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Periodically intermittent control strategies for \(\varvec{\alpha }\)-exponential stabilization of fractional-order complex-valued delayed neural networks

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Abstract

This paper studies the global \(\alpha \)-exponential stabilization of a kind of fractional-order neural networks with time delay in complex-valued domain. To end this, several useful fractional-order differential inequalities are set up, which generalize and improve the existing results. Then, a suitable periodically intermittent control scheme with time delay is put forward for the global \(\alpha \)-exponential stabilization of the addressed networks, which include feedback control as a special case. Utilizing these useful fractional-order differential inequalities and combining with the Lyapunov approach and other inequality techniques, some novel delay-independent criteria in terms of real-valued algebraic inequalities are obtained to ensure global \(\alpha \)-exponential stabilization of the discussed networks, which are very simple to implement in practice and avert to calculate the complex matrix inequalities. Finally, the availability of the theoretical criteria is verified by an illustrative example with simulations.

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References

  1. Diethelm, K.: The analysis of fractional differential equations. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Elsevier, New York (2006)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Delavari, H., Baleanu, D., Sadati, J.: Stability analysis of caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67, 2433–2439 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Stamova, I., Stamov, G.: Stability analysis of impulsive functional systems of fractional order. Commun. Nonlinear Sci. Numer. Simul. 19, 702–709 (2014)

    Article  MathSciNet  Google Scholar 

  8. Fernandez-Anaya, G., Nava-Antonio, G., Jamous-Galante, J., Muñoz-Vega, R., Hernández-Martínez, E.G.: Lyapunov functions for a class of nonlinear systems using Caputo derivative. Commun. Nonlinear Sci. Numer. Simul. 43, 91–99 (2017)

    Article  MathSciNet  Google Scholar 

  9. Daftardar-Gejji, V., Bhalekar, S., Gade, P.: Dynamics of fractional-ordered Chen system with delay. Pramana 79, 61–69 (2012)

    Article  Google Scholar 

  10. Yu, J., Hu, C., Jiang, H.J.: \(\alpha \)-stability and \(\alpha \)-synchronization for fractional-order neural networks. Neural Netw. 35, 82–87 (2012)

    Article  MATH  Google Scholar 

  11. Wu, H.Q., Zhang, X.X., Xue, S.H., Wang, L.F., Wang, Y.: LMI conditions to global Mittag-Leffler stability of fractional-order neural networks with impulses. Neurocomputing 193, 148–154 (2016)

    Article  Google Scholar 

  12. Song, C., Cao, J.D.: Dynamics in fractional-order neural networks. Neurocomputing 142, 494–498 (2014)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. Stamova, I.: Global Mittag-Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays. Nonlinear Dyn. 77, 1251–1260 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, C.N., Chu, R.T., Ma, J.: Controlling a chaotic resonator by means of dynamic track control. Complexity 21, 370–378 (2015)

    Article  MathSciNet  Google Scholar 

  16. Park, J.H.: Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters. J. Comput. Appl. Math. 213, 288–293 (2008)

    Article  MATH  Google Scholar 

  17. Bao, H.B., Park, J.H., Cao, J.D.: Matrix measure strategies for exponential synchronization and anti-synchronization of memristor-based neural networks with time-varying delays. Appl. Math. Comput. 270, 543–556 (2015)

    MathSciNet  Google Scholar 

  18. Chen, W.H., Jiang, Z.Y., Lu, X.M., Luo, S.X.: \(H_{\infty }\) synchronization for complex dynamical networks with coupling delays using distributed impulsive control. Nonlinear Anal. Hybrid Syst. 17, 111–127 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mei, J., Jiang, M.H., Wu, Z., Wang, X.H.: Periodically intermittent controlling for finite-time synchronization of complex dynamical networks. Nonlinear Dyn. 79, 295–305 (2015)

    Article  MATH  Google Scholar 

  20. Wei, L.N., Chen, W.H., Huang, G.J.: Globally exponential stabilization of neural networks with mixed time delays via impulsive control. Appl. Math. Comput. 260, 10–26 (2015)

    MathSciNet  Google Scholar 

  21. Ye, Z.Y., Zhang, H., Zhang, H.Y., Zhang, H., Lu, G.C.: Mean square stabilization and mean square exponential stabilization of stochastic BAM neural networks with Markovian jumping parameters. Chaos Soliton Fractals 73, 156–165 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. He, H.L., Yan, L., Tu, J.J.: Guaranteed cost stabilization of time-varying delay cellular neural networks via Riccati inequality approach. Neural Process. Lett. 35, 151–158 (2012)

    Article  Google Scholar 

  23. Liu, X.Y., Jiang, N., Cao, J.D., Wang, S.M., Wang, Z.X.: Finite-time stochastic stabilization for BAM neural networks with uncertainties. J. Frankl. Inst. 350, 2109–2123 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  25. Chen, W.H., Zhong, J.C., Zheng, W.X.: Delay-independent stabilization of a class of time-delay systems via periodically intermittent control. Automatica 71, 89–97 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhu, X.L., Wang, Y.Y.: Stabilization for sampled-data neural-network-based control systems. IEEE Trans. Syst. Man Cybern. 41, 210–221 (2011)

