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New Results on Stability for a Class of Fractional-Order Static Neural Networks

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Abstract

This paper investigates the stability of a class of fractional-order static neural networks. Two new Lyapunov functions with proper integral terms are constructed. These integrals with variable upper limit are convex functions. Based on the fractional-order Lyapunov direct method and some inequality skills, several novel stability sufficient conditions which ensure the global Mittag–Leffler stability of fractional-order projection neural networks (FPNNs) are presented in the forms of linear matrix inequalities (LMIs). Two LMI-based Mittag–Leffler stability criteria with less conservativeness are given for a special kind of FPNNs. Finally, the effectiveness of the proposed method is demonstrated via four numerical examples.

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References

  1. S.M. Abedi Pahnehkolaei, A. Alfi, J.A.T. Machado, Dynamic stability analysis of fractional order leaky integrator echo state neural networks. Commun. Nonlinear Sci. Numer. Simul. 47, 328–333 (2017)

  2. S.M. Abedi Pahnehkolaei, A. Alfi, J.A.T. Machado, Delay-dependent stability analysis of the QUAD vector field fractional order quaternion-valued memristive uncertain neutral type leaky integrator echo state neural networks. Neural Netw. 117, 307–327 (2019)

  3. S.M. Abedi Pahnehkolaei, A. Alfi, J.A.T. Machado, Delay independent robust stability analysis of delayed fractional quaternion-valued leaky integrator echo state neural networks with QUAD condition. Appl. Math. Comput. 359, 278–293 (2019)

  4. S.M. Abedi Pahnehkolaei, A. Alfi, J.A.T. Machado, Stability analysis of fractional quaternion-valued leaky integrator echo state neural networks with multiple time-varying delays. Neurocomputing 331, 388–402 (2019)

  5. N. Aguila-Camacho, M. Duarte-Mermoud, J. Gallegos, Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19, 2951–2957 (2014)

    MathSciNet  MATH  Google Scholar 

  6. P. Balasubramaniam, M.Syed Ali, Robust exponential stability of uncertain fuzzy Cohen–Grossberg neural networks with time-varying delays. Fuzzy Set Syst. 161(4), 608–618 (2010)

    MathSciNet  MATH  Google Scholar 

  7. D. Baleanu, A.N. Ranjbar, S.J. Sadati, H. Delavari, Lyapunov–Krasovskill stability theorem for fractional systems with delay. Rom. J. Phys. 56, 5–6 (2011)

    MATH  Google Scholar 

  8. H. Bao, J. Cao, Projective synchonization of fractional-order memristor-based neural networks. Neural Netw. 63, 1–9 (2015)

    MATH  Google Scholar 

  9. L. Chen, Y. Chai, R. Wu, J. Yang, Stability and stabilization of a class of nonlinear fractional-order systems with Caputo derivative. IEEE Trans. Circuits Syst. 59, 602–606 (2012)

    Google Scholar 

  10. L. Chen, Y. Chai, R. Wu, T. Ma, H. Zhai, Dynamic analysis of a class of fractional-order neural networks with delay. Neurocomputing 111, 190–194 (2013)

    Google Scholar 

  11. W. Chen, H. Dai, Y. Song, Z. Zhang, Convex Lyapunov functions for stability analysis of fractional order systems. IET Control Theory. Appl. 11, 1070–1074 (2017)

    MathSciNet  Google Scholar 

  12. N. Cui, H. Jiang, C. Hu, A. Abdurahman, Global asymptotic and robust stability of inertial neural networks with proportional delays. Neurocomputing 272, 326–333 (2018)

    Google Scholar 

  13. K. Diethelm, The Analysis of Fractional Differential Equations (Springer, New York, 2010), pp. 195–211

    MATH  Google Scholar 

  14. J. Di, Y. He, C. Zhang, M. Wu, Novel stability criteria for recurrent neural networks with time-varying delay. Neurocomputing 138, 383–391 (2014)

    Google Scholar 

  15. M. Duarte-Mermoud, N. Aguila-Camacho, J. Gallegos, R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22, 650–659 (2015)

