Kybernetika 60 no. 3, 317-356, 2024

Delay dependent complex-valued bidirectional associative memory neural networks with stochastic and impulsive effects: an exponential stability approach

Chinnamuniyandi Maharajan, Chandran Sowmiya and Changjin XuDOI: 10.14736/kyb-2024-3-0317

Abstract:

This paper investigates the stability in an exponential sense of complex-valued Bidirectional Associative Memory (BAM) neural networks with time delays under the stochastic and impulsive effects. By utilizing the contracting mapping theorem, the existence and uniqueness of the equilibrium point for the proposed complex-valued neural networks are verified. Moreover, based on the Lyapunov\,--\,Krasovskii functional construction, matrix inequality techniques and stability theory, some novel time-delayed sufficient conditions are attained in linear matrix inequalities (LMIs) form, which ensure the exponential stability of the trivial solution for the addressed neural networks. Finally, to illustrate the superiority and effects of our theoretical results, two numerical examples with their simulations are provided via MATLAB LMI control toolbox.

Keywords:

Lyapunov-Krasovskii functional, Exponential stability, distributed delays, Complex-valued neural networks, Linear matrix inequality, BAM neural networks, Impulsive effects, Stochastic noise, discrete delays, leakage delays, mixed time delays

Classification:

34Dxx, 92B20, 93Exx

paper.pdf

References:

