Stability of nonlinear stochastic systems under event-triggered impulsive control with distributed-delay impulses

Bing Shang; Jin-E Zhang; Bing Shang; Jin-E Zhang

doi:10.3934/mbe.2025085

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 9: 2339-2359. doi: 10.3934/mbe.2025085

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Research article

Stability of nonlinear stochastic systems under event-triggered impulsive control with distributed-delay impulses

Bing Shang ,
Jin-E Zhang ^,

School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

Received: 19 April 2025 Revised: 12 June 2025 Accepted: 09 July 2025 Published: 18 July 2025

This paper investigated the stability of nonlinear stochastic systems with distributed-delay impulses within the framework of event-triggered impulsive control (ETIC). A continuous event-triggered mechanism (ETM) with a fixed waiting time and a periodic ETM with a fixed sampling period were proposed, effectively eliminating the occurrence of Zeno behavior. By employing the Lyapunov method and mathematical induction, a set of sufficient conditions was established to ensure the p-th moment uniform stability (p-US) and p-th moment exponential stability (p-ES) of the considered system. Furthermore, the theoretical results were applied to a class of nonlinear stochastic systems. Utilizing the linear matrix inequality (LMI) approach, a joint design of the ETM and impulsive control gains was achieved. Finally, numerical examples were provided to demonstrate the effectiveness and feasibility of the proposed theoretical results.
- nonlinear stochastic system,
- event-triggered impulsive control,
- distributed-delay impulses,
- stability
Citation: Bing Shang, Jin-E Zhang. Stability of nonlinear stochastic systems under event-triggered impulsive control with distributed-delay impulses[J]. Mathematical Biosciences and Engineering, 2025, 22(9): 2339-2359. doi: 10.3934/mbe.2025085

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Abstract

This paper investigated the stability of nonlinear stochastic systems with distributed-delay impulses within the framework of event-triggered impulsive control (ETIC). A continuous event-triggered mechanism (ETM) with a fixed waiting time and a periodic ETM with a fixed sampling period were proposed, effectively eliminating the occurrence of Zeno behavior. By employing the Lyapunov method and mathematical induction, a set of sufficient conditions was established to ensure the p-th moment uniform stability (p-US) and p-th moment exponential stability (p-ES) of the considered system. Furthermore, the theoretical results were applied to a class of nonlinear stochastic systems. Utilizing the linear matrix inequality (LMI) approach, a joint design of the ETM and impulsive control gains was achieved. Finally, numerical examples were provided to demonstrate the effectiveness and feasibility of the proposed theoretical results.

