Skip to main content
SpringerLink
Log in

Controllability of fractional neutral stochastic integrodifferential inclusions of order \(p \in (0,1]\), \(q \in (1,2]\) with fractional Brownian motion

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract.

This paper investigates the complete and approximate controllability of nonlinear fractional neutral stochastic integrodifferential inclusions of order \(0 < p\leq 1 < q\leq 2\) with fractional Brownian motion (fBm) in finite dimensional space. By using fixed-point theorems, namely the Bohnenblust-Karlin and Covitz-Nadler for the convex and nonconvex cases, a new set of sufficient conditions are formulated which guarantees each type of controllability. The result is established under the assumption that the associated linear stochastic system is completely and approximately controllable. Finally, two examples are presented to illustrate the efficiency of the obtained theoretical 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. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)

  2. K. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Academic press, New York, 1974)

  3. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (Academic press, California, 1998)

  4. S.G. Samko, A.A. Kilbas, I. Maricev, Fractional Integrals and Derivatives

  5. J. Sabatier, Om.P. Agrawal, J.A. Tenreiro Machado, Advances in Fractional Calculus (Springer, Dordrecht, 2007)

  6. K. Balachandran, V. Govindaraj, M. Rivero, J.J. Trujillo, Appl. Math. Comput. 257, 66 (2015)

    MathSciNet  Google Scholar 

  7. K. Balachandran, S. Karthikeyan, J. Franklin Inst. 345, 382 (2008)

    Article  MathSciNet  Google Scholar 

  8. N.I. Mahmudov, A. Denker, Int. J. Control 73, 144 (2000)

    Article  MathSciNet  Google Scholar 

  9. N.I. Mahmudov, S. Zorlu, Int. J. Control 76, 95 (2003)

    Article  MathSciNet  Google Scholar 

  10. T. Sathiyaraj, P. Balasubramaniam, Eur. Phys. J. ST 225, 83 (2016)

    Article  Google Scholar 

  11. T. Sathiyaraj, P. Balasubramaniam, Differ. Equ. Dyn. Syst., DOI:10.1007/s12591-016-0277-y:1-14 (2016)

  12. S. Abbas, B. Mouffak, Opusc. Math. 33, 209 (2013)

    Article  Google Scholar 

  13. P. Balasubramaniam, Tamkang J. Math. 33, 25 (2002)

    MathSciNet  Google Scholar 

  14. W. Dai, C.C. Heyde, J. Appl. Math. Stoch. Anal. 9, 439 (1996)

    Article  MathSciNet  Google Scholar 

  15. D. Feyel, A. De La Pradelle, Potential Anal. 10, 273 (1999)

    Article  MathSciNet  Google Scholar 

  16. E. Hernandez, D.N. Keck, M.A. McKibben, J. Appl. Math. Stoch. Anal 26, 69747 (2007)

    MathSciNet  Google Scholar 

  17. B.B. Mandelbrot, J.W. Van Ness, SIAM Rev. 10, 422 (1968)

    Article  ADS  MathSciNet  Google Scholar 

  18. B. Boufoussi, S. Hajji, Stat. Probab. Lett. 82, 1549 (2012)

    Article  MathSciNet  Google Scholar 

  19. T. Caraballo, M.J. Garrido-Atienza, T. Taniguchi, Nonlinear Anal.: Theory, Methods Appl. 74, 3671 (2011)

    Article  MathSciNet  Google Scholar 

  20. S. Zhang, J. Sun, Int. J. Robust Nonlinear Control 25, 2196 (2015)

    Google Scholar 

  21. L. Shen, S. Junping, S. Jitao, Automatica 46, 1068 (2010)

    Article  Google Scholar 

  22. R. Sakthivel, Y. Ren, A. Debbouche, N.I. Mahmudov, Appl. Anal. 95, 1 (2015)

    MathSciNet  Google Scholar 

  23. J. Kexue, P. Jigen, Appl. Math. Lett. 24, 2019 (2011)

    Article  MathSciNet  Google Scholar 

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

  25. N.I. Mahmudov, SIAM J. Control Optim. 42, 1604 (2003)

    Article  MathSciNet  Google Scholar 

  26. J.B. Aubin, A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory (Springer-Verlag, New York, 1984)

  27. L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings (Kluwer Academic Publishers, Dordrecht, 1999)

  28. S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis: Volume II: Applications (Springer Science & Business Media, Dordrecht, 2013)

  29. M. Kisielewicz, Differential Inclusions and Optimal Control (Kluwer Academic Publishers, Dordrecht, 1991)

  30. A. Lasota, Z. Opial, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 13, 781 (1965)

    MathSciNet  Google Scholar 

  31. C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions (Springer, New York, 2006)

  32. H.F. Bohnenblust, S. Karlin, On a Theorem of Ville, Contributions to the Theory of Games (Annals of Mathematics Studies, Princeton University Press, Princeton, 1950)

  33. H. Covitz, S.B. Nadler, Isr. J. Math. 8, 5 (1970)

    Article  MathSciNet  Google Scholar 

  34. R. Sakthivel, J.H. Kim, N.I. Mahmudov, Rep. Math. Phys. 58, 433 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  35. A.A. Chikriy, I.I. Matichin, J. Autom. Inf. Sci. 40, 1 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Gandhigram Rural Institute, Deemed University, 624 302, Gandhigram, Tamil Nadu, India

    T. Sathiyaraj & P. Balasubramaniam

Authors
  1. T. Sathiyaraj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Balasubramaniam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Balasubramaniam.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sathiyaraj, T., Balasubramaniam, P. Controllability of fractional neutral stochastic integrodifferential inclusions of order \(p \in (0,1]\), \(q \in (1,2]\) with fractional Brownian motion. Eur. Phys. J. Plus 131, 357 (2016). https://doi.org/10.1140/epjp/i2016-16357-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2016-16357-2

Search

Navigation