Exponential input-to-state stability for delayed inertial BAM neural networks

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

Focusing on delayed inertial bidirectional associative memory (BAM) neural networks, this paper addressed their exponential input-to-state stability (EIS). Rather than relying on conventional reduced-order methods and Lyapunov-Krasovskii functionals (LKFs), we established EIS via the characteristics method by introducing three sufficient conditions cast as linear scalar inequalities. Three illustrative numerical examples, backed by simulation results, confirmed the theoretical conclusions.

Keywords

delayed inertial BAM neural networks / exponential input-to-state stability / global exponential stability / characteristics method / reduced-order method

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] B. Kosko, Adaptive bi-directional associative memories, Appl. Opt., 26 (1987), 4947-4960. https://doi.org/10.1364/AO.26.004947

[2] B. Kosko, Bi-directional associative memories, IEEE Trans. Syst. Man Cybern., 18 (1988), 49-60. https://doi.org/10.1109/21.87054

[3] B. Kosko, Neural Networks and Fuzzy Systems-A Dynamical System Approach Machine Intelligence, Englewood Cliffs, Prentice-Hall, 1992.

[4] K. Babcock, R. Westervelt, Stability and dynamics of simple electronic neural networks with added inertia, Phys. D Nonlinear Phenom., 23 (1986), 464-469. https://doi.org/10.1016/0167-2789(86)90152-1

[5] K. Babcock, R. Westervelt, Dynamics of simple electronic neural networks, Phys. D Nonlinear Phenom., 28 (1987), 305-316. https://doi.org/10.1016/0167-2789(87)90021-2

[6] Y. Ke, C. Miao, Stability analysis of BAM neural networks with inertial term and time delay, Wseas. Trans. Syst., 10 (2011), 425-438. https://dl.acm.org/doi/10.5555/2190008.2190011

[7] Y. Ke, C. Miao, Stability and existence of periodic solutions in inertial BAM neural networks with time delay, Neural Comput. Appl., 23 (2013), 1089-1099. https://doi.org/10.1007/s00521-012-1037-8

[8] W. Wang, Mean-square exponential input-to-state stability of stochastic fuzzy delayed Cohen-Grossberg neural networks, J. Exp. Theor. Artif. Intell., 36 (2024), 1823-1836. https://doi.org/10.1080/0952813X.2023.2165725

[9] J. Liu, J. Jian, B. Wang, Stability analysis for BAM quaternion-valued inertial neural networks with time delay via nonlinear measure approach, Math. Comput. Simulation, 174 (2020), 134-152. https://dx.doi.org/10.1016/j.matcom.2020.03.002

[10] Y. Zhang, Z. Li, W. Jiang, W. Liu, The stability of anti-periodic solutions for fractional-order inertial BAM neural networks with time-delays, AIMS Math., 8 (2023), 6176-6190. https://doi.org/10.3934/math.2023312

[11] H. Liao, Z. Zhang, L. Ren, W. Peng, Global asymptotic stability of periodic solutions for inertial delayed BAM neural networks via novel computing method of degree and inequality techniques, Chaos Solitons Fractals, 104 (2017), 785-787. https://doi.org/10.1016/j.chaos.2017.09.035

[12] H. Lin, H. Zeng, X. Zhang, Stability analysis for delayed neural networks via a generalized reciprocally convex inequality, IEEE Trans. Neural Networks Learn. Syst., 34 (2023), 7491-7499. https://doi.org/10.1109/TNNLS.2022.3144032

[13] H. Zeng, Z. Zhu, W. Wang, X. Zhang, Relaxed stability criteria of delayed neural networks using delay-parameters-dependent slack matrices, Neural Networks, 180 (2024), 106676. https://doi.org/10.1016/j.neunet.2024.106676

[14] Y. Chen, C. Lu, X. Zhang, Allowable delay set flexible fragmentation approach to passivity analysis of delayed neural networks, Neurocomputing, 629 (2025), 129730. https://doi.org/10.1016/j.neucom.2025.129730

[15] Z. Zhang, Z. Quan, Global exponential stability via inequality technique for inertial BAM neural networks with time delays, Neurocomputing, 151 (2015), 1316-1326. https://doi.org/10.1016/j.neucom.2014.10.072

[16] H. Zeng, Z. Zhu, S. Xiao, X. Zhang, A switched system model for exponential stability and dissipativity of delayed neural networks, IEEE Trans. Neural Networks Learn. Syst., 36 (2025), 19708-19717. https://doi.org/10.1109/TNNLS.2025.3590251

[17] N. Cheng, W. Wang, H. Zeng, X. Liu, X. Zhang, Novel exponential-weighted integral inequality for exponential stability analysis of time-varying delay systems, Appl. Math. Lett., 172 (2026), 109730. https://doi.org/10.1016/j.aml.2025.109730

[18] H. Zeng, Y. Chen, Y. He, X. Zhang, A delay-derivative-dependent switched system model method for stability analysis of linear systems with time-varying delay, Automatica, 175 (2025), 112183. https://doi.org/10.1016/j.automatica.2025.112183

[19] R. Zhao, B. Wang, J. Jian, Lagrange Stability of BAM quaternion-valued inertial neural networks via auxiliary function-based integral inequalities, Neural Process. Lett., 54 (2022), 1351-1369. https://doi.org/10.1007/s11063-021-10685-6

[20] R. Zhao, B. Wang, J. Jian, Global µ-stabilization of quaternion-valued inertial BAM neural networks with time-varying delays via time-delayed impulsive control, Math. Comput. Simul., 202 (2022), 223-245. https://doi.org/10.1016/j.matcom.2022.05.036