References

[1]	X. Q. Yao, S. M. Zhang, Y. H. Du, Hybrid impulsive control based synchronization of leakage and multiple delayed fractional-order neural networks with parameter mismatch, Neural Process. Lett., 55 (2023), 11371–11395. https://doi.org/10.1007/s11063-023-11380-4 doi: 10.1007/s11063-023-11380-4
[2]	N. Zhang, S. J. Jiang, W. X. Li, Stability of stochastic state-dependent delayed complex networks under stochastic hybrid impulsive control, Syst. Control Lett., 174 (2023), 105494. https://doi.org/10.1016/j.sysconle.2023.105494 doi: 10.1016/j.sysconle.2023.105494
[3]	Z. S. Shuai, P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513–1532. https://doi.org/10.1137/120876642 doi: 10.1137/120876642
[4]	C. Z. Yuan, F. Wu, Delay scheduled impulsive control for networked control systems, IEEE Trans. Control Netw. Syst., (2016), 587–597. http://doi.org/10.1109/TCNS.2016.2541341 doi: 10.1109/TCNS.2016.2541341
[5]	J. C. Sanchez, C. Louembet, F. Gavilan, R. Vazquez, Event-based impulsive control for spacecraft rendezvous hovering phases, J. Guid. Control Dyn., 44 (2021), 1794–1810. https://doi.org/10.2514/1.G005507 doi: 10.2514/1.G005507
[6]	J. D. Cao, T. Stamov, G. Stamov, I. Stamova, Impulsive controllers design for the practical stability analysis of gene regulatory networks with distributed delays, Fractal Fract., 7 (2023), 847. https://doi.org/10.3390/fractalfract7120847 doi: 10.3390/fractalfract7120847
[7]	W. X. Shi, S. Lu, J. Z. Zhang, Impulsive control for a plant-pest-natural enemy model with stage structure, J. Appl. Anal. Comput., 15 (2025), 261–285. http://doi.org/10.11948/20240083 doi: 10.11948/20240083
[8]	G. Stamov, I. Stamova, C. Spirova, Impulsive reaction-diffusion delayed models in biology: Integral manifolds approach, Entropy, 23 (2021), 1631. https://doi.org/10.3390/e23 doi: 10.3390/e23
[9]	Y. B. Wu, Z. R Guo, L. Xue, C. K. Ahn, J. Liu, Stabilization of complex networks under asynchronously intermittent event-triggered control, Automatica, 161 (2024), 111493. https://doi.org/10.1016/j.automatica.2023.111493 doi: 10.1016/j.automatica.2023.111493
[10]	P. J. Ning, C. C. Hua, K. Li, R. Meng, Event-triggered control for nonlinear uncertain systems via a prescribed-time approach, IEEE Trans. Autom. Control, 68 (2023), 6975–6981. https://doi.org/10.1109/TAC.2023.3243863 doi: 10.1109/TAC.2023.3243863
[11]	W. Yao, C. H. Wang, Y. C. Sun, S. Q. Gong, H. R. Lin, Event-triggered control for robust exponential synchronization of inertial memristive neural networks under parameter disturbance, Neural Networks, 164 (2023), 67–80. https://doi.org/10.1016/j.neunet.2023.04.024 doi: 10.1016/j.neunet.2023.04.024
[12]	X. T. Liu, Y. Guo, M. Z. Li, Y. F. Zhang, Exponential synchronization of complex networks via intermittent dynamic event-triggered control, Neurocomputing, 581 (2024), 127478. https://doi.org/10.1016/j.neucom.2024.127478 doi: 10.1016/j.neucom.2024.127478
[13]	Z. H. Hu, X. W. Mu, Event-triggered impulsive control for nonlinear stochastic systems, IEEE Trans. Cybern., 52 (2021), 7805–7813. https://doi.org/10.1109/TCYB.2021.3052166 doi: 10.1109/TCYB.2021.3052166
[14]	P. L. Yu, F. Q. Deng, X. Y. Zhao, Y. J. Huang, Stability analysis of nonlinear systems in the presence of event-triggered impulsive control, Int. J. Robust Nonlinear Control, 34 (2024), 3835–3853. https://doi.org/10.1002/rnc.7165 doi: 10.1002/rnc.7165
[15]	L. N. Liu, C. L. Pan, J. Y. Fang, Event-triggered impulsive control of nonlinear stochastic systems with exogenous disturbances, Int. J. Robust Nonlinear Control, 35 (2025), 1654–1665. https://doi.org/10.1002/rnc.7746 doi: 10.1002/rnc.7746
[16]	Y. L. Zhang, L. Q. Yang, K. I. Kou, Y. Liu, Synchronization of fractional-order quaternion-valued neural networks with image encryption via event-triggered impulsive control, Knowl. Based Syst., 296 (2024), 111953. https://doi.org/10.1016/j.knosys.2024.111953 doi: 10.1016/j.knosys.2024.111953
[17]	X. D. Li, D. X. Peng, J. D. 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
[18]	K. X. Zhang, E. Braverman, Event-triggered impulsive control for nonlinear systems with actuation delays, IEEE Trans. Autom. Control, 68 (2022), 540–547. http://doi.org/10.1109/TAC.2022.3142127 doi: 10.1109/TAC.2022.3142127
[19]	X. D. Li, W. L. Liu, S. Gorbachev, J. D. Cao, Event-triggered impulsive control for input-to-state stabilization of nonlinear time-delay systems, IEEE Trans. Cybern., 54 (2023), 2536–2544. https://doi.org/10.1109/TCYB.2023.3270487 doi: 10.1109/TCYB.2023.3270487
[20]	D. X. Peng, X. D. Li, R. Rakkiyappan, Y. H. Ding, Stabilization of stochastic delayed systems: Event-triggered impulsive control, Appl. Math. Comput., 401 (2021), 126054. https://doi.org/10.1016/j.amc.2021.126054 doi: 10.1016/j.amc.2021.126054
