Skip to main content
SpringerLink
Log in

Controllability Results of Hilfer Fractional Derivative Through Integral Contractors

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

The paper sheds light on Hilfer’s controllability facts of neutral fractional system. Originally, the mild solution is derived using semigroup theory and the Laplace transform approach. Controllability of the Hilfer fractional system in non-dense domain using integral contractor which employs the sequence technique with the advantage that the nonlinear function does not meet the Lipschitz condition. To support the computed results, an appropriate examples are discussed.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Agarwal, P., Baleanu, D., Quan, Y., Momani, C.S., Machado, J.A.: Fractional Calculus- Models. Algorithms Technology. Springer, Singapore (2018)

    Google Scholar 

  2. Altman, M.: Contractors and Contractor Directions: Theory and Applications. Dekker, New York (1977)

    MATH  Google Scholar 

  3. Almalahi, M.A., Panchal, S.K., Aldwoah, K.: On the explicit solution of \(\Psi -\)Hilfer integro-differential nonlocal cauchy problem. Progr. Fract. Differ. Appl. 9(1), 65–67 (2023)

    Article  Google Scholar 

  4. Almalahi, M.A., Bazighifan, O., Panchal, S.K., Askar, S.S., Oros, G.I.: Analytical study of two nonlinear coupled hybrid systems involving generalized Hilfer fractional operators. Fractal Fract. 178(5), 1–21 (2021)

    Google Scholar 

  5. Beyrouthy, T., Fesquet, L.: An event-driven FIR filter: design and implementation, An event-driven FIR filter: design and implementation. Science 5, 59–65 (2011)

    Google Scholar 

  6. Chandra, A., Chattopadhyay, S.: Design of hardware efficient FIR filter: a review of the state of the art approaches, engineering science and technology. Int. J. 19, 212–226 (2016)

    Google Scholar 

  7. Diop, A., Frederico, G.S.F., Sousa, J.V.D.C.: On controllability for a class of multi-term time-fractional random differential equations with state-dependent delay. Ann. Funct. Anal. (2022). https://doi.org/10.1007/s43034-022-00165-w

    Article  MathSciNet  MATH  Google Scholar 

  8. Du, J., Jiang, W., Pang, D., Niazi, A.U.K.: Exact controllability for Hilfer fractional differential inclusions involving nonlocal initial conditions. Complexity 2018, 1–13 (2018)

    MATH  Google Scholar 

  9. Fu, X.L.: On solutions of neutral nonlocal evolution equations with non-dense domain. J. Math. Anal. Appl. 299, 392–410 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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  MATH  Google Scholar 

  11. George, R.K.: Approximate controllability of semilinear systems using integral contractors. Numer. Funct. Anal. Optim. 16, 127–138 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gu, H., Zhou, Y., Ahmad, B., Alsaedi, A.: Integral solutions of fractional evolution equations with non-dense domain. Electron. J. Differ. Equ. 145, 1–15 (2017)

    Google Scholar 

  13. 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 

  14. Haiping, Y., Jianming, G., Yongsheng, D.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hilal, K., Kajouni, A., Lmou, H.: Boundary value problem for the Langevin equation and inclusion with the Hilfer Fractional Derivative. Int. J. Differ. Equ. 2022, 1–12 (2022)

    MathSciNet  MATH  Google Scholar 

  16. Hilal, K., Kajouni, A., Lmou, H.: Existence and stability results for a coupled system of Hilfer fractional Langevin equation with non local integral boundary value conditions. Arxiv 2, 1241–1259 (2020) https://doi.org/10.48550/arXiv.2206.07457

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

    Book  MATH  Google Scholar 

  18. Hilfer, R.: Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284, 399–408 (2002)

    Article  Google Scholar 

  19. 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, 1–9 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations In: North-Holland Mathematics Studies, 204 Elsevier Science, Amsterdam (2006)

  21. Kumar, S., Sukavanam, N.: Controllability of fractional order system with nonlinear term having integral contractor. Fract. Calc. Appl. Anal. 16(4), 791–801 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, (2009)

