This paper investigates the dynamic behavior of Caputo-type fractional stochastic delay differential equations. We construct an Itô-Caputo differential formula associated with the Mittag-Leffler type matrix polynomial function. Utilizing this formula along with tools such as Hölder’s inequality, the Burkholder-Davis-Gundy (BDG) inequality, and Gronwall’s inequality, the p-th moment estimation bounds for the system’s solution are rigorously derived. Furthermore, by constructing stochastic differential operators and imposing several assumptions, a criterion for the p-th moment exponential stability of the system’s solution is established. Finally, numerical examples are provided to verify the effectiveness of the proposed theoretical results.
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Exponential Stability in the p-th Moment for Caputo Fractional Stochastic Delay Differential Systems
- Yun Zhong 1,
- Mengmeng Li 1,2,*
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Received: 04 Mar 2026 | Revised: 25 Apr 2026 | Accepted: 06 May 2026 | Published: 13 Jul 2026
Abstract
References
- 1.
Sadek, L.; Algefary, A. On quantum trigonometric fractional calculus. Alex. Eng. J. 2025, 120, 371–377.
- 2.
Yang, G.; Wu, G.; Hui, F. Discrete fractional calculus with exponential memory: Propositions, numerical schemes and asymptotic stability. Nonlinear Anal. Model. Control 2024, 29, 32–52.
- 3.
Baleanu, D.; Diethelm, K.; Scalas, E.; et al. Fractional Calculus: Models and Numerical Methods; World Scientific: Singapore, 2012.
- 4.
Sabatier, J.; Agrawal, O.; Machado, J. Advances in Fractional Calculus; Springer: Dordrecht, The Netherlands, 2007.
- 5.
You, Z.; Feckan, M.; Wang, J. Relative controllability of fractional delay differential equations via delayed perturbation of Mittag-Leffler functions. J. Comput. Appl. Math. 2020, 378, 112939.
- 6.
Zhou, Y.; Wang, J.; Zhang, L. Basic Theory of Fractional Differential Equations; World Scientific: Singapore, 2016.
- 7.
Ma, Y.-K.; Raja, M.M.; Shukla, A.; et al. New results on approximate controllability of fractional delay integrodifferential systems of order 1 < r < 2 with Sobolev-type. Alex. Eng. J. 2023, 81, 501–518.
- 8.
Li, M.; Wang, J. Finite time stability of fractional delay differential equations. Appl. Math. Lett. 2017, 64, 170–176.
- 9.
Li, M.; Debbouche, A.; Wang, J. Relative controllability in fractional differential equations with pure delay. Math. Methods Appl. Sci. 2018, 41, 8906–8914.
- 10.
Li, M.; Wang, J. Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations. Appl. Math. Comput. 2018, 324, 254–265.
- 11.
Mahmudov, N. Delayed perturbation of Mittag-Leffler functions and their applications to fractional linear delay differential equations. Math. Methods Appl. Sci. 2019, 42, 5489–5497.
- 12.
Si, Y.; Feckan, M.; Wang, J.; et al. Relative controllability of delay multi-agent systems. Int. J. Robust Nonlinear Control 2021, 31, 4965–4993.
- 13.
Xiao, G.; Wang, J. Representation of solutions of linear conformable delay differential equations. Appl. Math. Lett. 2021, 117, 107088.
- 14.
Zhao, D.; Liu, Y.; Li, H. Fast-time complete controllability of nonlinear fractional delay integrodifferential evolution equations with nonlocal conditions and a parameter. Math. Methods Appl. Sci. 2022, 45, 5649–5669.
- 15.
Ren, W.; Xiong, J. Stability and stabilization of switched stochastic systems under asynchronous switching. Syst. Control Lett. 2016, 97, 184–192.
- 16.
Mao, X. Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control. Automatica 2013, 49, 3677–3681.
- 17.
Wang, B.; Zhu, Q. Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems. Syst. Control Lett. 2017, 105, 55–61.
- 18.
Briat, C. Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints. Automatica 2016, 74, 279–287.
- 19.
Ito, H.; Nishimura, Y. Stability of stochastic nonlinear systems in cascade with not necessarily unbounded decay rates. Automatica 2015, 62, 51–64.
- 20.
Zhao, X.; Deng, F. A new type of stability theorem for stochastic systems with application to stochastic stabilization. IEEE Trans. Autom. Control 2016, 61, 240–245.
- 21.
Zhu, Q. Stability analysis of stochastic delay differential equations with Levy noise. Syst. Control Lett. 2018, 118, 62–68.
- 22.
Song, R.; Zhao, J.; Zhu, Q. Boundedness and stability of nonlinear hybrid neutral stochastic delay differential equation with Levy jumps under different structures. J. Franklin Inst. 2024, 361, 106803.
- 23.
Li, M.; Wang, J. The existence and averaging principle for Caputo fractional stochastic delay differential systems. Fract. Calc. Appl. Anal. 2023, 26, 893–912.
- 24.
Xiao, G.; Wang, J. Stability of solutions of Caputo fractional stochastic differential equations. Nonlinear Anal. Model. Control 2021, 26, 581–596.
- 25.
Mao, X.; Yuan, C. Stochastic Differential Delay Equations with Markovian Switching; Imperial College Press: London, UK, 2006.
- 26.
Mao, X. Stochastic Differential Equations and Their Applications, 1st ed.; Horwood Publisher: Chichester, UK, 1997.
- 27.
Teel, A.; Subbaraman, A.; Sferlazza, A. Stability analysis for stochastic hybrid systems: A survey. Automatica 2014, 50, 2435–2456.
- 28.
Kilbas, A.; Srivastava, H.; Trujillo, J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006.
- 29.
Gorenflo, R.; Kilbas, A.; Mainardi, F.; et al. Mittag-Leffler Functions, Related Topics and Applications; Springer: Berlin, Germany, 2020.
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