Skip to main content
Springer Nature Link
Log in

Approximate Controllability of Fractional Order Semilinear Delay Systems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, the approximate controllability for a class of semilinear delay control systems of fractional order is proved under the natural assumption that the linear system is approximately controllable. The existence and uniqueness of the mild solution is also proved under suitable assumptions. An example is given to illustrate our main results.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Abd El-Ghaffar, A., Moubarak, M.R.A., Shamardan, A.B.: Controllability of fractional nonlinear control system. J. Fractional Calc. 17, 59–69 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Balachandran, K., Park, J.Y.: Controllability of fractional integrodifferential systems in Banach spaces. Nonlinear Anal. Hybrid Syst 3, 363–367 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bragdi, M., Hazi, M.: Existence and controllability result for an evolution fractional integrodifferential systems. Int. J. Contemp. Math. Sci. 5(19), 901–910 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Tai, Z., Wang, X.: Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces. Appl. Math. Lett. 22, 1760–1765 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Triggiani, R.: A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control Optim. 15, 407–411 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dauer, J.P., Mahmudov, N.I.: Approximate controllability of semilinear functional equations in Hilbert spaces. J. Math. Anal. Appl. 273, 310–327 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Sukavanam, N., Tomar, N.K.: Approximate controllability of semilinear delay control systems. Nonlinear Funct. Anal. Appl. 12(1), 53–59 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Jeong, J.M., Kim, J.R., Roh, H.H.: Controllability for semilinear retarded control systems in Hilbert spaces. J. Dyn. Control Syst. 13(4), 577–591 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jeong, J.M., Kim, H.G.: Controllability for semilinear functional integrodifferential equations. Bull. Korean Math. Soc. 46(3), 463–475 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bian, W.: Approximate controllability for semilinear systems. Acta Math. Hung. 81(1–2), 41–57 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chukwu, E.N., Lenhart, S.M.: Controllability question for nonlinear systems in abstract space. J. Optim. Theory Appl. 68(3), 437–462 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sukavanam, N.: Divya: Approximate controllability of abstract semilinear deterministic control system. Bull. Calcutta Math. Soc. 96(3), 195–202 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Wang, L.W.: Approximate controllability for integrodifferential equations with multiple delays. J. Optim. Theory Appl. 143, 185–206 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Naito, K., Park, J.Y.: Approximate controllability for trajectories of a delay Volterra control system. J. Optim. Theory Appl. 61(2), 271–279 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  18. Hu, L., Ren, Y., Sakthivel, R.: Existence and uniqueness of mild solution for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays. Semigroup Forum 79, 507–514 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tomara, N.K., Sukavanama, N.: Exact controllability of a semilinear thermoelastic system with control solely in thermal equation. Numer. Funct. Anal. Optim. 29(9–10), 1171–1179 (2008)

    Article  MathSciNet  Google Scholar 

  20. Naito, K.: Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim. 25(3), 715–722 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, IIT Roorkee, Roorkee, 247667, India

    N. Sukavanam & Surendra Kumar

Authors
  1. N. Sukavanam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Surendra Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Sukavanam.

Additional information

Communicated by Felix L. Chernousko.

The authors thank the referee for valuable comments and suggestions. The second author is thankful to Council of Scientific & Industrial Research for the financial support with grant number 09/143(0621)/2008-EMR-I.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sukavanam, N., Kumar, S. Approximate Controllability of Fractional Order Semilinear Delay Systems. J Optim Theory Appl 151, 373–384 (2011). https://doi.org/10.1007/s10957-011-9905-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9905-4

Keywords