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Qing Zhu, Aiguo Song, Shumin Fei, Yuequan Yang, Zhiqiang Cao, "Synchronization Control for Stochastic Neural Networks with Mixed Time-Varying Delays", *The Scientific World Journal*, vol. 2014, Article ID 840185, 10 pages, 2014. https://doi.org/10.1155/2014/840185

# Synchronization Control for Stochastic Neural Networks with Mixed Time-Varying Delays

**Academic Editor:**Guanghui Wen

#### Abstract

Synchronization control of stochastic neural networks with time-varying discrete and continuous delays has been investigated. A novel control scheme is proposed using the Lyapunov functional method and linear matrix inequality (LMI) approach. Sufficient conditions have been derived to ensure the global asymptotical mean-square stability for the error system, and thus the drive system synchronizes with the response system. Also, the control gain matrix can be obtained. With these effective methods, synchronization can be achieved. Simulation results are presented to show the effectiveness of the theoretical results.

#### 1. Introduction

In recent years, stochastic neural networks (NNs) have gained particular research interests. These systems represent a class of stochastic systems that is popular in modeling practical systems which may experience random disturbances and parameters varying. Such a system can be found in biology systems, social systems, and wireless communication networks. Numerous results on stochastic neural network have been reported in the literature [1–5]. As one of the mostly investigated dynamical behaviors, the synchronization in stochastic NNs with or without time delays has drawn significant research interest; see, for example, [6–10] and the references therein.

On the other hand, time delays are frequently encountered in many practical control systems, such as aircraft, chemical, or process control systems. The existence of the time delays may be the source of instability of serious deterioration in the performance of the closed-loop systems. Thus, stability of delayed NNs has been a focal subject for research [11–15]. Most recently, significant and substantial progresses have been achieved in the synchronization of stochastic NNs. These include synchronization of randomly coupled neural networks with Markovian jumping and time-delay [15], adaptive synchronization for stochastic NNs with time-varying delays and distributed delays [16], passivity analysis for discrete time stochastic Markovian jump NNs with mixed time delays [17], adaptive synchronization for stochastic T-S fuzzy NNs with time-delay and Markovian jumping parameters [18], nonfragile synchronization of NNs with time-varying delay and randomly occurring controller gain fluctuation synchronization of biological NN systems with stochastic perturbations and time delays [19], global exponential adaptive synchronization of complex dynamical networks with neutral-type NN nodes and stochastic disturbances [20], adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer [21], the effects of time delay on the stochastic resonance in feed-forward-loop NN motifs [22], state estimation for wireless network control system with stochastic uncertainty and time delay based on sliding mode observer [23], pinning synchronization in fixed and switching directed networks of Lorenz-type nodes [24], consensus tracking for higher-order multiagent systems with switching directed topologies and occasionally missing control inputs [25], and consensus tracking of multiagent systems with Lipschitz-type node dynamics and switching topologies [26].

Although substantial progresses have been made in the synchronization control of delayed stochastic neural network systems [16–32], there are still some problems which have not been fully studied. For example, many control schemes of time-varying delayed NNs assume that the derivative of the time-delay function is less than one. Also, few works have been done on the synchronization of mixed delayed stochastic NNs. In this paper, we propose sufficient conditions of the synchronization of the stochastic NN system, as well as the state feedback control design. At first, sufficient conditions are proposed in terms of LMIs to guarantee the stochastic asymptotical stability of the error system. The assumption that the derivative of the time-delay function is less than one is eliminated. Then, we give some corollaries to solve the problem in some special cases.

This paper is organized as follows. After the introduction in Section 1, the problem statement and preliminaries are presented in Section 2. Next, some sufficient conditions are presented for the synchronization of the delayed drive and response system. Also, some corollaries and remarks are given to show the advantages of this paper in Section 3. Then, the numerical simulation result is given in Section 4. A conclusion is drawn in Section 5.

#### 2. Problem Statement and Preliminaries

Before proceeding, we introduce some notations which will be used later for derivations and discussions. denotes the Euclidean norm of a vector or the Frobenius norm of a matrix. (<0) denotes that matrix is a positive (negative) definite matrix. denotes the mathematical expectation.

The recurrent network under investigation is where is the system state associated with the neurons, denotes the number of neurons in the network, corresponds to the activation functions of neurons, and and are the time-varying discrete delay and continuous delay, respectively; the initial conditions are given by , where denotes the set of all continuous functions from to . is a diagonal matrix, , , and are the connection weight matrix, discrete time-delay, and continuous time-delay connection weight matrices, respectively.

In this paper, we consider model (1) as the master system. The response system is where , and are matrices which are the same as (1) and is the controller. It has the same structure as the drive system. is the error state. is a dimension Brownian motion defined on a complete probability space .

