Stabilization of stochastic systems: event-triggered impulsive control

Huiling Li; Jin-E Zhang; Ailong Wu; Huiling Li; Jin-E Zhang; Ailong Wu

doi:10.3934/math.2025825

AIMS Mathematics

2025, Volume 10, Issue 8: 18475-18493. doi: 10.3934/math.2025825

Previous Article Next Article

Research article

Stabilization of stochastic systems: event-triggered impulsive control

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

Received: 29 May 2025 Revised: 06 July 2025 Accepted: 28 July 2025 Published: 14 August 2025
MSC : 93E03

This paper systematically establishes criteria for ensuring $ r $-th moment asymptotic stability and $ r $-th moment exponential stability in stochastic systems via an event-triggered impulsive control (ETIC) strategy, considering both scenarios with and without impulsive delay. By constructing appropriate Lyapunov functions and applying stochastic analysis methods, the intrinsic relationships among event-triggered parameters, impulsive control (IC) strength, and system stability are established. Moreover, this paper thoroughly investigates how the selection of event-triggered parameters, the length of the inspection interval, and the magnitude of impulsive delay influence the convergence rate of the system. Finally, two numerical examples are presented to verify the effectiveness of the proposed ETIC method, one of which focuses on the consensus problem of stochastic multi-agent systems.
- asymptotic stability,
- exponential stability,
- event-triggered impulsive control,
- impulsive delay,
- stochastic systems
Citation: Huiling Li, Jin-E Zhang, Ailong Wu. Stabilization of stochastic systems: event-triggered impulsive control[J]. AIMS Mathematics, 2025, 10(8): 18475-18493. doi: 10.3934/math.2025825

Related Papers:

Abstract

This paper systematically establishes criteria for ensuring $ r $-th moment asymptotic stability and $ r $-th moment exponential stability in stochastic systems via an event-triggered impulsive control (ETIC) strategy, considering both scenarios with and without impulsive delay. By constructing appropriate Lyapunov functions and applying stochastic analysis methods, the intrinsic relationships among event-triggered parameters, impulsive control (IC) strength, and system stability are established. Moreover, this paper thoroughly investigates how the selection of event-triggered parameters, the length of the inspection interval, and the magnitude of impulsive delay influence the convergence rate of the system. Finally, two numerical examples are presented to verify the effectiveness of the proposed ETIC method, one of which focuses on the consensus problem of stochastic multi-agent systems.

