Advertisement

Nonlinear Dynamics

, Volume 62, Issue 3, pp 609–614 | Cite as

Fractional variational optimal control problems with delayed arguments

  • Fahd Jarad
  • Thabet Abdeljawad
  • Dumitru BaleanuEmail author
Original Paper

Abstract

The paper deals with optimal control problems in the presence of delay in the state variables as well as the presence of the Riemann–Liouville fractional derivatives of the state variables. One example is analyzed in detail.

Keywords

Fractional derivative Fractional optimal control Delay 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kilbas, A.A., Srivastava, H.H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) zbMATHGoogle Scholar
  2. 2.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives—Theory and Applications. Gordon and Breach, New York (1993) zbMATHGoogle Scholar
  3. 3.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) zbMATHGoogle Scholar
  4. 4.
    Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishing, Reading (2006) Google Scholar
  5. 5.
    West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, New York (2003) Google Scholar
  6. 6.
    Mainardi, F., Luchko, Y., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4(2), 153–192 (2001) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Scalas, E., Gorenflo, R., Mainardi, F.: Uncoupled continuous-time random walks: solution and limiting behavior of the master equation. Phys. Rev. E 69, 011107 (2004) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Klimek, M.: Fractional sequential mechanics—models with symmetric fractional derivative. Czechoslov. J. Phys. 51, 1348–1354 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2006) CrossRefGoogle Scholar
  11. 11.
    Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, S.I., Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327, 891–897 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Baleanu, D., Muslih, S.I.: Lagrangian formulation of classical fields within Riemann–Liouville fractional derivatives. Phys. Scr. 72(2–3), 119–121 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Muslih, S.I., Baleanu, D.: Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives. J. Math. Anal. Appl. 304(3), 599–606 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Baleanu, D., Avkar, T.: Lagrangians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cimento B 119, 73–79 (2004) Google Scholar
  15. 15.
    Jesus, I.S., Machado, J.A.T.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263–282 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Agrawal, O.P., Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for Fractional Optimal Control Problems. J. Vib. Control 13(9–10), 1269–1281 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Chen, Y.Q., Vinagre, B.M., Podlubny, I.: Continued fraction expansion approaches to discretizing fractional order derivatives—an expository review. Nonlinear Dyn. 38(1–4), 155–170 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Bohannan, G.W.: Analog fractional order controller in temperature and motor control applications. J. Vib. Control 14, 1487–1498 (2008) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Suárez, I.J., Vinagre, B.M., Chen, Y.Q.: A fractional adaptation scheme for lateral control of an AGV. J. Vib. Control 14, 1499–1511 (2008) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Rosenblueth, J.F.: Systems with time delay in the calculus of variations: a variational approach. J. Math. Control Inf. 5, 125–145 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004) zbMATHCrossRefGoogle Scholar
  22. 22.
    Dumitru, B., Maraaba, T., Jarad, F.: Fractional principles with delay. J. Phys. A Math. Theor. 41(31), 315403 (2008) CrossRefGoogle Scholar
  23. 23.
    Lima, F., Machado, J., Crisostomo, M.: Pseudo phase plane, delay and fractional dynamics. JESA 42, 1037–1051 (2008) CrossRefGoogle Scholar
  24. 24.
    Deng, W., Li, C., Lu, J.: Stability analysis of linear fractional differential system with multiple time scales. Nonlinear Dyn. 48, 409–416 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Driver, R.D.: Ordinary and Delay Differential Equations. Applied Mathematical Sciences, vol. 20. Springer, New York (1977) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Fahd Jarad
    • 1
  • Thabet Abdeljawad
    • 1
  • Dumitru Baleanu
    • 1
    • 2
    Email author
  1. 1.Department of Mathematics and Computer Sciences, Faculty of Arts and SciencesÇankaya UniversityAnkaraTurkey
  2. 2.Institute of Space SciencesMagurele-BucharestRomania

Personalised recommendations