. In [27], the impulsive stabilization results for neural network without uncertainty were given by using Razumikhin technique, but the authors did not take discrete-time analogs into account. The authors in [28] investigated the existence and global exponential stability of periodic solution for delayed neural networks with impulsive and stochastic effects. The robust exponential stability of discrete-time switched Hopfield neural networks with time delay was studied in [29]. In [30], some new stability results for discrete-time recurrent neural networks with time-varying delay were proposed. And some conditions for robust stability of discrete-time uncertain neural networks with leakage time-varying delay were obtained in [31]. In [32], a new delay-independent condition for global robust stability of neural networks with time delays was given. In [18], although exponential stability for discrete-time impulsive delay neural networks with and without uncertainty was investigated by using Lyapunov functionals, the results were complicated. Consequently, we consider globally exponential stability of discrete-time impulsive delayed neural networks with uncertainty via Razumikhin-type theorems. Besides, different from articles [1, 2, 4, 16, 18-22], we do not require that activation functions in this paper are bounded on [figure omitted; refer to PDF] .

The rest of this paper is organized as follows. In Section 2, the model description and some useful lemmas are provided. In Section 3, the main results are derived to ensure the exponential stability analysis of discrete-time impulsive delayed neural network with certainties and uncertainties. In Section 4, some numerical examples are presented to demonstrate the effectiveness of the obtained results. The conclusion is given in Section 5.

2. Preliminary

Consider the following discrete-time neural network with delay and impulses: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] denotes the set of nonnegative integers, that is, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] denotes the state associated with the [figure omitted; refer to PDF] th neuron at time [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are the interconnection matrices representing the weight coefficients of the neurons, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] denote the neuron activation functions, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] denotes the set of positive integers, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] means a constant input vector, [figure omitted; refer to PDF] determines the size of the jump, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are impulsive moments satisfying [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

A vector [figure omitted; refer to PDF] is said to be an equilibrium point of system (1) if it satisfies [figure omitted; refer to PDF]

In this paper, we assume that there exists a unique equilibrium point for system (1), and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . Set [figure omitted; refer to PDF] ; then system (1) becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

When parameter uncertainty is considered in system (3), the system becomes an impulsive discrete-time neural network with delay and uncertainty, which can be formulated as follows: [figure omitted; refer to PDF] For interval matrices [figure omitted; refer to PDF] , there exist known real constant matrices [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

In order to obtain our main results, the following assumptions are always made through this paper.

[figure omitted; refer to PDF] The activation function [figure omitted; refer to PDF] is a continuous function on [figure omitted; refer to PDF] and there exist scalars [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that, for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

[figure omitted; refer to PDF] The activation function [figure omitted; refer to PDF] is a continuous function on [figure omitted; refer to PDF] and there exist scalars [figure omitted; refer to PDF] [figure omitted; refer to PDF] such that, for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

For convenience, the following notations are defined: [figure omitted; refer to PDF]

We introduce the following lemmas.

Lemma 1 (see [11]).

Let [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] be real matrices of appropriate dimensions with [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] if and only if there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Lemma 2 (Schur's complement).

The linear matrix inequality [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , is equivalent to either of the following conditions:

(1) [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

(2) [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Lemma 3 (see [17]).

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; suppose there exists a Lyapunov function [figure omitted; refer to PDF] , such that

(1) [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;

(2) [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] ;

(3) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Then, if [figure omitted; refer to PDF] , the trivial solution of system (3) is globally exponentially stable.

Lemma 4 (see [17]).

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] be all positive numbers; suppose there exists a Lyapunov function [figure omitted; refer to PDF] , such that

(1) [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;

(2) [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] ;

(3) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Then, if [figure omitted; refer to PDF] , the trivial solution of system (3) is globally exponentially stable.

3. Main Results

Theorem 5.

Suppose that [figure omitted; refer to PDF] hold. If there exist positive numbers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and positive diagonal matrices [figure omitted; refer to PDF] such that

(1) [figure omitted; refer to PDF] where the symbol [figure omitted; refer to PDF] represents the elements below the main diagonal of the symmetric matrix;

(2) [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] ,

then the trivial solution of system (3) is globally exponentially stable.

Proof.

Consider the Lyapunov function [figure omitted; refer to PDF] When [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] By using [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we obtain [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] Substituting these into (14), [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

From condition ( [figure omitted; refer to PDF] ), we know [figure omitted; refer to PDF]

Moreover, we have [figure omitted; refer to PDF]

From the above proving procedure, one observes that all conditions of Lemma 3 are satisfied, which implies that the trivial solution of system (3) is globally exponentially stable. This completes the proof.

Remark 6.

