Finite-time synchronization and approximate controllability of neutral fractional stochastic systems with non-instantaneous impulses driven by Rosenblatt process

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper investigates practical finite-time synchronization and approximate controllability of neutral fractional stochastic differential systems driven by the Rosenblatt process. The considered systems involve multiple time-varying delays, non-instantaneous impulses, and superlinear growth conditions. First, the existence and uniqueness of mild solutions are established by applying the Krasnoselskii fixed-point theorem and the Banach contraction principle. Second, a nonlinear finite-time feedback controller is designed for the drive-response systems, and sufficient conditions for practical finite-time synchronization under non-instantaneous impulses are derived. A key technical contribution is the development of a generalized Bihari-type inequality for fractional integral inequalities with weakly singular kernels and multiple time-varying delays, which provides explicit estimates for the synchronization settling time. Furthermore, based on the synchronization analysis, the approximate controllability of the response systems is investigated by employing the controllability Gramian and the resolvent operator approach. Finally, numerical simulations are presented to illustrate the effectiveness of the theoretical results.

Keywords

neutral fractional stochastic differential systems / Rosenblatt process / non-instantaneous impulses / practical finite-time synchronization / approximate controllability

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations , Elsevier, 2006.

[2] V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. Theory Methods Appl., 69 (2008), 2677-2682. https://doi.org/10.1016/j.na.2007.08.042

[3] R. Agarwal, D. O’Regan, S. Hristova, Stability of Caputo fractional differential equations by Lyapunov functions, Appl. Math., 60 (2015), 653-676. https://doi.org/10.1007/s10492-015-0116-4

[4] J. Zou, D. Luo, On the averaging principle of Caputo type neutral fractional stochastic differential equations, Qual. Theory Dyn. Syst., 23 (2024), 82. https://doi.org/10.1007/s12346-023-00916-7

[5] X. Mao, Stochastic Differential Equations and Applications , Amsterdam, Elsevier, 2007. https://doi.org/10.1533/9780857099402

[6] C. A. Tudor, Analysis of the Rosenblatt process, ESAIM: Probab. Stat., 12 (2008), 230-257. https://doi.org/10.1051/ps:2007037

[7] D. Luo, Z. Luo, Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses, Adv. Differ. Equ., 2019 (2019), 155. https://doi.org/10.1186/s13662-019-2101-9

[8] R. Gan, C. Li, Finite-time stability of nonlinear time-varying systems with saturated impulse inputs, Nonlinear Dyn., 111 (2023), 3497-3507. https://doi.org/10.1007/s11071-022-08024-y

[9] X. Xiao, J. Xie, Asymptotic stability of Hilfer fractional neutral stochastic differential equations with time-varying delays and non-instantaneous impulses, Nonlinear Dyn., 113 (2025), 21339-21358. https://doi.org/10.1007/s11071-025-11282-1

[10] J. Pradeesh, V. Vijayakumar, On the asymptotic stability of Hilfer fractional neutral stochastic differential systems with infinite delay, Qual. Theory Dyn. Syst., 23 (2024), 153. https://doi.org/10.1007/s12346-024-01007-x

[11] J. A. Asadzade, N. I. Mahmudov, Finite time stability analysis for fractional stochastic neutral delay differential equations, J. Appl. Math. Comput., 70 (2024), 5293-5317. https://doi.org/10.1007/s12190-024-02174-5

[12] R. Agarwal, S. Hristova, D. O’Regan, Non-instantaneous Impulses in Differential Equations , Springer Cham, 2017. https://doi.org/10.1007/978-3-319-66384-5

[13] M. Y. Guo, P. G. Wang, Finite-time stability of non-instantaneous impulsive systems with double state-dependent delays, Appl. Math. Comput., 477 (2024), 128824. https://doi.org/10.1016/j.amc.2024.128824

[14] J. Wu, L. Jiao, Synchronization in complex delayed dynamical networks with nonsymmetric coupling, Physica A, 386 (2007), 513-530. https://doi.org/10.1016/j.physa.2007.07.052

[15] L. Zhang, X. Yang, C. Xu, J. Feng, Exponential synchronization of complex-valued complex networks with time-varying delays and stochastic perturbations via time-delayed impulsive control, Appl. Math. Comput., 306 (2017), 22-30. https://doi.org/10.1016/j.amc.2017.02.004

[16] R. Surendar, M. Muthtamilselvan, K. Ahn, Stochastic disturbance with finite-time chaos stabilization and synchronization for a fractional-order nonautonomous hybrid nonlinear complex system via a sliding mode control, Chaos Solitons Fractals, 181 (2024), 114659. https://doi.org/10.1016/j.chaos.2024.114659

[17] B. Zheng, W. Dai, Z. Wang, Further results on distributed finite-time synchronization for fractional-order coupled neural networks, Commun. Nonlinear Sci. Numer. Simulat., 147 (2025), 108832. https://doi.org/10.1016/j.cnsns.2025.108832

