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Numerical Solution of 1D and 2D Fractional Optimal Control of System via Bernoulli Polynomials

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Abstract

In this current paper, we present a numerical method to solve a class of fractional optimal control problems of system described by integer-fractional derivative. The fractional derivative is explained in the Caputo sense and a new formulation for the Bernoulli Caputo fractional derivative operational matrix has been achieved for the first time. Then this matrix and operational matrix of multiplication of these polynomials are applied to solve the fractional-order optimal control problems by the direct method. As a matter of fact, the functions of the problem are approximated by Bernoulli polynomials with unknown coefficients in the functional and conditions. Therefore, a fractional optimal control problem is converted to an unconstrained optimization problem and then it could be solved via Newton’s iterative method. Also we demonstrate that if the number of Bernoulli basis is increased, this method is convergent. The effectiveness and validity of the new methodology are illustrated by four examples, in addition our findings in comparison with the existing results show the preference of presented method.

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References

  1. Benson, D.A., Meerschaert, M.M., Revielle, J.: Fractional calculus in hydrologic modeling: a numerical perspective. Adv. Water Resour. 51, 479–497 (2013)

    Article  Google Scholar 

  2. Larsson, S., Racheva, M., Saedpanah, F.: Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity. Comput. Method. Appl. Mech. Eng. 283, 196–209 (2015)

    Article  MathSciNet  Google Scholar 

  3. Baillie, R.T.: Long memory processes and fractional integration in econometrics. J. Econom. 73, 5–59 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alkahtani, B., Gulati, V., Klman, A.: Application of Sumudu transform in generalized fractional reaction–diffusion equation. Int. J. Appl. Comput. Math. 2, 387–394 (2016)

    Article  MathSciNet  Google Scholar 

  5. Shen, J., Cao, J.: Necessary and sufficient conditions for consensus of delayed fractional-order systems. Asian J Control 14(6), 1690–1697 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Rossikhin, Y.A., Shitikova, M.V.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50, 15–67 (1997)

    Article  Google Scholar 

  7. Magin, R.L.: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32, 1–104 (2004)

    Article  Google Scholar 

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

  9. Zamani, M., Karimi-Ghartemani, M., Sadati, N.: FOPID controller design for robust performance using particle swarm optimization. J. Fract. Calc. Appl. Anal. 10, 169–188 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Jesus, I.S., Machado, J.A.T.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263–282 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Suarez, 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Agrawal, O.P.: Fractional optimal control of a distributed system using eigenfunctions. ASME J. Comput. Nonlinear Dyn. (2008). https://doi.org/10.1115/1.2833873

  14. Rahimkhani, P., Ordokhani, Y., Babolian, E.: An efficient approximate method for solving delay fractional optimal control problems. Nonlinear Dyn. 86(3), 1649–1661 (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. Keshavarz, E., Ordokhani, Y., Razzaghi, M.: A numerical solution for fractional optimal control problems via Bernoulli polynomials. J. Vib. Control 22(18), 3889–3903 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rabiei, K., Ordokhani, Y., Babolian, E.: The Boubaker polynomials and their application to solve fractional optimal control problems. Nonlinear Dyn. 88, 1013–1026 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ozdemir, N., Agrawal, O.P., Iskender, B.B., Karadeniz, B.: Fractional optimal control of a 2-dimensional distributed system using eigenfunctions. Nonlinear Dyn. 55(3), 251–260 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Povstenko, Y.: Time-fractional radial diffusion in sphere. Nonlinear Dyn. 53, 55–65 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Idiri, G., Djennoune, S., Bettayeb, M.: Solving fixed final time fractional optimal control problems using the parametric optimization method. Asian J Control 18(4), 1524–1536 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sweilam, N.H., Alajmi, T.M.: Legendre spectral collocation method for solving some type of fractional optimal control problem. J Adv Res 6, 393–403 (2015)

    Article  Google Scholar 

  22. Podlubny, I.: Fractional Diferential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  23. Diethelm, K., Ford, N.J., Feed, A.D., Luchko, Y.U.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194, 743–773 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mashayekhi, S., Ordokhani, Y., Razzaghi, M.: Hybrid functions approach for nonlinear constrained optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 17, 1831–1843 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Arfken, G.: Mathematical Methods for Physicists, 3rd edn. Academic Press, San diego (1985)

    MATH  Google Scholar 

  26. Kreyszig, E.: Intoductory Functional Analysis with Applications. Wiley, New York (1987)

    Google Scholar 

  27. Tohidi, E., Bhrawy, A.H., Erfani, K.: A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Appl. Math. Model. 37(6), 4283–4294 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Almeida, R., Torres, D.F.M.: A discrete method to solve fractional optimal control problems. Nonlinear Dyn. 80(2), 1811–1816 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nemati, A., Yousefi, S.A., 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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lancaster, P.: Theory of Matrices. Academic Press, New York (1969)

    MATH  Google Scholar 

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Acknowledgements

We would like to thank the referees for their helpful suggestions to improve the earlier version of this article.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

    Kobra Rabiei & Yadollah Ordokhani

  2. Department of Computer Science, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran

    Esmaeil Babolian

Authors
  1. Kobra Rabiei
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  2. Yadollah Ordokhani
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  3. Esmaeil Babolian
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Correspondence to Yadollah Ordokhani.

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Rabiei, K., Ordokhani, Y. & Babolian, E. Numerical Solution of 1D and 2D Fractional Optimal Control of System via Bernoulli Polynomials. Int. J. Appl. Comput. Math 4, 7 (2018). https://doi.org/10.1007/s40819-017-0435-0

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  • DOI: https://doi.org/10.1007/s40819-017-0435-0

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