The parameter [figure omitted; refer to PDF] in inequality [figure omitted; refer to PDF] describes the influence of impulses on the stability of discrete-time neural networks. In Theorem 5, when [figure omitted; refer to PDF] , this means the impulses have destabilizing effects, and we give the bound of [figure omitted; refer to PDF] which keep the stability property of the system. In reality, [figure omitted; refer to PDF] is not a necessary condition, and if [figure omitted; refer to PDF] , it means that both impulse-free system and its effects are stabilizing the dynamic system.

Theorem 7.

Suppose that [figure omitted; refer to PDF] hold. If there exist positive numbers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and positive diagonal matrices [figure omitted; refer to PDF] , such that

(1) [figure omitted; refer to PDF] where the symbol [figure omitted; refer to PDF] represents the elements below the main diagonal of the symmetric matrix;

(2) [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] ,

then the trivial solution of system (3) is globally exponentially stable.

Proof.

Consider the Lyapunov function [figure omitted; refer to PDF]

Similar to the proof of Theorem 5, for all [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Moreover, we get [figure omitted; refer to PDF]

Then all conditions of Lemma 4 are satisfied, which indicates that the trivial solution of system (3) is globally exponentially stable. This completes the proof.

Remark 8.

In [18], in order to obtain the condition of globally exponential stability of the equilibrium point of (1), Theorems 5 and 7 are given. In these theorems, the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and positive definite symmetric matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are introduced. And in Theorem 5, the matrix [figure omitted; refer to PDF] , and under some other conditions, the equilibrium point of (1) is globally exponentially stable. The conditions in Theorem 7 are similar. However, in this paper, we only introduce the positive parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and positive diagonal matrices [figure omitted; refer to PDF] . In Theorem 5, we choose parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , in which [figure omitted; refer to PDF] is known, and [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . In Theorem 7, we choose parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . Comparing the conditions of Theorems 5 and 7 of this paper with those given in [18], we find that the conditions in this paper are easier implementation in practice.

Theorem 9.

Suppose that [figure omitted; refer to PDF] hold. If there exist constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] and positive diagonal matrices [figure omitted; refer to PDF] , such that

(1) [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and the symbol [figure omitted; refer to PDF] represents the elements below the main diagonal of the symmetric matrix;

(2) [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] ,

then the trivial solution of system (4) is globally exponentially stable.

Proof.

Consider the Lyapunov function [figure omitted; refer to PDF]

Similar to the proof of Theorem 5, for all [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

By using Lemma 2, the condition [figure omitted; refer to PDF] can be equivalently expressed as [figure omitted; refer to PDF]

Inequality (28) can be rewritten in the following form: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] According to Lemma 1, inequality (29) holds if and only if there exists a scalar [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and again by using Schur's complement, inequality (31) is equivalent to condition ( [figure omitted; refer to PDF] ). Thus, we obtain [figure omitted; refer to PDF] Moreover, we get [figure omitted; refer to PDF]

Then all conditions of Lemma 3 are satisfied, which implies that the trivial solution of system (4) is globally exponentially stable. This completes the proof.

Theorem 10.

Suppose that [figure omitted; refer to PDF] hold. If there exist constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and positive diagonal matrices [figure omitted; refer to PDF] , such that

(1) [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and the symbol [figure omitted; refer to PDF] represents the elements below the main diagonal of a symmetric matrix.

(2) [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] ,

then the trivial solution of system (4) is globally exponentially stable.

Proof.

Consider the Lyapunov function [figure omitted; refer to PDF]

Similar to the proof of Theorem 5, Theorem 10 can be similarly proved; thus we omit it here.

Remark 11.

From the conditions of Theorem 10, the original impulse-free neural network may not be globally stable. Theorem 10 actually derives the impulsive stabilization result for discrete-time delayed neural networks with uncertainty, and the impulses are key in stabilizing the uncertain system.

Remark 12.

Compared with the existing works in [18], the conditions of Theorems 9 and 10 in this paper are easier implementation.

4. Numerical Examples

Example 1.

Consider the following discrete-time impulsive neural network: [figure omitted; refer to PDF]

The matrices are given as follows: [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

It is easy to obtain that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] We choose [figure omitted; refer to PDF] , and solving LMI ( [figure omitted; refer to PDF] ) in Theorem 7, we get [figure omitted; refer to PDF] Then condition ( [figure omitted; refer to PDF] ) of Theorem 7 holds. Thus, the trivial solution of system (36) is globally exponentially stable. Figure 1 depicts the state trajectories [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of system (36) without impulsive effects. Figure 2 depicts the state variables [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with impulsive effects.

Figure 1: The response curves of system (36) without impulsive effects.

[figure omitted; refer to PDF]

Figure 2: The response curves of system (36) with impulsive effects ( [figure omitted; refer to PDF] ).

[figure omitted; refer to PDF]

Example 2.

