Stability of neutral highly nonlinear hybrid stochastic delayed systems with delayed impulses

Jia Li; Jin-E Zhang; Jia Li; Jin-E Zhang

doi:10.3934/math.2026642

AIMS Mathematics

2026, Volume 11, Issue 6: 15626-15648. doi: 10.3934/math.2026642

Previous Article Next Article

Research article

Stability of neutral highly nonlinear hybrid stochastic delayed systems with delayed impulses

Jia Li ,
Jin-E Zhang ^,

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

Received: 18 March 2026 Revised: 20 May 2026 Accepted: 26 May 2026 Published: 03 June 2026
MSC : 93C10, 93D40

This paper investigates the stability of a class of neutral highly nonlinear hybrid stochastic delayed systems with delayed impulses. Based on the comparison principle and stochastic analysis techniques, the lth moment asymptotic stability and the almost sure asymptotic stability of the argumented system are proven. Finally, the effectiveness of the theoretical results is demonstrated through two examples.
- delayed impulses,
- high nonlinearity,
- hybrid stochastic system,
- neutral term,
- almost sure asymptotic stability
Citation: Jia Li, Jin-E Zhang. Stability of neutral highly nonlinear hybrid stochastic delayed systems with delayed impulses[J]. AIMS Mathematics, 2026, 11(6): 15626-15648. doi: 10.3934/math.2026642

Related Papers:

Abstract

This paper investigates the stability of a class of neutral highly nonlinear hybrid stochastic delayed systems with delayed impulses. Based on the comparison principle and stochastic analysis techniques, the lth moment asymptotic stability and the almost sure asymptotic stability of the argumented system are proven. Finally, the effectiveness of the theoretical results is demonstrated through two examples.