[18] E. H. Lakhel, M. A. McKibben, Controllability for time-dependent neutral stochastic functional differential equations with Rosenblatt process and impulses, Int. J. Control Autom. Syst., 17 (2019), 286-297. https://doi.org/10.1007/s12555-016-0363-5

[19] S. Hussain, M. Sarwar, K. Abodayeh, C. Promsakon, T. Sitthiwirattham, Controllability of Hilfer fractional neutral impulsive stochastic delayed differential equations with nonlocal conditions, Chaos Solitons Fractals, 183 (2024), 114876. https://doi.org/10.1016/j.chaos.2024.114876

[20] G. Gokul, R. Udhayakumar, Approximate controllability for Hilfer fractional stochastic non-instantaneous impulsive differential system with Rosenblatt process and Poisson jumps, Qual. Theory Dyn. Syst., 23 (2024), 56. https://doi.org/10.1007/s12346-023-00912-x

[21] A. E. Ofem, M. O. Udo, O. Joseph, R. George, C. F. Chikwe, Convergence analysis of a new implicit iterative scheme and its application to delay Caputo fractional differential equations, Fractal Fract., 7 (2023), 212. https://doi.org/10.3390/fractalfract7030212

[22] J. A. Ugboh, J. Oboyi, M. O. Udo, H. A. Nabwey, A. E. Ofem, O. K. Narain, On a faster iterative method for solving fractional delay differential equations in Banach spaces, Fractal Fract., 8 (2024), 166. https://doi.org/10.3390/fractalfract8030166

[23] J. A. Ugboh, J. Oboyi, A. E. Ofem, G. C. Ugwunnadi, O. K. Narain, A novel fixed point iteration procedure for approximating the solution of impulsive fractional differential equations, Nonlinear Funct. Anal. Appl., 29 (2024), 841-865. https://doi.org/10.22771/nfaa.2024.29.03.12

[24] J. O. Achuobi, E. P. Akpan, R. George, A. E. Ofem, Stability analysis of Caputo fractional time-dependent systems with delay using vector Lyapunov functions, AIMS Math., 9 (2024), 28079-28099. https://doi.org/10.3934/math.20241362

[25] W. J. Li, W. H. Bai, J. D. Cao, Turing instability and pattern formation in a diffusive predator-prey system with opportunistic predators and weak Allee effect, Phys. Rev. E, 113 (2026). https://doi.org/10.1103/5fx7-q41y

[26] W. H. Li, L. T. Zhang, J. D. Cao, A note on Turing-Hopf bifurcation in a diffusive Leslie-Gower model with weak Allee effect on prey and fear effect on predator, Appl. Math. Lett., 172 (2025), 109741. https://doi.org/10.1016/j.aml.2025.109741

[27] W. H. Li, L. T. Zhang, J. D. Cao, F. Xu, Z. W. Cai, Finite time attractivity and exponentially stable of a multi-stage epidemic system with discontinuous incidence, Qual. Theory Dyn. Syst., 24 (2025), 199. https://doi.org/10.1007/s12346-025-01358-z

[28] J. Mei, H. Jian, Y. Li, W. F. Wang, D. Lin, Deep neural networks-based output-dependent intermittent control for a class of uncertain nonlinear systems, Chaos Solitons Fractals, 185 (2024), 114999. https://doi.org/10.1016/j.chaos.2024.114999

[29] J. Mei, R. R. Ye, Z. R. Xiang, J. H. Hu, W. F. Wang, A direct data-driven intermittent control and learning via Lyapunov-guided attraction region estimation with neural feedback loop design, IEEE Trans. Autom. Sci. Eng., 23 (2026), 6504-6519. https://doi.org/10.1109/TASE.2026.3669581

[30] H. M. Ahmed, Controllability for Sobolev type fractional integro-differential systems in a Banach space, Adv. Differ. Equ., 2012 (2012), 167. https://doi.org/10.1186/1687-1847-2012-167

[31] H. M. Ahmed, M. M. El-Borai, M. E. Ramadan, Boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and Poisson jumps, Adv. Differ. Equ., 2019 (2019), 82. https://doi.org/10.1186/s13662-019-2028-1

[32] H. M. Ahmed, M. M. El-Borai, A. S. O. El Bab, M. E. Ramadan, Approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion, Boundary Value Probl., 2020 (2020), 120. https://doi.org/10.1186/s13661-020-01418-0

[33] H. M. Ahmed, Noninstantaneous impulsive conformable fractional stochastic delay integro-differential system with Rosenblatt process and control function, Qual. Theory Dyn. Syst., 21 (2022), 15. https://doi.org/10.1007/s12346-021-00544-z

[34] P. Balasubramaniam, Solvability of Atangana-Baleanu-Riemann (ABR) fractional stochastic differential equations driven by Rosenblatt process via measure of noncompactness, Chaos Solitons Fractals, 157 (2022), 111960. https://doi.org/10.1016/j.chaos.2022.111960

[35] T. B. Ngoc, N. H. Tuan, Existence of mild solutions to semilinear fractional evolution equation using Krasnoselskii fixed point theorem, Filomat, 36 (2022), 1099-1112. https://doi.org/10.2298/FIL2204099N