Consider the following discrete-time impulsive neural network: [figure omitted; refer to PDF]

The matrices are given as follows: [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

It is easy to see that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . We choose [figure omitted; refer to PDF] , and solving LMI ( [figure omitted; refer to PDF] ) in Theorem 9, we get [figure omitted; refer to PDF] By choosing [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] . Also by choosing [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we derive [figure omitted; refer to PDF] . Whether [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , condition ( [figure omitted; refer to PDF] ) of Theorem 9 always holds. Thus the trivial solution of system (39) is globally exponentially stable. Figure 3 depicts the state trajectories [figure omitted; refer to PDF] and [figure omitted; refer to PDF] of system (39) without impulsive effects. Figures 4 and 5 depict the state variables [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with different impulsive effects, respectively.

Figure 3: The response curves of system (39) without impulsive effects.

[figure omitted; refer to PDF]

Figure 4: The response curves of system (39) with impulsive effects ( [figure omitted; refer to PDF] ).

[figure omitted; refer to PDF]

Figure 5: The response curves of system (39) with impulsive effects ( [figure omitted; refer to PDF] ).

[figure omitted; refer to PDF]

Remark 13.

In Example 1, compared with Figures 1 and 2, we get that impulse effects will make unstable system stable if conditions of Theorem 7 hold. Similarly, according to Example 2, uncertain systems with delay and impulse will also be stable if the conditions of Theorem 9 hold. Consequently, through two examples, we have the following results. On the one hand, when the original dynamic system is stable, we figure out feasible regions of impulses that can keep the stability property of the system. On the other hand, when the original dynamic system is unstable, we use impulses to stabilize it.

5. Conclusions

In this paper, the stability of discrete-time impulsive delayed neural network with and without uncertainty has been investigated. Uncertain parameters take values in some intervals. By applying Lyapunov functions and Razumikhin-type theorems, as well as linear matrix inequalities (LMIs), some new results for achieving exponential stability have been derived. Meanwhile, two examples together with their simulations have been presented to show the effectiveness and the advantage of the presented results.

Acknowledgments

This work was supported by the National Natural Science Foundation of Peoples Republic of China (Grants no. 61164004, no. 61473244, and no. 11402223), Natural Science Foundation of Xinjiang (Grant no. 2013211B06), Project Funded by China Postdoctoral Science Foundation (Grants no. 2013M540782 and no. 2014T70953), Natural Science Foundation of Xinjiang University (Grant no. BS120101), and Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20136501120001).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] H. Lu, R. Shen, F.-L. Chung, "Global exponential convergence of Cohen-Grossberg neural networks with time delays," IEEE Transactions on Neural Networks , vol. 16, no. 6, pp. 1694-1696, 2005.

[2] S. Guo, L. Huang, "Stability analysis of Cohen-Grossberg neural networks," IEEE Transactions on Neural Networks , vol. 17, no. 1, pp. 106-117, 2006.

[3] W. Su, Y. Chen, "Global robust stability criteria of stochastic Cohen-Grossberg neural networks with discrete and distributed time-varying delays," Communications in Nonlinear Science and Numerical Simulation , vol. 14, no. 2, pp. 520-528, 2009.

[4] W. Luo, S. Zhong, J. Yang, "Global exponential stability of impulsive Cohen-Grossberg neural networks with delays," Chaos, Solitons & Fractals , vol. 42, no. 2, pp. 1084-1091, 2009.

[5] Z. Orman, "New sufficient conditions for global stability of neutral-type neural networks with time delays," Neurocomputing , vol. 97, pp. 141-148, 2012.

[6] S. Mohamad, "Exponential stability in Hopfield-type neural networks with impulses," Chaos, Solitons and Fractals , vol. 32, no. 2, pp. 456-467, 2007.

[7] Z. Chen, J. Ruan, "Global dynamic analysis of general Cohen-Grossberg neural networks with impulse," Chaos, Solitons and Fractals , vol. 32, no. 5, pp. 1830-1837, 2007.

[8] Z. Chen, J. Ruan, "Global stability analysis of impulsive Cohen-Grossberg neural networks with delay," Physics Letters A , vol. 345, no. 1-3, pp. 101-111, 2005.

[9] Z. Yang, D. Xu, "Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays," Applied Mathematics and Computation , vol. 177, no. 1, pp. 63-78, 2006.

[10] Q. Song, J. Zhang, "Global exponential stability of impulsive Cohen-Grossberg neural network with time-varying delays," Nonlinear Analysis: Real World Applications , vol. 9, no. 2, pp. 500-510, 2008.

[11] B. Liu, D. J. Hill, "Impulsive consensus for complex dynamical networks with nonidentical nodes and coupling time-delays," SIAM Journal on Control and Optimization , vol. 49, no. 2, pp. 315-338, 2011.

