Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2018, Article ID 1603629, 8 pages
https://doi.org/10.1155/2018/1603629
Research Article

Adaptive Synchronization for Uncertain Delayed Fractional-Order Hopfield Neural Networks via Fractional-Order Sliding Mode Control

1College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China
2College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Bo Meng; nc.ude.tsuds@2290bm

Received 17 May 2018; Accepted 10 July 2018; Published 18 July 2018

Academic Editor: Xue-Jun Xie

Copyright © 2018 Bo Meng and Xiaohong Wang. 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.

Linked References

  1. G. Velmurugan, R. Rakkiyappan, V. Vembarasan, J. Cao, and A. Alsaedi, “Dissipativity and stability analysis of fractional-order complex-valued neural networks with time delay,” Neural Networks, vol. 86, pp. 42–53, 2017. View at Publisher · View at Google Scholar · View at Scopus
  2. I. Aizenberg, “Complex-valued neural networks with multi-valued neurons,” Springer, pp. 39–62, 2011. View at Google Scholar
  3. R. Rakkiyappan, G. Velmurugan, and J. Cao, “Multiple μ-stability analysis of complex-valued neural networks with unbounded time-varying delays,” Neurocomputing, vol. 149, pp. 594–607, 2015. View at Publisher · View at Google Scholar · View at Scopus
  4. R. Guo, Z. Zhang, X. Liu, and C. Lin, “Existence, uniqueness, and exponential stability analysis for complex-valued memristor-based BAM neural networks with time delays,” Applied Mathematics and Computation, vol. 311, pp. 100–117, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  5. W. Lv and F. Wang, “Adaptive tracking control for a class of uncertain nonlinear systems with infinite number of actuator failures using neural networks,” Advances in Difference Equations, Paper No. 374, 16 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  6. L. Li, Z. Wang, Y. Li, H. Shen, and J. Lu, “Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays,” Applied Mathematics and Computation, vol. 330, pp. 152–169, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  7. I. Podlubny, “Fractional differential equations,” Mathematics in Science and Engineering, 1998. View at Google Scholar · View at MathSciNet
  8. Y. Cui, “Uniqueness of solution for boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 51, pp. 48–54, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Z. Bai, X. Dong, and C. Yin, “Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions,” Boundary Value Problems, vol. 1, pp. 63–71, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Z. Bai, Y. Chen, H. Lian, and S. Sun, “On the existence of blow up solutions for a class of fractional differential equations,” Fractional Calculus and Applied Analysis, vol. 17, no. 4, pp. 1175–1187, 2014. View at Publisher · View at Google Scholar · View at Scopus
  11. R. Pu, X. Zhang, Y. Cui, P. Li, and W. Wang, “Positive solutions for singular semipositone fractional differential equation subject to multipoint boundary conditions,” Journal of Function Spaces, vol. 2017, pp. 1–7, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  12. Y. Zou and G. He, “On the uniqueness of solutions for a class of fractional differential equations,” Applied Mathematics Letters, vol. 74, pp. 68–73, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  13. Y. Cu, W. Ma, Q. Sun, and X. Su, “New uniqueness results for boundary value problem of fractional differential equation,” Nonlinear Analysis: Modelling and Control, pp. 31–39, 2018. View at Publisher · View at Google Scholar
  14. J. Sabatier, O. P. Agrawal, and A. J. T. Machado, Advances in fractional calculus, Springer Netherlands, 2007.
  15. Z. Wang, “A numerical method for delayed fractional-order differential equations,” Journal of Applied Mathematics, vol. 2013, Article ID 256071, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  16. Z. Wang, X. Huang, and J. Zhou, “A numerical method for delayed fractional-order differential equations: based on G-L definition,” Applied Mathematics & Information Sciences, vol. 7, no. 2, pp. 525–529, 2013. View at Google Scholar
  17. B. Xin and Y. Li, “0-1 test for chaos in a fractional order financial system with investment incentive,” Abstract and Applied Analysis, Art. ID 876298, 10 pages, 2013. View at Google Scholar · View at MathSciNet
  18. Z. Wang, X. Wang, Y. Li, and X. Huang, “Stability and Hopf bifurcation of fractional-order complex-valued single neuron model with time delay,” International Journal of Bifurcation and Chaos, vol. 27, no. 13, 1750209, 13 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Z. Wang, X. Huang, and G. D. Shi, “Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1531–1539, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. I. Petras, “Modeling and numerical analysis of fractional-order bloch equations,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 341–356, 2011. View at Publisher · View at Google Scholar
  21. N. Aguila-Camacho, M. A. Duarte-Mermoud, and J. A. Gallegos, “Lyapunov functions for fractional order systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 9, pp. 2951–2957, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. Y. Xi, Y. Yu, S. Zhang, and X. Hai, “Finite-time robust control of uncertain fractional-order Hopfield neural networks via sliding mode control,” Chinese Physics B, vol. 27, no. 1, 010202, 2018. View at Publisher · View at Google Scholar
  23. L. Chen, Y. Chai, R. Wu, T. Ma, and H. Zhai, “Dynamic analysis of a class of fractional-order neural networks with delay,” Neurocomputing, vol. 111, pp. 190–194, 2013. View at Publisher · View at Google Scholar · View at Scopus
  24. H. Liu, S. Li, H. Wang, Y. Huo, and J. Luo, “Adaptive synchronization for a class of uncertain fractional-order neural networks,” Entropy, vol. 17, no. 10, pp. 7185–7200, 2015. View at Publisher · View at Google Scholar
  25. G. A. Anastassiou, “Fractional neural network approximation,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1655–1676, 2012. View at Publisher · View at Google Scholar · View at Scopus
  26. H. Wang, Y. Yu, G. Wen, and S. Zhang, “Stability analysis of fractional-order neural networks with time delay,” Neural Processing Letters, vol. 42, no. 2, pp. 479–500, 2015. View at Publisher · View at Google Scholar · View at Scopus
  27. Y. Fan, X. Huang, Z. Wang, and Y. Li, “Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with discontinuous memductance function,” Nonlinear Dynamics, vol. 93, no. 2, pp. 611–627, 2018. View at Publisher · View at Google Scholar
  28. C. Song and J. Cao, “Dynamics in fractional-order neural networks,” Neurocomputing, vol. 142, pp. 494–498, 2014. View at Publisher · View at Google Scholar · View at Scopus
  29. J. Zhou, C. Sang, X. Li, M. Fang, and Z. Wang, “H∞ consensus for nonlinear stochastic multi-agent systems with time delay,” Applied Mathematics and Computation, vol. 325, pp. 41–58, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  30. J. Wang, K. Liang, X. Huang, Z. Wang, and H. Shen, “Dissipative fault-tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback,” Applied Mathematics and Computation, vol. 328, pp. 247–262, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  31. H. Gao, J. Xia, G. Zhuang, Z. Wang, and Q. Sun, “Nonfragile finite-time extended dissipative control for a class of uncertain switched neutral systems,” Complexity, vol. 2017, pp. 1–22, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  32. Y. Li, Y. Chen, and I. Podlubny, “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1810–1821, 2010. View at Publisher · View at Google Scholar · View at Scopus
  33. Y. Li, Y. Chen, and I. Podlubny, “Mittag-Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. H. Ma and Y. Jia, “Stability analysis for stochastic differential equations with infinite Markovian switchings,” Journal of Mathematical Analysis and Applications, vol. 435, no. 1, pp. 593–605, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. Z. Yan and W. Zhang, “Finite-time stability and stabilization of Itô-type stochastic singular systems,” Abstract and Applied Analysis, vol. 2014, pp. 1–10, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  36. Y. Li, W. Zhang, and X. Liu, “Stability of nonlinear stochastic discrete-time systems,” Journal of Applied Mathematics, vol. 2013, Article ID 356746, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  37. Y. Li, Y. Sun, and F. Meng, “New criteria for exponential stability of switched time-varying systems with delays and nonlinear disturbances,” Nonlinear Analysis: Hybrid Systems, vol. 26, pp. 284–291, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. Y. Lin and W. Zhang, “Necessary/sufficient conditions for Pareto optimum in cooperative difference game,” Optimal Control Applications and Methods, vol. 39, no. 2, pp. 1043–1060, 2018. View at Publisher · View at Google Scholar
  39. Y. Lin, T. Zhang, and W. Zhang, “Pareto-based guaranteed cost control of the uncertain mean-field stochastic systems in infinite horizon,” Automatica, vol. 92, pp. 197–209, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  40. L. Yao and W. Zhang, “Adaptive tracking control for a class of random pure-feedback nonlinear systems with Markovian switching,” International Journal of Robust and Nonlinear Control, vol. 28, no. 8, pp. 3112–3126, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  41. W. Zhang, Y. Lin, and L. Xue, “Linear quadratic Pareto optimal control problem of stochastic singular systems,” Journal of The Franklin Institute, vol. 354, no. 2, pp. 1220–1238, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  42. V. I. Utkin and A. S. Poznyak, “Adaptive sliding mode control, advances in sliding mode control,” Springer Berlin Heidelberg, pp. 21–53, 2013. View at Google Scholar
  43. M. Ö. Efe, “Fractional fuzzy adaptive sliding-mode control of a 2-DOF direct-drive robot arm,” IEEE Transactions on Systems Man and Cybernetics Part B Cybernetics A Publication of the IEEE Systems Man and Cybernetics Society, vol. 38, no. 6, pp. 1561–1570, 2008. View at Publisher · View at Google Scholar · View at Scopus
  44. M. P. Aghababa, “Design of a chatter-free terminal sliding mode controller for nonlinear fractional-order dynamical systems,” International Journal of Control, vol. 86, no. 10, pp. 1744–1756, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. K. Liang, M. Dai, H. Shen, J. Wang, Z. Wang, and B. Chen, “L2-L∞ synchronization for singularly perturbed complex networks with semi-Markov jump topology,” Applied Mathematics and Computation, vol. 321, pp. 450–462, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  46. Z. Zhang, H. Shao, Z. Wang, and H. Shen, “Reduced-order observer design for the synchronization of the generalized Lorenz chaotic systems,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7614–7621, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  47. X. Huang, Y. Fan, J. Jia, Z. Wang, and Y. Li, “Quasi-synchronisation of fractional-order memristor-based neural networks with parameter mismatches,” IET Control Theory & Applications, vol. 11, no. 14, pp. 2317–2327, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  48. J. Chen, Z. Zeng, and P. Jiang, “Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks,” Neural Networks, vol. 51, pp. 1–8, 2014. View at Publisher · View at Google Scholar · View at Scopus
  49. A. Wu and Z. Zeng, “Global Mittag–Leffler stabilization of fractional-order memristive neural networks,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 1, pp. 206–217, 2017. View at Publisher · View at Google Scholar