References

[1]	P. Talatchian, M. W. Daniels, A. Madhavan, M. R. Pufall, E. Jué, W. H. Rippard, et al., Mutual control of stochastic switching for two electrically coupled superparamagnetic tunnel junctions, Phys. Rev. B, 104 (2021), 054427. https://doi.org/10.1103/PhysRevB.104.054427 doi: 10.1103/PhysRevB.104.054427
[2]	M. S. Krause, Measurement validity is fundamentally a matter of definition, not correlation, Rev. Gen. Psychol., 16 (2012), 391–400. https://doi.org/10.1037/a0027701 doi: 10.1037/a0027701
[3]	C. Y. Ji, D. Q. Jiang, N. Z. Shi, Two-group SIR epidemic model with stochastic perturbation, Acta Math. Sin., 28 (2012), 2545–2560. https://doi.org/10.1007/s10114-012-9668-3 doi: 10.1007/s10114-012-9668-3
[4]	X. R. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica, 49 (2013), 3677–3681. https://doi.org/10.1016/j.automatica.2013.09.005 doi: 10.1016/j.automatica.2013.09.005
[5]	S. R. You, W. Liu, J. Q. Lu, X. R. Mao, Q. W. Qiu, Stabilization of hybrid systems by feedback control based on discrete-time state observations, SIAM J. Control Optim., 53 (2015), 905–925. https://doi.org/10.1137/140985779 doi: 10.1137/140985779
[6]	Y. Chen, W. X. Zheng, Stability analysis and control for switched stochastic delayed systems, Int. J. Robust Nonlinear Control, 26 (2016), 303–328. https://doi.org/10.1002/rnc.3314 doi: 10.1002/rnc.3314
[7]	Y. L. Wu, M. X. Liu, X. T. Wu, W. B. Zhang, J. D. Cao, Input-to-state stability analysis for stochastic delayed systems with Markovian switching, IEEE Access, 5 (2017), 23663–23671. https://doi.org/10.1109/ACCESS.2017.2759304 doi: 10.1109/ACCESS.2017.2759304
[8]	W. Y. Fei, L. J. Hu, X. R. Mao, M. X. Shen, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 82 (2017), 165–170. https://doi.org/10.1016/j.automatica.2017.04.050 doi: 10.1016/j.automatica.2017.04.050
[9]	H. Q. Peng, Q. X. Zhu, Finite-time stability and stabilization of highly nonlinear stochastic systems via the noise control, IEEE Trans. Autom. Control, 2026. https://doi.org/10.1109/TAC.2026.3660601
[10]	F. Chen, W. Y. Fei, X. R. Mao, M. X. Shen, L. T. Yan, Stability analysis of highly nonlinear hybrid multiple-delay stochastic differential equations, J. Appl. Anal. Comput., 9 (2019), 1053–1070. https://doi.org/10.11948/2156-907X.20180257 doi: 10.11948/2156-907X.20180257
[11]	H. L. Dong, J. Tang, X. R. Mao, Stabilization of highly nonlinear hybrid stochastic differential delay equations with Lévy noise by delay feedback control, SIAM J. Control Optim., 6 (2022), 3302–3325. https://doi.org/10.1137/22M1480392 doi: 10.1137/22M1480392
[12]	Y. Zhao, Q. X. Zhu, Stabilization by delay feedback control for highly nonlinear switched stochastic systems with time delays, Int. J. Robust Nonlinear Control, 31 (2021), 3070–3089. https://doi.org/10.1002/rnc.5434 doi: 10.1002/rnc.5434
[13]	R. L. Song, B. Wang, Q. X. Zhu, Stabilization by variable-delay feedback control for highly nonlinear hybrid stochastic differential delay equations, Syst. Control Lett., 157 (2021), 105041. https://doi.org/10.1016/j.sysconle.2021.105041 doi: 10.1016/j.sysconle.2021.105041
[14]	W. N. Zhou, Q. Y. Zhu, P. Shi, H. Y. Su, J. A. Fang, L. W. Zhou, Adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching parameters, IEEE Trans. Cybern., 44 (2014), 2848–2860. https://doi.org/10.1109/TCYB.2014.2317236 doi: 10.1109/TCYB.2014.2317236
[15]	Y. L. Huang, Q. X. Zhu, $pth$ moment exponential stability of highly nonlinear neutral hybrid stochastic delayed systems with impulsive effect, IEEE Trans. Syst. Man Cybern. Syst., 55 (2025), 6486–6493. https://doi.org/10.1109/TSMC.2025.3579565 doi: 10.1109/TSMC.2025.3579565
[16]	R. A. El-Nabulsi, O. Moaaz, O. Bazighifan, New results for oscillatory behavior of fourth-order differential equations, Symmetry, 12 (2020), 136. https://doi.org/10.3390/sym12010136 doi: 10.3390/sym12010136
[17]	A. Al-Jaser, F. Alharbi, D. Chalishajar, B. Qaraad, Functional differential equations with non-canonical operator: Oscillatory features of solutions, Axioms, 14 (2025), 588. https://doi.org/10.3390/axioms14080588 doi: 10.3390/axioms14080588
[18]	H. B. Chen, P. Shi, C. C. Lim, P. Hu, Exponential stability for neutral stochastic markov systems with time-varying delay and its applications, IEEE Trans. Cybern., 46 (2016), 1350–1362. https://doi.org/10.1109/TCYB.2015.2442274 doi: 10.1109/TCYB.2015.2442274
