## Abstract and Applied Analysis

### Synchronization of Chaotic Neural Networks with Leakage Delay and Mixed Time-Varying Delays via Sampled-Data Control

#### Abstract

This paper investigates the synchronization problem for neural networks with leakage delay and both discrete and distributed time-varying delays under sampled-data control. By employing the Lyapunov functional method and using the matrix inequality techniques, a delay-dependent LMIs criterion is given to ensure that the master systems and the slave systems are synchronous. An example with simulations is given to show the effectiveness of the proposed criterion.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 290574, 10 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449806

Digital Object Identifier
doi:10.1155/2013/290574

Mathematical Reviews number (MathSciNet)
MR3132554

Zentralblatt MATH identifier
1291.93189

#### Citation

Lei, Ting; Song, Qiankun; Zhao, Zhenjiang; Yang, Jianxi. Synchronization of Chaotic Neural Networks with Leakage Delay and Mixed Time-Varying Delays via Sampled-Data Control. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 290574, 10 pages. doi:10.1155/2013/290574. https://projecteuclid.org/euclid.aaa/1393449806

#### References

• L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990.
• 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.
• F. Zou and J. A. Nossek, “A chaotic attractor with cellular neural networks,” IEEE Transactions on Circuits and Systems I, vol. 38, no. 7, pp. 811–812, 1991.
• D. Hansel and H. Sompolinsky, “Synchronization and computation in a chaotic neural network,” Physical Review Letters, vol. 68, no. 5, pp. 718–721, 1992.
• T. B. Luzyanina, “Synchronization in an oscillator neural network model with time-delayed coupling,” Network, vol. 6, no. 1, pp. 43–59, 1995.
• J. Karbowski and N. Kopell, “Multispikes and synchronization in a large neural network with temporal delays,” Neural Computation, vol. 12, no. 7, pp. 1573–1606, 2000.
• W. Lu and T. Chen, “Synchronization of coupled connected neural networks with delays,” IEEE Transactions on Circuits and Systems I, vol. 51, no. 12, pp. 2491–2503, 2004.
• H. Zhang, Y. Xie, Z. Wang, and C. Zheng, “Adaptive synchronization between two different chaotic neural networks with time delay,” IEEE Transactions on Neural Networks, vol. 18, no. 6, pp. 1841–1845, 2007.
• T. Chen, W. Wu, and W. Zhou, “Global $\mu$-synchronization of linearly coupled unbounded time-varying delayed neural networks with unbounded delayed coupling,” IEEE Transactions on Neural Networks, vol. 19, no. 10, pp. 1809–1816, 2008.
• J. Cao, G. Chen, and P. Li, “Global synchronization in an array of delayed neural networks with hybrid coupling,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 38, no. 2, pp. 488–498, 2008.
• J. Liang, Z. Wang, and P. Li, “Robust synchronisation of delayed neural networks with both linear and non-linear couplings,” International Journal of Systems Science, vol. 40, no. 9, pp. 973–984, 2009.
• F. O. Souza and R. M. Palhares, “Synchronisation of chaotic delayed artificial neural networks: an ${H}_{\infty }$ control approach,” International Journal of Systems Science, vol. 40, no. 9, pp. 937–944, 2009.
• J. H. Park and 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.
• H. R. Karimi and H. Gao, “New delay-dependent exponential ${H}_{\infty }$ synchronization for uncertain neural networks with mixed time delays,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 40, no. 1, pp. 173–185, 2010.
• H. Wang and Q. Song, “Synchronization for an array of coupled stochastic discrete-time neural networks with mixed delays,” Neurocomputing, vol. 74, no. 10, pp. 1572–1584, 2011.
• J. Lu, D. W. C. Ho, J. Cao, and J. Kurths, “Exponential synchronization of linearly coupled neural networks with impulsive disturbances,” IEEE Transactions on Neural Networks, vol. 22, no. 2, pp. 329–335, 2011.
• P. Balasubramaniam and V. Vembarasan, “Synchronization of recurrent neural networks with mixed time-delays via output coupling with delayed feedback,” Nonlinear Dynamics, vol. 70, no. 1, pp. 677–691, 2012.
