Skip to main content
SpringerLink
Log in

Fractional optimal control problem for ordinary differential equation in weighted Lebesgue spaces

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, a necessary and sufficient condition, such as the Pontryagin’s maxi-mum principle for a fractional optimal control problem with concentrated parameters, is given by the ordinary fractional differential equation with a coefficient in weighted Lebesgue spaces. We discuss a formulation of fractional optimal control problems by a fractional differential equation in the sense of Caputo fractional derivative. The statement of the fractional optimal control problem is studied by using a new version of the increment method that essentially uses the concept of an adjoint equation of the integral form.

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. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)

    Article  MathSciNet  Google Scholar 

  2. Agrawal, O.P., Defterli, O., Baleanu, D.: Fractional optimal control problems with several state and control variables. J. Vib. Control 16(13), 1967–1976 (2012)

    Article  MathSciNet  Google Scholar 

  3. Ali, H.M., Lobo Pereira, F., Gama, S.M.A.: A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems. Math. Methods Appl. Sci. 39(13), 3640–3649 (2016)

    Article  MathSciNet  Google Scholar 

  4. Alsaedi, A., Alghamdi, N., Agrawal, R.P., Ntouyas, S.K., Ahmad, B.: Multi-term fractional-order boundary-value problems with nonlocal integral boundary conditions. Electron. J. Differ. Equ. 2018(87), 1–16 (2018)

    MathSciNet  Google Scholar 

  5. Bachar, I., Mâagli, H., Rădulescu, V.D.: Positive solutions for superlinear Riemann–Liouville fractional boundary-value problems. Electron. J. Differ. Equ. 2017(240), 1–16 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Baleanu, D., Machado, J.T., Luo, A.C.J.: Fractional Dynamics and Control. Springer, New York (2012)

    Book  Google Scholar 

  7. Bandaliyev, R.A., Guliyev, V.S., Mamedov, I.G., Sadigov, A.B.: The optimal control problem in the processes described by the Goursat problem for a hyperbolic equation in variable exponent Sobolev spaces with dominating mixed derivatives. J. Comput. Appl. Math. 305, 11–17 (2016)

    Article  MathSciNet  Google Scholar 

  8. Bandaliyev, R.A., Guliyev, V.S., Mamedov, I.G., Rustamov, Y.I.: Optimal control problem for Bianchi equation in variable exponent Sobolev spaces. J. Optim. Theory. Appl. 180, 303 (2019). https://doi.org/10.1007/s10957-018-1290-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Castillo, R.E., Rafeiro, H.: An Introductory Course in Lebesgue Spaces. CMS Books in Mathematics/Ouvrages de Mathèmatiques de la SMC. Springer, New York (2016)

    Google Scholar 

  10. Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)

    Book  Google Scholar 

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

    Book  Google Scholar 

  12. Kamocki, R.: Pontryagin maximum principle for fractional ordinary optimal control problems. Math. Methods Appl. Sci. 37(11), 1668–1686 (2014)

    Article  MathSciNet  Google Scholar 

  13. Kamocki, R.: On the existence of optimal solutions to fractional optimal control problems. Appl. Math. Comput. 35, 94–104 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Kazem, S.: Exact solution of some linear fractional differential equations by Laplace transform. Int. J. Nonlinear Sci. 16, 3–11 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Kexue, L., Jigen, P.: Laplace transform and fractional differential equations. Appl. Math. Lett. 24, 2019–2023 (2011)

    Article  MathSciNet  Google Scholar 

  16. Kilbas, A.A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Google Scholar 

  17. Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16, 1140–1153 (2011)

    Article  MathSciNet  Google Scholar 

  18. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelaticity. Imperial College Press, London (2010)

    Book  Google Scholar 

  19. Malinowska, A.B., Torres, D.F.M.: Introduction to the Fractional Calculus of Variations. Imperial College Press, London (2012)

    Book  Google Scholar 

  20. Mardanov, M.J., Sharifov, Y.A.: Pontryagin’s maximum principle for the optimal control problems with multipoint boundary conditions. Abstr. Appl. Anal. Article ID 428042, 1–6 (2015)

  21. Maz’ya, V.G.: Sobolev Spaces. Springer, Berlin (1985)

    Book  Google Scholar 

  22. Mu, P., Wang, L., Liu, C.: A control parametrization method to solve the fractional-order optimal control problem. J. Optim. Theory. Appl. (2017). https://doi.org/10.1007/s10957-017-1163-7

    Article  Google Scholar 

  23. Mu, P., Wang, L., An, Y., et al.: A novel fractional microbial batch culture process and parameter identification. Differ. Equ. Dyn. Syst. 26, 265 (2018). https://doi.org/10.1007/s12591-017-0381-7

    Article  MathSciNet  MATH  Google Scholar 

  24. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  25. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishenko, E.F.: Mathematical Theory of Optimal Processes. Nauka, Moscow (1969). (in Russian)

    Google Scholar 

  26. Pooseh, S., Almeida, R., Torres, D.F.M.: Fractional order optimal control problems with free terminal time. J. Ind. Manag. Optim. 10(2), 363–381 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Samko, S.G., Kilbas, A.A., Marichev, D.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, London (1993)

    MATH  Google Scholar 

  28. Tarasov, V.E.: Fractional Dynamics: Fractional Dynamics Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, New York (2011)

    MATH  Google Scholar 

  29. Wen, L., Wang, S., Rehbock, V.: Numerical solution of fractional optimal control. J. Optim. Theory. Appl. 180, 556 (2019). https://doi.org/10.1007/s10957-018-1418-y

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the grant of Presidium of Azerbaijan National Academy of Sciences 2018 in the framework of funding of research programs and by the Ministry of Education and Science of the Russian Federation (Agreement Number: 02.a03.21.0008). We would like to thank both reviewers for their valuable comments on the paper.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan

    R. A. Bandaliyev, M. J. Mardanov & T. K. Melikov

  2. Institute of Control Systems of NAS of Azerbaijan, Baku, Azerbaijan

    I. G. Mamedov & T. K. Melikov

  3. S.M. Nikolskii Institute of Mathematics, RUDN University, Moscow, Russia, 117198

    R. A. Bandaliyev

  4. Baku State University, Baku, Azerbaijan

    M. J. Mardanov

Authors
  1. R. A. Bandaliyev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. G. Mamedov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. J. Mardanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. T. K. Melikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. A. Bandaliyev.

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

Bandaliyev, R.A., Mamedov, I.G., Mardanov, M.J. et al. Fractional optimal control problem for ordinary differential equation in weighted Lebesgue spaces. Optim Lett 14, 1519–1532 (2020). https://doi.org/10.1007/s11590-019-01518-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-019-01518-6

Keywords

Search

Navigation