    Article  Google Scholar 

  27. Zhang, G.D., Shen, Y.: Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control. Neural Netw. 55, 1–10 (2014)

    Article  MATH  Google Scholar 

  28. Yang, X.S., Cao, J.D.: Stochastic synchronization of coupled neural networks with intermittent control. Phys. Lett. A 373, 3259–3272 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, S.J., Li, C.D., Huang, T.W.: Exponential stabilization and synchronization for fuzzy model of memristive neural networks by periodically intermittent control. Neural Netw. 75, 162–172 (2016)

    Article  Google Scholar 

  30. Zhang, Z.M., He, Y., Zhang, C.K., Wu, M.: Exponential stabilization of neural networks with time-varying delay by periodically intermittent control. Neurocomputing 207, 469–475 (2016)

    Article  Google Scholar 

  31. Nitta, T.: Solving the XOR problem and the detection of symmetry using a single complex-valued neuron. Neural Netw. 16, 1101–1105 (2003)

    Article  Google Scholar 

  32. Song, Q.K., Yan, H., Zhao, Z.J., Liu, Y.R.: Global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects. Neural Netw. 79, 108–116 (2015)

    Article  Google Scholar 

  33. Song, Q.K., Zhao, Z.J., Liu, Y.R.: Stability analysis of complex-valued neural networks with probabilistic time-varying delays. Neurocomputing 159, 96–104 (2015)

    Article  Google Scholar 

  34. Rakkiyappan, R., Cao, J.D., Velmurugan, G.: Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans. Neural Netw. Learn. Syst. 26, 84–97 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rakkiyappan, R., Sivaranjani, R., Velmurugan, G., Cao, J.D.: Analysis of global \(o(t^{-\alpha })\) stability and global asymptotical periodicity for a class of fractional-order complex-valued neural networks with time varying delays. Neural Netw. 77, 51–69 (2016)

    Article  Google Scholar 

  36. Rakkiyappan, R., Velmurugan, G., Cao, J.D.: Stability analysis of fractional-order complex-valued neural networks with time delays. Chaos Soliton Fractals 78, 297–316 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Jian, J.G., Wan, P.: Lagrange \(\alpha \)-exponential stability and \(\alpha \)-exponential convergence for fractional-order complex-valued neural networks. Neural Netw. 91, 1–10 (2017)

    Article  Google Scholar 

  38. Liu, L., Wu, A.L., Song, X.G.: Global \(o(t^{-\alpha })\) stabilization of fractional-order memristive neural networks with time delays. Springerplus 5, 1–22 (2016)

    Article  Google Scholar 

  39. Wu, A.L., Zeng, Z.G., Song, X.G.: Global Mittag-Leffler stabilization of fractional-order bidirectional associative memory neural networks. Neurocomputing 177, 489–496 (2016)

    Article  Google Scholar 

  40. Su, K.L., Li, C.L.: Control chaos in fractional-order system via two kinds of intermittent schemes. Optik 126, 2671–2673 (2015)

    Article  Google Scholar 

  41. Bao, H.B., Park, J.H., Cao, J.D.: Synchronization of fractional-order complex-valued neural networks with time delay. Neural Netw. 81, 16–28 (2016)

    Article  Google Scholar 

  42. Ji, Y.D., Qiu, J.Q.: Stabilization of fractional-order singular uncertain systems. ISA Trans. 56, 53–64 (2015)

    Article  Google Scholar 

  43. Wan, P., Jian, J.G.: Global convergence analysis of impulsive inertial neural networks with time-varying delays. Neurocomputing 245, 68–76 (2017)

    Article  Google Scholar 

  44. Bhalekar, S., Daftardar-Gejji, V.: A predictor–corrector scheme for solving nonlinear delay differential equations of fractional order. J. Fract. Calc. Appl. 1, 1–9 (2011)

    MATH  Google Scholar 

  45. Yuan, L.G., Yang, Q.G., Zeng, C.B.: Chaos detection and parameter identification in fractional-order chaotic systems with delay. Nonlinear Dyn. 73, 439–448 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported partially by the National Natural Science Foundation of China (11601268).

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Authors and Affiliations

  1. College of Science, China Three Gorges University, Yichang, Hubei, 443002, China

    Peng Wan & Jigui Jian

  2. Department of Electrical, Electric and Computer Engineering, Centre of New Energy Systems, University of Pretoria, Pretoria, 0002, South Africa

    Jun Mei

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  1. Peng Wan
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  2. Jigui Jian
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  3. Jun Mei
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Correspondence to Jigui Jian.

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Wan, P., Jian, J. & Mei, J. Periodically intermittent control strategies for \(\varvec{\alpha }\)-exponential stabilization of fractional-order complex-valued delayed neural networks. Nonlinear Dyn 92, 247–265 (2018). https://doi.org/10.1007/s11071-018-4053-0

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