Kybernetika 53 no. 3, 513-529, 2017

Stability analysis for neutral-type impulsive neural networks with delays

Bo Du, Yurong Liu and Dan CaoDOI: 10.14736/kyb-2017-3-0513

Abstract:

By using linear matrix inequality (LMI) approach and Lyapunov functional method, we obtain some new sufficient conditions ensuring global asymptotic stability and global exponential stability of a generalized neutral-type impulsive neural networks with delays. A simulation example is provided to demonstrate the usefulness of the main results obtained. The main contribution in this paper is that a new neutral-type impulsive neural networks with variable delays is studied by constructing a novel Lyapunov functional and LMI approach.

Keywords:

stability, neural networks, neutral-type, Lyapunov functional method

Classification:

34G20, 35B40

References:

  1. A. Berman and R. J. Plemmons: Nonnegative Matrices in Mathematical Sciences. Academic Press, New York 1979.   CrossRef
  2. A. Bouzerdoum and T. R. Pattison: Neural networks for quadratic optimization with bound constraints. IEEE Trans. Neural Networks 4 (1993), 293-303.   DOI:10.1109/72.207617
  3. P. P. Civalleri, M. Gilli and L. Pandolfi: On stability of cellular neural networks with delay. IEEE Trans. Circuits Syst. I 40 (1993), 157-165.   DOI:10.1109/81.222796
  4. C. Cheng, T. Liao, J. Yan and C. Hwang: Globally asymptotic stability of a class of neutral-type neural networks with delays. IEEE Trans. Syst. Man Cybern. 36 (2006), 1191-1195.   DOI:10.1109/tsmcb.2006.874677
  5. L. Cheng, Z. Hou and M. Tan: A neutral-type delayed projection neural network for solving nonlinear variational inequalities. IEEE Trans. Circuits Syst. II-Express Brief 55 (2008), 806-810.   DOI:10.1109/tcsii.2008.922472
  6. L. O. Chua and L. Yang: Cellular neural networks: applications. IEEE Trans. Circuits Syst. 35 (1988), 1273-1290.   DOI:10.1109/31.7601
  7. Z. Gui, W. Ge and X. Yang: Periodic oscillation for a Hopfield neural networks with neutral delays. Phys. Lett. A 364 (2007), 267-273.   DOI:10.1016/j.physleta.2006.12.013
  8. Z. Guan, G. Chen and Y. Qin: On equilibria, stability and instability of Hopfield neural networks. IEEE Trans. Neural Networks 2 (2000), 534-540.   DOI:10.1109/72.839023
  9. J. Hale: Theory of Functional Differential Equations. Applied Mathematical Springer-Verlag, New York 1977.   DOI:10.1007/978-1-4612-9892-2
  10. X.-M. Hang and Q.-L. Han: Event-based H$_\infty$ filtering for sampled-data systems. Automatica 51 (2015), 55-69.   CrossRef
  11. W. He, G. Chen, Q.-L. Han and F. Qian: Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control. Inform. Sci. 20 (2017), 145-158.   DOI:10.1016/j.ins.2015.06.005
  12. W. He, F. Qian and J. Cao: Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control. Neural Networks 85 (2017), 1-9.   DOI:10.1016/j.neunet.2016.09.002
  13. W. He, F. Qian, J. Lam, G. Chen, Q.-L. Han and J. Kurths: Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control: Error estimation, optimization and design. Automatica 62 (2015), 249-262.   DOI:10.1016/j.automatica.2015.09.028
  14. D. H. Ji, J. H. Koo, S. C. Won, S. M. Lee and J. H. Park: Passivity-based control for Hopfield neural networks using convex representation. Appl. Math. Comput. 217 (2011), 6168-6175.   DOI:10.1016/j.amc.2010.12.100
  15. S. K. Kaul and X. Z. Liu: Vector Lyapunov functions for impulsive differential systems with variable times. Dyn. Continuous Discrete Impulsive Systems 6 (1999), 25-38.   CrossRef
  16. M. P. Kennedy and L. O. Chua: Neural networks for non-linear programming. IEEE Trans. Circuits. Syst. 35 (1988), 554-562.   DOI:10.1109/31.1783
  17. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov: Theory of Impulsive Differential Equations. World Scientific, Singapore 1989.   DOI:10.1142/0906
  18. V. Lakshmikantham, S. Leela and S. K. Kaul: Comparison principle for impulsive differential equations with variable times and stability theory. Nonlinear Anal. 22 (1994), 499-503.   DOI:10.1016/0362-546x(94)90170-8
  19. V. Lakshmikantham, N. S. Papageorgiou and J. Vasundhara: The method of upper and lower solutions and monotone technique for impulsive differential equations with variable moments. Appl. Anal. 15 (1993), 41-58.   DOI:10.1080/00036819308840203
  20. S. Lakshmanan, J. H. Park, H. Y. Jung and P. Balasubramaniam: Design of state estimator for neural networks with leakage, discrete and distributed delays. Appl. Math. Comput. 218 (2012), 11297-11310.   DOI:10.1016/j.amc.2012.05.022
  21. T. Li, W. Zheng and C. Lin: Delay-slope dependent stability results of recurrent neural networks. IEEE Trans. Neural Networks 22 (2011), 2138-2143.   DOI:10.1109/tnn.2011.2169425
  22. C. Lien, K. Yu, Y. Lin, Y. Chung and L. Chung: Exponential convergence rate estimation for uncertain delayed neural networks of neutral type. Chao. Solit. Fract. 40 (2009), 2491-2499.   DOI:10.1016/j.chaos.2007.10.043