    MathSciNet  MATH  Google Scholar 

  16. M. Faieghi, S. Kuntanapreeda, H. Delavari, D. Baleanu, LMI-based stabilization of a class fractional-order chaotic systems. Nonliear Dyn. 1–2, 301–309 (2013)

    MathSciNet  MATH  Google Scholar 

  17. F. Fracchinei, J. Pang, Finite Dimensional Variational Inequalities and Complementarity Problems (Springer, New York, 2003), pp. 2–12

    Google Scholar 

  18. T.L. Friesz, D.H. Bernstein, N.J. Mehta, R.L. Tobin, S. Ganjlizadeh, Day-to-Day dynamic network disequilibria and idealized traveler information systems. Oper. Res. 42, 1120–1136 (1994)

    MathSciNet  MATH  Google Scholar 

  19. J. Gallegos, M. Duarte-Mermoud, Convergence of fractional adaptive systems using gradient approach. ISA. Trans. 69, 31–42 (2017)

    MATH  Google Scholar 

  20. X. Ge, Q. Han, D. Ding, X. Zhang, B. Ning, A survey on recent advances in distributed sampled-data cooperative control of multi-agent systems. Neurocomputing 275, 1684–1701 (2017)

    Google Scholar 

  21. X. Ge, Q. Han, Z. Wang, A dynamic event-triggered transmission scheme for distributed set-membership estimation over wireless sensor networks. IEEE Trans. Cybern. 99, 1–13 (2017)

    Google Scholar 

  22. C. Hsu, C. Chang, Intelligent dynamic sliding-mode neural control using recurrent perturbation fuzzy neural networks. Neurocomputing 173(P3), 734–743 (2016)

    Google Scholar 

  23. B. Huang, G. Hui, D. Gong, Z. Wang, X. Meng, A projection neural networks with mixed delays for solving linear variational inequality. Neurocomputing 125, 28–32 (2014)

    Google Scholar 

  24. S. Javad, S. Effati, M. Pakdaman, A neural network approach for solving a class of fractional optimal control problems. Neural Process. Lett. 45, 1–16 (2016)

    Google Scholar 

  25. G. Joya, M.A. Atencia, F. Sandoval, Hopfield neural networks for optimization: study of the different dynamics. Neurocomputing 43(1–4), 219–237 (2002)

    MATH  Google Scholar 

  26. R. Krishnasamy, P. Balasubramaniam, A descriptor system approach to the delay-dependent exponential stability analysis for switched neutral systems with nonlinear perturbations. Nonlinear Anal. Hybrid Syst. 15, 23–36 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Y. Li, Y. Chen, I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

    MathSciNet  MATH  Google Scholar 

  29. H. Li, Y. Jiang, Z. Wang, C. Hu, Global stability problem for feedback control systems of impulsive fractional differential equations on networks. Neurocomputing 161, 155–161 (2015)

    Google Scholar 

  30. W. Lin, Y. He, C. Zhang, M. Wu, M. Ji, Stability analysis of recurrent neural networks with interval time-varying delay via free-matrix-based integral inequality. Neurocomputing 205, 490–497 (2016)

    Google Scholar 

  31. X. Liu, F. Wang, Y. Shu, A novel summation inequality for stability analysis of discrete-time neural networks. J. Comput. Appl. Math. 304, 160–171 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Y. Liu, Z. Wang, X. Liu, Asymptotic stability for neural networks with mixed time-delays: the discrete-time case. Neural Netw. 22, 67–74 (2009)

    MATH  Google Scholar 

  33. Y.V. Pershin, M.D. Ventra, Experimental demonstration of associative memory with memristive neural networks. Neural Netw. 23(7), 881–886 (2010)

    Google Scholar 

  34. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999), pp. 261–275

    MATH  Google Scholar 

  35. R. Rakkiyappan, P. Balasubramaniam, On exponential stability results for fuzzy impulsive neural networks. Fuzzy Set Syst. 161(13), 1823–1835 (2010)

    MathSciNet  MATH  Google Scholar 

  36. R. Saravanakumar, M.Syed Ali, C.K. Ahn, H.R. Karimi, P. Shi, Stability of Markovian jump generalized neural networks with interval time-varying delays. IEEE Trans. Neural Netw. Learn. Syst 28(8), 1840–1850 (2016)