To propose our main results, it is necessary to make the following assumptions.(A1)Each function is nondecreasing and globally Lipschitz with a constant : (A2)Matrix function satisfies where and are matrices with appropriate dimensions.(A3)Discrete delay and continuous delay are both differential functions of time, and the following conditions hold: where , and are positive constants. It is worth to emphasis that the constraint, the derivative of time-delay function is less than one [1], is eliminated in this paper, while a more general upper boundary constraint is utilized instead of it. This improvement makes our results applicable for a wide range of time-delayed stochastic neural networks.

Let error state be ; subtracting (1) from (2) yields the synchronization error dynamical system as follows: where .

In this paper, we design a memoryless state feedback controller: where is a constant gain matrix.

Substituting the controller into the error system (6), we get It is well known that system (8) has a unique solution [33].

*Definition 1. *The system (8) is called globally asymptotically mean-square stable, if the following holds:
where is the mathematical expectation. The following Lemma is given.

Lemma 2 (Schur complement lemma). *For matrices , and with compatible dimensions, the following three conditions are equivalent:
*

Lemma 3. *For vector function and symmetric matrix , the following holds:
*

#### 3. Criteria of Synchronization

In this section, new criteria are presented for the global asymptotical stability of the equilibrium point of the neural network defined by (8), and thus the drive system (1) synchronizes with the response system (2). Its proof is based on a new Lyapunov functional method and linear matrix inequality (LMI) approach.

Theorem 4. *Under the assumptions A1–A3, the equilibrium point of model (8) is called globally asymptotically stable in mean-square, if there exist symmetric positive-definite matrices , diagonal matrices , and general matrix such that the following inequalities hold:**where
*

*Proof. *The Lyapunov function candidate is given as
where
The weak infinitesimal operator of the process is given by
By (5) and (14), we have
By Lemma 3, we obtain
By Newton-Leibniz formula, we have
From assumption A1, we get
where , and are positive-definite diagonal matrices. By (27) and (28), we define several nonnegative expressions as follows:
Then expressions , and are to be added to the inequality of to facilitate the proof.

Therefore, combining (19)–(29) we have
where
We introduce matrices and which satisfy
where
By Lemma 2, we get that . Thus . From (24) and the Ito formula, it is obvious that
From the definition of in (15), there exists positive constant such that
where is the maximal eigenvalue of and it is negative.

Therefore, from (26) and the discussion in [34], we know that the equilibrium of (11) is globally asymptotically stable in mean-square. This completes the proof.

Corollary 5. *Under the assumptions A1–A3, the equilibrium point of model (8) is called globally asymptotically stable in mean-square, if there exist symmetric positive-definite matrices , diagonal matrices , and general matrices and , such that the following inequalities hold:**where
**
Furthermore, the control gain matrix is given as
*

*Proof. *Let in Theorem 4; it is easy to get the result.

Corollary 6. *If the systems (1) and (2) have no integral time-delay items (e.g., ), the result can be simplified as follows. Under assumptions A1–A3 and , the equilibrium point of model (8) is called globally asymptotically stable in mean-square, if there exist symmetric positive-definite matrices , diagonal matrices , and general matrices and , such that the following inequalities hold:**where
**
Furthermore, the control gain matrix is given as
*

*Proof. *Let in system (8). It is much similar to the proof of Theorem 4 and is then omitted.

*Remark 7. *In this paper, we use a memoryless state feedback control to achieve the synchronization of the stochastic delayed neural network system. Compared with the delayed feedback control [1], it is easy to implement. The biggest advantage of this approach is that we only need to know the boundaries of time delays, instead of exact values of time delays.

*Remark 8. *In Corollary 5, we give an approach to choose the control gain matrix and it is helpful for the design of the controller to let the drive system synchronize with the response system.

#### 4. Numerical Simulations

In this section, we present numerical simulations to show the effectiveness of the theoretical results.

Consider the drive system (1) of a delayed Hopfield neural network with coefficient matrices as The corresponding response system refers to (2), where It is easy to see that , where is the identity matrix. By Theorem 4 and Corollary 5, we can get the following feasible solutions: The initial states of drive system and response system are and .

The results are shown in Figures 1, 2, 3, and 4.

**(a)**

**(b)**

**(a)**

**(b)**

**(a)**

**(b)**

**(a)**

**(b)**

#### 5. Conclusion

In this paper, we considered synchronization control of stochastic neural networks with time-varying delays. We use Lyapunov functional method and linear matrix inequality (LMI) technique to solve this problem. Several sufficient conditions have been derived to ensure the global asymptotical stability for the error system, and thus the drive system synchronizes with the response system. Also, the control gain can be obtained. The results are novel since there are few works about the synchronization of mixed delayed system and the constraint of the derivative of the time-delay function is relaxed. It is easy to apply these sufficient conditions to the real networks. Finally, a numerical simulation is presented to verify the theoretical results.

