Abstract
This paper discusses the approximate controllability of Hilfer fractional stochastic differential system involving non-instantaneous impulses with Rosenblatt process and Poisson jumps. By utilising stochastic analysis, semigroup theory, fractional calculus, and Krasnoselskii’s fixed point theorem, we prove our primary outcomes. Firstly, we prove the approximate controllability of the Hilfer fractional system. As a final step, we provide an example to highlight our discussion.
Similar content being viewed by others
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Abbreviations
- FDEs:
-
Fractional differential equations
- HFSNIIDS:
-
Hilfer fractional stochastic non-instantaneous impulsive differential system
- R–L:
-
Riemann–Liouville
- HF:
-
Hilfer fractional
- HFD:
-
Hilfer fractional derivative
- SDEs:
-
Stochastic differential equations
- fBm:
-
Fractional Brownian motion
References
Agarwal, R., Hristova, S., O’Regan D.: Non-instantaneous Impulses in Differential Equations, Non-Instantaneous Impulses in Differential Equations, pp. 1–72. Springer (2017)
Ahmed, H.M., El-Borai, M.M.: Hilfer fractional stochastic integro-differential equations. Appl. Math. Comput. 331, 182–189 (2018)
Ahmed, H.M., El-Borai, M.M., El Bab, A.S.O., Ramadan, M.E.: Approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion. Bound. Value Probl. 2020, 120 (2020)
Ali, A., Shah, K., Abdeljawad, T., Khan, H., Khan, A.: Study of fractional order pantograph type impulsive antiperiodic boundary value problem. Adv. Differ. Equ. 2020, 572 (2020)
Alkhazzan, A., Ziang, P., Baleanu, D., Khan, H., Khan, A., Jarad, F., Shah, A.: Stability and existence results for a class of nonlinear fractional differential equations with singularity. Math. Methods Appl. Sci. 41(18), 9321–9334 (2018)
Balasubramaniam, P., Saravanakumar, S., Ratnavelu, K.: Study a class of Hilfer fractional stochastic integrodifferential equations with Poisson jumps. Stoch. Anal. Appl. 36(6), 1021–1036 (2018)
Bedi, P., Kumar, A., Abdeljawad, T., Khan, Z.A., Khan, A.: Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators. Adv. Differ. Equ. 2020, 615 (2020)
Bedi, P., Kumar, A., Khan, A.: Controllability of neutral impulsive fractional differential equations with Atangana–Baleanu–Caputo derivatives. Chaos, Solitons Fractals 150, 111153 (2021)
Dhayal, R., Malik, M., Abbas, S., Debbouche, A.: Optimal controls for second-order stochastic differential equations driven by mixed-fractional Brownian motion with impulses. Math. Methods Adv. Sci. 43(7), 4107–4124 (2020)
Dhayal, R., Malik, M., Abbas, S.: Approximate and trajectory controllability of fractional stochastic differential equation with non-instantaneous impulses and poisson jumps. Asian J. Control 23(6), 2669–2680 (2021)
Dhayal, R., Malik, M., Abbas, S.: Approximate controllability for a class of non-instantaneous impulsive stochastic fractional differential equation driven by fractional brownian motion. Differ. Equ. Dyn. Syst. 29(1), 175–191 (2021)
Dineshkumar, C., Udhayakumar, R.: New results concerning to approximate controllability of Hilfer fractional neutral stochastic delay integro-differential systems. Numer. Methods Part. Differ. Equ. 37(2), 1072–1090 (2020)
Dineshkumar, C., Udhayakumar, R., Vijayakumar, V., Nisar, K.S.: A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systems. Chaos, Solitons Fractals 142, 110472 (2021)
Gu, H.B., Trujillo, J.J.: Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 257, 344–354 (2015)
Hakkar, N., Dhayal, R., Debbouche, A., Torres, D.F.M.: Approximate controllability of delayed fractional stochastic differential systems with mixed noise and impulsive effects. Fractal Fract. 7(2), 104 (2023)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Karthikeyan, K., Debbouche, A., Torres, D.F.M.: Analysis of Hilfer fractional integro-differential equations with almost sectorial operators. Fractal Fract. 5(1), 22 (2021)
Khan, H., Tunc, C., Chen, W., Khan, A.: Existence theorems and Hyers–Ulam stability for a class of hybrid fractional differential equations with \(p\)-Laplacian operator. J. Appl. Anal. Comput. 8(4), 1211–1226 (2018)
Khan, A., Khan, H., Gomez-Aguilar, J.F., Abdeljawad, T.: Existence and Hyers–Ulam stability for a nonlinear singular fractional differential equations with Mittag–Leffler kernel. Chaos, Solitons Fractals 127, 422–427 (2019)
Khan, H., Khan, A., Jarad, F., Shah, A.: Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system. Chaos, Solitons Fractals 131, 109477 (2020)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science Inc, New York (2006)
Kunita, H.: Stochastic Differential Equations Based on L\(\grave{\rm e }\)vy Processes and Stochastic Flows of Diffeomorphisms, Real and Stochastic Analysis, pp. 305–373. Boston, Birkhauser (2004)