[21] C. Aouiti, Q. Hui, H. Jallouli, Fixed-time stabilization of fuzzy neutral-type inertial neural networks with time-varying delay, Fuzzy Sets Syst., 411 (2021), 48-67. https://doi.org/10.1016/j.fss.2020.10.018

[22] L. Duan, J. Li, Fixed-time synchronization of fuzzy neutral-type BAM memristive inertial neural networks with proportional delays, Inf. Sci., 576 (2021), 522-541. https://doi.org/10.1016/j.ins.2021.06.093

[23] H. Wu, X. Zhao, L. Wang, J. Cao, Observer-based fixed-time topology identification and synchronization for complex networks via quantized pinning control strategy, Appl. Math. Comput., 507 (2025), 129568. https://doi.org/10.1016/j.amc.2025.129568

[24] D. Chen, Z. Zhang, Finite-time synchronization for delayed BAM neural networks by the approach of the same structural functions, Chaos Solitons Fractals, 164 (2022), 112655. https://doi.org/10.1016/j.chaos.2022.112655

[25] L. Duan, J. Li, Finite-time synchronization for a fully complex-valued BAM inertial neural network with proportional delays via non-reduced order and non-separation approach, Neurocomputing, 611 (2025), 128648. https://doi.org/10.1016/j.neucom.2024.128648

[26] X. Hou, H. Wu, J. Cao, Practical finite-time synchronization for Lur’e systems with performance constraint and actuator faults: A memory-based quantized dynamic event-triggered control strategy, Appl. Math. Comput., 487 (2025), 129108. https://doi.org/10.1016/j.amc.2024.129108

[27] Q. Peng, J. Jian, Synchronization analysis of fractional-order inertial type neural networks with time delays, Math. Comput. Simul., 205 (2023), 62-77. https://doi.org/10.1016/j.matcom.2022.09.023

[28] J. Cao, Y. Wan, Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays, Neural Networks, 53 (2014), 165-172. https://doi.org/10.1016/j.neunet.2014.02.003

[29] X. Ma, B. Liu, X. Jia, X. Zhang, Synchronization control of chaotic neural networks subject to actuator saturation under event-triggered sampling scheme, Neurocomputing, 677 (2026), 133038. https://doi.org/10.1016/j.neucom.2026.133038

[30] X. Zhao, H. Wu, J. Cao, L. Wang, Prescribed-time synchronization for complex dynamic networks of piecewise smooth systems: A hybrid event-triggering control approach, Qual. Theory Dyn. Syst., 24 (2025), 11. https://doi.org/10.1007/s12346-024-01166-x

[31] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443. https://doi.org/10.1109/9.28018

[32] Y. Jiang, S. Zhu, X. Liu, S. Wen, C. Mu, Input-to-state stability of delayed memristor-based inertial neural networks via non-reduced order method, Neural Networks, 178 (2024), 106545. https://doi.org/10.1016/j.neunet.2024.106545

[33] W. Zhang, J. Qi, X. He, Input-to-state stability of impulsive inertial memristive neural networks with time-varying delayed, J. Franklin Inst., 355 (2018), 8971-8988. https://doi.org/10.1016/j.jfranklin.2018.10.008

[34] M. Iswarya, R. Raja, J. Cao, M. Niezabitowski, J. Alzabut, C. Maharajan, New results on exponential input-to-state stability analysis of memristor based complex-valued inertial neural networks with proportional and distributed delays, Math. Comput. Simul., 201 (2022), 440-461. https://doi.org/10.1016/j.matcom.2021.01.020

[35] S. Chang, Y. Wang, X. Zhang, X. Wang, A new method to study global exponential stability of inertial neural networks with multiple time-varying transmission delays, Math. Comput. Simul., 211 (2023), 329-340. https://doi.org/10.1016/j.matcom.2023.04.008

[36] W. Wang, J. Wu, W. Chen, The characteristics method to study global exponential stability of delayed inertial neural networks, Math. Comput. Simul., 232 (2025), 91-101. https://doi.org/10.1016/j.matcom.2024.12.021

[37] W. Wang, W. Zeng, W. Chen, Global exponential stability of periodic solutions for inertial delayed BAM neural networks, Commun. Nonlinear Sci. Numer. Simul., 145 (2025), 108728. https://doi.org/10.1016/j.cnsns.2025.108728

[38] W. Wang, W. Zeng, W. Chen, New sufficient conditions on the global exponential stability of delayed inertial neural networks, Neurocomputing, 622 (2025), 129301. https://doi.org/10.1016/j.neucom.2024.129301

[39] Q. Wang, L. Duan, L. Huang, Z. Su, Global exponential stability of a periodic inertial memristive neural networks with time delays: characteristic approach, Neurocomputing, 668 (2026), 132408. https://doi.org/10.1016/j.neucom.2025.132408

[40] W. Wang, W. Chen, New study on neutral-type inertial BAM neural networks via the characteristic method, J. Math. Anal. Appl., 557 (2026), 130325. https://doi.org/10.1016/j.jmaa.2025.130325

[41] W. Wang, Further results on mean-square exponential input-to-state stability of stochastic delayed Cohen-Grossberg neural networks, Neural Process. Lett., 55 (2023), 3953-3965. https://doi.org/10.1007/s11063-022-10974-8

[42] W. Wang, Mean-square exponential input-to-state stability of stochastic delayed recurrent neural networks with local Lipschitz condition, Math. Meth. Appl. Sci., 46 (2023), 17788-17797. https://doi.org/10.1002/mma.9531