Advertisement

Approximate Controllability via Resolvent Operators of Sobolev-Type Fractional Stochastic Integrodifferential Equations with Fractional Brownian Motion and Poisson Jumps

  • Hamdy M. AhmedEmail author
Original Paper
  • 22 Downloads

Abstract

Using fractional calculus, stochastic analysis theory, and fixed point theorems with the properties of analytic \(\alpha \)-resolvent operators, sufficient conditions for approximate controllability of Sobolev-type fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps are established. Finally, an example is given to illustrate the obtained results.

Keywords

Fractional Brownian motion Poisson jumps Sobolev-type fractional stochastic integrodifferential equations Approximate controllability Resolvent operators 

Mathematics Subject Classification

60H15 34A08 60G22 93B05 

Notes

Acknowledgements

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

References

  1. 1.
    Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Boufoussi, B., Hajji, S.: Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space. Statistics and Probability Letters 82, 1549–1558 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arthi, G., Park, JuH, Jung, H.Y.: Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion. Communications in Nonlinear Science and Numerical Simulation 32, 145–157 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Diop, M.A., Ezzinbi, K., Mbaye, M.M.: Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion. Stochastics 87, 1061–1093 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Boudaoui, A., Caraballo, T., Ouahab, A.: Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay. Applicable Analysis 95, 2039–2062 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Tamilalagan, P., Balasubramaniam, P.: Moment stability via resolvent operators of fractional stochastic differential inclusions driven by fractional Brownian motion. Applied Mathematics and Computation 305, 299–307 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bashirov, A.E., Mahmudov, N.I.: On concepts of controllability for linear deterministic and stochastic systems. SIAM J. Control Optim. 37, 1808–1821 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hamdy, M.: Ahmed, Controllability of fractional stochastic delay equations. Lobachevskii Journal of Mathematics 30, 195–202 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sathya, R., Balachandran, K.: Controllability of Sobolev-Type neutral stochastic mixed integrodifferential systems. European journal of mathematicall sciences 1, 68–87 (2012)Google Scholar
  10. 10.
    Karthikeyan, S., Balachandran, K., Sathya, M.: Controllability of nonlinear stochastic systems with multiple time-varying delays in control. International Journal of Applied Mathematics and Computer Science 25, 207–215 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hamdy, M.: Ahmed, Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion. IMA Journal of Mathematical Control and Information 32, 781–794 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Balachandran, K., Matarand, M., Trujillo, J.J.: Note on controllability of linear fractional dynamical systems. Journal of control and decision 3, 267–279 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mabel Lizzy, R., Balachandran, K., Suvinthra, M.: Controllability of nonlinear stochastic fractional systems with distributed delays in control, Journal of control and decision. 1-16, (2017)  https://doi.org/10.1080/23307706.2017.1297690
  14. 14.
    Debbouche, A., Torres, Delfim F.M.: Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions. Applied Mathematics and Computation 243, 161–175 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Muthukumar, P., Rajivganthi, C.: Approximate controllability of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. Journal of Systems Science and Complexity 28, 1033–1048 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Muthukumar, P., Thiagu, K.: Existence of solutions and approximate controllability of fractional nonlocal neutral impulsive stochastic differential equations of order \(1 < q < 2\) with infinite delay and Poisson jumps. Journal of Dynamical and Control Systems 23, 213–235 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sakthivel, R., Ganesh, R., Ren, Y., Anthoni, S.M.: Approximate controllability of nonlinear fractional dynamical systems. Communications in Nonlinear Science and Numerical Simulation 18, 3498–3508 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kerboua, Mourad, Debbouche, Amar, Baleanu, Dumitru: Approximate controllability of Sobolev Type nonlocal fractional stochastic dynamics systems in Hilbert space. Abstract and applied analysis 2013, 1–10 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mokkedem, Fatima Zahra, Fu, Xianlong: Approximate controllability of semi-linear neutral integro-differential systems with finite delay. Applied Mathematics and Computation 242, 202–215 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yan, Zuomao, Jia, Xiumei: Approximate controllability of impulsive fractional stochastic partial neutral integrodifferential inclusions with infinite delay. Advances in Difference Equations 2015, 1–31 (2015)zbMATHGoogle Scholar
  21. 21.
    Rajivganthi, C., Muthukumar, P., Ganesh Priya, B.: Approximate controllability of fractional stochastic integrodifferential equations with infinite delay of order \(1<\alpha <2\), IMA Journal of Mathematical Control and Information. 1–15, (2015)Google Scholar
  22. 22.
    Tamilalagan, P., Balasubramaniam, P.: Approximate controllability of fractional stochastic differential equations driven by mixed fractional Brownian motion via resolvent operators. International Journal of Control 90, 1713–1727 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Muthukumar, P., Thiagu, K.: Existence of solutions and approximate controllability of fractional nonlocal neutral impulsive stochastic differential equations of order \(1 < q < 2\) with infinite delay and Poisson Jumps. J. Dyn. Control Syst. 23, 213–235 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yan, Zuomao, Fangxia, Lu: the approximate controllability of a multi-valued fractional impulsive stochastic partial integro-differential equation with infinite delay. Applied Mathematics and Computation 292, 425–447 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  26. 26.
    Agarwal, R.P., Santos, J.P.C., Cuevas, C.: Analytic resolvent operator and existence results for fractional order evolutionary integral equations. J. Abstr. Differ. Equ. Appl. 2, 26–47 (2012)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Santos, J.P.C., Cuevas, C., Andrade, B.: Existence results for a fractional equations with state dependent delay, Advances in Difference Equations, 2011 (2011), Article ID 642013Google Scholar
  28. 28.
    Dauer, P.J., Mahmudov, N.I.: Controllability of stochastic semilinear functional differential equations in Hilbert spaces. J. Math. Anal. Appl. 290, 373–394 (2004)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Andrade, B.D., Santos, J.P.C.: Existence of solutions for a fractional neutral integro differential equation with unbounded delay. Electron. J 2012, 1–13 (2012)Google Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Higher Institute of EngineeringEl-Shorouk AcademyEl-Shorouk City, CairoEgypt

Personalised recommendations