Skip to main content
SpringerLink
Log in

A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems

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

Abstract

We develop a simple and accurate method to solve fractional variational and fractional optimal control problems with dependence on Caputo and Riemann–Liouville operators. Using known formulas for computing fractional derivatives of polynomials, we rewrite the fractional functional dynamical optimization problem as a classical static optimization problem. The method for classical optimal control problems is called Ritz’s method. Examples show that the proposed approach is more accurate than recent methods available in the literature.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

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

    Google Scholar 

  2. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives. Translated from the 1987 Russian original. Gordon and Breach, Yverdon (1993)

  3. Valério, D., Tenreiro Machado, J., Kiryakova, V.: Some pioneers of the applications of fractional calculus. Fract. Calc. Appl. Anal. 17(2), 552–578 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. de Oliveira, E.C., Machado, J.A.T.: A review of definitions for fractional derivatives and integral. Math. Probl. Eng. 2014, 238459 (2014)

    Article  MathSciNet  Google Scholar 

  5. Ortigueira, M.D., Trujillo, J.J.: A unified approach to fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 17(12), 5151–5157 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ortigueira, M.D.: Fractional calculus for scientists and engineers. Lecture Notes in Electrical Engineering, vol. 84. Springer, Dordrecht (2011)

  7. Tenreiro Machado, J.A., Baleanu, D., Chen, W., Sabatier, J.: New trends in fractional dynamics. J. Vib. Control 20(7), 963 (2014)

    Article  MathSciNet  Google Scholar 

  8. Almeida, R., Pooseh, S., Torres, D.F.M.: Computational Methods in the Fractional Calculus of Variations. Imperial College Press, London (2015)

    Book  MATH  Google Scholar 

  9. Malinowska, A.B., Odzijewicz, T., Torres, D.F.M.: Advanced Methods in the Fractional Calculus of Variations. Springer, Cham (2015)

    Book  MATH  Google Scholar 

  10. Malinowska, A.B., Torres, D.F.M.: Introduction to the fractional calculus of variations. Imperial College Press, London (2012)

    Book  MATH  Google Scholar 

  11. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E (3) 53(2), 1890–1899 (1996)

    Article  MathSciNet  Google Scholar 

  12. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E (3) 55(3), part B, 3581–3592 (1997)

  13. Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A 40(24), 6287–6303 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Almeida, R., Torres, D.F.M.: Leitmann’s direct method for fractional optimization problems. Appl. Math. Comput. 217(3), 956–962 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Almeida, R., Torres, D.F.M.: Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1490–1500 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Atanacković, T.M., Janev, M., Konjik, S., Pilipović, S., Zorica, D.: Expansion formula for fractional derivatives in variational problems. J. Math. Anal. Appl. 409(2), 911–924 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Baleanu, D., Garra, R., Petras, I.: A fractional variational approach to the fractional Basset-type equation. Rep. Math. Phys. 72(1), 57–64 (2013)

    Article  MathSciNet  Google Scholar 

  18. Bourdin, L., Odzijewicz, T., Torres, D.F.M.: Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition–application to fractional variational problems. Differ. Integral Equ. 27(7–8), 743–766 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Odzijewicz, T., Torres, D.F.M.: The generalized fractional calculus of variations. Southeast Asian Bull. Math. 38(1), 93–117 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Almeida, R., Khosravian-Arab, H., Shamsi, M.: A generalized fractional variational problem depending on indefinite integrals: Euler–Lagrange equation and numerical solution. J. Vib. Control 19(14), 2177–2186 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Blaszczyk, T., Ciesielski, M.: Numerical solution of fractional Sturm–Liouville equation in integral form. Fract. Calc. Appl. Anal. 17(2), 307–320 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Almeida, R., Torres, D.F.M.: A discrete method to solve fractional optimal control problems. Nonlinear Dyn. 80(4), 1811–1816 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pooseh, S., Almeida, R., Torres, D.F.M.: Numerical approximations of fractional derivatives with applications. Asian J. Control 15(3), 698–712 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dehghan, M., Hamedi, E.-A., Khosravian-Arab, H.: A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. J. Vib. Control. (2014). doi:10.1177/1077546314543727

  25. Caputo, M.: Linear models of dissipation whose \(Q\) is almost frequency independent. II. Fract. Calc. Appl. Anal. 11(1), 4–14 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38(1–4), 323–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Frederico, G.S.F., Torres, D.F.M.: Fractional conservation laws in optimal control theory. Nonlinear Dyn. 53(3), 215–222 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Frederico, G.S.F., Torres, D.F.M.: Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. Int. Math. Forum 3(9–12), 479–493 (2008)

    MathSciNet  MATH  Google Scholar 

  29. 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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sweilam, N.H., Al-Ajami, T.M., Hoppe, R.H.W.: Numerical solution of some types of fractional optimal control problems. Sci. World J. 2013, 306237 (2013)

    Article  Google Scholar 

Download references

Acknowledgments

This work is part of first author’s PhD project. It was partially supported by Islamic Azad University, Tehran, Iran, and CIDMA-FCT, Portugal, within project UID/MAT/04106/2013. Jahanshahi was also supported by a scholarship from the Ministry of Science, Research and Technology of the Islamic Republic of Iran, to visit the University of Aveiro, Portugal, and work with Professor Torres. The hospitality and the excellent working conditions at the University of Aveiro are here gratefully acknowledged. The authors are indebted to an anonymous referee for a careful reading of the original manuscript and for providing several suggestions, questions, and remarks. They are also grateful to the Editor-in-Chief, Professor Giannessi, and Ryan Loxton, for English improvements.

Author information

Authors and Affiliations

  1. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

    Salman Jahanshahi

  2. Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193, Aveiro, Portugal

    Delfim F. M. Torres

Authors
  1. Salman Jahanshahi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Delfim F. M. Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Delfim F. M. Torres.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jahanshahi, S., Torres, D.F.M. A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems. J Optim Theory Appl 174, 156–175 (2017). https://doi.org/10.1007/s10957-016-0884-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-016-0884-3

Keywords

Mathematics Subject Classification

Search

Navigation