#### Conflict of Interests

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

#### Acknowledgments

The authors would like to thank the editor and the reviewers for their helpful and valuable comments and suggestions, which have improved the presentation. This work is supported by National Natural Science Foundation of China under Grant nos. 61273352, 61175111, 61174046, 60904030, 60874045, and 60874030, Natural Science Foundation of the Jiangsu Higher Education Institutions (nos. 10KJB510027 and 09KJB510019), Postdoctoral Science Research Project of Jiangsu Province (no. 1102167C).

#### References

- W. Yu and J. Cao, “Synchronization control of stochastic delayed neural networks,”
*Physica A: Statistical Mechanics & Its Applications*, vol. 373, no. 1, pp. 252–260, 2007. View at: Publisher Site | Google Scholar - Z. G. Wu, P. Shi, H. Su, and J. Chu, “Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled data,”
*IEEE Transactions on Cybernetics*, vol. 43, no. 6, pp. 1796–1806, 2013. View at: Publisher Site | Google Scholar - Z. Wang, Y. Wang, and Y. Liu, “Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays,”
*IEEE Transactions on Neural Networks*, vol. 21, no. 1, pp. 11–25, 2010. View at: Publisher Site | Google Scholar - M. J. Park, O. M. Kwon, J. H. Park, S. M. Lee, and E. J. Cha, “Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay,”
*Journal of the Franklin Institute*, vol. 349, no. 5, pp. 1699–1720, 2012. View at: Publisher Site | Google Scholar - Z. Wu, J. H. Park, H. Su, and J. Chu, “Discontinuous lyapunov functional approach to synchronization of time-delay neural networks using sampled-data,”
*Nonlinear Dynamics*, vol. 69, no. 4, pp. 2021–2030, 2012. View at: Publisher Site | Google Scholar - B. Shen, Z. Wang, and X. Liu, “Bounded H∞ synchronization and state estimation for discrete time-varying stochastic complex networks over a finite horizon,”
*IEEE Transactions on Neural Networks*, vol. 22, no. 1, pp. 145–157, 2011. View at: Publisher Site | Google Scholar - H. R. Karimi and H. Gao, “New delay-dependent exponential H∞ synchronization for uncertain neural networks with mixed time delays,”
*IEEE Transactions on Systems, Man, and Cybernetics B: Cybernetics*, vol. 40, no. 1, pp. 173–185, 2010. View at: Publisher Site | Google Scholar - C. Li, W. Yu, and T. Huang, “Impulsive synchronization schemes of stochastic complex networks with switching topology: average time approach,”
*Neural Networks*, vol. 54, pp. 85–94, 2014. View at: Publisher Site | Google Scholar - C. Li, C. Li, X. Liao, and T. Huang, “Impulsive effects on stability of high-order BAM neural networks with time delays,”
*Neurocomputing*, vol. 74, no. 10, pp. 1541–1550, 2011. View at: Publisher Site | Google Scholar - Q. Zhu, S. Fei, T. Zhang, and T. Li, “Adaptive RBF neural-networks control for a class of time-delay nonlinear systems,”
*Neurocomputing*, vol. 71, no. 16–18, pp. 3617–3624, 2008. View at: Publisher Site | Google Scholar - W. Zhou, D. Tong, Y. Gao, C. Ji, and H. Su, “Mode and delay-dependent adaptive exponential synchronization in pth moment for stochastic delayed neural networks with markovian switching,”
*IEEE Transactions on Neural Networks & Learning Systems*, vol. 23, no. 4, pp. 662–668, 2012. View at: Publisher Site | Google Scholar - J. Cao, Z. Wang, and Y. Sun, “Synchronization in an array of linearly stochastically coupled networks with time delays,”
*Physica A: Statistical Mechanics and its Applications*, vol. 385, no. 2, pp. 718–728, 2007. View at: Publisher Site | Google Scholar | MathSciNet - J. Cao and Y. Wan, “Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays,”
*Neural Networks*, vol. 53, no. 5, pp. 165–172, 2014. View at: Google Scholar - G. Wen, Z. Duan, Z. Li, and G. Chen, “Stochastic consensus in directed networks of agents with non-linear dynamics and repairable actuator failures,”
*IET Control Theory & Applications*, vol. 6, no. 11, pp. 1583–1593, 2012. View at: Publisher Site | Google Scholar - X. Yang, J. Cao, and J. Lu, “Synchronization of randomly coupled neural networks with markovian jumping and time-delay,”
*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 60, no. 2, pp. 363–376, 2013. View at: Publisher Site | Google Scholar - Q. Zhu and J. Cao, “Adaptive synchronization under almost every initial data for stochastic neural networks with time-varying delays and distributed delays,”
