Existence and asymptotic properties of global solution for hybrid neutral stochastic differential delay equations with colored noise

Siru Li; Tian Xu; Ailong Wu; Siru Li; Tian Xu; Ailong Wu

doi:10.3934/math.2025291

AIMS Mathematics

2025, Volume 10, Issue 3: 6379-6405. doi: 10.3934/math.2025291

Previous Article Next Article

Research article Special Issues

Existence and asymptotic properties of global solution for hybrid neutral stochastic differential delay equations with colored noise

1.
School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China
2.
Research Institute for Smart Cites, Shenzhen University, Shenzhen 518060, China

Received: 13 January 2025 Revised: 10 March 2025 Accepted: 14 March 2025 Published: 21 March 2025
MSC : 93E15

In this paper, stability of highly nonlinear hybrid neutral stochastic differential delay equations (NSDDEs) is discussed. In contrast to the white noise examined in previous literature, we incorporate colored noise into the highly nonlinear hybrid NSDDEs. Under some assumptions, we can show that highly nonlinear hybrid NSDDEs have a unique global solution. Meanwhile, we establish some criteria related to noise-to-state stability (NSS) of global solutions. Additionally, some theorems are given to guarantee asymptotic stability in $ L^{\hat{\alpha}} $ and almost surely asymptotic stability of global solution. These related discriminant rules are delay-dependent. Finally, an example is provided to demonstrate the validity of theoretical results.
- noise-to-state stability,
- asymptotic stability,
- almost surely stability,
- neutral stochastic differential delay equations
Citation: Siru Li, Tian Xu, Ailong Wu. Existence and asymptotic properties of global solution for hybrid neutral stochastic differential delay equations with colored noise[J]. AIMS Mathematics, 2025, 10(3): 6379-6405. doi: 10.3934/math.2025291

Related Papers:

Abstract

In this paper, stability of highly nonlinear hybrid neutral stochastic differential delay equations (NSDDEs) is discussed. In contrast to the white noise examined in previous literature, we incorporate colored noise into the highly nonlinear hybrid NSDDEs. Under some assumptions, we can show that highly nonlinear hybrid NSDDEs have a unique global solution. Meanwhile, we establish some criteria related to noise-to-state stability (NSS) of global solutions. Additionally, some theorems are given to guarantee asymptotic stability in $ L^{\hat{\alpha}} $ and almost surely asymptotic stability of global solution. These related discriminant rules are delay-dependent. Finally, an example is provided to demonstrate the validity of theoretical results.