Lakhel, E.H., McKibben, M.A.: Controllability for time-dependent neutral stochastic fractional differential equations with Rosenblatt process and impulses. Int. J. Control Autom. Syst. 17, 286–297 (2019)
Liu, J., Wei, W., Xu, W.: Approximate controllability of non-instantaneous impulsive Stochatic evolution systems driven by fractional Brownian motion with Hurst parameter \(H \in (0, \frac{1}{2})\). Fractal and Fractional 6(8), 440 (2022)
Lv, J., Yang, X.: Approximate controllability of Hilfer fractional differential equations. Math. Methods Appl. Sci. 43(1), 242–254 (2020)
Maejima, M., Tudor, C.A.: On the distribution of the Rosenblatt process. Stat. Probab. Lett. 83(6), 1490–1495 (2013)
Mahmudov, N.I., Denker, A.: Approximate controllability of linear stochastic systems. Int. J. Control 73(2), 144–151 (2000)
Mao, X.: Stochastic Differential Equations and applications. Horwood, Chichester, Elsevier (2007)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Ramkumar, K., Ravikumar, K., Anguraj, A.: Hilfer fractional neutral stochastic differential equations with non-instantaneous impulses. AIMS Math. 6(5), 4474–4491 (2021)
Rihan, F.A., Rajivgandhi, C., Muthukumar, P.: Fractional stochastic differential equations with Hilfer fractional derivative: Poisson jumps and optimal control. Discret. Dyn. Nat. Soc. 2017, 5394528 (2017)
Sakthivel, R.: Approximate controllability of impulsive stochastic evolution equations. Funkc. Ekvac. 52(3), 381–393 (2009)
Sakthivel, R., Kim, J.H., Mahmudov, N.I.: On controllability of nonlinear stochastic systems. Rep. Math. Phys. 58(3), 433–443 (2006)
Sakthivel, R., Ganesh, R., Anthoni, S.M.: Approximate controllability of fractional nonlinear differential inclusions. Appl. Math. Comput. 225, 708–717 (2013)
Saravanakumar, S., Balasubramaniam, P.: On impulsive Hilfer fractional stochastic differential system driven by Rosenblatt process. Stoch. Anal. Appl. 37(6), 955–976 (2019)
Saravanakumar, S., Balasubramaniam, P.: Approximate controllability of nonlinear Hilfer fractional stochastic differential system with Rosenblatt process and Poisson jumps. Int. J. Nonlinear Sci. Numer. Simul. 21(7–8), 727–737 (2020)
Saravanakumar, S., Balasubramaniam, P.: Non-instantaneous impulsive Hilfer fractional stochastic differential equations driven by fractional Brownian motion. Stoch. Anal. Appl. 39(3), 549–566 (2021)
Sivasankar, S., Udhayakumar, R.: Discussion on existence of mild solutions for Hilfer fractional neutral stochastic evolution equations via almost sectorial operators with delay. Qual. Theory Dyn. Syst. 22(2), 67 (2023)
Sivasankar, S., Udhayakumar, R., Muthukumaran, V.: A new conversation on the existence of Hilfer fractional stochastic Volterra–Fredholm integro-differential inclusions via almost sectorial operators. Nonlinear Anal.: Model. Control 28(2), 288–307 (2023)
Tajadodi, H., Khan, A., Gomez-Aguilar, J.F., Khan, H.: Optimal control problems with Atangana–Baleanu fractional derivative. Optim. Control Appl. Methods 42(1), 96–109 (2021)
Tudor, C.A.: Analysis of the Rosenblatt process. ESAIM-Prob. Stat. 12, 230–257 (2008)
Varshini, S., Banupriya, K., Ramkumar, K., Ravikumar, K.: Existence and Stability results of stochastic differential equations with non-instantaneous impulse and Poisson jumps. Nonauton. Dyn. Syst. 9(1), 256–271 (2022)
Varun Bose, C.S., Udhayakumar, R., Elshenhab, A.M., Kumar, M.S., Ro, J.S.: Discussion on the approximate controllability of Hilfer fractional neutral integro-differential inclusions via almost sectorial operators. Fractal Fract 6(10), 607 (2022)
Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)
Acknowledgements
The authors also would like to thank the reviewer and the editor for their valuable comments and suggestions which improved the quality of the paper.
Author information
Authors and Affiliations
Contributions
Conceptualisation, G.Gokul(G.G) and R.Udhayakumar (R.U.); methodology, G.G.; validation, G.G. and R.U.; formal analysis, G.G.; investigation, R.U.; resources, G.G.; writing original draft preparation, G.G.; writing review and editing, R.U.; visualisation, R.U.; supervision, R.U.; project administration, R.U. All authors have read and agreed to the published version of the paper. All the authors are equally contributed to this paper.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Gokul, G., Udhayakumar, R. Approximate Controllability for Hilfer Fractional Stochastic Non-instantaneous Impulsive Differential System with Rosenblatt Process and Poisson Jumps. Qual. Theory Dyn. Syst. 23, 56 (2024). https://doi.org/10.1007/s12346-023-00912-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-023-00912-x