[19]	C. Tan, Q. X. Zhu, On exponential stability in pth moment of neutral Markov switched stochastic time-delay systems, J. Franklin Inst., 360 (2023), 12855–12874. https://doi.org/10.1016/j.jfranklin.2023.09.051 doi: 10.1016/j.jfranklin.2023.09.051
[20]	M. Milošević, Convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay, J. Comput. Appl. Math., 280 (2015), 248–264. https://doi.org/10.1016/j.cam.2014.12.002 doi: 10.1016/j.cam.2014.12.002
[21]	X. R. Mao, Y. Shen, C. G. Yuan, Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stoch. Process. Appl., 118 (2008), 1385–1406. https://doi.org/10.1016/j.spa.2007.09.005 doi: 10.1016/j.spa.2007.09.005
[22]	X. Y. Li, X. R. Mao, A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica, 48 (2012), 2329–2334. https://doi.org/10.1016/j.automatica.2012.06.045 doi: 10.1016/j.automatica.2012.06.045
[23]	Y. Zhao, Q. X. Zhu, Stabilization of stochastic highly nonlinear delay systems with neutral term, IEEE Trans. Autom. Control, 68 (2023), 2544–2551. https://doi.org/10.1109/TAC.2022.3186827 doi: 10.1109/TAC.2022.3186827
[24]	Y. Zhao, Q. X. Zhu, Stabilization of highly nonlinear neutral stochastic systems with Markovian switching by periodically intermittent feedback control, Int. J. Robust Nonlinear Control, 32 (2022), 10201–10214. https://doi.org/10.1002/rnc.6403 doi: 10.1002/rnc.6403
[25]	F. Y. Zhao, Y. J. Li, H. T. Fan, S. D. Lei, Stabilization of highly nonlinear neutral multiple delays hybrid systems with time-changed Lévy noise by delay feedback control, Phys. Scr., 100 (2025), 065201. https://doi.org/10.1088/1402-4896/adcb73 doi: 10.1088/1402-4896/adcb73
[26]	Z. G. Liu, Q. X. Zhu, Delay feedback control of highly nonlinear neutral stochastic delay differential equations driven by G-Brownian motion, Syst. Control Lett., 181 (2023), 105640. https://doi.org/10.1016/j.sysconle.2023.105640 doi: 10.1016/j.sysconle.2023.105640
[27]	C. Anusha, C. Ravichandran, F. Aljuaydi, K. S. Nisar, D. Chalishajar, Stability and solutions of adjoint nonlinear impulsive neutral mixed integral equations in dynamic systems, J. Appl. Math. Comput., 71 (2025), 7521–7541. https://doi.org/10.1007/s12190-025-02579-w doi: 10.1007/s12190-025-02579-w
[28]	H. F. Xu, Q. X. Zhu, Stability of discrete-time impulsive stochastic systems with hybrid non-deterministic delays, IEEE Trans. Autom. Control, 2026. https://doi.org/10.1109/TAC.2026.3666702
[29]	P. Cheng, F. Q. Deng, F. Q. Yao, Exponential stability analysis of impulsive stochastic functional differential systems with delayed impulses, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2104–2114. https://doi.org/10.1016/j.cnsns.2013.10.008 doi: 10.1016/j.cnsns.2013.10.008
[30]	H. F. Xu, Q. X. Zhu, New criteria on pth moment exponential stability of stochastic delayed differential systems subject to average-delay impulses, Syst. Control Lett., 164 (2022), 105234. https://doi.org/10.1016/j.sysconle.2022.105234 doi: 10.1016/j.sysconle.2022.105234
[31]	Y. Lu, D. H. Ruan, Q. X. Zhu, Stability of nonlinear systems with multi-delayed random impulses: Average estimation and delay approach, AIMS Math., 10 (2025), 22075–22091. https://doi.org/10.3934/math.2025982 doi: 10.3934/math.2025982
[32]	J. Zhang, S. Zhu, Y. Che, Stability of highly nonlinear hybrid stochastic delayed systems with multiple periodic delayed impulses, Syst. Control Lett., 212 (2026), 106418. https://doi.org/10.1016/j.sysconle.2026.106418 doi: 10.1016/j.sysconle.2026.106418
[33]	X. D. Li, S. J. Song, Impulsive systems with delays, Springer, 2022. https://doi.org/10.1007/978-981-16-4687-4
[34]	V. Kolmanovskii, N. Koroleva, T. Maizenberg, X. R. Mao, A. Matasov, Neutral stochastic differential delay equations with Markovian switching, Stochastic Anal. Appl., 21 (2003), 819–847. https://doi.org/10.1081/SAP-120022865 doi: 10.1081/SAP-120022865
[35]	L. J. Hu, X. R. Mao, Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Syst. Control Lett., 62 (2013), 178–187. https://doi.org/10.1016/j.sysconle.2012.11.009 doi: 10.1016/j.sysconle.2012.11.009
[36]	X. R. Mao, C. G. Yuan, Stability of stochastic differential equations with Markovian switching, World Scientific, 2006. https://doi.org/10.1142/p473
[37]	R. H. Wu, X. L. Zou, K. Wang, Asymptotic properties of a stochastic Lotka–Volterra cooperative system with impulsive perturbations, Nonlinear Dyn., 77 (2014), 807–817. https://doi.org/10.1007/s11071-014-1343-z doi: 10.1007/s11071-014-1343-z

Reader Comments

Your name:*

Email:*
© 2026 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)