• D. Ghosh and S. Banerjee, “Projective synchronization of time-varying delayed neural network with adaptive scaling factors,” Chaos, Solitons & Fractals, vol. 53, pp. 1–9, 2013.
• T. Botmart and W. Weera, “Guaranteed cost control for exponential synchronization of cellular neural networks with mixed time-varying delays via hybrid feedback control,” Abstract and Applied Analysis, vol. 2013, Article ID 175796, 12 pages, 2013.
• 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, vol. 60, no. 2, pp. 363–376, 2013.
• Q. Zhu and J. Cao, “Adaptive synchronization of chaotic Cohen-Crossberg neural networks with mixed time delays,” Nonlinear Dynamics, vol. 61, no. 3, pp. 517–534, 2010.
• Q. Zhu and J. Cao, “$p$th moment exponential synchronization for stochastic delayed Cohen-Grossberg neural networks with Markovian switching,” Nonlinear Dynamics, vol. 67, no. 1, pp. 829–845, 2012.
• E. Fridman, A. Seuret, and J.-P. Richard, “Robust sampled-data stabilization of linear systems: an input delay approach,” Automatica, vol. 40, no. 8, pp. 1441–1446, 2004.
• H. K. Lam and L. D. Seneviratne, “Chaotic synchronization using sampled-data fuzzy controller based on fuzzy-model-based approach,” IEEE Transactions on Circuits and Systems I, vol. 55, no. 3, pp. 883–892, 2008.
• J.-G. Lu and D. J. Hill, “Global asymptotical synchronization of chaotic Lur'e systems using sampled data: a linear matrix inequality approach,” IEEE Transactions on Circuits and Systems II, vol. 55, no. 6, pp. 586–590, 2008.
• C.-K. Zhang, Y. He, and M. Wu, “Improved global asymptotical synchronization of chaotic lur'e systems with sampled-data control,” IEEE Transactions on Circuits and Systems II, vol. 56, no. 4, pp. 320–324, 2009.
• W. Chen, Z. Wang, and X. Lu, “On sampled-data control for master-slave synchronization of chaotic Lur'e systems,” IEEE Transactions on Circuits and Systems II, vol. 59, no. 8, pp. 515–519, 2012.
• N. Li, Y. Zhang, J. Hu, and Z. Nie, “Synchronization for general complex dynamical networks with sampled-data,” Neurocomputing, vol. 74, no. 5, pp. 805–811, 2011.
• T. H. Lee, J. H. Park, O. M. Kwon, and S. M. Lee, “Synchronization of chaos systems via sampled-data control,” Transactions of the Korean Institute of Electrical Engineers, vol. 61, no. 4, pp. 617–621, 2012.
• B. Shen, Z. Wang, and X. Liu, “Sampled-data synchronization control of dynamical networks with stochastic sampling,” Institute of Electrical and Electronics Engineers, vol. 57, no. 10, pp. 2644–2650, 2012.
• T. H. Lee, Z.-G. Wu, and J. H. Park, “Synchronization of a complex dynamical network with coupling time-varying delays via sampled-data control,” Applied Mathematics and Computation, vol. 219, no. 3, pp. 1354–1366, 2012.
• S. J. S. Theesar, S. Banerjee, and P. Balasubramaniam, “Synchronization of chaotic systems under sampled-data control,” Nonlinear Dynamics, vol. 70, no. 3, pp. 1977–1987, 2012.
• Z.-G. Wu, J. H. Park, H. Su, B. Song, and J. Chu, “Exponential synchronization for complex dynamical networks with sampled-data,” Journal of the Franklin Institute, vol. 349, no. 9, pp. 2735–2749, 2012.
• H. Zhang and J. Zhou, “Synchronization of sampled-data coupled harmonic oscillators with control inputs missing,” Systems & Control Letters, vol. 61, no. 12, pp. 1277–1285, 2012.
• Z. Wu, P. Shi, H. Su, and J. Chu, “Sampled-data exponential synchronization of complex dynamical networks with time-varying coupling delay,” IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 8, pp. 1177–1187, 2013.
• C.-K. Zhang, Y. He, and M. Wu, “Exponential synchronization of neural networks with time-varying mixed delays and sampled-data,” Neurocomputing, vol. 74, no. 1–3, pp. 265–273, 2010.
• Z.-G. 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.
• Q. Gan and Y. Liang, “Synchronization of chaotic neural networks with time delay in the leakage term and parametric uncertainties based on sampled-data control,” Journal of the Franklin Institute, vol. 349, no. 6, pp. 1955–1971, 2012.
• K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Birkhauser, Boston, Mass, USA, 2003.
• Y. Liu, Z. Wang, and 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.