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Sampled-data synchronization of randomly coupled reaction–diffusion neural networks with Markovian jumping and mixed delays using multiple integral approach

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Abstract

This paper is devoted to investigate the problem of global asymptotic synchronization of an array of N randomly coupled reaction–diffusion neural networks with Markovian jumping parameters and mixed delays using sampled-data control technique. The jump parameters are determined by a continuous-time, discrete-state Markovian chain, and the mixed time delays under consideration comprise both discrete and distributed delays. A multiple integral inequality is proposed firstly in Markovian jump reaction–diffusion neural networks with mixed delays. Through constructing appropriate Lyapunov–Krasovskii functional including multiple integral terms, some novel synchronization criteria in terms of linear matrix inequalities are derived. The obtained LMIs can be easily verified for feasibility through any of the available softwares. Finally, numerical examples with simulations are provided to illustrate the effectiveness of the proposed theoretical results.

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References

  1. Young S, Scott P, Nasrabadi N (1997) Object recognition using multilayer Hopfield neural network. IEEE Trans Image Process 6:357–372

    Article  Google Scholar 

  2. Atencia M, Joya G, Sandoval F (2005) Dynamical analysis of continuous higher order Hopfield networks for combinatorial optimization. Neural Comput 17:1802–1819

    Article  MathSciNet  MATH  Google Scholar 

  3. Sheikhan M, Shahnazi R, Garoucy S (2013) Synchronization of general chaotic systems using neural controllers with application to secure communication. Neural Comput Appl 22:361–373

    Article  Google Scholar 

  4. Huang H, Feng G (2009) Synchronization of nonidentical chaotic neural networks with time delays. Neural Netw 22:1841–1845

    Article  MATH  Google Scholar 

  5. Li X, Ding C, Zhu Q (2010) Synchronization of stochastic perturbed chaotic neural networks with mixed delays. J Franklin Inst 347:1266–1280

    Article  MathSciNet  MATH  Google Scholar 

  6. Liang J, Wang Z, Li P (2009) Robust Synchronization of delayed neural networks with both linear and non-linear couplings. Int J Syst Sci 40:973–984

    Article  MATH  Google Scholar 

  7. Grassi G, Mascolo S (1999) Synchronizing high dimensional chaotic systems via eigenvalue placement in application to cellular neural networks. Int J Bifurc Chaos 9:705–711

    Article  MATH  Google Scholar 

  8. Zhang L (2009) \(H_\infty\) estimation for discrete-time piecewise homogeneous Markov jump linear systems. Automatica 45:2570–2576

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhang W, Li C, Huang T, Qi J (2014) Global exponential synchronization for coupled switched delayed recurrent neural networks with stochastic perturbation and impulsive effects. Neural Comput Appl 25:1275–1283

    Article  Google Scholar 

  10. Cao J, Wan Y (2014) Matrix measure strategies for stability and synchronization for inertial BAM neural network with time delay. Neural Netw 53:165–172

    Article  MATH  Google Scholar 

  11. Haken H (2006) Pattern recognition and synchronization in pulse-coupled neural networks. Nonlinear Dyn 44:269–276

    Article  MathSciNet  MATH  Google Scholar 

  12. Gong D, Liu J, Zhou S (2015) Chaotic synchronization for coupled neural networks based on T-S fuzzy theory. Int J Syst Sci 46:681–689

    Article  MATH  Google Scholar 

  13. Liu X (2010) Synchronization of linearly coupled neural networks with reaction–diffusion terms and unbounded time delays. Neurocomputing 73:2681–2888

    Article  Google Scholar 

  14. Wang K, Teng Z, Jiang H (2012) Adaptive synchronization in an array of linearly coupled neural networks with reaction–diffusion terms and time delays. Commun Nonlinear Sci Numer Simul 17:3866–3875

    Article  MathSciNet  MATH  Google Scholar 

  15. Yang X, Cao J, Yang Z (2013) Synchronization of coupled reaction–diffusion neural networks with time-varying delays via pinning-impulsive controller. SIAM J Control Optim 51:3486–3510

    Article  MathSciNet  MATH  Google Scholar 

  16. Hu C, Jiang H, Teng Z (2010) Impulsive control and synchronization for delayed neural networks with reaction–diffusion terms. IEEE Trans Neural Netw 21:67–81

    Article  Google Scholar 

  17. Liu Z, Peng J (2010) Delay-independent stability of stochastic reaction–diffusion neural networks with Dirichlet boundary conditions. Neural Comput Appl 19:151–158

    Article  Google Scholar 

  18. Wang Y, Cao J (2007) Synchronization of a class of delayed neural networks with reaction–diffusion terms. Phys Lett A 369:201–211

    Article  Google Scholar 

  19. Krasovskii NN, Lidskii EA (1961) Analysis and design of controllers in systems with random attributes. Autom Remote Control 22:1021–1025

