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Exponential synchronization of discontinuous neural networks with time-varying mixed delays via state feedback and impulsive control

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Abstract

This paper investigates drive-response synchronization for a class of neural networks with time-varying discrete and distributed delays (mixed delays) as well as discontinuous activations. Strict mathematical proof shows the global existence of Filippov solutions to neural networks with discontinuous activation functions and the mixed delays. State feedback controller and impulsive controller are designed respectively to guarantee global exponential synchronization of the neural networks. By using Lyapunov function and new analysis techniques, several new synchronization criteria are obtained. Moreover, lower bound on the convergence rate is explicitly estimated when state feedback controller is utilized. Results of this paper are new and some existing ones are extended and improved. Finally, numerical simulations are given to verify the effectiveness of the theoretical results.

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References

  • Aubin J, Cellina A (1984) Differential inclusions: set-valued maps and viability theory. Springer, New York

    Book  Google Scholar 

  • Balasubramaniam P, Ntouyas SK, Vinayagam D (2005) Existence of solutions of semilinear stochastic delay evolution inclusions in a Hilbert space. J Math Anal Appl 305(2):438–451

    Article  Google Scholar 

  • Benchohra M, Ntouyas SK (2000) Existence of mild solutions of semilinear evolution inclusions with nonlocal conditions. Georgian Math J 7(2):221–230

    Google Scholar 

  • Cai Z, Huang L (2011) Existence and global asymptotic stability of periodic solution for discrete and distributed time-varying delayed neural networks with discontinuous activations. Neurocomputing 74:3170–3179

    Article  Google Scholar 

  • Cao J, Alofi A, Al-Mazrooei A, Elaiw A (2013) Synchronization of switched interval networks and applications to chaotic neural networks. Abstr Appl Anal. Article ID 940573, 11 p

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

    Article  PubMed  Google Scholar 

  • Cao J, Wang Z, Sun Y (2007) Synchronization in an array of linearly stochastically coupled networks with time delays. Physica A 385(2):718–728

    Article  Google Scholar 

  • Clarke F (1983) Optimization and nonsmooth analysis. Wiley, New York

    Google Scholar 

  • Danca M (2004) Controlling chaos in discontinuous dynamical systems. Chaos Solitons Fractals 22:605–612

    Article  Google Scholar 

  • Di Marco M, Forti M, Grazzini M, Pancioni L (2005) Fourth-order nearly-symmetric cnns exhibiting complex dynamics. Int J Bifurc Chaos 15(5):1579–1587

    Article  Google Scholar 

  • Di Marco M, Forti M, Grazzini M, Pancioni L (2010) Limit set dichotomy and convergence of semiflows defined by cooperative standard cnns. Int J Bifurc Chaos 20(11):3549–3563

    Article  Google Scholar 

  • Filippov A (1960) Differential equations with discontinuous right-hand side. Matematicheskii Sb 93(1):99–128

    Google Scholar 

  • Forti M, Nistri P (2003) Global convergence of neural networks with discontinuous neuron activations. IEEE Trans Circuts Syst I 50(11):1421–1435

    Article  Google Scholar 

  • Forti M, Grazzini M, Nistri P, Pancioni L (2006) Generalized lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations. Physica D 214(1):88–99

    Article  Google Scholar 

  • Forti M, Nistri P, Papini D (2005) Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain. IEEE Trans Neural Netw 16(6):1449–1463

    Article  PubMed  Google Scholar 

  • Haddad G (1981) Monotone viable trajectories for functional differential inclusions. J Differ Eq 42(1):1–24

    Article  Google Scholar 

  • Kamel MS, Xia Y (2009) Cooperative recurrent modular neural networks for constrained optimization: a survey of models and applications. Cogn Neurodyn 3(1):47–81

    Article  PubMed Central  PubMed  Google Scholar 

  • Li JH, Michel AN, Porod W (1989) Analysis and synthesis of a class of neural networks: variable structure systems with infinite grain. IEEE Trans Circuts Syst 36(5):713–731

    Article  Google Scholar 

  • Li Y, Liu Z, Luo J, Wu H (2013) Coupling-induced synchronization in multicellular circadian oscillators of mammals. Cogn Neurodyn 7(1):59–65

    Article  PubMed Central  PubMed  Google Scholar 

  • Liao CW, Lu CY (2011) Design of delay-dependent state estimator for discrete-time recurrent neural networks with interval discrete and infinite-distributed time-varying delays. Cogn Neurodyn 5(2):133–143

    Article  PubMed Central  PubMed  Google Scholar 

  • Liao T, Huang NS (1999) An observer-based approach for chaotic synchronization with applications to secure communications. IEEE Trans Circuts Syst I 46(9):1144–1150

    Article  Google Scholar 

  • Liu J, Liu X, Xie W (2012) Global convergence of neural networks with mixed time-varying delays and discontinuous neuron activations. Inf Sci 183:92–105

    Article  Google Scholar 

  • Liu X, Park JH, Jiang N, Cao J (2014) Nonsmooth finite-time stabilization of neural networks with discontinuous activations. Neural Netw 52:25–32