    MathSciNet  Google Scholar 

  37. M. Syed Ali, P. Balasubramaniam, Stability analysis of uncertain fuzzy Hopfield neural networks with time delays. Commun. Nonlinear Sci. Numer. Simul. 14(6), 2776–2783 (2009)

    MathSciNet  MATH  Google Scholar 

  38. M.Syed Ali, S. Saravanan, J. Cao, Finite-time boundedness, \(L_2\)-gain analysis and control of Markovian jump switched neural networks with additive time-varying delays. Nonlinear Anal. Hybrid Syst 23, 27–43 (2017)

    MathSciNet  MATH  Google Scholar 

  39. M. Syed Ali, M. Usha, Z. Orman, S. Arik, Improved result on state estimation for complex dynamical networks with time varying delays and stochastic sampling via sampled-data control. Neural Netw. 114, 28–37 (2019)

    MATH  Google Scholar 

  40. H. Wang, Y. Yu, G. Wen, S. Zhang, J. Yu, Global stability analysis of fractional-order Hopfield neural networks with time delay. Neurocomputing 154, 15–23 (2015)

    Google Scholar 

  41. Z. Wu, J. Lam, H. Su, C. Jian, Stability and dissipativity analysis of static neural networks with time delay. IEEE Trans. Neural Netw. Learn. Syst. 23, 199–210 (2012)

    Google Scholar 

  42. Z. Wu, Y. Zou, Global fractional-order projective dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 19, 2811–2819 (2014)

    MathSciNet  MATH  Google Scholar 

  43. Z. Wu, Y. Zou, N. Huang, A system of fractional-order interval projection neural networks. J. Comput. Appl. Math. 294, 389–402 (2016)

    MathSciNet  MATH  Google Scholar 

  44. Z. Wu, J. Li, N. Huang, A new system of global fractional-order interval implict projection neural networks. Neurocomputing 282, 111–121 (2018)

    Google Scholar 

  45. F. Wang, X. Liu, M. Tang, L. Chen, Further results on stability and synchronization of fractional-order Hopfield neural networks. Neurocomputing 346, 12–19 (2019)

    Google Scholar 

  46. Y. Yang, Y. He, Y. Wang, M. Wu, Stability analysis of fractional-order neural networks: an LMI approach. Neurocomputing 285, 82–93 (2018)

    Google Scholar 

  47. X. Yang, C. Li, T. Huang, Q. Song, J. Huang, Global Mittag–Leffler synchronization of fractional-order neural networks via impulsive control. Neural Process Lett. 48, 459–479 (2018)

    Google Scholar 

  48. X. Zhang, Q. Han, New Lyapunov–Krasovskii functionals for global asymptotic stability of delayed neural networks. IEEE Trans. Neural Netw. 20, 533–539 (2009)

    Google Scholar 

  49. S. Zhang, Y. Yu, H. Wang, Mittag–Leffler stability of fractional-order Hopfield neural networks. Nonlinear Anal. Hybrid Syst. 16, 104–121 (2015)

    MathSciNet  MATH  Google Scholar 

  50. S. Zhang, Y. Yu, Q. Wang, Stability analysis of fractional-order Hopfield neural networks with discontinuous activation functions. Neurocomputing 171, 1075–1084 (2016)

    Google Scholar 

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Acknowledgements

The authors would like to thank the editor and anonymous reviewers for their valuable comments and suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Xiangqian Yao, Meilan Tang, Zhijian Ye & Xinge Liu

  2. School of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou, 450002, China

    Fengxian Wang

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  1. Xiangqian Yao
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  2. Meilan Tang
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This work is partly supported by the National Science Foundation of China under Grants 61773404 and 11601104 and Fundamental Research Funds for the Central Universities of Central South University 2018zzts316.

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Yao, X., Tang, M., Wang, F. et al. New Results on Stability for a Class of Fractional-Order Static Neural Networks. Circuits Syst Signal Process 39, 5926–5950 (2020). https://doi.org/10.1007/s00034-020-01451-5

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