    Google Scholar 

  20. Liu Y, Wang Z, Liu X (2008) Exponential synchronization of complex networks with Markovian jump and mixed delays. Phys Lett A 372:3986–3998

    Article  MathSciNet  MATH  Google Scholar 

  21. Shen H, Xu S, Zhou J, Lu J (2011) Fuzzy \(H_\infty\) filtering for nonlinear Markovian jump neutral systems. Int J Syst Sci 42:767–780

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang Z, Liu L, Liu X (2010) Exponential stability of a class of stochastic system with Markovian jump parameters and mode-dependent mixed time-delays. IEEE Trans Autom Control 55:1656–1662

    Article  Google Scholar 

  23. Chen B, Huang J, Niu Y (2012) Sliding mode control for Markovian jumping systems with actuator nonlinearities. Int J Syst Sci 43:656–664

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu S, Chen T, Lam J (2003) Robust \(H_\infty\) filtering for uncertain Markovian jump systems with mode-dependent time delays. IEEE Trans Autom Control 48:900–908

    Article  Google Scholar 

  25. Liu Y (2009) Stochastic asymptotic stability of Markovian jumping neural networks with Markov mode estimation and mode-dependent delays. Phys Lett A 373:3741–3742

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang L, Boukas E, Lam J (2008) Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities. IEEE Trans Autom Control 53:2458–2464

    Article  MathSciNet  Google Scholar 

  27. Balasubramaniam P, Vembarasan V, Rakkiyappan R (1999) Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying. IEEE Trans Circuits Syst I Fundam Theory Appl 46:1144–1150

    Article  MATH  Google Scholar 

  28. Wang L, Zhang Z, Wang Y (2008) Stochastic exponential stability of the delayed reaction–diffusion recurrent neural networks with Markovian jumping parameters. Phys Lett A 372:3201–3209

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang H, Wang J, Wang Z, Liang H Mode-dependent stochastic synchronization for Markovian coupled neural networks with time-varying mode-delays. IEEE Trans Neural Netw Learn Syst. doi:10.1109/TNNLS/2014.2387885

  30. Zheng CD, Zhang H, Wang Z (2014) Exponential synchronization of stochastic chaotic neural networks with mixed time delays and Markovian switching. Neural Comput Appl 25:429–442

    Article  Google Scholar 

  31. Yang X, Cao J, Lu J (2013) Synchronization of randomly coupled neural networks with Markovian jumping and time-delay. IEEE Trans Circuits Syst I Regul Pap 60:363–376

    Article  MathSciNet  Google Scholar 

  32. Kao YG, Guo JF, Wang CH, Sun XQ (2012) Delay-dependent robust exponential stability of Markovian jumping reaction–diffusion Cohen-Grossberg neural networks with mixed delays. J Franklin Inst 349:1972–1988

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang L, Zhu YZ, Zheng WX Synchronization and state estimation of a class of hierarchical hybrid neural networks with time-varying delays. IEEE Trans Neural Netw Learn Syst. doi:10.1109/TNNLS.2015.2412676

  34. Shi G, Ma Q (2011) Synchronization of stochastic Markovian jump neural networks with reaction–diffusion terms. Neurocomputing 77:275–280

    Article  Google Scholar 

  35. Zhou J, Shen H, Wang Z (2012) Synchronization of delayed reaction–diffusion neural networks with markovian jumping parameters. In: The second international conference on computer application and system modeling

  36. Wang X, Li C, Huang T, Chen L (2014) Impulsive exponential synchronization of randomly coupled neural networks with Markovian jumping and mixed model-dependent time delays. Neural Netw 60:25–32

    Article  MATH  Google Scholar 

  37. Zhang CK, He Y, Wu M (2010) Exponential synchronization of neural networks with time-varying mixed delays and sampled-data. Neurocomputing 74:265–273

    Article  Google Scholar 

  38. Li N, Zhang Y, Hu J, Nie Z (2011) Synchronization for general complex dynamical networks with sampled-data. Neurocomputing 74:805–811

    Article  Google Scholar 

  39. Liu YY, Yang GH (2012) Sampled-data \(H_\infty\) control for networked control systems with digital control inputs. Int J Syst Sci 43:1728–1740

    Article  MathSciNet  MATH  Google Scholar 

  40. Li L, Yang Y, Lin G The stabilization of BAM neural networks with time-varying delays in the leakage terms via sampled-data control. Neural Comput Appl. doi:10.1007/s00521-015-1865-4

  41. Kalpana M, Balasubramaniam P (2013) Stochastic asymptotical synchronization of chaotic Markovian jumping fuzzy cellular neural networks with mixed delays and the Wiener process based on sampled-data control. Chin Phys B 22:078401

