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Journal of Optimization Theory and Applications

, Volume 180, Issue 2, pp 536–555 | Cite as

Numerical Solution of Two-Dimensional Variable-Order Fractional Optimal Control Problem by Generalized Polynomial Basis

  • Fakhrodin Mohammadi
  • Hossein HassaniEmail author
Article
  • 52 Downloads

Abstract

This paper deals with an efficient numerical method for solving two-dimensional variable-order fractional optimal control problem. The dynamic constraint of two-dimensional variable-order fractional optimal control problem is given by the classical partial differential equations such as convection–diffusion, diffusion-wave and Burgers’ equations. The presented numerical approach is essentially based on a new class of basis functions with control parameters, called generalized polynomials, and the Lagrange multipliers method. First, generalized polynomials are introduced and an explicit formulation for their variable-order fractional operational matrix is obtained. Then, the state and control functions are expanded in terms of generalized polynomials with unknown coefficients and control parameters. By using the residual function and its 2-norm, the under consideration problem is transformed into an optimization one. Finally, the necessary conditions of optimality results in a system of algebraic equations with unknown coefficients and control parameters can be simply solved. Some illustrative examples are given to demonstrate accuracy and efficiency of the proposed method.

Keywords

Two-dimensional variable-order fractional optimal control problem Generalized polynomials Operational matrix Lagrange multipliers Optimization method 

