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Synchronization Control of Riemann-Liouville Fractional Competitive Network Systems with Time-varying Delay and Different Time Scales

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  • Intelligent Control and Applications
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Abstract

This paper is concerned with a class of Riemann-Liouville fractional-order competitive neural networks with time-varying delay and different time scales. Based on delay-partitioning approach, we construct two suitable Lyapunov functionals including fractional integral terms, respectively, and avoid computing their fractional-order derivatives to derive the synchronization conditions. The sufficient conditions are proposed to ensure the complete synchronization between fractional-order response system and fractional-order derive system. By solving the algebraic equalities or linear matrix inequalities (LMIs), the design of the gain matrix of the linear feedback controller can be realized. An illustrative example is also presented to show the validity and feasibility of the theoretical results.

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References

  1. A. Meyer-Bäese, F. Ohl, and H. Scheich, “Singular perturbation analysis of competitive neural networks with different time-scales,” Neural Computation, vol. 8, pp. 1731–1742, 1996. [click]

    Article  Google Scholar 

  2. X. Lou and B. Cui, “Synchronization of competitive neural networks with different time scales,” Physica A, vol. 380, pp. 563–576, 2007. [click]

    Article  Google Scholar 

  3. H. Gu, “Adaptive synchronization for competitive neural networks with different time scales and stochastic perturbation,” Neurocomputing, vol. 73, pp. 350–356, 2009. [click]

    Article  Google Scholar 

  4. Q. Gan, R. Xu, and X. Kang, “Synchronization of unknown chaotic delayed competitive neural networks with different time scales based on adaptive control and parameter identification,” Nonlinear Dynamics, vol. 67, pp. 1893–1902, 2012. [click]

    Article  MathSciNet  MATH  Google Scholar 

  5. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland, 2006.

    MATH  Google Scholar 

  6. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus, Springer, Dordrecht, 2007.

    Book  MATH  Google Scholar 

  7. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Heidelberg, 2010.

    Book  MATH  Google Scholar 

  8. S. Liu, X. Wu, X. F. Zhou, and W. Jiang, “Asymptotical stability of Riemann-Liouville fractional nonlinear systems,” Nonlinear Dynamics, vol. 86, no. 1, pp. 65–71, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Marir, M. Chadli, and D. Bouagada, “A novel approach of admissibility for singular linear continuous-time fractional-order systems,” International Journal of Control, Automation and Systems, vol. 15, no. 2, pp. 959–964, 2017. [click]

    Article  MATH  Google Scholar 

  10. S. Marir, M. Chadli, and D. Bouagada. “New admissibility conditions for singular linear continuous-time fractionalorder systems,” Journal of the Franklin Institute, vol. 354, no. 2, pp. 752–766, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  11. I. Stamova, “Global Mittag-Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays,” Nonlinear Dynamics, vol. 77, no. 4, pp. 1251–1260, 2014. [click]

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Rakkiyappan, J. Cao, and G. Velmurugan, “Existence and uniform stability analysis of fractional-order complexvalued neural networks with time delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 1, pp. 84–97, 2015. [click]

    Article  MathSciNet  Google Scholar 

  13. R. Rakkiyappan, R. Sivaranjani, G. Velmurugan G, and J. Cao, “Analysis of global o(t -α) stability and global asymptotical periodicity for a class of fractional-order complexvalued neural networks with time varying delays,” Neural Networks, vol. 77, pp. 51–69, 2016. [click]

    Article  Google Scholar 

  14. H. Zhang, R. Ye, J. Cao, and A. Alsaedi, “Existence and globally asymptotic stability of equilibrium solution for fractional-order hybrid BAM neural networks with distributed delays and impulses,” Complexity, vol. 2017, pp. 1–13, 2017.

    MathSciNet  MATH  Google Scholar 

  15. H. Zhang, R. Ye, J. Cao, and A. Alsaedi, “Delayindependent stability of Riemann-Liouville fractional neutral-type delayed neural networks,” Neural Processing Letters, doi:10.1007/s11063-017-9658-7, 2017.