References

[1]	E. Allen, Modeling with Itô Stochastic Differential Equations, Dordrecht: Springer, 2007. https://doi.org/10.1007/978-1-4020-5953-7
[2]	M. X. Shen, C. Fei, W. Y. Fei, X. R. Mao, Stabilisation by delay feedback control for highly nonlinear neutral stochastic differential equations, Syst. Control Lett., 137 (2020), 104645. https://doi.org/10.1016/j.sysconle.2020.104645 doi: 10.1016/j.sysconle.2020.104645
[3]	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
[4]	Y. Xu, Z. M. He, Exponential stability of neutral stochastic delay differential equations with Markovian switching, Appl. Math. Lett., 52 (2016), 64–73. https://doi.org/10.1016/j.aml.2015.08.019 doi: 10.1016/j.aml.2015.08.019
[5]	B. L. Lu, Q. X. Zhu, P. He, Exponential stability of highly nonlinear hybrid differently structured neutral stochastic differential equations with unbounded delays, Fractal. Fract., 6 (2022), 385. https://doi.org/10.3390/fractalfract6070385 doi: 10.3390/fractalfract6070385
[6]	X. R. Mao, Stochastic Differential Equations and Applications, Amsterdam: Elsevier, 2007. https://doi.org/10.1016/C2013-0-07332-X
[7]	R. Khasminskii, Stochastic Stability of Differential Equations, Berlin: Springer, 2012,385. https://doi.org/10.1007/978-3-642-23280-0
[8]	X. Y. Li, X. R. Mao, Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control, Automatica, 112 (2020), 108657. https://doi.org/10.1016/j.automatica.2019.108657 doi: 10.1016/j.automatica.2019.108657
[9]	S. Mekki, T. Blouhi, J. J. Nieto, A. Ouahab, Some existence results for systems of impulsive stochastic differential equations, Ann. Math. Siles., 35 (2021), 260–281. https://doi.org/10.2478/amsil-2020-0028 doi: 10.2478/amsil-2020-0028
[10]	N. Yang, J. Hu, D. Y. Chen, Noise-to-state and delay-dependent stability of highly nonlinear hybrid stochastic differential equation with multiple delays, Int. J. Control, 97 (2024), 577–588. https://doi.org/10.1080/00207179.2022.2160824 doi: 10.1080/00207179.2022.2160824
[11]	L. K. Feng, W. H. Zhang, Z. J. Wu, Noise-to-state stability of random impulsive delay systems with multiple random impulses, Appl. Math. Comput., 436 (2023), 127517. https://doi.org/10.1016/j.amc.2022.127517 doi: 10.1016/j.amc.2022.127517
[12]	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
[13]	C. Fei, W. Y. Fei, L. T. Yan, Existence and stability of solutions to highly nonlinear stochastic differential delay equations driven by G-Brownian motion, Appl. Math. - J. Chinese Univ., 34 (2019), 184–204. https://doi.org/10.1007/s11766-019-3619-x doi: 10.1007/s11766-019-3619-x
[14]	Y. Zhao, Q. X. Zhu, Stabilization of stochastic highly nonlinear delay systems with neutral term, IEEE Trans. Autom. Control, 68 (2022), 2544–2551. https://doi.org/10.1109/TAC.2022.3186827 doi: 10.1109/TAC.2022.3186827
[15]	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
[16]	C. Fei, 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
[17]	M. X. Shen, W. Y. Fei, X. R. Mao, Y. Liang, Stability of highly nonlinear neutral stochastic differential delay equations, Syst. Control Lett., 115 (2018), 1–8. https://doi.org/10.1016/j.sysconle.2018.02.013 doi: 10.1016/j.sysconle.2018.02.013
[18]	W. Y. Fei, L. J. Hu, X. R. Mao, M. X. Shen, Generalized criteria on delay‐dependent stability of highly nonlinear hybrid stochastic systems, Int. J. Robust Nonlinear Control, 29 (2019), 1201–1215. https://doi.org/10.1002/rnc.4402 doi: 10.1002/rnc.4402
[19]	C. Fei, W. Y. Fei, X. R. Mao, L. T. Yan, Delay-dependent asymptotic stability of highly nonlinear stochastic differential delay equations driven by G-Brownian motion, J. Franklin Inst., 359 (2022), 4366–4392. https://doi.org/10.1016/j.jfranklin.2022.03.027 doi: 10.1016/j.jfranklin.2022.03.027
[20]	A. Zouine, H. Bouzahir, C. Imzegouan, Delay-dependent stability of highly nonlinear hybrid stochastic systems with Lévy noise, Nonlinear Stud., 27 (2020), 879–896. https://doi.org/10.1016/j.jfranklin.2022.03.027 doi: 10.1016/j.jfranklin.2022.03.027
[21]	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
[22]	R. L. Song, B. Wang, Q. X. Zhu, Delay-dependent stability of nonlinear hybrid neutral stochastic differential equations with multiple delays, Int. J. Robust Nonlinear Control, 31 (2021), 250–267. https://doi.org/10.1002/rnc.5275 doi: 10.1002/rnc.5275
[23]	M. X. Shen, C. Fei, W. Y. Fei, X. R. Mao, C. H. Mei, Delay‐dependent stability of highly nonlinear neutral stochastic functional differential equations, Int. J. Robust Nonlinear Control, 32 (2022), 9957–9976. https://doi.org/10.1002/rnc.6384 doi: 10.1002/rnc.6384
[24]	W. Lu, Noise-to-state exponential stability of neutral random nonlinear systems, Adv. Differ. Equ., 2018 (2018), 313. https://doi.org/10.1186/s13662-018-1739-z doi: 10.1186/s13662-018-1739-z
[25]	Z. J. Wu, Stability criteria of random nonlinear systems and their applications, IEEE Trans. Autom. Control, 60 (2014), 1038–1049. https://doi.org/10.1109/TAC.2014.2365684 doi: 10.1109/TAC.2014.2365684
[26]	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., 60 (2022), 3302–3325. https://doi.org/10.1137/22M1480392 doi: 10.1137/22M1480392
[27]	F. A. Rihan, H. J. Alsakaji, Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species, Discrete Cont. Dyn. Syst. Ser. S., 15 (2022), 245–263. https://doi.org/10.3934/dcdss.2020468 doi: 10.3934/dcdss.2020468
[28]	F. A. Rihan, H. J. Alsakaji, S. Kundu, O. Mohamed, Dynamics of a time-delay differential model for tumour-immune interactions with random noise, Alex. Eng. J., 61 (2022), 11913–11923. https://doi.org/10.1016/j.aej.2022.05.027 doi: 10.1016/j.aej.2022.05.027

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)