References

[1]	J. Xiao, Z. G. Zeng, A. L. Wu, S. P. Wen, Fixed-time synchronization of delayed Cohen-Grossberg neural networks based on a novel sliding mode, Neural Netw., 128 (2020), 1–12. https://doi.org/10.1016/j.neunet.2020.04.020 doi: 10.1016/j.neunet.2020.04.020
[2]	W. Zhu, D. D. Wang, L. Liu, G. Feng, Event-based impulsive control of continuous-time dynamic systems and its application to synchronization of memristive neural networks, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 3599–3609. https://doi.org/10.1109/TNNLS.2017.2731865 doi: 10.1109/TNNLS.2017.2731865
[3]	H. S. Yan, G. B. Wang, Adaptive tracking control for stochastic nonlinear systems with time-varying delays using multi-dimensional Taylor network, ISA Trans., 132 (2023), 246–257. https://doi.org/10.1016/j.isatra.2022.06.004 doi: 10.1016/j.isatra.2022.06.004
[4]	X. F. Xing, H. Q. Wu, J. D. Cao, Event-triggered impulsive control for synchronization in finite time of fractional-order reaction-diffusion complex networks, Neurocomputing, 557 (2023), 126703. https://doi.org/10.1016/j.neucom.2023.126703 doi: 10.1016/j.neucom.2023.126703
[5]	X. Y. Guo, C. L. Wang, L. Liu, Adaptive fault-tolerant control for a class of nonlinear multi-agent systems with multiple unknown time-varying control directions, Automatica, 167 (2024), 111802. https://doi.org/10.1016/j.automatica.2024.111802 doi: 10.1016/j.automatica.2024.111802
[6]	J. L. Yu, X. W. Dong, Q. D. Li, J. H. Lü, Z. Ren, Adaptive practical optimal time-varying formation tracking control for disturbed high-order multi-agent systems, IEEE Trans. Circuits Syst. I Regul. Pap., 69 (2022), 2567–2578. https://doi.org/10.1109/TCSI.2022.3151464 doi: 10.1109/TCSI.2022.3151464
[7]	T. Dong, R. He, H. Q. Li, W. J. Hu, T. W. Huang, Exponential stabilization of phase-change inertial neural networks with time-varying delays, IEEE Trans. Syst. Man Cybern. Syst., 55 (2025), 2659–2669. https://doi.org/10.1109/TSMC.2024.3525038 doi: 10.1109/TSMC.2024.3525038
[8]	K. J. Åström, B. Bernhardsson, Comparison of periodic and event based sampling for first-order stochastic systems, IFAC Proc. Vol., 32 (1999), 5006–5011. https://doi.org/10.1016/S1474-6670(17)56852-4 doi: 10.1016/S1474-6670(17)56852-4
[9]	K. E. Åarzén, A simple event-based PID controller, IFAC Proc. Vol., 32 (1999), 8687–8692. https://doi.org/10.1016/S1474-6670(17)57482-0 doi: 10.1016/S1474-6670(17)57482-0
[10]	J. J. Chen, B. S. Chen, Z. G. Zeng, Exponential quasi-synchronization of coupled delayed memristive neural networks via intermittent event-triggered control, Neural Netw., 141 (2021), 98–106. https://doi.org/10.1016/j.neunet.2021.01.013 doi: 10.1016/j.neunet.2021.01.013
[11]	Y. B. Wu, J. Sun, N. Gunasekaran, J. Kurths, J. Liu, Exponential output synchronization of complex networks with output coupling via intermittent event-triggered control, IEEE Trans. Control Netw. Syst., 11 (2024), 284–294. https://doi.org/10.1109/TCNS.2023.3280457 doi: 10.1109/TCNS.2023.3280457
[12]	X. F. Xing, H. Q. Wu, J. D. Cao, Event-triggered impulsive control for synchronization in finite time of fractional-order reaction-diffusion complex networks, Neurocomputing, 557 (2023), 126703. https://doi.org/10.1016/j.neucom.2023.126703 doi: 10.1016/j.neucom.2023.126703
[13]	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
[14]	W. Sun, H. N. Zheng, W. L. Guo, Y. H. Xu, J. D. Cao, M. Abdel-Aty, et al., Quasisynchronization of heterogeneous dynamical networks via event-triggered impulsive controls, IEEE Trans. Cybern., 52 (2022), 228–239. https://doi.org/10.1109/TCYB.2020.2975234 doi: 10.1109/TCYB.2020.2975234
[15]	B. X. Jiang, J. F. Wei, Y. Liu, W. H. Gui, Periodic event-triggered impulsive control of linear uncertain systems, IEEE Trans. Autom. Sci. Eng., 22 (2025), 6876–6883. https://doi.org/10.1109/TASE.2024.3456125 doi: 10.1109/TASE.2024.3456125
[16]	Z. H. Hu, X. W. Mu, Event-triggered impulsive control for nonlinear stochastic systems, IEEE Trans. Cybern., 52 (2022), 7805–7813. https://doi.org/10.1109/TCYB.2021.3052166 doi: 10.1109/TCYB.2021.3052166
[17]	B. Liu, Z. J. Sun, M. Li, D. N. Liu, Stabilization via event-triggered impulsive control with constraints for switched stochastic systems, IEEE Trans. Cybern., 52 (2022), 11834–11846. https://doi.org/10.1109/TCYB.2021.3073023 doi: 10.1109/TCYB.2021.3073023