    Google Scholar 

  16. F. Wang, Y. Yang, M. Hu, and X. Xu, “Projective cluster synchronization of fractional-order coupled-delay complex network via adaptive pinning control,” Physica A, vol. 434, pp. 134–143, 2015. [click]

    Article  MathSciNet  Google Scholar 

  17. Q. Song, X. Yang, C. Li, T. Huang, and X. Chen, “Stability analysis of nonlinear fractional-order systems with variable-time impulses,” Journal of the Franklin Institute, vol. 354, vo. 7, pp. 2959–2978, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. B. Stojanovic, D. L. Debeljkovic, and M. A. Misic, “Finite-time stability for a linear discrete-time delay systems by using discrete convolution: An LMI approach,” International Journal of Control, Automation and Systems, vol. 14, no. 4, pp. 1144–1151, 2016.

    Article  Google Scholar 

  19. W. Gong, J. Liang, C. Zhang, and J. Cao, “Nonlinear measure approach for the stability analysis of complex-valued neural networks,” Neural Processing Letters, vol. 44, no. 2, pp. 539–554, 2016. [click]

    Article  Google Scholar 

  20. R. Samidurai, S. Rajavel, J. Cao, A. Alsaedi, F. Alsaadi, and B. Ahmad, “Delay-partitioning approach to stability analysis of state estimation for neutral-type neural networks with both time-varying delays and leakage term via sampled-data control,” International Journal of Systems Science, vol. 48, no. 8, pp. 1752–1765, 2017. [click]

    Article  MathSciNet  MATH  Google Scholar 

  21. X. Yang and J. Cao, “Finite-time stochastic synchronization of complex networks,” Applied Mathematical Modelling, vol. 34, no. 11, pp. 3631–3641, 2010. [click]

    Article  MathSciNet  MATH  Google Scholar 

  22. X. Yang, J. Cao, and J. Liang, “Exponential synchronization of memristive neural networks with delays: interval matrix method,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 8, pp. 1878–1888, 2017. [click]

    Article  MathSciNet  Google Scholar 

  23. E. Wu and X. Yang, “Adaptive synchronization of coupled nonidentical chaotic systems with complex variables and stochastic perturbations” Nonlinear Dynamics, vol. 84, no. 1, pp. 261–269, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  24. X. Yang, Z. Feng, J. Feng, and J. Cao, “Synchronization of discrete-time neural networks with delays and Markov jump topologies based on tracker information,” Neural Networks, vol. 85, pp. 157–164, 2017. [click]

    Article  Google Scholar 

  25. H. Bao and J. Cao, “Finite-time generalized synchronization of nonidentical delayed chaotic systems,” Nonlinear Analysis: Modelling and Control, vol. 21, no. pp. 306–324, 2016. [click]

    MathSciNet  Google Scholar 

  26. X. Yang and J. Lu, “Finite-time synchronization of coupled networks with Markovian topology and impulsive effects,” IEEE Transactions on Automatic Control, vol. 61, no. 8, pp. 2256–2261, 2016. [click]

    Article  MathSciNet  MATH  Google Scholar 

  27. L. Shi, X. Yang, Y. Li, and Z. Feng, “Finite-time synchronization of nonidentical chaotic systems with multiple time-varying delays and bounded perturbations,” Nonlinear Dynamics, vol. 83, no. 1–2, pp. 75–87, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Wu, J. Cao, Q. Li, A. Alsaedi, and F. E. Alsaadi, “Finitetime synchronization of uncertain coupled switched neural networks under asynchronous switching,” Neural Networks, vol. 85, pp. 128–139, 2017. [click]

    Article  Google Scholar 

  29. J. Cao and R. Li, “Fixed-time synchronization of delayed memristor-based recurrent neural networks,” Science China Information Sciences, vol. 60, no. 3, 032201, 2017.