  23. Nisar, K.S., Jothimani, K., Kaliraj, K., Ravichandran, C.: An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain. Chaos Solitons & Fractals 146, 110915 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nisar, K.S., Jothimani, K., Ravichandran, C., Baleanu, D., Kumar, D.: New approach on controllability of Hilfer fractional derivatives with nondense domain. AIMS Math. 7(6), 10079–10095 (2022)

    Article  MathSciNet  Google Scholar 

  25. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  26. Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  27. Prato, G.D., Sinestrari, E.: Differential operators with non-dense domain. Annali Della Scuola Normale Superiore di Pisa 14, 285–344 (1987)

    MathSciNet  MATH  Google Scholar 

  28. Ravichandran, C., Jothimani, K., Nisar, K.S., Mahmoud, E.E., Yahia, I.S.: An interpretation on controllability of Hilfer fractional derivative with nondense domain. Alexandria Engineering Journal 61(12), 9941–9948 (2022)

    Article  Google Scholar 

  29. Chaudhary, R., Reich, S.: Existence and controllability results for Hilfer fractional evolution equations via integral contractors, Fractional Calculus and Applied Analysis 25, 2400–2419 (2022)

  30. Vijayakumar, V., Udhayakumar, R.: Results on approximate controllability for non-densely defined Hilfer fractional differential system with infinite delay. Chaos Solitons & Fractals 139, 1–9 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Vellappandi, M., Govindaraj, V.: Operator theoretic approach in fractional-order delay optimal control problems. Math. Methods Appl. Sci. 46(6), 6529–6544 (2022)

    Article  MathSciNet  Google Scholar 

  32. Sousa, J.V.d.C., Capelas de Oliveira, E.: On the \(\Psi \)-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)

  33. Sousa, J.V.d.C., Capelas de Oliveira, E.: Leibniz type rule: \(\Psi \)-Hilfer fractional operator, Commun. Nonlinear Sci. Numer. Simul. 77: 305–311 (2019)

  34. Wang, J.R., Ibrahim, G., O’Regan, D.: Controllability of Hilfer fractional noninstantaneous impulsive semilinear differential inclusions with nonlocal conditions. Nonlinear Anal. Model. Control 24(6), 743–762 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Zahoor, S., Naseem, S.: Design and implementation of an efficient FIR digital filter. Cogent Eng. 4, 1323373 (2017)

    Article  Google Scholar 

  36. Zhang, Z., Liu, B.: Controllability results for fractional functional differential equations with non-dense domain. Numer. Funct. Anal. Optim. 35(4), 443–460 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. Real World Appl. 11, 4465–4475 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Book  MATH  Google Scholar 

  39. Zufeng, Z., Liu, B.: Controllability results for fractional functional differential equations with non-dense domain. Numer. Funct. Anal. Optim. 35, 443–460 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632 014, Tamilnadu, India

    K. Jothimani

  2. Department of Mathematics, Sri Eshwar College of Engineering, Coimbatore, 641 202, Tamilnadu, India

    N. Valliammal

  3. Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj, Saudi Arabia

    S. Alsaeed & Kottakkaran S. Nisar

  4. Applied Sciences College, Department of Mathematical Sciences, Umm Al-Qura University, P.O. Box 715, 21955, Makkah, Saudi Arabia

    S. Alsaeed

  5. Department of Mathematics, Kongunadu Arts and Science College, Coimbatore, 641 029, Tamilnadu, India

    C. Ravichandran

Authors
  1. K. Jothimani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. Valliammal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Alsaeed
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kottakkaran S. Nisar
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. C. Ravichandran
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

onceptualization, KJ, NV; Formal analysis, SA, KSN; Investigation, KJ, NV, CR; Software, SA, CR, KSN; Validation, KSN, CR; Writing – original draft, KJ, NV, KSN, CR; Revision, Review and editing, SA, KSN, CR.

Corresponding authors

Correspondence to S. Alsaeed or Kottakkaran S. Nisar.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jothimani, K., Valliammal, N., Alsaeed, S. et al. Controllability Results of Hilfer Fractional Derivative Through Integral Contractors. Qual. Theory Dyn. Syst. 22, 137 (2023). https://doi.org/10.1007/s12346-023-00833-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00833-9

Keywords

Mathematics Subject Classification

Search

Navigation