[18]	Z. H. Hu, X. W. Mu, Event-triggered impulsive control for stochastic networked control systems under cyber attacks, IEEE Trans. Syst. Man Cybern. Syst., 52 (2022), 5636–5645. https://doi.org/10.1109/TSMC.2021.3130614 doi: 10.1109/TSMC.2021.3130614
[19]	H. H. Guo, J. Liu, C. K. Ahn, Y. B. Wu, W. X. Li, Dynamic event-triggered impulsive control for stochastic nonlinear systems with extension in complex networks, IEEE Trans. Circuits Syst. I Regul. Pap., 69 (2022), 2167–2178. https://doi.org/10.1109/TCSI.2022.3141583 doi: 10.1109/TCSI.2022.3141583
[20]	Y. Liu, J. Y. Xu, J. Q. Lu, W. H. Gui, Stability of stochastic time-delay systems involving delayed impulses, Automatica, 152 (2023), 110955. https://doi.org/10.1016/j.automatica.2023.110955 doi: 10.1016/j.automatica.2023.110955
[21]	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
[22]	W. Zhang, J. J. Chen, S. P. Wen, T. W. Huang, Event-triggered random delayed impulsive consensus of multi-agent systems with time-varying delay, IEEE Trans. Emerg. Top. Comput. Intell., 9 (2025), 2059–2064. https://doi.org/10.1109/TETCI.2024.3377507 doi: 10.1109/TETCI.2024.3377507
[23]	Z. D. Ai, G. D. Zong, W. X. Zheng, Stabilization of a class of Lipschitz nonlinear systems through an event-triggered impulsive controller, IEEE Trans. Autom. Sci. Eng., 22 (2025), 4145–4153. https://doi.org/10.1109/TASE.2024.3407956 doi: 10.1109/TASE.2024.3407956
[24]	S. X. Luo, Stability and $L_{2}$-gain analysis of linear periodic event-triggered systems with large delays, IEEE Trans. Circuits Syst. II Express Briefs, 70 (2023), 2117–2121. https://doi.org/10.1109/TCSII.2022.3229132 doi: 10.1109/TCSII.2022.3229132
[25]	K. P. Gao, J. Q. Lu, W. X. Zheng, X. Y. Chen, Synchronization in coupled neural networks with hybrid delayed impulses: average impulsive delay-gain method, IEEE Trans. Neural Netw. Learn. Syst., 36 (2025), 3608–3617. https://doi.org/10.1109/TNNLS.2024.3357515 doi: 10.1109/TNNLS.2024.3357515
[26]	H. F. Xu, Z. X. Zhu, W. X. Zheng, Exponential stability of stochastic nonlinear delay systems subject to multiple periodic impulses, IEEE Trans. Autom. Control, 69 (2024), 2621–2628. https://doi.org/10.1109/TAC.2023.3335005 doi: 10.1109/TAC.2023.3335005
[27]	X. Y. Zhang, C. D. Li, H. F. Li, Finite-time stabilization of nonlinear systems via impulsive control with state-dependent delay, J. Franklin Inst., 359 (2022), 1196–1214. https://doi.org/10.1016/j.jfranklin.2021.11.013 doi: 10.1016/j.jfranklin.2021.11.013
[28]	S. C. Wu, X. D. Li, Finite-time stability of nonlinear systems with delayed impulses, IEEE Trans. Syst. Man Cybern. Syst., 53 (2023), 7453–7460. https://doi.org/10.1109/TSMC.2023.3298071 doi: 10.1109/TSMC.2023.3298071
[29]	S. Dashkovskiy, M. Kosmykov, A. Mironchenko, L. Naujok, Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods, Nonlinear Anal. Hybrid Syst., 6 (2012), 899–915. https://doi.org/10.1016/j.nahs.2012.02.001 doi: 10.1016/j.nahs.2012.02.001
[30]	L. G. Xu, B. Z. Bao, H. X. Hu, Stability of impulsive delayed switched systems with conformable fractional-order derivatives, Int. J. Syst. Sci., 56 (2025), 1271–1288. https://doi.org/10.1080/00207721.2024.2421454 doi: 10.1080/00207721.2024.2421454
[31]	M. Y. Li, X. Y. Yang, X. D. Li, Delayed impulsive control for lag synchronization of delayed neural networks involving partial unmeasurable states, IEEE Trans. Neural Netw. Learn. Syst., 35 (2024), 783–791. https://doi.org/10.1109/TNNLS.2022.3177234 doi: 10.1109/TNNLS.2022.3177234
[32]	S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, Philadelphia: SIAM, 1994. https://doi.org/10.1137/1.9781611970777
[33]	B. Liu, D. J. Hill, C. F. Zhang, Z. J. Sun, Stabilization of discrete-time dynamical systems under event-triggered impulsive control with and without time-delays, J. Syst. Sci. Complex., 31 (2018), 130–146. https://doi.org/10.1007/s11424-018-7135-7 doi: 10.1007/s11424-018-7135-7
[34]	L. J. Pan, J. D. Cao, Exponential stability of impulsive stochastic functional differential equations, J. Math. Anal. Appl., 382 (2011), 672–685. https://doi.org/10.1016/j.jmaa.2011.04.084 doi: 10.1016/j.jmaa.2011.04.084
[35]	F. Q. Yao, L. Qiu, 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
[36]	X. Mao, Stochastic differential equations and applications, 2 Eds., Elsevier, 2007.
[37]	F. C. Klebaner, Introduction to stochastic calculus with applications, 3 Eds., London: Imperial College Press, 2012. https://doi.org/10.1142/p821

Reader Comments

Your name:*

Email:*
© 2025 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)