Research article

Controlling stability through the rate of decay of the delay feedback kernels

  • Received: 02 August 2023 Revised: 30 August 2023 Accepted: 07 September 2023 Published: 14 September 2023
  • MSC : 34C11, 34G20, 92B20

  • Of concern is the Hopfield neural network system comprising discrete as well as distributed delays in the form of a convolution. For a desired convergence rate of the solution to the equilibrium state, we establish sufficient conditions on the delay kernels ensuring this matter. Our result improves an existing one in the literature. The adopted approach is completely different. It relies on a judicious choice of a Lyapunov-like function and careful manipulations.

    Citation: Mohammed D. Kassim. Controlling stability through the rate of decay of the delay feedback kernels[J]. AIMS Mathematics, 2023, 8(11): 26343-26356. doi: 10.3934/math.20231344

    Related Papers:

  • Of concern is the Hopfield neural network system comprising discrete as well as distributed delays in the form of a convolution. For a desired convergence rate of the solution to the equilibrium state, we establish sufficient conditions on the delay kernels ensuring this matter. Our result improves an existing one in the literature. The adopted approach is completely different. It relies on a judicious choice of a Lyapunov-like function and careful manipulations.



    加载中


    [1] A. Bouzerdoum, T. Pattison, Neural network for quadratic optimization with bound constraints, IEEE Trans. Neural Networ., 4 (1993), 293–304. http://dx.doi.org/10.1109/72.207617 doi: 10.1109/72.207617
    [2] L. Chua, T. Roska, Stability of a class of nonreciprocal cellular neural networks, IEEE Trans. Circuits-I, 37 (1990), 1520–1527. http://dx.doi.org/10.1109/31.101272 doi: 10.1109/31.101272
    [3] B. Crespi, Storage capacity of non-monotonic neurons, Neural Networks, 12 (1999), 1377–1389. http://dx.doi.org/10.1016/S0893-6080(99)00074-X doi: 10.1016/S0893-6080(99)00074-X
    [4] H. Cui, J. Guo, J. Feng, T. Wang, Global $\mu$-stability of impulsive reaction-diffusion neural networks with unbounded time-varying delays and bounded continuously distributed delays, Neurocomputing, 157 (2015), 1–10. http://dx.doi.org/10.1016/j.neucom.2015.01.044 doi: 10.1016/j.neucom.2015.01.044
    [5] C. Feng, R. Plamondon, On the stability analysis of delayed neural network systems, Neural Networks, 14 (2001), 1181–1188. http://dx.doi.org/10.1016/S0893-6080(01)00088-0 doi: 10.1016/S0893-6080(01)00088-0
    [6] Y. Guo, Global asymptotic stability analysis for integro-differential systems modeling neural networks with delays, Z. Angew. Math. Phys., 61 (2010), 971–978. http://dx.doi.org/10.1007/s00033-009-0057-4 doi: 10.1007/s00033-009-0057-4
    [7] J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, PNAS, 79 (1982), 2554–2558. http://dx.doi.org/10.1073/pnas.79.8.2554 doi: 10.1073/pnas.79.8.2554
    [8] J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, PNAS, 81 (1984), 3088–3092. http://dx.doi.org/10.1073/pnas.81.10.3088 doi: 10.1073/pnas.81.10.3088
    [9] J. Hopfield, D. Tank, Computing with neural circuits: a model, Science, 233 (1986), 625–633. http://dx.doi.org/10.1126/science.3755256 doi: 10.1126/science.3755256
    [10] J. Inoue, Retrieval phase diagrams of non-monotonic Hopfield networks, J. Phys. A: Math. Gen., 29 (1996), 4815–4826. http://dx.doi.org/10.1088/0305-4470/29/16/008 doi: 10.1088/0305-4470/29/16/008
    [11] M. Kennedy, L. Chua, Neural networks for non-linear programming, IEEE Trans. Circuits-I, 35 (1998), 554–562. http://dx.doi.org/10.1109/31.1783 doi: 10.1109/31.1783
    [12] B. Kosko, Neural networks and fuzzy systems: a dynamical systems approach to machine intelligence, New Jersey: Prentice-Hall, 1991.