[21]	M. Z. Wang, P. Li, X. D. Li, Event-triggered delayed impulsive control for input-to-state stability of nonlinear impulsive systems, Nonlinear Anal. Hybrid Syst., 47 (2023), 101277. https://doi.org/10.1016/j.nahs.2022.101277 doi: 10.1016/j.nahs.2022.101277
[22]	P. F. Wang, W. Y. Guo, H. Su, Finite-time stability via event-triggered delayed impulse control for time-varying nonlinear impulsive systems, J. Franklin Inst., 361 (2024), 107152. https://doi.org/10.1016/j.jfranklin.2024.107152 doi: 10.1016/j.jfranklin.2024.107152
[23]	M. Y. Shi, L. L. Li, J. D. Cao, L. Hua, M. Abdel-Aty, Stability analysis of inertial delayed neural network with delayed impulses via dynamic event-triggered impulsive control, Neurocomputing, 626 (2025), 129573. https://doi.org/10.1016/j.neucom.2025.129573 doi: 10.1016/j.neucom.2025.129573
[24]	M. Z. Wang, S. C. Shu, X. D. Li, Event-triggered delayed impulsive control for nonlinear systems with applications, J. Franklin Inst., 358 (2021), 4277–4291. https://doi.org/10.1016/j.jfranklin.2021.03.021 doi: 10.1016/j.jfranklin.2021.03.021
[25]	Z. L. Liu, M. Z. Luo, J. Cheng, K. B. Shi, Synchronization control of partial-state-based neural networks: event-triggered impulsive control with distributed actuation delay, Phys. A, 657 (2025), 130228. https://doi.org/10.1016/j.physa.2024.130228 doi: 10.1016/j.physa.2024.130228
[26]	X. Z. Liu, K. X. Zhang, Stabilization of nonlinear time-delay systems: Distributed-delay dependent impulsive control, Syst. Control Lett., 120 (2018), 17–22. https://doi.org/10.1016/j.sysconle.2018.07.012 doi: 10.1016/j.sysconle.2018.07.012
[27]	Q. Fang, M. Z. Wang, X. D. Li, Event-triggered distributed delayed impulsive control for nonlinear systems with applications to complex networks, Chaos Solitons Fractals, 175 (2023), 113943. https://doi.org/10.1016/j.chaos.2023.113943 doi: 10.1016/j.chaos.2023.113943
[28]	X. Y. Zhang, C. D. Li, H. F. Li, Z. G. 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 (2021), 3624–3635. https://doi.org/10.1109/TNNLS.2021.3116069 doi: 10.1109/TNNLS.2021.3116069
[29]	D. P. Kuang, D. D. Gao, J. L. Li, Stabilization of nonlinear stochastic systems via event-triggered impulsive control, Math. Comput. Simul., 233 (2025), 389–399. https://doi.org/10.1016/j.matcom.2025.01.025 doi: 10.1016/j.matcom.2025.01.025
[30]	K. It$\hat{o}$, On Stochastic Differential Equations, American Mathematical Society, 1951.
[31]	J. Y. Xu, Y. Liu, J. L. Qiu, J. Q. Lu, Event-triggered impulsive control for nonlinear stochastic delayed systems and complex networks, Commun. Nonlinear Sci. Numer. Simul., 140 (2025), 108305. https://doi.org/10.1016/j.cnsns.2024.108305 doi: 10.1016/j.cnsns.2024.108305
[32]	M. L. Xia, L. N. Liu, J. Y. Fang, B. Y. Qu, Exponentially weighted input-to-state stability of stochastic differential systems via event-triggered impulsive control, Chaos Solitons Fractals, 182 (2024), 114836. https://doi.org/10.1016/j.chaos.2024.114836 doi: 10.1016/j.chaos.2024.114836
[33]	J. Li, Q. X. Zhu, Event-triggered impulsive control of stochastic functional differential systems, Chaos Solitons Fractals, 170 (2023), 113416. https://doi.org/10.1016/j.chaos.2023.113416 doi: 10.1016/j.chaos.2023.113416
[34]	F. Q. Yao, Q. Li, H. Shen, On input-to-state stability of impulsive stochastic systems, J. Franklin Inst., 351 (2014), 4636–4651. https://doi.org/10.1016/j.jfranklin.2014.06.011 doi: 10.1016/j.jfranklin.2014.06.011
[35]	T. Caraballo, M. A. Hammami, L. Mchiri, On the practical global uniform asymptotic stability of stochastic differential equations, Stochastics, 88 (2016), 45–56. https://doi.org/10.1080/17442508.2015.1029719 doi: 10.1080/17442508.2015.1029719
[36]	W. H. Li, L. Shi, M. J. Shi, J. F. Yue, B. X. Lin, K. Y. Qin, Analyzing containment control performance for fractional-order multi-mgent systems via a delay margin perspective, IEEE Trans. Netw. Sci. Eng., 11 (2024), 2810–2821. http://doi.org/10.1109/TNSE.2024.3350122 doi: 10.1109/TNSE.2024.3350122
[37]	D. B. Fan, X. F. Zhang, C. Y. Wen, Exponential regulation of uncertain nonlinear triangular impulsive systems: A logic-based switching gain approach, IEEE Trans. Autom. Control, 99 (2025), 1–8. http://doi.org/10.1109/TAC.2025.3545697 doi: 10.1109/TAC.2025.3545697
[38]	S. Z. Wu, X. Liang, Y. C. Fang, W. He, Geometric maneuvering for underactuated VTOL vehicles, IEEE Trans. Autom. Control, 69 (2023), 1507–1519. http://doi.org/10.1109/TAC.2023.3324268 doi: 10.1109/TAC.2023.3324268

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