  23. X. Liu and G. Ballinger: Existence and continuability of solutions for differential equations with delays and state-dependent impulses. Nonlinear Anal. 51 (2002), 633-647.   DOI:10.1016/s0362-546x(01)00847-1
  24. Y. Liu, Z. Wang and X. Liu: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Networks 19 (2006), 667-675.   DOI:10.1016/j.neunet.2005.03.015
  25. S. Long and D. Xu: Delay-dependent stability analysis for impulsive neural networks with time varying delays. Neurocomputing 71 (2008), 1705-1713.   DOI:10.1016/j.neucom.2007.03.010
  26. C. M. Marcus and R. M. Westervelt: Stability of analog neural networks with delay. Phys. Rev. A 39 (1989), 347-359.   DOI:10.1103/physreva.39.347
  27. Y. Niu, J. Lam and X. Wang: Sliding-mode control for uncertain neutral delay systems. IEE Proc. Part D: Control Theory Appl. 151 (2004), 38-44.   DOI:10.1049/ip-cta:20040009
  28. J. H. Park: Further result on asymptotic stability criterion of cellular neural networks with time-varying discrete and distributed delays. Appl. Math. Comput. 182 (2006), 1661-1666.   DOI:10.1016/j.amc.2006.06.005
  29. J. H. Park, O. Kwon and S. Lee: LMI optimization approach on stability for delayed neural networks of neutral-type. Appl. Math. Comput. 196 (2008), 236-244.   DOI:10.1016/j.amc.2007.05.047
  30. J. Qin and J. Cao: Delay-dependent robust stability of neutral-type neural networks with time delays. J. Math. Cont. Sci. Appl. 1 (2007), 179-188.   CrossRef
  31. R. Rakkiyappan, P. Balasubramaniama and J. Cao: Global exponential stability results for neutral-type impulsive neural networks. Nonlinear Anal. RWA 11 (2010), 122-130.   DOI:10.1016/j.nonrwa.2008.10.050
  32. T. Roska and L. O. Chua: Cellular neural networks with nonlinear and delay-type templates. Int. J. Circuit Theory Appl. 20 (1992), 469-481.   DOI:10.1002/cta.4490200504
  33. A. M. Samoilenko and N. A. Perestyuk: Impulsive Differential Equations. World Scientific, Singapore 1995.   DOI:10.1142/9789812798664
  34. V. Singh: On global robust stability of interval Hopfield neural networks with delay. Chao. Solit. Fract. 33 (2007), 1183-1188.   DOI:10.1016/j.chaos.2006.01.121
  35. C. C. Travis and G. F. Webb: Existence and stability for partial functional differential equations. Trans. Amer. Math. Soc. 200 (1974), 395-418.   DOI:10.1090/s0002-9947-1974-0382808-3
  36. Z. Wang, Y. Wang and Y. Liu: Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays. IEEE Trans. Neural Networks 21 (2010), 11-25.   DOI:10.1109/tnn.2009.2033599
  37. J. Wang, X.-M. Zhang and Q.-L. Han: Event-triggered generalized dissipativity filtering for neural networks with time-varying delays. IEEE Trans. Neural Networks and Learning Systems 27 (2016), 77-88.   DOI:10.1109/tnnls.2015.2411734
  38. G. F. Webb: Autonomos nonlinear functional differential equations and nonlinear semigroups. J. Math. Anal. Appl. 46 (1974), 1-12.   DOI:10.1016/0022-247x(74)90277-7
  39. S. Xu, J. Lam, D. Ho and Y. Zou: Delay-dependent exponential stability for class of neural networks with time delays. J. Comput. Appl. Math. 183 (2005), 16-28.   DOI:10.1016/j.cam.2004.12.025
  40. D. Xu and Z. Yang: Impulsive delay differential inequality and stability of neural networks. J. Math. Anal. Appl. 305 (2005), 107-120.   DOI:10.1016/j.jmaa.2004.10.040
  41. Y. Yang and J. Cao: Stability and periodicity in delayed cellular neural networks with impulsive effects. Nonlinear Anal. RWA 8 (2007), 362-374.   DOI:10.1016/j.nonrwa.2005.11.004
  42. J. Zhang: Global stability analysis in delayed cellular neural networks. Comput. Math. Appl. 45 (2003), 1707-1720.   DOI:10.1016/s0898-1221(03)00149-4
  43. X.-M. Zhang and Q.-L. Han: New Lyapunov-Krasovskii functionals for global asymptotic stability of delayed neural networks. IEEE Trans. Neural Networks 20 (2009), 533-539.   DOI:10.1109/tnn.2009.2014160
  44. X.-M. Zhang and Q.-L. Han: Global asymptotic stability for a class of generalized neural networks with interval time-varying delays. IEEE Trans. Neural Networks 22 (2011), 1180-1192.   DOI:10.1109/tnn.2011.2147331
  45. X.-M. Zhang and Q.-L. Han: Global asymptotic stability analysis for delayed neural networks using a matrix-based quadratic convex approach. Neural Networks 54 (2014), 57-69.   DOI:10.1016/j.neunet.2014.02.012
  46. X.-M. Zhang and Q.-L. Han: Event-triggered H$_\infty$ control for a class of nonlinear networked control systems using novel integral inequalities. Int. J. Robust Nonlinear Control 27 (2016), 4, 679-700.   DOI:10.1002/rnc.3598
  47. Y. Zhang and J. T. Sun: Boundedness of the solutions of impulsive differential systems with time-varying delay. Appl. Math. Comput. 154 (2004), 279-288.   DOI:10.1016/s0096-3003(03)00712-4
  48. Y. Zhang, S. Xu, Y. Chu and J. Lu: Robust global synchronization of complex networks with neutral-type delayed nodes. Appl. Math. Comput. 216 (2010), 768-778.   DOI:10.1016/j.amc.2010.01.075