Stability analysis of time-varying neutral stochastic pantograph equations with Markovian switching and multiple proportional delays

Yuhan Yu; Yinfang Song; Yuhan Yu; Yinfang Song

doi:10.3934/math.20251081

AIMS Mathematics

2025, Volume 10, Issue 10: 24371-24388. doi: 10.3934/math.20251081

Previous Article Next Article

Research article Special Issues

Stability analysis of time-varying neutral stochastic pantograph equations with Markovian switching and multiple proportional delays

Yuhan Yu ,
Yinfang Song ^,

School of Information and Mathematics, Yangtze University, Jingzhou, Hubei 434023, China

Received: 22 August 2025 Revised: 12 October 2025 Accepted: 16 October 2025 Published: 24 October 2025
MSC : 34K40, 37H30, 93E15

This paper rigorously investigates the stochastic stability of time-varying neutral stochastic pantograph systems with Markovian switching. Primarily, multiple proportional delays and time-varying coefficients are taken into account in the neutral hybrid stochastic systems. In order to surmount the complexities stemming from those factors, the Lyapunov functional approach and several stochastic analytical techniques are utilized, and a modified version of pantograph delays integral inequality is developed. Moreover, various stability criteria such as $ q $th moment stability, $ q $th moment asymptotic stability, $ q $th exponential stability, almost sure asymptotic stability and almost sure exponential stability are put forward, where the upper bound of the diffusion operator can be a function with sign-changed time-varying coefficients rather than negative constants. Eventually, through several numerical examples, the theoretical results are strictly verified, demonstrating the feasibility and effectiveness of the proposed method.
- neutral stochastic pantograph systems,
- multiple proportional delays,
- time-varying coefficients,
- Markovian switching,
- stability
Citation: Yuhan Yu, Yinfang Song. Stability analysis of time-varying neutral stochastic pantograph equations with Markovian switching and multiple proportional delays[J]. AIMS Mathematics, 2025, 10(10): 24371-24388. doi: 10.3934/math.20251081

Related Papers:

Abstract

This paper rigorously investigates the stochastic stability of time-varying neutral stochastic pantograph systems with Markovian switching. Primarily, multiple proportional delays and time-varying coefficients are taken into account in the neutral hybrid stochastic systems. In order to surmount the complexities stemming from those factors, the Lyapunov functional approach and several stochastic analytical techniques are utilized, and a modified version of pantograph delays integral inequality is developed. Moreover, various stability criteria such as $ q $th moment stability, $ q $th moment asymptotic stability, $ q $th exponential stability, almost sure asymptotic stability and almost sure exponential stability are put forward, where the upper bound of the diffusion operator can be a function with sign-changed time-varying coefficients rather than negative constants. Eventually, through several numerical examples, the theoretical results are strictly verified, demonstrating the feasibility and effectiveness of the proposed method.