    Article  MathSciNet  Google Scholar 

  30. X. Ding, J. Cao, A. Alsaedi, F. E. Alsaadi, and T. Hayat, “Robust fixed-time synchronization for uncertain complexvalued neural networks with discontinuous activation functions,” Neural Networks, vol. 90, pp. 42–55, 2017. [click]

    Article  Google Scholar 

  31. H. R. Karimi and P. Maass, “Delay-range-dependent exponential H∞ synchronization of a class of delayed neural networks,” Chaos, Solitons and Fractals, vol. 41, no. 3, pp. 1125–1135, 2009. [click]

    Article  MathSciNet  MATH  Google Scholar 

  32. 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, vol. 40, no. 1, pp. 173–185, 2010. [click]

    Article  Google Scholar 

  33. H. R. Karimi, “A sliding mode approach to H∞ synchronization of master-slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties,” Journal of the Franklin Institute, vol. 349, no. 4, pp. 1480–1496, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  34. Y. Zhao, B. Li, J. Qin, H. Gao, and H. R. Karimi, “H∞ consensus and synchronization of nonlinear systems based on a novel fuzzy model,” IEEE Transactions on Cybernetics, vol. 43, no. 6, pp. 2157–2169, 2013. [click]

    Article  Google Scholar 

  35. W. He, F. Qian, and J. Cao, “Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control,” Neural Networks, vol. 85, pp. 1–9, 2017. [click]

    Article  Google Scholar 

  36. H. Bao, J. H. Park, and J. Cao, ”Synchronization of fractional-order complex-valued neural networks with time delay,” Neural Networks, vol. 81, pp. 16–28, 2016. [click]

    Article  Google Scholar 

  37. Y. Gu, Y. Yu, and H. Wang, “Synchronization for fractional-order time-delayed memristor-based neural networks with parameter uncertainty,” Journal of the Franklin Institute, vol. 353, no. 15, pp. 3657–3684, 2016. [click]

    Article  MathSciNet  MATH  Google Scholar 

  38. M. Zarefard and S. Effati, “Adaptive synchronization between two non-identical BAM neural networks with unknown parameters and time-varying delays,” International Journal of Control Automation and Systems, vol. 15, no. 4, pp. 1877–1887, 2017. [click]

    Article  Google Scholar 

  39. X. Yang, J. Lam, D. W. C. Ho, and Z. Feng, “Fixedtime synchronization of complex networks with impulsive effects via non-chattering control,” IEEE Transactions on Automatic Control, vol. 62, no. 11, pp. 5511–5521, 2017.

    Article  MathSciNet  Google Scholar 

  40. Q. Song, J. Cao, and F. Liu, “Pinning-controlled synchronization of hybrid-coupled complex dynamical networks with mixed time-delays,” International Journal of Robust and Nonlinear Control, vol. 22, pp. 690–706, 2012. [click]

    Article  MathSciNet  MATH  Google Scholar 

  41. X. Yang,and D. W. C. Ho, “Synchronization of delayed memristive neural networks: Robust analysis approach,” IEEE Transactions on Cybernetics, vol. 46, no. 12, pp. 3377–3387, 2016. [click]

    Article  Google Scholar 

  42. H. Zhang, J. Cao, and W. Jiang, “Controllability criteria for linear fractional differential systems with state delay and impulses,” Journal of Applied Mathematics, vol. 2013, pp. 1–9, 2013.

    MathSciNet  MATH  Google Scholar 

  43. S. Mou, H. Gao, W. Qiang, and K. Chen, “New delaydependent exponential stability for neural networks with time delay,” IEEE Transactions on Systems Man and Cybernetics B, vol. 38, pp. 535–571, 2008.

    Article  Google Scholar 

  44. Z. Li, H. Liu, and H. Gao, “A delay partitioning approach to H∞ filtering for continuous time-delay systems, Circuits, Systems, and Signal Processing, vol. 30, pp. 501–513, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  45. Y. Wei, M. Wang, and J. Qiu, “New approach to delay-dependent H ∞ filtering for discrete-time Markovian jump systems with time-varying delay and incomplete transition descriptions,” IET Control Theory and Applications, vol. 7, no. 5, pp. 684–696, 2013.

    Article  MathSciNet  Google Scholar 

  46. Y. Wei, J. Qiu, H. R. Karimi, and M. Wang, “New results on H ∞ dynamic output feedback control for Markovian jump systems with time-varying delay and defective mode information,” Optimal Control Applications and Methods, vol. 35, no. 6, pp. 656–675, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  47. Y. Wei, J. Qiu, and H. R. Karimi, “Fuzzy-affine-modelbased memory filter design of nonlinear systems with time-varying delay,” IEEE Transactions on Fuzzy Systems, doi:10.1109/TFUZZ.2017.2686352, 2017.

