Skip to main content
SpringerLink
Log in

Noninstantaneous Impulsive Conformable Fractional Stochastic Delay Integro-Differential System with Rosenblatt Process and Control Function

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, noninstantaneous impulsive conformable fractional stochastic delay integro-differential system driven by Rosenblatt process is studied. Sufficient conditions for approximate controllability and null controllability for the considered problem are established. Finally, an example is introduced to explain the obtained results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, Amsterdam (2006)

    Book  Google Scholar 

  2. Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheol. Acta 45(5), 765–771 (2006)

    Article  Google Scholar 

  3. Heymans, N.: Fractional calculus description of non-linear viscoelastic behaviour of polymers. Nonlinear Dyn. 38(1), 221–231 (2004)

    Article  Google Scholar 

  4. Paola, M.D., Zingales, M.: Exact mechanical models of fractional hereditary materials. J. Rheol. 56(5), 983–1004 (2012)

    Article  Google Scholar 

  5. Ballinger, G., Liu, X.: Boundedness for impulsive delay differential equations and applications in populations growth models. Nonlinear Anal. 53, 1041–1062 (2003)

    Article  MathSciNet  Google Scholar 

  6. Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141, 1641–1649 (2013)

    Article  MathSciNet  Google Scholar 

  7. Pierri, M., O’Regan, D., Rolnik, V.: Existence of solutions for semi-linear abstract differential equations with non instantaneous impulses. Appl. Math. Comput. 219, 6743–6749 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Gautam, G.R., Dabas, J.: Mild solutions for class of neutral fractional functional differential equations with not instantaneous impulses. Appl. Math. Comput. 259, 480–489 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Hernández, E., Pierri, M., O’Regan, D.: On abstract differential equations with noninstantaneous impulses. Topol. Methods Nonlinear Anal. 46, 1067–1085 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Tudor, C.A.: Analysis of the Rosenblatt process. ESAIM Probab. Stat. 12, 230–257 (2008)

    Article  MathSciNet  Google Scholar 

  11. Maejima, M., Tudor, C.A.: On the distribution of the Rosenblatt process. Stat. Probab. Lett. 83, 1490–1495 (2013)

    Article  MathSciNet  Google Scholar 

  12. Shen, G.J., Ren, Y.: Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space. J. Korean Stat. Soc. 44, 123–133 (2015)

    Article  MathSciNet  Google Scholar 

  13. Rathinasamy, S., Yong, R.: Approximate controllability of fractional differential equations with state-dependent delay. RM 63(3), 949–963 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Rajivganthi, C., Muthukumar, P., Ganesh Priya, B.: Approximate controllability of fractional stochastic integro-differential equations with infinite delay of order \(1\le \alpha \le 2\). IMA J. Math. Control. Inf. 33(3), 685–699 (2016)

    Article  Google Scholar 

  15. Zhang, X., Zhu, C., Yuan, C.: Approximate controllability of impulsive fractional stochastic differential equations with state-dependent delay. Adv. Differ. Equ. 2015(1), 1–12 (2015)

    Article  MathSciNet  Google Scholar 

  16. Yan, Z., Lu, F.: On approximate controllability of fractional stochastic neutral integro-differential inclusions with infinite delay. Appl. Anal. 94(6), 1235–1258 (2015)

    Article  MathSciNet  Google Scholar 

  17. Debbouche, A., Torres, D.F.M.: Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions. Appl. Math. Comput. 243, 161–175 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Ahmed, H.M.: Approximate controllability of impulsive neutral stochastic differential equations with fractional Brownian motion in a Hilbert space. Adv. Differ. Equ. 113, 1–11 (2014)

    MathSciNet  Google Scholar 

  19. Yan, Z., Jia, X.: Approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay. Collect. Math. 66, 93–124 (2015)

    Article  MathSciNet  Google Scholar 

  20. Sathiyaraj, T., Feckan, M., Wang, J.: Null controllability results for stochastic delay systems with delayed perturbation of matrices. Chaos Solit. Fractals 138, 109927 (2020)

    Article  MathSciNet  Google Scholar 

  21. Wang, J., Ahmed, H.M.: Null controllability of nonlocal Hilfer fractional stochastic differential equations. Miskolc Math. Notes 18(2), 1073–1083 (2017)

    Article  MathSciNet  Google Scholar 

  22. Ahmed, H.M., El-Borai, M.M., El-Owaidy, H.M., Ghanem, A.S.: Null controllability of fractional stochastic delay integro-differential equations. J. Math. Comput. Sci. 19, 143–150 (2019)

    Article  Google Scholar 

  23. Ahmed, H.M.: Hilfer fractional neutral stochastic partial differential equations with delay driven by Rosenblatt process. J. Control Decis. 1–18 (2021)

  24. Khalil, R., Al, Horani M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  Google Scholar 

  25. Lakhel, E.H., McKibben, M.: Controllability for time-dependent neutral stochastic functional differential equations with Rosenblatt process and impulses. Int. J. Control Autom. Syst. 17, 286–297 (2019)

    Article  Google Scholar 

  26. Hannabou, M., Hilal, K., Kajouni, A.: Existence and uniqueness of mild solutions to impulsive nonlocal Cauchy problems. J. Math. 2020, Article ID 5729128 (2020)

  27. Sakthivel, R., Ganesh, R., Anthoni, S.M.: Approximate controllability of fractional nonlinear differential inclusions. Appl. Math. Comput. 225, 708–717 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Dauer, J.P., Mahmudov, N.I.: Controllability of stochastic semilinear functional differential equations in Hilbert spaces. J. Math. Anal. Appl. 290, 373–394 (2004)

    Article  MathSciNet  Google Scholar 

  29. Dauer, J.P., Mahmudov, N.I.: Exact null controllability of semilinear integrodifferential systems in Hilbert spaces. J. Math. Anal. Appl. 299, 322–332 (2004)

    Article  MathSciNet  Google Scholar 

  30. Park, J.Y., Balasubramaniam, P.: Exact null controllability of abstract semilinear functional integrodifferential stochastic evolution equations in Hilbert space. Taiwan J. Math. 13, 2093–2103 (2009)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank the referees and the editor for their important comments and suggestions, which have significantly improved the paper.

Author information

Authors and Affiliations

  1. Higher Institute of Engineering, El Shorouk Academy, El Shorouk, Cairo, Egypt

    Hamdy M. Ahmed

Authors
  1. Hamdy M. Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

Ethics declarations

Availability of data and material

Not applicable

Conflict of interests

The author declare that they he has no competing interests.

Funding

Not applicable

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmed, H.M. Noninstantaneous Impulsive Conformable Fractional Stochastic Delay Integro-Differential System with Rosenblatt Process and Control Function. Qual. Theory Dyn. Syst. 21, 15 (2022). https://doi.org/10.1007/s12346-021-00544-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-021-00544-z

Keywords

Mathematics Subject Classification

Search

Navigation