    Article  Google Scholar 

  42. Zhang CK, He Y, Wu M (2009) Improved global asymptotical synchronization of chaotic Lur’e systems with sampled-data control. IEEE Trans Circuits Syst II Express Briefs 56:320–324

    Article  Google Scholar 

  43. Wu Z, Shi P, Su H, Chu J (2013) Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled-data. IEEE Trans Cybern 43:1796–1806

    Article  Google Scholar 

  44. Astrom K, Wittenmark B (1989) Adaptive control. Addison-Wesley, Reading

    Google Scholar 

  45. Mikheev Y, Sobolev V, Fridman E (1988) Asymptotic analysis of digital control systems. Autom Remote Control 49:1175–1180

    MathSciNet  MATH  Google Scholar 

  46. Fang M, Park JH (2013) A multiple integral approach to stability of neutral time-delay systems. Appl Math Comput 224:714–718

    MathSciNet  MATH  Google Scholar 

  47. Wang J, Zhang H, Wang Z, Wang B (2013) Local exponential synchronization in complex dynamical networks with time-varying delay and hybrid coupling. Appl Math Comput 225:16–32

    MathSciNet  MATH  Google Scholar 

  48. Park JH, Lee TH (2015) Synchronization of complex dynamical networks with discontinuous coupling signals. Nonlinear Dyn 79:1353–1362

    Article  MathSciNet  MATH  Google Scholar 

  49. Fridman E, Seuret A, Richard JP (2004) Robust sampled-data stabilization of linear systems: an input delay approach. Automatica 40:1441–1446

    Article  MathSciNet  MATH  Google Scholar 

  50. Lu JG (2008) Global exponential stability and periodicity of reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions. Chaos Solitons Fractals 35:116–125

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhang HG, Ma TD, Huang GB, Wang CX (2010) Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control. IEEE Trans Syst Man Cybern B Cybern 40:831–844

    Article  Google Scholar 

  52. Yu W, Cao J, Chen G, Lu J, Han J, Wei W (2009) Local synchronization of a complex network model. IEEE Trans Syst Man Cybern B Cybern 39:230–241

    Article  Google Scholar 

  53. Chen J, Chen X (2001) Special matrices. Tsinghua University Press, Beijing

    Google Scholar 

  54. Tahara S, Fujii T, Yokoyama T (2007) Variable sampling quasi multirate deadbeat control method for single phase PWM inverter in low carrier frequency. In: Power conversion conference (PCC), pp 804–809

  55. Hu B, Michel AN (2000) Stability analysis of digital feedback control systems with time-varying sampling periods. Automatica 36:897–905

    Article  MathSciNet  MATH  Google Scholar 

  56. Sala A (2005) Computer control under time-varying sampling period: an LMI gridding approach. Automatica 41:2077–2082

    Article  MathSciNet  MATH  Google Scholar 

  57. Gao H, Wu J, Shi P (2009) Robust sampled-data \(H_\infty\) control with stochastic sampling. Automatica 45:1729–1736

    Article  MathSciNet  MATH  Google Scholar 

  58. Lee T, Park J, Lee S, Kwon O (2013) Robust synchronization of chaotic systems with randomly occurring uncertainties via stochastic sampled-data control. Int J Control 86:107–119

    Article  MATH  Google Scholar 

  59. Lee T, Park J, Kwon O, Lee S (2013) Stochastic sampled-data control for state estimation of time-varying delayed neural networks. Neural Netw 46:99–108

    Article  MATH  Google Scholar 

  60. Ding S, Wang Z (2015) Stochastic exponential synchronization control of Memristive neural networks with multiple time-varying delays. Neurocomputing 162:16–25

    Article  Google Scholar 

  61. Wang Z, Zhang H, Jiang B (2011) LMI-based approach for global asymptotic stability analysis of recurrent neural networks with various delays and structures. IEEE Trans Neural Netw 22:1032–1045

    Article  Google Scholar 

  62. Wang Z, Liu L, Shan Q-H, Zhang H Stability criteria for recurrent neural networks with time-varying delay based on secondary delay partitioning method. IEEE Trans Neural Netw Learn Syst. doi:10.1109/TNNLS.2014.2387434

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  1. Department of Mathematics, Bharathiar University, Coimbatore, Tamilnadu, 641 046, India

    R. Rakkiyappan & S. Dharani

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  1. R. Rakkiyappan
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  2. S. Dharani
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Correspondence to R. Rakkiyappan.

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Rakkiyappan, R., Dharani, S. Sampled-data synchronization of randomly coupled reaction–diffusion neural networks with Markovian jumping and mixed delays using multiple integral approach. Neural Comput & Applic 28, 449–462 (2017). https://doi.org/10.1007/s00521-015-2079-5

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  • DOI: https://doi.org/10.1007/s00521-015-2079-5

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