References

[1]	V. Kolmanovskii, N. Koroleva, T. Maizenberg, X. R. Mao, A. Matasov, Neutral stochastic differential delay equations with Markovian switching, Stoch. Anal. Appl., 21 (2003), 819–847. https://doi.org/10.1081/SAP-120022865 doi: 10.1081/SAP-120022865
[2]	C. G. Yuan, X. R. Mao, Robust stability and controllability of stochastic differential delay equations with Markovian switching, Automatica, 40 (2004), 343–354. https://doi.org/10.1016/j.automatica.2003.10.012 doi: 10.1016/j.automatica.2003.10.012
[3]	X. R. Mao, C. G. Yuan, Stochastic differential equations with Markovian switching, Imperial college press, 2006. https://doi.org/10.1142/p473
[4]	Q. Luo, X. R. Mao, Y. Shen, New criteria on exponential stability of neutral stochastic differential delay equations, Syst. Control Lett., 55 (2006), 826–834. https://doi.org/10.1016/j.sysconle.2006.04.005 doi: 10.1016/j.sysconle.2006.04.005
[5]	X. R. Mao, Y. Shen, C. G. Yuan, Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stoch. Proc. Appl., 118 (2008), 1385–1406. https://doi.org/10.1016/j.spa.2007.09.005 doi: 10.1016/j.spa.2007.09.005
[6]	Y. Chen, W. X. Zheng, A. K. Xue, A new result on stability analysis for stochastic neutral systems, Automatica, 46 (2010), 2100–2104. https://doi.org/10.1016/j.automatica.2010.08.007 doi: 10.1016/j.automatica.2010.08.007
[7]	L. Shaikhet, Some new aspects of Lyapunov-type theorems for stochastic differential equations of neutral type, SIAM J. Control Optim., 48 (2010), 4481–4499. https://doi.org/10.1137/080744165 doi: 10.1137/080744165
[8]	S. G. Peng, Y. Zhang, Some new criteria on $\mathit{p}$th moment stability of stochastic functional differential equations with Markovian switching, IEEE T. Automat. Contr., 55 (2010), 2886–2890. https://doi.org/10.1109/TAC.2010.2074251 doi: 10.1109/TAC.2010.2074251
[9]	B. Wang, Q. X. Zhu, Stability analysis of semi-Markov switched stochastic systems, Automatica, 94 (2018), 72–80. https://doi.org/10.1016/j.automatica.2018.04.016 doi: 10.1016/j.automatica.2018.04.016
[10]	B. Wang, Q. X. Zhu, S. B. Li, Stabilization of hidden Markov jump singular systems with limit mode switching information, IEEE T. Automat. Contr., 70 (2024), 1558–2523. https://doi.org/10.1109/TAC.2024.3518418 doi: 10.1109/TAC.2024.3518418
[11]	X. H. Liu, H. S. Xi, Synchronization of neutral complex dynamical networks with Markovian switching based on sampled-data controller, Neurocomputing, 139 (2014), 163–179. https://doi.org/10.1016/j.neucom.2014.02.048 doi: 10.1016/j.neucom.2014.02.048
[12]	H. B. Chen, P. Shi, C. C. Lim, Exponential synchronization for Markovian stochastic coupled neural networks of neutral-type via adaptive feedback control, IEEE T. Neur. Net. Lear., 28 (2016), 1618–1632. https://doi.org/10.1109/TNNLS.2016.2546962 doi: 10.1109/TNNLS.2016.2546962
[13]	H. B. Chen, P. Shi, C. C. Lim, Stability analysis for neutral stochastic delay systems with Markovian switching, Syst. Control Lett., 110 (2017), 38–48. https://doi.org/10.1016/j.sysconle.2017.10.008 doi: 10.1016/j.sysconle.2017.10.008
[14]	Y. C. Liu, C. Tan, H. Q. Peng, H. Y. Yuan, Q. X. Zhu, Stability of neutral Markov switching stochastic delay systems via the generalised Halanay inequality, Commun. Nonlinear Sci., 152 (2026), 109126. https://doi.org/10.1016/j.cnsns.2025.109126 doi: 10.1016/j.cnsns.2025.109126
[15]	B. Zhou, Stability analysis of non‐linear time‐varying systems by Lyapunov functions with indefinite derivatives, IET Control Theory A., 11 (2017), 1434–1442. https://doi.org/10.1049/iet-cta.2016.1538 doi: 10.1049/iet-cta.2016.1538
[16]	W. B. Zhang, Q. L. Han, Y. Tang, Y. R. Liu, Sampled-data control for a class of linear time-varying systems, Automatica, 103 (2019), 126–134. https://doi.org/10.1016/j.automatica.2019.01.027 doi: 10.1016/j.automatica.2019.01.027
[17]	X. T. Wu, Y. Tang, W. B. Zhang, Stability analysis of stochastic delayed systems with an application to multi-agent systems, IEEE T. Automat. Contr., 61 (2016), 4143–4149. https://doi.org/10.1109/TAC.2016.2548559 doi: 10.1109/TAC.2016.2548559
[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 T. Cybernetics, 46 (2015), 1350–1362. https://doi.org/10.1109/TCYB.2015.2442274 doi: 10.1109/TCYB.2015.2442274
[19]	X. L. Jiang, G. H. Xia, Z. G. Feng, Z. Y. Jiang, J. B. Qiu, Reachable set estimation for Markovian jump neutral-type neural networks with time-varying delays, IEEE T. Cybernetics, 52 (2020), 1150–1163. https://doi.org/10.1109/TCYB.2020.2985837 doi: 10.1109/TCYB.2020.2985837
[20]	L. C. Feng, L. Liu, J. D. Cao, L. Rutkowski, G. P. Lu, General decay stability for nonautonomous neutral stochastic systems with time-varying delays and Markovian switching, IEEE T. Cybernetics, 52 (2020), 5441–5453. https://doi.org/10.1109/TCYB.2020.3031992 doi: 10.1109/TCYB.2020.3031992