  1. C. Aouiti, E. A. Assali, I. B. Gharbia and Y. E. Foutayeni: Existence and exponential stability of piecewise pseudo almost periodic solution of neutral-type inertial neural networks with mixed delay and impulsive perturbations. Neurocomputing 357 (2019), 292-309.   DOI:10.1016/j.neucom.2019.04.077
  2. X. Chen, Q. Song, Y. Liu and Z. Zhao: Global $\mu$-stability of impulsive complex-valued neural networks with leakage delay and mixed delays. Abstract Appl. Anal. 12 (2014), 1-14.   DOI:10.1155/2014/397532
  3. X. Chen, Z. Zhao, Q. Song and J. Hu: Multistability of complex-valued neural networks with time-varying delays. Appl. Math. Comput. 294 (2017), 18-35.   DOI:10.1016/j.amc.2016.08.054
  4. Q. Duan, J. Park and Z. Wu: Exponential state estimator design for discrete-time neural networks with discrete and distributed time-varying delays. Complexity 20 (2014), 38-48.   DOI:10.1002/cplx.21494
  5. T. G. Kumar, M. S. Ali, B. Priya, V. Gokulakrishnan and S. A. Kauser: Impulsive effects on stochastic bidirectional associative memory neural networks with reaction-diffusion and leakage delays. Int. J. Computer Math. 99 (2022), 8, 1669-1686.   DOI:10.1080/00207160.2021.1999428
  6. Y. Gholami: Existence and global asymptotic stability criteria for nonlinear neutral-type neural networks involving multiple time delays using a quadratic-integral Lyapunov functional. Adv. Differ. Eqns, 112 (2021).   DOI:10.1186/s13662-021-03274-3
  7. K. Gopalsamy: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers, Dordrecht 1992.   CrossRef
  8. K. Gu: A integral inequality in the stability problem of time-delay systems. In: Proc. 39th IEEE Conference on Decision Control, Sydney 3 (200), pp. 2805-2810.   DOI:10.1109/CDC.2000.914233
  9. S. Haykin: Neural Networks. Prentice Hall, New Jersey 1999.   CrossRef
  10. D. W. C. Ho, J. Liang and J. Lam: Global exponential stability of impulsive highorder BAM neural networks with time-varying delays. Neural Networks 19 (2006), 10, 1581-1590.   DOI:10.1016/j.neunet.2006.02.006
  11. J. Hu, H. Tan and C. Zeng: Global exponential stability of delayed complex-valued neural networks with discontinuous activation functions. Neurocomputing 416 (2020), 8, 1-11.   DOI:10.1016/j.neucom.2020.02.006
  12. J. Hu and J. Wang: Global stability of complex-valued recurrent neural networks with time-delays. IEEE Trans. Neur. Netw. Learn. Systems 23 (2012), 853-865.   DOI:10.1109/TNNLS.2012.2195028
  13. N. Huo, B. Li and Y. Li: Global exponential stability and existence of almost periodic solutions in distribution for Clifford-valued stochastic high-order Hopfield neural networks with time-varying delays. AIMS Math. 7 (2022), 3, 3653-3679.   DOI:10.3934/math.2022202
  14. M. Hymavathi, G. Muhiuddin, M. S. Ali, F. Jehad, Al-Amri, N. Gunasekaran and R. Vadivel: Global exponential stability of Fractional order complex-valued neural networks with leakage delay and mixed time varying delays. Fractal and Fractional 6 (2022), 3.   DOI:10.3390/fractalfract6030140
  15. M. D. Ji, Y. He, M. Wu and C. K. Zhang: New exponential stability criterion for neural networks with time-varying delay. In Proc. 33rd Chinese control conference; Nanjing 2014, pp. 6119-6123.   DOI:10.1109/ChiCC.2014.6895991
  16. M. D. Ji, Y. He, M. Wu and C. K. Zhang: Further results on exponential stability of neural networks with time-varying delay. Appl. Math. Comput. 256 (2015), 175-182.   DOI:10.1016/j.amc.2015.01.004
  17. S. Jia and Y. Chen: Global exponential asymptotic stability of RNNs with mixed asynchronous time-varying delays. Adv. Diff. Eqns 200 (2020).   DOI:10.1186/s13662-020-02648-3
  18. B. Kosko: Adaptive bi-directional associative memories. Appl. Optim. 26 (1987), 4947-4960.   DOI:10.1364/AO.26.004947
  19. B. Kosko: Bidirectional associative memories. IEEE Trans. System Man Cybernet. 18 (1988), 49-60.   DOI:10.1109/21.87054
  20. B. Li, F. Liu, Q. Song, D. Zhang and H. Qiu: State estimation of complex-valued neural networks with leakage delay: A dynamic event-triggered approach. Neurocomputing 520 (2023), 230-239.   DOI:10.1016/j.neucom.2022.11.079
  21. B. Li and B. Tang: New stability criterion for Fractional-order Quaternion-valued neural networks involving discrete and leakage delays. J. Math. Article ID 9988073 (2021), 20 pages.   DOI:10.1155/2021/9988073
  22. C. Li, J. Lian and Y. Wang: Stability of switched memristive neural networks with impulse and stochastic disturbance. Neurocomputing {\mi275} (2018), 2565-2573.   DOI:10.1016/j.neucom.2017.11.031