[12] Y. Shao, "Existence of exponential periodic attractor of BAM neural networks with time-varying delays and impulses," Neurocomputing , vol. 93, pp. 1-9, 2012.

[13] S. Mohamad, K. Gopalsamy, "Dynamics of a class of discrete-time neural networks and their continuous-time counterparts," Mathematics and Computers in Simulation , vol. 53, no. 1-2, pp. 1-39, 2000.

[14] W. Li, L. Pang, H. Su, K. Wang, "Global stability for discrete Cohen-Grossberg neural networks with finite and infinite delays," Applied Mathematics Letters , vol. 25, no. 12, pp. 2246-2251, 2012.

[15] S. Mohamad, K. Gopalsamy, "Exponential stability of continuous-time and discrete-time cellular neural networks with delays," Applied Mathematics and Computation , vol. 135, no. 1, pp. 17-38, 2003.

[16] Z. Chen, D. Zhao, X. Fu, "Discrete analogue of high-order periodic Cohen-Grossberg neural networks with delay," Applied Mathematics and Computation , vol. 214, no. 1, pp. 210-217, 2009.

[17] S. Wu, C. Li, X. Liao, S. Duan, "Exponential stability of impulsive discrete systems with time delay and applications in stochastic neural networks: a Razumikhin approach," Neurocomputing , vol. 82, pp. 29-36, 2012.

[18] Y. Zhang, "Exponential stability analysis for discrete-time impulsive delay neural networks with and without uncertainty," Journal of the Franklin Institute , vol. 350, no. 4, pp. 737-756, 2013.

[19] Y. Sun, J. Cao, Z. Wang, "Exponential synchronization of stochastic perturbed chaotic delayed neural networks," Neurocomputing , vol. 70, no. 13-15, pp. 2477-2485, 2007.

[20] S. Boyed, L. Ghaoui, E. Feron, V. Balakrishnan Linear Matrix Inequalities in System and Control Theory , SIAM, Philadelphia, Pa, USA, 1994.

[21] Q. Song, J. Cao, "Dynamical behaviors of discrete-time fuzzy cellular neural networks with variable delays and impulses," Journal of the Franklin Institute , vol. 345, no. 1, pp. 39-59, 2008.

[22] S.-L. Wu, K.-L. Li, J.-S. Zhang, "Exponential stability of discrete-time neural networks with delay and impulses," Applied Mathematics and Computation , vol. 218, no. 12, pp. 6972-6986, 2012.

[23] Y. Li, C. Wang, "Existence and global exponential stability of equilibrium for discrete-time fuzzy BAM neural networks with variable delays and impulses," Fuzzy Sets and Systems , vol. 217, pp. 62-79, 2013.

[24] W. Xiong, J. Cao, "Global exponential stability of discrete-time Cohen-Grossberg neural networks," Neurocomputing , vol. 64, pp. 433-446, 2005.

[25] M. Luo, S. Zhong, "Global dissipativity of uncertain discrete-time stochastic neural networks with time-varying delays," Neurocomputing , vol. 85, pp. 20-28, 2012.

[26] C. Li, S. Wu, G. G. Feng, X. Liao, "Stabilizing effects of impulses in discrete-time delayed neural networks," IEEE Transactions on Neural Networks , vol. 22, no. 2, pp. 323-329, 2011.

[27] B. Liu, H. J. Marquez, "Uniform stability of discrete delay systems and synchronization of discrete delay dynamical networks via Razumikhin technique," IEEE Transactions on Circuits and Systems. I. Regular Papers , vol. 55, no. 9, pp. 2795-2805, 2008.

[28] X. Li, "Existence and global exponential stability of periodic solution for delayed neural networks with impulsive and stochastic effects," Neurocomputing , vol. 73, no. 4-6, pp. 749-758, 2010.

[29] L. Hou, G. Zong, Y. Wu, "Robust exponential stability analysis of discrete-time switched Hopfield neural networks with time delay," Nonlinear Analysis: Hybrid Systems , vol. 5, no. 3, pp. 525-534, 2011.

[30] X.-L. Zhu, Y. Wang, G.-H. Yang, "New delay-dependent stability results for discrete-time recurrent neural networks with time-varying delay," Neurocomputing , vol. 72, no. 13-15, pp. 3376-3383, 2009.

[31] L. Jarina Banu, P. Balasubramaniam, K. Ratnavelu, "Robust stability analysis for discrete-time uncertain neural networks with leakage time-varying delay," Neurocomputing , vol. 151, no. 2, pp. 808-816, 2015.

[32] R. Samli, "A new delay-independent condition for global robust stability of neural networks with time delays," Neural Networks , vol. 66, pp. 131-137, 2015.