*Communications in Nonlinear Science & Numerical Simulation*, vol. 16, no. 4, pp. 2139–2159, 2011. View at: Publisher Site | Google Scholar | MathSciNet - Z. Wu, P. Shi, H. Su, and J. Chu, “Passivity analysis for discrete-time stochastic markovian jump neural networks with mixed time delays,”
*IEEE Transactions on Neural Networks*, vol. 22, no. 10, pp. 1566–1575, 2011. View at: Publisher Site | Google Scholar - D. Tong, Q. Zhu, W. Zhou, Y. Xu, and J. A. Fang, “Adaptive synchronization for stochastic T-S fuzzy neural networks with time-delay and Markovian jumping parameters,”
*Neurocomputing*, vol. 117, pp. 91–97, 2013. View at: Publisher Site | Google Scholar - M. Fang and J. H. Park, “Non-fragile synchronization of neural networks with time-varying delay and randomly occurring controller gain fluctuation,”
*Applied Mathematics and Computation*, vol. 219, no. 15, pp. 8009–8017, 2013. View at: Publisher Site | Google Scholar - Y. Zhang, D. W. Gu, and S. Xu, “Global exponential adaptive synchronization of complex dynamical networks with neutral-type neural network nodes and stochastic disturbances,”
*IEEE Transactions on Circuits & Systems I: Regular Papers*, vol. 60, no. 10, pp. 2709–2718, 2013. View at: Publisher Site | Google Scholar | MathSciNet - S. C. Jeong, D. H. Ji, J. H. Park, and S. C. Won, “Adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer,”
*Applied Mathematics and Computation*, vol. 219, no. 11, pp. 5984–5995, 2013. View at: Publisher Site | Google Scholar - C. Liu, J. Wang, H. Yu et al., “The effects of time delay on the stochastic resonance in feed-forward-loop neuronal network motifs,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 19, no. 4, pp. 1088–1096, 2014. View at: Google Scholar - P. Guo, J. Zhang, H. R. Karimi, Y. Liu, M. Lyu, and Y. Bo, “State estimation for wireless network control system with stochastic uncertainty and time delay based on sliding mode observer,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 303840, 8 pages, 2014. View at: Publisher Site | Google Scholar - G. Wen, W. Yu, Y. Zhao, and J. Cao, “Pinning synchronisation in fixed and switching directed networks of Lorenz-type nodes,”
*IET Control Theory & Applications*, vol. 7, no. 10, pp. 1387–1397, 2013. View at: Publisher Site | Google Scholar | MathSciNet - G. Wen, G. Hu, W. Yu, J. Cao, and G. Chen, “Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs,”
*Systems & Control Letters*, vol. 62, no. 12, pp. 1151–1158, 2013. View at: Publisher Site | Google Scholar - G. Wen, Z. Duan, G. Chen, and W. Yu, “Consensus tracking of multi-agent systems with Lipschitz-type node dynamics and switching topologies,”
*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 61, no. 2, pp. 499–511, 2014. View at: Publisher Site | Google Scholar | MathSciNet - W. H. Chen, J. X. Xu, and Z. H. Guan, “Guaranteed cost control for uncertain Markovian jump systems with mode-dependent time-delays,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 12, pp. 2270–2277, 2003. View at: Publisher Site | Google Scholar | MathSciNet - S. Xu, J. Lam, and X. Mao, “Delay-dependent ${H}_{\infty}$ control and filtering for uncertain Markovian jump systems with time-varying delays,”
*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 54, no. 9, pp. 2070–2077, 2007. View at: Publisher Site | Google Scholar | MathSciNet - S. Xu, T. Chen, and J. Lam, “Robust ${H}^{\infty}$ filtering for uncertain Markovian jump systems with mode-dependent time delays,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 5, pp. 900–907, 2003. View at: Publisher Site | Google Scholar | MathSciNet - Z. Wang, Y. Liu, L. Yu, and 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. View at: Publisher Site | Google Scholar - X. Mao, “Exponential stability of stochastic delay interval systems with Markovian switching,”
*IEEE Transactions on Automatic Control*, vol. 47, no. 10, pp. 1604–1612, 2002. View at: Publisher Site | Google Scholar | MathSciNet - X. Zeng, Q. Hui, W. M. Haddad, T. Hayakawa, J. M. Bailey, and X. Zeng, “Synchronization of biological neural network systems with stochastic perturbations and time delays,”
*Journal of the Franklin Institute*, vol. 351, no. 3, pp. 1205–1225, 2013. View at: Google Scholar - V. B. Kolmanovskii and A. D. Myshkis,
*Introduction to the theory and applications of functional-differential equations*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. View at: Publisher Site | MathSciNet - Z. Schuss,
*Theory and Applications of Stochastic Differential Equations*, Wiley, New York, NY, USA, 1980. View at: MathSciNet

#### Copyright

Copyright © 2014 Qing Zhu et al. 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.