Mathematics Subject Classification

34A08 49J20 41A58 49J21 

References

  1. 1.
    Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations. In: To Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, New York (1998)Google Scholar
  2. 2.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Langhorne (1993)zbMATHGoogle Scholar
  3. 3.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2016)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cattani, C., Guariglia, E., Wang, S., Han, L.: On the critical strip of the Riemann zeta fractional derivative. Fundam. Inf. 151(1–4), 459–472 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Srivastava, M.H., Kuma, D., Singh, H.: An efficient analytical technique for fractional model of vibration equation. Appl. Math. Model. 45, 192–204 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Jajarmi, A., Baleanu, D.: Suboptimal control of fractional-order dynamic systems with delay argument. J. Vib. Control 24(12), 2430–2446 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mohammadi, F., Mohyud-Din, S.T.: A fractional-order Legendre collocation method for solving the Bagley-Torvik equations. Adv. Differ. Equ. 2016(1), 269 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dahaghina, MSh, Hassani, H.: A new optimization method for a class of time fractional convection–diffusion-wave equations with variable coefficients. Eur. Phys. J. Plus 132, 130 (2017)CrossRefGoogle Scholar
  9. 9.
    Dehghan, M., Abbaszadeh, M., Deng, W.: Fourth-order numerical method for the space–time tempered fractional diffusion-wave equation. Appl. Math. Lett. 73, 120–127 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ezz-Eldien, S.S., Hafez, R.M., Bhrawy, A.H., Baleanu, D., El-Kalaawy, A.A.: New numerical approach for fractional variational problems using shifted Legendre orthonormal polynomials. J. Optim. Theory Appl. 174(1), 295–320 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bhrawy, A.H., Zaky, M.A., Machado, J.A.T.: Numerical solution of the two-sided space–time fractional telegraph equation via Chebyshev tau approximation. J. Optim. Theory Appl. 174(1), 321–341 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fu, Z.J., Chen, W., Yang, H.T.: Boundary particle method for Laplace transformed time fractional diffusion equations. J. Comput. Phys. 235, 52–66 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wei, S., Chen, W., Hon, Y.C.: Implicit local radial basis function method for solving two-dimensional time fractional diffusion equations. Therm. Sci. 19, 59–67 (2015)CrossRefGoogle Scholar
  14. 14.
    Sweilam, N.H., Khader, M.M., Almarwm, H.M.: Numerical studies for the variable-order nonlinear fractional wave equation. Fract. Calc. Appl. Anal. 15, 669–683 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yang, X.J., Machado, J.A.T.: A new fractional operator of variable order: application in the description of anomalous diffusion. Phys. A 481, 276–283 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fu, Z.J., Chen, W., Ling, L.: Method of approximate particular solutions for constant- and variable-order fractional diffusion models. Eng. Anal. Bound. Elem. 57, 37–46 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dahaghin, MSh, Hassani, H.: An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation. Nonlinear Dyn. 88(3), 1587–1598 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bohannan, G.W.: Analog fractional order controller in temperature and motor control applications. J. Vib. Control. 14(9–10), 1487–1498 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zamani, M., Karimi-Ghartemani, M., Sadati, N.: FOPID controller design for robust performance using particle swarm optimization. Fract. Calcul. Appl. Anal. 10(2), 169–187 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Tripathy, M.C., Mondal, D., Biswas, K., Sen, S.: Design and performance study of phase-locked loop using fractional-order loop filter. Int. J. Circuit Theory Appl. 43(6), 776–792 (2015)CrossRefGoogle Scholar
  21. 21.
    Khader, M.M., Hendy, A.S.: An efficient numerical scheme for solving fractional optimal control problems. Int. J. Nonlinear Sci. 14(3), 287–296 (2012)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Biswas, R.K., Sen, S.: Fractional optimal control problems: a pseudo-state-space approach. J. Vib. Control. 17(7), 1034–1041 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jafari, H., Ghasempour, S., Baleanu, D.: On comparison between iterative methods for solving nonlinear optimal control problems. J. Vib. Control. 22(9), 2281–2287 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lotfi, A., Dehghan, M., Yousefi, S.A.: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 62(3), 1055–1067 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nemati, A., Yousefi, S.A.: A numerical method for solving fractional optimal control problems using Ritz method. J. Comput. Nonlinear Dyn. 11(5), 051015 (2016)CrossRefGoogle Scholar
  26. 26.
    Nemati, A., Yousefi, S., Soltanian, F., Ardabili, J.S.: An efficient numerical solution of fractional optimal control problems by using the Ritz method and Bernstein operational matrix. Asian J. Control 18(6), 2272–2282 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rabiei, K., Ordokhani, Y., Babolian, E.: The Boubaker polynomials and their application to solve fractional optimal control problems. Nonlinear Dyn. 88(2), 1013–1026 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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 22(6), 1547–1559 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ejlali, N., Hosseini, S.M.: A pseudospectral method for fractional optimal control problems. J. Optim. Theory Appl. 174(1), 1–25 (2016)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Heydari, M.H., Hooshmandasl, M.R., Maalek Ghaini, F.M., Cattani, C.: Wavelets method for solving fractional optimal control problems. Appl. Math. Comput. 286, 139–154 (2016)MathSciNetGoogle Scholar
  31. 31.
    Doha, E.H., Bhrawy, A.H., Baleanu, D., Ezz-Eldien, S.S., Hafez, R.M.: An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Adv. Differ. Equ. 1, 1–17 (2015)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Almeida, R., Torres, D.F.: A discrete method to solve fractional optimal control problems. Nonlinear Dyn. 80(4), 1811–1816 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S., Abdelkawy, M.A.: An accurate numerical technique for solving fractional optimal control problems. Proc. Roman. Acad. A 16, 47–54 (2015)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Gugat, M., Hante, F.M.: Lipschitz continuity of the value function in mixed-integer optimal control problems. Math. Control Signals Syst. 29, 3 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Alipour, M., Rostamy, D., Baleanu, D.: Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. J. Vib. Control 19(16), 2523–2540 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Tsai, J.S.H., Li, J., Shieh, L.S.: Discretized quadratic optimal control for continuous-time two-dimensional system. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49(1), 116–125 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hasan, M.M., Tangpong, X.W., Agrawal, O.P.: Fractional optimal control of distributed systems in spherical and cylindrical coordinates. J. Vib. Control 18, 1506–1525 (2009)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Özdemir, N., Agrawal, O.P., Iskender, B.B., Karadeniz, D.: Fractional optimal control of a 2-dimensional distributed system using eigenfunctions. Nonlinear Dyn. 55, 251–260 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Nemati, A., Yousefi, S.A.: A numerical scheme for solving two-dimensional fractional optimal control problems by the Ritz method combined with fractional operational matrix. IMA J. Math. Control Inf. 34(4), 1079–1097 (2017)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Nemati, A.: Numerical solution of 2D fractional optimal control problems by the spectral method along with Bernstein operational matrix. Int. J. Control. (2017).  https://doi.org/10.1080/00207179.2017.1334267
  41. 41.
    Mamehrashi, K., Yousefi, S.A.: A numerical method for solving a nonlinear 2-D optimal control problem with the classical diffusion equation. Int. J. Control 90(2), 298–306 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Rahimkhani, P., Ordokhani, Y.: Generalized fractional-order Bernoulli-Legendre functions: an effective tool for solving two-dimensional fractional optimal control problems. IMA J. Math. Control Inf. (2017).  https://doi.org/10.1093/imamci/dnx041 zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of HormozganBandar AbbasIran
  2. 2.Shahrekord UniversityShahrekordIran

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