  23. C. Li, C. Li, X. Liao and T. Huang: Impulsive effects on stability of high-order BAM neural networks with time delays. Neurocomputing 74 (2011), 1541-1550.   DOI:10.1016/j.neucom.2010.12.028
  24. D. Li, X. Wang and D. Xu: Existence and global image exponential stability of periodic solution for impulsive stochastic neural networks with delays. Nonlinear Anal.: Hybrid Sys. 6 (2012), 847-858.   CrossRef
  25. X. Li, M. Bohner and C. Wang: Impulsive differential equations: Periodic solutions and applications. Automatica 52 (2015), 173-178.   DOI:10.1016/j.automatica.2014.11.009
  26. X. Li and S. Song: Stabilization of delay systems: delay-dependent impulsive control. IEEE Trans. Automat. Control 62 (2017), 1, 406-411.   DOI:10.1109/TAC.2016.2530041
  27. X. Li, X. Zhang and S. Song: Effect of delayed impulses on input-to-state stability of nonlinear systems. Automatica 76 (2017), 378-382.   DOI:10.1016/j.automatica.2016.08.009
  28. X. Liu and T. Chen: Global exponential stability for complex-valued recurrent neural networks with asynchronous time delays. IEEE Trans. Neural Netw. Learn. Sys. 23 (2015), 3, 593-606.   DOI:10.1109/TNNLS.2015.2415496
  29. C. Maharajan, R. Raja, J. Cao, G. Rajchakit and A. Alsaedi: Novel results on passivity and exponential passivity for multiple discrete delayed neutral-type neural networks with leakage and distributed time-delays. Chaos, Solitons Fractals 115 (2018), 268-282.   DOI:10.1016/j.chaos.2018.07.008
  30. C. Maharajan, R. Raja, J. Cao and G. Rajchakit: Novel global robust exponential stability criterion for uncertain inertial-type BAM neural networks with discrete and distributed time-varying delays via Lagrange sense. J. Frankl. Inst. 355 (2018), 11, 4727-4754.   DOI:10.1016/j.jfranklin.2018.04.034
  31. C. Maharajan and C. Sowmiya: Exponential stability of delay dependent neutral-type descriptor neural networks with uncertain parameters. Franklin Open 5 (2023), 100042.   DOI:10.1016/j.fraope.2023.100042
  32. C. Maharajan, C. Sowmiya and C. J. Xu: Fractional order uncertain BAM neural networks with mixed time delays: An existence and Quasi-uniform stability analysis. J. Intel. Fuzzy Sys. 46 2024, 2, 4291-4313.   DOI:10.3233/JIFS-234744
  33. X. Mao, X. Wang and H. Qin: Stability analysis of quaternion-valued BAM neural networks fractional-order model with impulses and proportional delays. Neurocomputing 509 (2022), 206-220.   DOI:10.1016/j.neucom.2022.08.059
  34. W. Ou, C. J. Xu, Q. Cui, Z. Liu, Y. Pang, M. Farman, S. Ahmad and A. Zeb: Mathematical study on bifurcation dynamics and control mechanism of tri-neuron BAM neural networks including delay. Math. Meth. Appl. Sci. (2023), 1-25.   DOI:10.1002/mma.9347
  35. J. Park and O. Kwon: Global stability for neural networks of neutral-type with interval time-varying delays. Chaos, Solitons Fractals 41 (2009), 1174-1181.   DOI:10.1016/j.chaos.2008.04.049
  36. G. Rajchakit and R. Sriraman: Robust passivity and stability analysis of uncertain complex-valued impulsive neural networks with time-varying delays. Neur. Proces. Lett. 53 (2021), 581-606.   DOI:10.1007/s11063-020-10401-w
  37. R. Rakkiyappan, G. Velmurugan and X. Li: Complete stability analysis of complexvalued neural networks with time delays and impulses. Neur. Proces. Lett. 41 (2015), 435-468.   DOI:10.1007/s11063-014-9349-6
  38. H. Ren, H. R. Karimi, R. Lu and Y. Wu: Synchronization of network systems via aperiodic sampled-data control with constant delay and application to unmanned ground vehicles. IEEE Tans. Industr. Electronics 67 (2020), 6, 4980-4990.   DOI:10.1109/TIE.2019.2928241
  39. R. Samidurai and R. Sriraman: Non-fragile sampled-data stabilization analysis for linear systems with probabilistic time-varying delays. J. Franklin Inst. 356 (2019), 8, 4335-4357.   DOI:10.1016/j.jfranklin.2018.11.046
  40. R. Samidurai, R. Sriraman and S. Zhu: Stability and dissipativity analysis for uncertain Markovian jump systems with random delays via new approach. Int. J. Syst. Sci. 50 (2019), 8, 1609-1625.   DOI:10.1080/00207721.2019.1618942
  41. S. Shao and B. Du: Global asymptotic stability of competitive neural networks with reaction-diffusion terms and mixed delays. Symmetry 14 ID 2224 (2022), 1-11.   DOI:10.3390/sym14112224
  42. Q. Song, H. Shu, Z. Zhao, Y. Liu and F. E. Alsaadi: Lagrange stability analysis for complex-valued neural networks with leakage delay and mixed time-varying delays. Neurocomputing 244 (2017), 33-41.   DOI:10.1016/j.neucom.2017.03.015
