Skip to main content
Log in

Approximate Controllability for a Class of Non-instantaneous Impulsive Stochastic Fractional Differential Equation Driven by Fractional Brownian Motion

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

In this manuscript, we study the approximate controllability for a class of non-instantaneous impulsive stochastic fractional differential equation driven by fractional Brownian motion in Hilbert space. The results are obtained by using the \(\rho \)-resolvent family, and fixed point techniques. Finally, we have given an example to illustrate the applicability of our results.

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

Access this article

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. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Miller, K.S., Ross, B.: An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  4. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  5. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  6. Mao, X.: Stochastic Differential Equations and Applications. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  7. Sakthivel, R., Revathi, P., Ren, Y.: Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. 81, 70–86 (2013)

    Article  MathSciNet  Google Scholar 

  8. Benchaabane, A., Sakthivel, R.: Sobolev-type fractional stochastic differential equations with non-Lipschitz coefficients. J. Comput. Appl. Math. 312(1), 65–73 (2017)

    Article  MathSciNet  Google Scholar 

  9. Jingyun, L.v., Yang, X.: Nonlocal fractional stochastic differential equations driven by fractional Brownian motion. Adv. Differ. Equ. 2017(1), 198 (2017)

  10. Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141(5), 1641–1649 (2013)

    Article  MathSciNet  Google Scholar 

  11. Wang, J., Fečkan, M.: A general class of impulsive evolution equations. Topol. Methods Nonlinear Anal. 46(2), 915–933 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Malik, M., Kumar, A., Fečkan, M.: Existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses. J. King Saud Univ. Sci. 30(2), 204–213 (2018)

    Article  Google Scholar 

  13. Malik, M., Dhayal, R., Abbas, S., Kumar, A.: Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses. Rev. R. Acad. Cienc. Exactas F’ıs. Nat. Ser. A Math. 113(1), 103–118 (2019)

    Article  MathSciNet  Google Scholar 

  14. Malik, M., Dhayal, R., Abbas, S.: Exact Controllability of a Retarded Fractional Differential Equation with Non-instantaneous Impulses. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 26(1), 53–69 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Liu, S., Wang, J.: Optimal controls of systems governed by semilinear fractional differential equations with not instantaneous impulses. J. Optim. Theory Appl. 174(2), 455–473 (2017)

    Article  MathSciNet  Google Scholar 

  16. Klamka, J.: Stochastic controllability of linear systems with delay in control. Bull. Polish Acad. Sci. Tech. Sci. 55(1), 23–29 (2007)

    MathSciNet  MATH  Google Scholar 

  17. Klamka, J.: Stochastic controllability of linear systems with state delays. Int. J. Appl. Math. Comput. Sci. 17(1), 5–13 (2017)

    Article  MathSciNet  Google Scholar 

  18. Klamka, J.: Controllability of dynamical systems: a survey. Bull. Polish Acad. Sci. Tech. Sci. 61(2), 335–342 (2013)

    Google Scholar 

  19. Abid, S.H., Hasan, S.Q., Quaez, U.J.: Approximate controllability of fractional Sobolev type stochastic differential equations driven by mixed fractional Brownian motion. J. Math. Sci. Appl. 3(1), 3–11 (2015)

    Google Scholar 

  20. Chadha, A., Bora, S.N., Sakthivel, R.: Approximate controllability of impulsive stochastic fractional differential equations with nonlocal conditions. Dyn. Syst. Appl. 27(1), 1–29 (2018)

    Google Scholar 

  21. Sakthivel, R., Ganesh, R., Suganya, S.: Approximate controllability of fractional neutral stochastic system with infinite delay. Rep. Math. Phys. 70(3), 291–311 (2012)

    Article  MathSciNet  Google Scholar 

  22. Sakthivel, R., Suganya, S., Anthoni, S.M.: Approximate controllability of fractional stochastic evolution equations. Comput. Math. Appl. 63(3), 660–668 (2012)

    Article  MathSciNet  Google Scholar 

  23. Tamilalagan, P., Balasubramaniam, P.: Approximate controllability of fractional stochastic differential equations driven by mixed fractional Brownian motion via resolvent operators. Int. J. Control 90(8), 1713–1727 (2016)

    Article  MathSciNet  Google Scholar 

  24. Sakthivel, R., Ren, Y., Debbouche, A., Mahmudov, N.I.: Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions. Appl. Anal. 95(11), 2361–2382 (2016)

    Article  MathSciNet  Google Scholar 

  25. Rajivganthi, C., Thiagu, K., Muthukumar, P., Balasubramaniam, P.: Existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and poisson jumps. Appl. Math. 60(4), 395–419 (2015)

    Article  MathSciNet  Google Scholar 

  26. Sakthivel, R., Ganesh, R., Ren, Y., Anthoni, S.M.: Approximate controllability of nonlinear fractional dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 18(12), 3498–3508 (2013)

    Article  MathSciNet  Google Scholar 

  27. Boufoussi, B., Hajji, S.: Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space. Stat. Probab. Lett. 82(8), 1549–1558 (2012)

    Article  MathSciNet  Google Scholar 

  28. Hasse, M.: The Functional Calculus for Sectorial Operators, Operator Theory : Advances and Applications, vol. 196. Birkhauser, Basel (2006)

    Book  Google Scholar 

  29. Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69(11), 3692–3705 (2008)

    Article  MathSciNet  Google Scholar 

  30. Dabas, J., Chauhan, A., Kumar, M.: Existence of the mild solutions for impulsive fractional equations with infinite delay. Int. J. Differ. Equ. 2011, 793023 (2011)

    MathSciNet  MATH  Google Scholar 

  31. Sakthivel, R., Ren, Y., Mahmudov, N.I.: On the approximate controllability of semilinear fractional differential systems. Comput. Math. Appl. 62(3), 1451–1459 (2011)

    Article  MathSciNet  Google Scholar 

  32. Radhakrishnan, B., Balachandran, K.: Controllability of impulsive neutral functional evolution integro-differential systems with infinite delay. Nonlinear Anal. 5(4), 655–670 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Farahi, S., Guendouzi, T.: Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions. Results Math. 65(3–4), 501–521 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are very thankful to the anonymous reviewers and editor for their constructive comments and suggestions which help us to improve the manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rajesh Dhayal.

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

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, 175–191 (2021). https://doi.org/10.1007/s12591-019-00463-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-019-00463-1

Keywords

Mathematics Subject Classification

Navigation