Reachable set bounding for delayed memristive neural networks via adaptive control

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Abstract

This article is concerned with the reachable set estimation (RSE) for delayed memristive neural networks (MNNs). By exploiting the differential inclusion theory and inequality techniques, the RSE problem of MNNs was investigated. A memoryless adaptive controller was designed to realize that states of MNNs converge to a bounded region. Based on this result, an updated memoryless adaptive controller was designed, which further removed the restriction that the delay derivative must be less than 1, leading to a more general result. The new results were presented in the form of algebraic criteria, which were straightforward to verify. Ultimately, the effectiveness of the proposed criteria was demonstrated through two numerical simulations.

Keywords

memristive neural networks / reachable set / adaptive control / time delay / Filippov solution

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References

[1] S. Zhao, L. Zhao, S. Wen, L. Cheng, Secure synchronization control of Markovian jump neural networks under dos attacks with memory-based adaptive event-triggered mechanism, Artif. Intell. Sci. Eng., 1(2025), 64-78. https://doi.org/10.23919/AISE.2025.000006

[2] G. Chen, Optimal synchronization of higher-order dynamical networks, Artif. Intell. Sci. Eng., 1(2025), 31-36. https://doi.org/10.23919/AISE.2025.000003

[3] I. Boybat, M. Le Gallo, S. R. Nandakumar, T. Moraitis, T. Parnel, T. Tuma, et al., Neuromorphic computing with multi-memristive synapses, Nat. Commun., 9(2018), 2514. https://doi.org/10.1038/s41467-018-04933-y

[4] H. Lin, C. Wang, Q. Deng, C. Xu, Z. Deng, C. Zhou, Review on chaotic dynamics of memristive neuron and neural network, Nonlinear Dyn., 106(2021), 959-973. https://doi.org/10.1007/s11071-021-06853-x

[5] S. Wen, T. Huang, Z. Zeng, Y. Chen, P. Li, Circuit design and exponential stabilization of memristive neural networks, Neural Networks, 63(2015), 48-56. https://doi.org/10.1016/j.neunet.2014.10.011

[6] H. Liu, L. Ma, Z. Wang, Y. Liu, F. E. Alsaadi, An overview of stability analysis and state estimation for memristive neural networks, Neurocomputing, 391(2020), 1-12. https://doi.org/10.1016/j.neucom.2020.01.066

[7] M. Roohi, S. Mirzajani, A. R. Haghighi, A. Basse-O’Connor, Robust design of two-level non-integer SMC based on deep soft actor-critic for synchronization of chaotic fractional order memristive neural networks, Fractal Fractional, 8(2024), 548. https://doi.org/10.3390/fractalfract8090548

[8] J. Zhao, F. Wang, Synchronization of reaction-diffusion delayed inertial memristive neural networks via adaptive pinning control, Neural Networks, 194(2026), 108183. https://doi.org/10.1016/j.neunet.2025.108183

[9] Y. Sheng, T. Huang, Z. Zeng, X. Miao, Global exponential stability of memristive neural networks with mixed time-varying delays, IEEE Trans. Neural Networks Learn. Syst., 32(2020), 3690-3699. https://doi.org/10.1109/TNNLS.2020.3015944

[10] G. Zhang, S. Wen, New approximate results of fixed-time stabilization for delayed inertial memristive neural networks, IEEE Trans. Circuits Syst. II: Express Briefs, 71(2024), 3428-3432. https://doi.org/10.1109/TCSII.2024.3361670

[11] H. Brahmi, B. Ammar, A. Ksibi, F. Cherif, G. Aldehim, A. M. Alimi, Finite-time complete periodic synchronization of memristive neural networks with mixed delays, Sci. Rep., 13(2023), 12545. https://doi.org/10.1038/s41598-023-37737-2

[12] L. Wang, Y. Wang, Y. Li, Finite-time synchronization of memristive neural networks modeling in terms of voltage-flux-time, IEEE Trans. Circuits Syst. II: Express Briefs, 70(2023), 3469-3473. https://doi.org/10.1109/TCSII.2023.3256004

[13] D. Li, H. Li, C. Hu, H. Jiang, J. Cao, Quasi-projective synchronization of discrete-time fractional-order delayed memristive neural networks with uncertainties, IEEE Trans. Cybern, 56(2026), 414-426. https://doi.org/10.1109/TCYB.2025.3610333

[14] H. Zhang, Y. Huang, N. Zhao, P. Shi, Improved event-triggered adaptive neural network control for multi-agent systems under denial-of-service attacks, Artif. Intell. Sci. Eng., 1(2025), 122-133. https://doi.org/10.23919/AISE.2025.000009

[15] S. N. Khoubravan, S. Mirzajani, A. Heydari, M. Roohi, Fuzzy Gain-Composite approach for robust control of chaotic fractional-order optical systems with guaranteed performance, AIMS Math., 10(2025), 27103-27128. https://doi.org/10.3934/math.20251191

[16] J. Zhao, A new result on reachable set estimation for time-varying delay singular systems, Int. J. Robust Nonlinear Control, 31(2021), 806-816. https://doi.org/10.1002/rnc.5311

[17] X. Zhang, Z. Feng, X. Zhang, On reachable set problem for impulse switched singular systems with mixed delays, IET Control Theory Appl., 17(2023), 628-638. https://doi.org/10.1049/cth2.12390

[18] L. Zhang, Z. Shen, B. Niu, N. Zhao, Dynamic-event-based reachable set synthesis for nonlinear delayed hidden semi-Markov jump systems under multiple cyber-attacks, IEEE Trans. Syst., Man, Cybern: Syst., 55(2025), 587-598. https://doi.org/10.1109/TSMC.2024.3485651

[19] C. Fan, J. Lam, X. Xie, K. Kwok, Reachable set estimation of switched systems with all subsystems unstable, IEEE Trans. Autom. Control, 70(2025), 7620-7627. https://doi.org/10.1109/TAC.2025.3575685

[20] Z. Fei, C. Guan, P. Shi, Reachable set estimation for discrete-time switched system with application to time-delay system, Int. J. Robust Nonlinear Control, 28(2018), 2468-2483. https://doi.org/10.1002/rnc.4028

[21] Y. Xue, M. Su, X. Yang, X. Zhang, State bounding and controller design for genetic regulatory networks with multiple delays and bounded disturbances, Int. J. Robust Nonlinear Control, 32(2022), 8032-8051. https://doi.org/10.1002/rnc.6262

[22] Y. Xue, L. Zhang, X. Zhang, Reachable set estimation for genetic regulatory networks with time-varying delays and bounded disturbances, Neurocomputing, 403(2020), 203-210. https://doi.org/10.1016/j.neucom.2020.03.113

[23] R. Vadivel, S. Sabarathinam, G. Zhai, N. Gunasekaran, Event-triggered reachable set estimation for synchronization of Markovian jump complex-valued delayed neural networks under cyber-attacks, Eur. Phys. J. Spec. Top., 234(2025), 3683-3703. https://doi.org/10.1140/epjs/s11734-024-01372-3

[24] L. Zhang, Q. Yin, B. Niu, X. Zhao, N. Zhao, Bumpless transfer control and reachable set estimation for Markovian networked systems against DoS attacks, Inf. Sci., 718(2025), 122396. https://doi.org/10.1016/j.ins.2025.122396

[25] Y. Shen, J. Zhao, L. Yu, Reachable set estimation of delayed second-order memristive neural networks, Appl. Math. Comput., 484(2025), 128994. https://doi.org/10.1016/j.amc.2024.128994

[26] J. Zhao, Y. Shen, L. Wang, L. Yu, Reachable set estimation of inertial complex-valued Memristive neural networks, IEEE Trans. Circuits Syst. II: Express Briefs, 72(2025), 213-217. https://doi.org/10.1109/TCSII.2024.3486746

[27] S. Zhu, Y. Gao, Y. Hou, C. Yang, Reachable set estimation for Memristive complex-valued neural networks with disturbances, IEEE Trans. Neural Networks Learn. Syst., 34(2023), 11029-11034. https://doi.org/10.1109/TNNLS.2022.3167117

[28] Y. Jiang, S. Zhu, S. Wen, C. Mu, Reachable set estimation of Memristive inertial neural networks, IEEE Trans. Circuits Syst. II: Express Briefs, 72(2025), 903-907. https://doi.org/10.1109/TCSII.2025.3572855

[29] Y. Gao, S. Zhu, J. Li, Reachable set bounding for a class of Memristive complex-valued neural networks with disturbances, Neurocomputing, 385(2020), 368-377. https://doi.org/10.1016/j.neucom.2019.12.085

[30] G. Zhang, Y. Shen, New algebraic criteria for synchronization stability of chaotic Memristive neural networks with time-varying delays, IEEE Trans. Neural Networks Learn. Syst., 24(2013), 1701-1707. https://doi.org/10.1109/TNNLS.2013.2264106

[31] E. Fridman, U. Shaked, On reachable sets for linear systems with delay and bounded peak inputs, Automatica, 39(2003), 2005-2010. https://doi.org/10.1016/S0005-1098(03)00204-8

[32] Y. Sheng, Y. Shen, Improved reachable set bounding for linear time-delay systems with disturbances, J. Franklin Institute, 353(2016), 2708-2721. https://doi.org/10.1016/j.jfranklin.2016.05.013