    [13] B. Liu, W. Lu, T. Chen, New criterion of asymptotic stability for delay systems with time-varying structures and delays, Neural Networks, 54 (2014), 103–111. http://dx.doi.org/10.1016/j.neunet.2014.03.003 doi: 10.1016/j.neunet.2014.03.003
    [14] T. Loan, D. Tuan, Global exponential stability of a class of neural networks with unbounded delays, Ukr. Math. J., 60 (2008), 1633–1649. http://dx.doi.org/10.1007/s11253-009-0155-7 doi: 10.1007/s11253-009-0155-7
    [15] S. Mohamad, Exponential stability in Hopfield-type neural networks with impulses, Chaos Soliton. Fract., 32 (2007), 456–467. http://dx.doi.org/10.1016/j.chaos.2006.06.035 doi: 10.1016/j.chaos.2006.06.035
    [16] S. Mohamed, K. Gopalsamy, Continuous and discrete Halanay-type inequalities, Bull. Aust. Math. Soc., 61 (2000), 371–385. http://dx.doi.org/10.1017/S0004972700022413 doi: 10.1017/S0004972700022413
    [17] S. Mohamad, K. Gopalsamy, H. Akca, Exponential stability of artificial neural networks with distributed delays and large impulses, Nonlinear Anal.-Real, 9 (2008), 872–888. http://dx.doi.org/10.1016/j.nonrwa.2007.01.011 doi: 10.1016/j.nonrwa.2007.01.011
    [18] H. Qiao, J. Peng, Z. Xu, Nonlinear measures: a new approach to exponential stability analysis for Hopfield-type neural networks, IEEE Trans. Neural Networ., 12 (2001), 360–370. http://dx.doi.org/10.1109/72.914530 doi: 10.1109/72.914530
    [19] Q. Song, Z. Zhao, Global dissipativity of neural networks with both variable and unbounded delays, Chaos Soliton. Fract., 25 (2005), 393–401. http://dx.doi.org/10.1016/j.chaos.2004.11.035 doi: 10.1016/j.chaos.2004.11.035
    [20] S. Sudharsanan, M. Sundareshan, Exponential stability and a systematic synthesis of a neural network for quadratic minimization, Neural Networks, 4 (1991), 599–613. http://dx.doi.org/10.1016/0893-6080(91)90014-V doi: 10.1016/0893-6080(91)90014-V
    [21] P. van den Driessche, X. Zou, Global attractivity in delayed Hopfield neural network models, SIAM J. Appl. Math., 58 (1998), 1878–1890. http://dx.doi.org/10.1137/S0036139997321219 doi: 10.1137/S0036139997321219
    [22] Y. Wang, W. Xiong, Q. Zhou, B. Xiao, Y. Yu, Global exponential stability of cellular neural networks with continuously distributed delays and impulses, Phys. Lett. A, 350 (2006), 89–95. http://dx.doi.org/10.1016/j.physleta.2005.10.084 doi: 10.1016/j.physleta.2005.10.084
    [23] H. Yanai, S. Ammari, Auto-associative memory with two stage dynamics of non-monotonic neurons, IEEE Trans. Neural Networ., 7 (1996), 803–815. http://dx.doi.org/10.1109/72.508925 doi: 10.1109/72.508925
    [24] L. Yin, Y. Chen, Y. Zhao, Global exponential stability for a class of neural networks with continuously distributed delays, Advances in Dynamical Systems and Applications, 4 (2009), 221–229.
    [25] L. Yin, X. Fu, $\mu$-stability of impulsive neural networks with unbounded time-varying delays and continuously distributed delays, Adv. Differ. Equ., 2011 (2011), 437842. http://dx.doi.org/0.1155/2011/437842
    [26] J. Zhang, X. Jin, Global stability analysis in delayed hopfield neural network models, Neural Networks, 13 (2000), 745–753. http://dx.doi.org/10.1016/S0893-6080(00)00050-2 doi: 10.1016/S0893-6080(00)00050-2
    [27] J. Zhang, Y. Suda, T. Iwasa, Absolutely exponential stability of a class of neural networks with unbounded delay, Neural Networks, 17 (2004), 391–397. http://dx.doi.org/10.1016/j.neunet.2003.09.005 doi: 10.1016/j.neunet.2003.09.005
    [28] J. Zhou, S. Li, Z. Yang, Global exponential stability of Hopfield neural networks with distributed delays, Appl. Math. Model., 33 (2009), 1513–1520. http://dx.doi.org/10.1016/j.apm.2008.02.006 doi: 10.1016/j.apm.2008.02.006
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1026) PDF downloads(48) Cited by(0)

Article outline

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog