Pinning synchronization of neural networks under denial-of-service attacks

Shuo Zhou; Xiaoyu Zhang; Hongfei Li; Shuo Zhou; Xiaoyu Zhang; Hongfei Li

doi:10.3934/mmc.2026019

Mathematical Modelling and Control

2026, Volume 6, Issue 3: 252-262. doi: 10.3934/mmc.2026019

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Research article

Pinning synchronization of neural networks under denial-of-service attacks

1.
School of Information Science and Technology, Beijing Foreign Studies University, Beijing 100081, China
2.
College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China

Received: 07 November 2025 Revised: 26 January 2026 Accepted: 03 February 2026 Published: 01 September 2026

This work examined pinning synchronization in coupled networks with current-state coupling under denial-of-service (DoS) attacks, using a pinning impulsive control (PIC) scheme and the average-impulse-interval technique. Lyapunov-based sufficient conditions were established to guarantee stability for impulsive time-delay systems, and a comparison principle suitable for such systems was derived. To address the high dimensionality of stability criteria, inequality techniques and matrix decomposition were employed to obtain reduced-dimensional conditions. A numerical example illustrated the effectiveness and practicality of the proposed approach.
- synchronization,
- coupled networks,
- DoS Attacks,
- pinning impulsive control
Citation: Shuo Zhou, Xiaoyu Zhang, Hongfei Li. Pinning synchronization of neural networks under denial-of-service attacks[J]. Mathematical Modelling and Control, 2026, 6(3): 252-262. doi: 10.3934/mmc.2026019

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Abstract

This work examined pinning synchronization in coupled networks with current-state coupling under denial-of-service (DoS) attacks, using a pinning impulsive control (PIC) scheme and the average-impulse-interval technique. Lyapunov-based sufficient conditions were established to guarantee stability for impulsive time-delay systems, and a comparison principle suitable for such systems was derived. To address the high dimensionality of stability criteria, inequality techniques and matrix decomposition were employed to obtain reduced-dimensional conditions. A numerical example illustrated the effectiveness and practicality of the proposed approach.

References

[1]	J. Lü, G. Chen, A new chaotic attractor coined, Int. J. Bifurcat. Chaos, 12 (2002), 659–661. https://doi.org/10.1142/S0218127402004620 doi: 10.1142/S0218127402004620
[2]	A. C. Chiang, Elements of dynamic optimization, Waveland Press Inc., 2000.
[3]	R. Y. Zhong, X. Xu, E. Klotz, S. T. Newman, Intelligent manufacturing in the context of industry 4.0: a review, Engineering, 3 (2017), 616–630. https://doi.org/10.1016/j.eng.2017.05.015 doi: 10.1016/j.eng.2017.05.015
[4]	A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93–153. https://doi.org/10.1016/j.physrep.2008.09.002 doi: 10.1016/j.physrep.2008.09.002
[5]	M. Li, X. Li, Synchronization of neural networks involving unmeasurable states and impulsive disturbances by observer and feedback control, Commun. Nonlinear Sci. Numer. Simul., 125 (2023), 107396. https://doi.org/10.1016/j.cnsns.2023.107396 doi: 10.1016/j.cnsns.2023.107396
[6]	Y. Liu, H. Li, R. Lu, Z. Zuo, X. Li, An overview of finite/fixed-time control and its application in engineering systems, IEEE/CAA J. Autom. Sin., 9 (2022), 2106–2120. https://doi.org/10.1109/JAS.2022.105413 doi: 10.1109/JAS.2022.105413
[7]	X. Zhang, C. Li, H. Li, Z. Cao, Synchronization of uncertain coupled neural networks With time-varying delay of unknown bound via distributed delayed impulsive control, IEEE Trans. Neural Networks Learn. Syst., 34 (2023), 3624–3635. https://doi.org/10.1109/TNNLS.2021.3116069 doi: 10.1109/TNNLS.2021.3116069
[8]	J. Lu, D. W. C. Ho, J. Cao, A unified synchronization criterion for impulsive dynamical networks, Automatica, 46 (2010), 1215–1221. https://doi.org/10.1016/j.automatica.2010.04.005 doi: 10.1016/j.automatica.2010.04.005
[9]	K. Zhou, J. C. Doyle, Essentials of robust control, Upper Saddle River, NJ: Prentice Hall, 1998.
[10]	X. Li, C. Zhu, Saturated impulsive control of nonlinear systems with applications, Automatica, 142 (2022), 110375. https://doi.org/10.1016/j.automatica.2022.110375 doi: 10.1016/j.automatica.2022.110375
[11]	X. Li, D. Peng, J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Autom. Control, 65 (2020), 4908–4913. https://doi.org/10.1109/TAC.2020.2964558 doi: 10.1109/TAC.2020.2964558
[12]	Z. Wang, L. Yan, Y. Fan, F. Wang, H. Shen, Switching event-triggered-based gain-scheduled control for bipartite synchronization of coupled coopetitive memristive neural networks, IEEE T. Syst. Man Cybern.: Syst., 55 (2025), 6951–6963. https://doi.org/10.1109/TSMC.2025.3594542 doi: 10.1109/TSMC.2025.3594542
[13]	Z. Wang, X. Si, X. Huang, Event-triggered prescribed-time bipartite synchronization of T-S fuzzy coopetition neural networks: a parameters optimizing strategy, IEEE Trans. Ind. Inf., 22 (2026), 441–451 https://doi.org/10.1109/TII.2025.3611599 doi: 10.1109/TII.2025.3611599
[14]	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 doi: 10.1016/j.amc.2024.129108
[15]	H. Sang, C. Cui, P. Wang, Y. Liu, C. Huang, J. Cao, Further on pinning quasi-synchronization of switched directed networks with event-triggered communication, IEEE Trans. Autom. Control, 70 (2025), 6127–6134 https://doi.org/10.1109/TAC.2025.3552027 doi: 10.1109/TAC.2025.3552027
[16]	X. Zhang, C. Li, H. Li, J. Xu, Synchronization of neural networks involving distributed-delay coupling: a distributed-delay differential inequalities approach, IEEE Trans. Neural Networks Learn. Syst., 35 (2024), 8086–8096. https://doi.org/10.1109/TNNLS.2022.3224393 doi: 10.1109/TNNLS.2022.3224393
[17]	S. Zhu, J. Cao, L. Lin, J. Lam, S. Azuma, Toward stabilizable large-scale Boolean networks by controlling the minimal set of nodes, IEEE Trans. Autom. Control, 69 (2024), 174–188. https://doi.org/10.1109/TAC.2023.3269321 doi: 10.1109/TAC.2023.3269321
[18]	Y. Li, J. Feng, X. Li, S. Xu, Pinning controller design for set reachability of state-dependent impulsive Boolean networks, IEEE Trans. Neural Networks Learn. Syst., 34 (2023), 10838–10850. https://doi.org/10.1109/TNNLS.2022.3171576 doi: 10.1109/TNNLS.2022.3171576
[19]	M. Haripriya, A. Manivannan, S. Dhanasekar, S. Lakshmanan, Finite-time synchronization of delayed complex dynamical networks via sampled-data controller, Math. Model. Control, 5 (2025), 73–84. https://doi.org/10.3934/mmc.2025006 doi: 10.3934/mmc.2025006
[20]	J. Deng, H. Li, C. Hu, H. Jiang, J. Cao, State estimation of discrete-time fractional-order nonautonomous neural networks with time delays, IEEE T. Syst. Man Cybren.: Syst., 55 (2025), 3707–3719. https://doi.org/10.1109/TSMC.2025.3546945 doi: 10.1109/TSMC.2025.3546945
[21]	T. Zhang, J. Cao, M. Abdel-Aty, A. Kashkynbayev, Finite-time stability of impulsive and switched hybrid systems with delay-dependent impulses, IEEE/CAA J. Autom. Sin., 12 (2025), 466–468. https://doi.org/10.1109/JAS.2024.124758 doi: 10.1109/JAS.2024.124758
[22]	J. Yan, B. Hu, Z. H. Guan, Controllability criteria on discrete-time impulsive hybrid systems with input delay, IEEE T. Syst. Man Cybren.: Syst., 53 (2023), 2304–2316. https://doi.org/10.1109/TSMC.2022.3212533 doi: 10.1109/TSMC.2022.3212533
[23]	X. Zhang, D. Wang, B. Yang, K. Ota, M. Dong, H. Li, Exponentially synchronous results for delayed neural networks with leakage delay via switched delay idea and AED-ADT method, IEEE Trans. Neural Networks Learn. Syst., 35 (2024), 3713–3724. https://doi.org/10.1109/TNNLS.2022.3196402 doi: 10.1109/TNNLS.2022.3196402
[24]	H. L. Li, J. Cao, C. Hu, H. Jiang, A. Alsaedi, Synchronization analysis of nabla fractional-order fuzzy neural networks with time delays via nonlinear feedback control, Fuzzy Sets Syst., 475 (2024), 108750. https://doi.org/10.1016/j.fss.2023.108750 doi: 10.1016/j.fss.2023.108750
[25]	H. Zheng, Y. Tian, Exponential stability of time-delay systems with highly nonlinear impulses involving delays, Math. Model. Control, 5 (2025), 103–120. https://doi.org/10.3934/mmc.2025008 doi: 10.3934/mmc.2025008
[26]	R. Li, Q. Gan, H. Wu, J. Cao, Q. Kang, Resilient finite-time consensus of variable-order fractional multiagent systems under DoS attacks, IEEE T. Syst. Man Cybern.: Syst., 55 (2025), 3188–3201. https://doi.org/10.1109/TSMC.2025.3539656 doi: 10.1109/TSMC.2025.3539656
[27]	X. Cai, K. Shi, Y. Sun, J. Cao, S. Wen, C. Qiao, et al., Stability analysis of networked control systems under DoS attacks and security controller design with mini-batch machine learning supervision, IEEE Trans. Inf. Forensics Secur., 19 (2024), 3857–3865. https://doi.org/10.1109/TIFS.2023.3347889 doi: 10.1109/TIFS.2023.3347889
[28]	L. Zha, R. Liao, J. Liu, X. Xie, E. Tian, J. Cao, Outlier-resistant distributed filtering over sensor networks under dynamic event-triggered schemes and DoS attacks, IEEE Trans. Autom. Sci. Eng., 22 (2025), 1152–1162. https://doi.org/10.1109/TASE.2024.3360718 doi: 10.1109/TASE.2024.3360718

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