    Article  PubMed  Google Scholar 

  • Liu X, Chen T, Cao J, Lu W (2011) Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches. Neural Netw 24(10):1013–1021

    Article  PubMed  Google Scholar 

  • Lu W, Chen T (2008) Almost periodic dynamics of a class of delayed neural networks with discontinuous activations. Neural Comput 20(4):1065–1090

    Article  PubMed  Google Scholar 

  • Lu W, Chen T (2006) Dynamical behaviors of delayed neural network systems with discontinuous activation functions. Neural Comput 18(3):683–708

    Article  Google Scholar 

  • Martelli M (1975) A Rothe’s type theorem for non compact acyclic-valued maps. Boll Unione Mat Ital 4(3):70–76

    Google Scholar 

  • Paden B, Sastry S (1987) A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulators. IEEE Trans Circuts Syst 34(1):73–82

    Article  Google Scholar 

  • Pecora L, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64(8):821–824

    Article  PubMed  Google Scholar 

  • Rigatos G (2014) Robust synchronization of coupled neural oscillators using the derivative-free nonlinear kalman filter. Cogn Neurodyn. doi:10.1007/s11571-014-9299-8

  • Tank D, Hopfield JJ (1987) Neural computation by concentrating information in time. Proc Natl Acad Sci 84(7):1896–1900

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Wang J, Huang L, Guo Z (2009) Global asymptotic stability of neural networks with discontinuous activations. Neural Netw 22(7):931–937

    Article  CAS  PubMed  Google Scholar 

  • Wang T, Xie L, de Souza CE (1992) Robust control of a class of uncertain nonlinear systems. Syst Control Lett 19(2):139–149

    Article  Google Scholar 

  • Wang Y, Wang Z, Liang J, Li Y, Du M (2010) Synchronization of stochastic genetic oscillator networks with time delays and Markovian jumping parameters. Neurocomputing 73(13–15):2532–2539

    Article  Google Scholar 

  • Xu A, Du Y, Wang R (2014) Interaction between different cells in olfactory bulb and synchronous kinematic analysis. Discrete Dyn Nat Soc. Artical ID 808792

  • Yan C, Wang R (2014) Asymmetric neural network synchronization and dynamics based on an adaptive learning rule of synapses. Neurocomputing 125:41–45

    Article  Google Scholar 

  • Yang X, Cao J (2010) Finite-time stochastic synchronization of complex networks. Appl Math Model 34(11):3631–3641

    Article  Google Scholar 

  • Yang X, Cao J (2013) Exponential synchronization of delayed neural networks with discontinuous activations. IEEE Trans Circuts Syst I 60(9):2431–2439

    Google Scholar 

  • Yang X, Huang C, Zhu Q (2011a) Synchronization of switched neural networks with mixed delays via impulsive control. Chaos Solitons Fractals 44(10):817–826

    Article  Google Scholar 

  • Yang X, Cao J, Lu J (2011b) Synchronization of delayed complex dynamical networks with impulsive and stochastic effects. Nonlinear Anal Real World Appl 12:2252–2266

    Article  Google Scholar 

  • Yang X, Cao J, Lu J (2013) Synchronization of coupled neural networks with random coupling strengths and mixed probabilistic time-varying delays. Int J Robust Nonlinear Control 23(18):2060–2081

    Article  Google Scholar 

  • Yang X, Cao J, Yu W (2014) Exponential synchronization of memristive Cohen–Grossberg neural networks with mixed delays. Cogn Neurodyn 8(3):239–249

    Article  PubMed  Google Scholar 

  • Zhang Z, Cao J, Zhou D (2014) Novel lmi-based condition on global asymptotic stability for a class of Cohen–Grossberg BAM networks with extended activation functions. IEEE Trans Neural Netw Lear Syst 25(6):1161–1172

    Article  Google Scholar 

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Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China (NSFC) under Grants Nos. 61263020, 11101053, 61272530, and 11072059, and CityU Grants 7008188 and 7002868, and the Program of Chongqing Innovation Team Project in University under Grant No. KJTD201308, the Natural Science Foundation of Jiangsu Province of China under Grant BK2012741.

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Authors and Affiliations

  1. Department of Mathematics, Chongqing Normal University, Chongqing, 401331, China

    Xinsong Yang

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Jinde Cao

  3. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Jinde Cao

  4. Department of Mathematics, City University of Hong Kong, Hong Kong, China

    Daniel W. C. Ho

  5. School of Automation, Nanjing University of Science and Technology, Nanjing, 210096, China

    Daniel W. C. Ho

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  1. Xinsong Yang
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  2. Jinde Cao
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  3. Daniel W. C. Ho
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Correspondence to Jinde Cao.

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Yang, X., Cao, J. & Ho, D.W.C. Exponential synchronization of discontinuous neural networks with time-varying mixed delays via state feedback and impulsive control. Cogn Neurodyn 9, 113–128 (2015). https://doi.org/10.1007/s11571-014-9307-z

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  • DOI: https://doi.org/10.1007/s11571-014-9307-z

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