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Numerical Solution of Fractional Optimal Control

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Abstract

This paper presents a numerical algorithm for solving a class of nonlinear optimal control problems subject to a system of fractional differential equations. We first propose a robust second-order numerical integration scheme for the system. The objective is approximated by the trapezoidal rule. We then apply a gradient-based optimization method to the discretized problem. Formulas for calculating the gradients are derived. Computational results demonstrate that our method is able to generate accurate numerical approximations for problems with multiple states and controls. It is also robust with respect to the fractional orders of derivatives.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Agrawal, O.P.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13, 1269–1281 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bhrawy, A.H., Doha, E.H., Machado, J.A., Ezz-Eldien, S.S.: An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index. Asian J. Control 17, 2389–2402 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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, 1547–1559 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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. 2015, 15 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ezz-Eldien, S.S., Doha, E.H., Baleanu, D., Bhrawy, A.H.: A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. J. Vib. Control 23, 16–30 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lotfi, A., Dehghan, M., Yousefi, S.A.: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 62, 1055–1067 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Agrawal, O.P.: A quadratic numerical scheme for fractional optimal control problems. J. Dyn. Syst. Meas. Control 130, 011010 (2008)

    Article  Google Scholar 

  10. Alizadeh, A., Effati, S.: An iterative approach for solving fractional optimal control problems. J. Vib. Control 24, 18–36 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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, 2523–2540 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Singha, N., Nahak, C.: An efficient approximation technique for solving a class of fractional optimal control problems. J. Optim. Theory Appl. 174, 785–802 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lotfi, A., Yousefi, S.A., Dehghan, M.: Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. J. Comp. Appl. Math. 250, 143–160 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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, 2272–2282 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tricaud, C., Chen, Y.Q.: An approximation method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl. 59, 1644–1655 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yousefi, S.A., Lotfi, A., Dehghan, M.: The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problem. J. Vib. Control 17, 2059–2065 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sabeh, Z., Shamsi, M., Dehghan, M.: Distributed optimal control of the viscous Burgers equation via a Legendre pseudo-spectral approach. Math. Methods Appl. Sci. 39, 3350–3360 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Baleanu, D., Defterli, O., Agrawal, O.P.: A central difference numerical scheme for fractional optimal control problems. J. Vib. Control 15, 583–597 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

  20. Chen, W., Wang, S.: A penalty method for a fractional order parabolic variational inequality governing American put option valuation. Comput. Math. Appl. 67, 77–90 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, W., Wang, S.: A power penalty method for a 2D fractional partial differential linear complementarity problem governing two-asset American option pricing. Appl. Math. Comput. 305, 174–187 (2017)

    MathSciNet  Google Scholar 

  22. Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. 26, 448–479 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor corrector approach for the numerical solution of fractional differential equation. Nonlinear Dyn. 29, 2–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Diethelm, K., Ford, N.J., Free, A.D., Yu, L.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Method Appl. Mech. Eng. 194, 743–773 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cao, J., Xu, C.: A high order scheme for the numerical solution of the fractional ordinary differential equations. J. Comput. Phys. 238, 154–168 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Huang, H., Tang, Y., Vazquez, L.: Convergence analysis of a block-by block method for fractional differential equation. Numer. Math. Theor. Methods Appl. 5, 229–241 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kumar, K., Agrawal, O.P.: An approximate method for numerical solution of fractional differential equations. Signal Process. 86, 2602–2610 (2006)

    Article  MATH  Google Scholar 

  28. Li, C., Tao, C.: On the fractional Adams method. Comput. Math. Appl. 58, 1573–1588 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Saadatmandi, A., Dehghan, M.: A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 59, 1326–1336 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, W., Wang, S., Rehbock, V.: A 2nd-order one-step numerical integration scheme for a fractional differential equation. Numer. Algebra Control Optim. 7, 273–287 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Di Pillo, G., Grippo, L.: Exact penalty functions in constrained optimization. SIAM J. Control Optim. 27, 1333–1360 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xing, A.Q., Chen, Z.H., Wang, C.L., Yao, Y.Y.: Exact penalty function approach to constrained optimal control problems. Optim. Control Appl. Methods 10, 173–180 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, W., Wang, S.: Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. J. Ind. Manag. Optim. 9, 365–389 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Alt, W., Baier, R., Lempio, F., Gerdts, M.: Approximations of linear control problems with bang–bang solutions. Optimization 62, 9–32 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by US Air Force Office of Scientific Research Project FA2386-15-1-4095.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA, 6845, Australia

    Wen Li, Song Wang & Volker Rehbock

Authors
  1. Wen Li
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  2. Song Wang
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  3. Volker Rehbock
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Corresponding author

Correspondence to Song Wang.

Additional information

Communicated by Mouhacine Benosman.

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Li, W., Wang, S. & Rehbock, V. Numerical Solution of Fractional Optimal Control. J Optim Theory Appl 180, 556–573 (2019). https://doi.org/10.1007/s10957-018-1418-y

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  • DOI: https://doi.org/10.1007/s10957-018-1418-y

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