[21]	H. B. Chen, P. Shi, C. C. Lim, Stability analysis of time-varying neutral stochastic hybrid delay system, IEEE T. Automat. Contr., 68 (2023), 5576–5583. https://doi.org/10.1109/TAC.2022.3220517 doi: 10.1109/TAC.2022.3220517
[22]	Y. Kasimu, G. Maimaitiaili, Non-fragile $H_{\infty}$ filter design for uncertain neutral Markovian jump systems with time-varying delays, AIMS Math., 9 (2024), 15559–15583. https://doi.org/10.3934/math.2024752 doi: 10.3934/math.2024752
[23]	X. P. Zhang, C. C. Shen, D. J. Xu, Reachable set estimation for neutral semi-Markovian jump systems with time-varying delay, AIMS Math., 9 (2024), 8043–8062. https://doi.org/10.3934/math.2024391 doi: 10.3934/math.2024391
[24]	P. F. Wang, R. F. Wang, H. Su, Stability of time-varying hybrid stochastic delayed systems with application to aperiodically intermittent stabilization, IEEE T. Cybernetics, 52 (2022), 9026–9035. https://doi.org/10.1109/TCYB.2021.3052042 doi: 10.1109/TCYB.2021.3052042
[25]	Y. L. Huang, Q. X. Zhu, pth moment exponential stability of highly nonlinear neutral hybrid stochastic delayed systems with impulsive effect, IEEE T. Syst. Man Cy. Syst., 55 (2025), 6486–6493. https://doi.org/10.1109/TSMC.2025.3579565 doi: 10.1109/TSMC.2025.3579565
[26]	V. Lakshmikantham, L. Z. Wen, B. G. Zhang, Theory of differential equations with unbounded delay, Springer Science and Business Media, 1994. https://doi.org/10.1007/978-1-4615-2606-3
[27]	X. Xu, L. Liu, M. Krstic, G. Feng, Stability analysis and predictor feedback control for systems with unbounded delays, Automatica, 135 (2022), 109958. https://doi.org/10.1016/j.automatica.2021.109958 doi: 10.1016/j.automatica.2021.109958
[28]	S. R. You, W. Mao, X. R. Mao, L. J. Hu, Analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients, Appl. Math. Comput., 263 (2015), 73–83. https://doi.org/10.1016/j.amc.2015.04.022 doi: 10.1016/j.amc.2015.04.022
[29]	W. Mao, L. J. Hu, X. R. Mao, Almost sure stability with general decay rate of neutral stochastic pantograph equations with Markovian switching, Electron. J. Qual. Theory Differ. Equ., 52 (2019), 1–17. https://doi.org/10.14232/ejqtde.2019.1.52 doi: 10.14232/ejqtde.2019.1.52
[30]	M. X. Shen, W. Y Fei, X. R. Mao, S. N. Deng, Exponential stability of highly nonlinear neutral pantograph stochastic differential equations, Asian J. Control, 22 (2020), 436–448. https://doi.org/10.1002/asjc.1903 doi: 10.1002/asjc.1903
[31]	T. Caraballo, L. Mchiri, B. Mohsen, M. Rhaima, $\mathit{p}$th moment exponential stability of neutral stochastic pantograph differential equations with Markovian switching, Commun. Nonlinear Sci., 102 (2021), 105916. https://doi.org/10.1016/j.cnsns.2021.105916 doi: 10.1016/j.cnsns.2021.105916
[32]	L. Mchiri, T. Caraballo, M. Rhaima, Partial asymptotic stability of neutral pantograph stochastic differential equations with Markovian switching, Adv. Cont. Discr. Mod., 2022 (2022), 18. https://doi.org/10.1186/s13662-022-03692-x doi: 10.1186/s13662-022-03692-x
[33]	T. Caraballo, M. Belfeki, L. Mchiri, M. Rhaima, $\mathit{h}$-stability in $\mathit{p}$th moment of neutral pantograph stochastic differential equations with Markovian switching driven by Lévy noise, Chaos Soliton. Fract., 151 (2021), 111249. https://doi.org/10.1016/j.chaos.2021.111249 doi: 10.1016/j.chaos.2021.111249
[34]	Z. H. Zou, Y. F. Song, Q. X. Zhu, C. Zhao, Asymptotic stability of neutral stochastic pantograph systems with Markovian switching, Int. J. Control, 97 (2024), 1311–1324. https://doi.org/10.1080/00207179.2023.2204558 doi: 10.1080/00207179.2023.2204558
[35]	D. H. Ruan, Y. Lu, Generalized exponential stability of neutral stochastic quaternion-valued neural networks with variable coefficients and infinite delay, Syst. Control Lett., 191 (2024), 105869. https://doi.org/10.1016/j.sysconle.2024.105869 doi: 10.1016/j.sysconle.2024.105869
[36]	A. Ben Makhlouf, A. Ben Hamed, L. Mchiri, M. Rhaima, Exponential stability of highly nonlinear hybrid neutral pantograph stochastic systems with multiple delays, Acta Appl. Math., 195 (2025), 3. https://doi.org/10.1007/s10440-025-00705-1 doi: 10.1007/s10440-025-00705-1
[37]	L. C. Feng, L. Liu, J. D. Cao, F. E. Alsaadi, General stabilization of non-autonomous hybrid systems with delays and random noises via delayed feedback control, Commun. Nonlinear Sci., 117 (2023), 106939. https://doi.org/10.1016/j.cnsns.2022.106939 doi: 10.1016/j.cnsns.2022.106939
[38]	L. C. Feng, Z. H. Wu, J. D. Cao, S. Q. Zheng, F. E. Alsaadi, Exponential stability for nonlinear hybrid stochastic systems with time varying delays of neutral type, Appl. Math. Lett., 107 (2020), 106468. https://doi.org/10.1016/j.aml.2020.106468 doi: 10.1016/j.aml.2020.106468

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)