    Google Scholar 

  48. Q. Gan, “Synchronization of competitive neural networks with different time scales and time-varying delay based on delay partitioning approach,” International Journal of Machine Learning and Cybernetics, vol. 4, no. 4, pp. 327–337, 2013.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics and Computation Science, Anqing Normal University, Anqing, 246133, China

    Hai Zhang & Miaolin Ye

  2. School of Mathematics, and Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing, 210096, China

    Jinde Cao

  3. School of Electrical Engineering, Nantong University, Nantong, 226000, China

    Jinde Cao

  4. School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, China

    Jinde Cao

  5. Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Ahmed Alsaedi

Authors
  1. Hai Zhang
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  2. Miaolin Ye
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  3. Jinde Cao
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  4. Ahmed Alsaedi
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Corresponding author

Correspondence to Hai Zhang.

Additional information

Recommended by Editor Hamid Reza Karimi. This work is jointly supported by National Natural Science Fund of China (11301308, 61573096, 61272530), the 333 Engineering Fund of Jiangsu Province of China (BRA2015286), the Natural Science Fund of Anhui Province of China (1608085MA14), the Key Project of Natural Science Research of Anhui Higher Education Institutions of China (gxyqZD2016205, KJ2015A152), the Natural Science Youth Fund of Jiangsu Province of China (BK20160660), and the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence (BM2017002).

Hai Zhang is currently a Professor with the School of Mathematics and Computation Science, Anqing Normal University, China. He received the M.Sc. and Ph.D. degrees from Anhui University, China, in 2007 and 2010, respectively. From December 2012 to November 2014, he was a Postdoctoral Research Fellow at the Department of Mathematics, Southeast University, China. His current research interests include nonlinear systems, fractional differential equations, neural networks, control theory and stability theory.

Miaolin Ye is currently a Professor in Anqing Normal University, China. From September 1985 to July 1986, he was a Visiting Scholar with the Department of Mathematics, Huazhong University of Science and Technology, China. From September 1995 to July 1996, he was an Academic Scholar with the Department of Mathematics, Nanjing University, China. He received the Ph.D. degree from Anhui University, China, in 2010. His current research interests include graph theory, complex networks and network optimization.

Jinde Cao is a Distinguished Professor, the Dean of School of Mathematics and the Director of the Research Center for Complex Systems and Network Sciences at Southeast University. From March 1989 to May 2000, he was with the Yunnan University. In May 2000, he joined the Department of Mathematics, Southeast University, Nanjing, China. From July 2001 to June 2002, he was a Postdoctoral Research Fellow at the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Hong Kong. He is an Associate Editor of the IEEE Transactions on Cybernetics, IEEE Transactions on Cognitive and Developmental Systems, Mathematics and Computers in Simulation, and Neural Networks. He is a Fellow of IEEE, and a Member of the Academy of Europe. He has been named as Highly-Cited Researcher in Mathematics, Computer Science and Engineering by Thomson Reuters. He received the National Award for Excellence in Innovation (2017).

Ahmed Alsaedi is currently a Professor in King Abdulaziz University, Saudi Arabia. He obtained his Ph.D. degree from Swansea University (UK) in 2002. His fields of interest include dynamical systems, nonlinear analysis. He has published several articles in peer-reviewed journals. He served as the chairman of the mathematics department at KAU and presently he is serving as director of the research program at KAU. Under his great leadership, this program is running quite successfully and it has attracted a large number of highly rated researchers and distinguished professors from all over the world. He is also the head of NAAM international research group at KAU.

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Zhang, H., Ye, M., Cao, J. et al. Synchronization Control of Riemann-Liouville Fractional Competitive Network Systems with Time-varying Delay and Different Time Scales. Int. J. Control Autom. Syst. 16, 1404–1414 (2018). https://doi.org/10.1007/s12555-017-0371-0

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