Skip to main content
SpringerLink
Log in

Approximate Controllability of a Class of Semilinear Hilfer Fractional Differential Equations

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

This article studies the approximate controllability for a class of fractional control system with analytic semigroup governed by differential equations with Hilfer fractional derivatives of order \(\delta \in (0,1)\) and type \(\zeta \in [0,1]\) in a Banach space. The existence and uniqueness of the mild solution is established with the help of semigroup theory, fractional power of operators and a generalized contraction type fixed point theorem. Further, a set of sufficient conditions is formulated for the approximate controllability of the system under consideration. The result obtained holds irrespective of whether the generated semigroup is compact or non-compact.

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

Availability of data and materials

Not applicable.

References

  1. Chalishajar, D.N., George, R.K., Nandakumaran, A.K., Acharya, F.S.: Trajectory controllability of nonlinear integro-differential system. J. Frankl. Inst. 347, 1065–1075 (2010)

    Article  MathSciNet  Google Scholar 

  2. 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  Google Scholar 

  3. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, New York (2010)

    Book  Google Scholar 

  4. Dineshkumar, C., Udhayakumar, R., Vijayakumar, V., Nisar, K.S.: A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systems. Chaos Solitons Fractals 142, 1–12 (2021)

    Article  MathSciNet  Google Scholar 

  5. Furati, K.M., Kassim, M.D., Tatar, N.E.: Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 64, 1616–1626 (2012)

    Article  MathSciNet  Google Scholar 

  6. Gu, H., Trujillo, J.J.: Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 257, 344–354 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Gu, H., Sun, Y.: Nonlocal controllability of fractional measure evolution equation. J. Inequal. Appl. 2020, 1–18 (2020)

    Article  MathSciNet  Google Scholar 

  8. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Kalman, R.E.: Control of randomly varying linear dynamical systems. In: Proc. Sympos. Appl. Math., vol. XIII, pp. 287–298. American Mathematical Society, Providence (1962)

  12. Kalman, R.E.: Mathematical description of linear dynamical systems. J. SIAM Control Ser. A. 1, 152–192 (1963)

    MathSciNet  MATH  Google Scholar 

  13. Kavitha, K., Vijayakumar, V., Udhayakumar, R.: Results on controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncompactness. Chaos Solitons Fractals 139, 1–12 (2020)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  15. Liu, Z., Li, X.: Approximate controllability of fractional evolution systems with Riemann–Liouville fractional derivatives. SIAM J. Control. Optim. 53, 1920–1933 (2015)

    Article  MathSciNet  Google Scholar 

  16. Lv, J., Yang, X.: Approximate controllability of Hilfer fractional differential equations. Math. Methods Appl. Sci. 43, 242–254 (2020)

    Article  MathSciNet  Google Scholar 

  17. Mahmudov, N.I., Zorlu, S.: On the approximate controllability of fractional evolution equations with compact analytic semigroup. J. Comput. Appl. Math. 259, 194–204 (2014)

    Article  MathSciNet  Google Scholar 

  18. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Sukavanam, N., Kumar, S.: Approximate controllability of fractional order semilinear delay systems. J. Optim. Theory Appl. 151, 373–384 (2011)

    Article  MathSciNet  Google Scholar 

  21. Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.Q.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)

    Article  Google Scholar 

  22. Triggiani, R.: Controllability and observability in Banach space with bounded operators. SIAM J. Control. Optim. 13, 462–491 (1975)

    Article  MathSciNet  Google Scholar 

  23. Vijayakumar, V., Udhayakumar, R.: Results on approximate controllability for non-densely defined Hilfer fractional differential system with infinite delay. Chaos Solitons Fractals 139, 1–11 (2020)

    Article  MathSciNet  Google Scholar 

  24. Wang, J.R., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. Real World Appl. 12, 262–272 (2011)

    Article  MathSciNet  Google Scholar 

  25. Yang, M., Wang, Q.: Approximate controllability of Riemann–Liouville fractional differential inclusions. Appl. Math. Comput. 274, 267–281 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Yang, M., Wang, Q.: Existence of mild solutions for a class of Hilfer fractional evolution equations with nonlocal conditions. Fract. Calc. Appl. Anal. 20, 679–705 (2017)

    Article  MathSciNet  Google Scholar 

  27. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)

    Article  MathSciNet  Google Scholar 

  28. Zhou, H.X.: Approximate controllability for a class of semilinear abstract equations. SIAM J. Control. Optim. 21, 551–565 (1983)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The second author expresses her gratitude to Indian Institute of Technology Guwahati, India for providing her senior research fellowship to carry out research towards Ph.D. The authors are grateful to the esteemed Reviewer for his/her insightful comments which have resulted in an improved version of the manuscript and to the Associate Editor Prof. Ioan Rasa for allowing a revision.

Funding

The second author received senior research fellowship from Indian Institute of Technology Guwahati for the period 2018–2020.

Author information

Authors and Affiliations

  1. Indian Institute of Technology Guwahati, Guwahati, 781 039, India

    Swaroop Nandan Bora & Bandita Roy

Authors
  1. Swaroop Nandan Bora
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bandita Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Conceptualization: BR, SNB; Methodology: BR; Formal analysis and investigation: BR; Writing - original draft preparation: BR, SNB; Writing - review and editing: BR, SNB.

Corresponding author

Correspondence to Swaroop Nandan Bora.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest or competing interest.

Code availability

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

Bora, S.N., Roy, B. Approximate Controllability of a Class of Semilinear Hilfer Fractional Differential Equations. Results Math 76, 197 (2021). https://doi.org/10.1007/s00025-021-01507-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-021-01507-1

Keywords

Mathematics Subject Classification

Search

Navigation