  43. Q. Song, L. Yang, Y. Liu and F. Alsaadi: Stability of quaternion-valued neutral-type neural networks with leakage delay and proportional delays. Neurocomputing 521 (2023), 191-198.   DOI:10.1016/j.neucom.2022.12.009
  44. Q. Song, Z. Zhao and Y. Liu: Stability analysis of complex-valued neural networks with probabilistic time-varying delays. Neurocomputing 159 (2015), 96-104.   DOI:10.1016/j.neucom.2015.02.015
  45. R. Sriraman and A. Nedunchezhiyan: Global stability of Clifford-valued Takagi-Sugeno fuzzy neural networks with time-varying delays and impulses. Kybernetika 58 (2022), 4, 498-521.   DOI:10.14736/kyb-2022-4-0498
  46. R. Sriraman, Y. Cao and R. Samidurai: Global asymptotic stability of stochastic complexvalued neural networks with probabilistic time-varying delays. Math. and Comput. Simul. 171 (2020), 103-118.   DOI:10.1016/j.matcom.2019.04.001
  47. M. S. Ali, J. Yogambigai, S. Saravanan and S. Elakkia: Stochastic stability of neutraltype Markovian-jumping BAM neural networks with time varying delays. J. Comput. and Appl. Math. 349 (2019), 142-156.   DOI:10.1016/j.cam.2018.09.03
  48. Y. Tan, S. Tang and X. Chen: Robust stability analysis of impulsive complex-valued neural networks with mixed time delays and parameter uncertainties. Adv. Diff. Eqns 62 (2018).   DOI:10.1186/s13662-018-1521-2
  49. J. Tian and X. Xie: New asymptotic stability criteria for neural networks with timevarying delay. Phys. Lett. A 374 (2010), 7, 938-943.   DOI:10.1016/j.physleta.2009.12.020
  50. Z. Wang and L. Huang: Global stability analysis for delayed complex-valued BAM neural networks. Neurocomputing 173 (2016), 3, 2083-2089.   DOI:10.1016/j.neucom.2015.09.086
  51. M. Wu, F. Liu, P. Shi, Y. He and R. Yokoyama: Exponential stability analysis for neural networks with time-varying delay. IEEE Trans. Syst. Man Cybernet., Part B: Cybernetics 38 (2008), 4, 1152-1156.   DOI:10.1109/TSMCB.2008.915652
  52. D. Xie and Y. Jiang: Global exponential stability of periodic solution for delayed complexvalued neural networks with impulses. Neurocomputing 207 (2016), 528-538.   DOI:10.1016/j.neucom.2016.04.054
  53. C. J. Xu, D. Mu, Y. Pan, C. Aouiti and L. Yao: Exploring bifurcation in a fractional-order predator-prey system with mixed delays. J. Appl. Anal. Comput.13 (2023), 3, 1119-1136.   DOI:10.11948/20210313
  54. M. Xu and B. Du: Dynamic behaviors for reaction-diffusion neural networks with mixed delays. AIMS Math. 5 (2020), 6, 6841-6855.   DOI:10.3934/math.2020439
  55. J. Yu, K. Zhang and S. Fei: Mean square exponential stability of generalized stochastic neural networks with time-varying delays. Asian J. Control 11 (2009), 6, 633-642.   DOI:10.1002/asjc.144
  56. W. Zhang and C. Li: Global robust stability of complex-valued recurrent neural networks with time-delays and uncertainties. Int. J. Biomath. 7 (2014), 2, 24 pages.   DOI:10.1142/S1793524514500168
  57. W. Zhang, C. Li, T. Huang and J. Tan: Exponential stability of inertial BAM neural networks with time-varying delay via periodically intermittent control. Neural Computing Appl. 26 (2015), 1781-1787.   DOI:10.1007/s00521-015-1838-7
  58. X. Zhang and X. Li: Input-to-state stability of non-linear systems with distributeddelayed impulses. IET Control Theory Appl. 11 (2017), 1, 81-89.   DOI:10.1049/iet-cta.2016.0469
  59. Z. Zhang, C. Lin and B. Chen: Global stability criterion for delayed complex-valued recurrent neural networks. IEEE Trans. Neur. Netw. Learn. Systems 25 (2014), 9, 1704-1708.   DOI:10.1109/TNNLS.2013.2288943
  60. B. Zhou and Q. Song: Boundedness and complete stability of complex-valued neural networks with time delay. IEEE Trans. Neur. Netw. Learn. Systems 24 (2013), 1227-1238.   DOI:10.1109/TNNLS.2013.2247626
  61. Q. Zhou, L. Wan, H. Fu and Q. Zhang: Exponential stability of stochastic Hopfield neural network with mixed multiple delays. AIMS Math. 6 (2021), 4, 4142-4155.   DOI:10.3934/math.2021245
  62. W. Zhou, B. Li and J. E. Zhang: Matrix measure approach for stability and synchronization of complex-valued neural networks with deviating argument. Mathemat. Probl. Engrg. Article ID 8877129; (2020), 16 pages.   DOI:10.1155/2020/8877129
  63. K. Zou, X. Li, N. Wang, J. Lou and J. Lu: Stability and stabilization of delayed neural networks with hybrid impulses. Complexity Article ID 8712027; (2020), 9 pages.   DOI:10.1155/2020/8712027