10.57647/ijm2c.2025.150423

Solving fractional optimal control problems and costate estimation via a Muntz pseudospectral method

PDF
  • Hussein Ghassemi,
  • Mohammad Maleki*1,
  1. Department of Mathematics, Isf. C. Islamic Azad University, Isfahan, Iran.

Received: 09-07-2025

Revised: 05-08-2025

Accepted: 07-08-2025

Published in Issue 10-08-2025

Copyright (c) 2025 Mohammad Maleki, Hussein Ghassemi (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Ghassemi, H., & Maleki, M. (2025). Solving fractional optimal control problems and costate estimation via a Muntz pseudospectral method. International Journal of Mathematical Modelling & Computations, 15(4). https://doi.org/10.57647/ijm2c.2025.150423
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

In this paper, we first introduce the Muntz–Legendre polynomials and establish their relation to Jacobi polynomials. Then, a stable scheme is presented for approximating the Caputo fractional derivative of the Muntz–Legendre polynomials. Next, we introduce a Muntz–polynomial pseudospectral method with fractional power of Legendre–Gauss–
Radau mesh points for solving fractional optimal control problems. This method is particularly suitable for problems whose solutions contain non-integer exponent factors. We also construct a novel costate estimation procedure based on the first order optimality conditions and the structure of the underlying problem. Three numerical examples, including a fractional model for tumor burden under immune suppression, are presented to demonstrate the applicability and spectral accuracy of the proposed method.

Keywords

  • Fractional optimal control; Pseudospectral; Costate estimation; M¨untz polynomials; Fractional cancer model
PDF

References

  1. I.M. Ross, M. Karpenko, A review of pseudospectral optimal control: from theory to flight, Annu. Rev. Control. 36(2) (2012) 182–197.
  2. H. Ghassemi, M. Maleki, M. Allame, On the modification and convergence of unconstrained optimal control using pseudospectral methods, Optim Control Appl. Meth. 42(3) (2021) 717–743.
  3. X. Tang, Z. Liu, X. Wang, Integral fractional pseudospectral methods for solving fractional optimal control problems, Automatica 62 (2015) 304–311.
  4. N. Ejlali, S.M. Hosseini, A pseudospectral method for fractional optimal control problems, J. Optim. Theory. Appl. 174 (2017) 83–107.
  5. S. Li, Z. Zhou, Legendre pseudo-spectral method for optimal control problem governed by a time-fractional diffusion equation, Int. J. Comput. Math. 95 (6-7) (2018) 1308–1325.
  6. M. Habibli, M.H. Noori Skandari, Fractional Chebyshev pseudospectral method for fractional optimal control problems, Optim Control Appl. Meth. 40 (3) (2019) 558–572.
  7. Y. Yang, M. H. Noori Skandari, Pseudospectral method for fractional infinite horizon optimal control problems, Optim Control Appl. Meth. 41 (6) (2020) 2201–2212.
  8. K. T. Elgindy, Fourier-Gegenbauer pseudospectral method for solving periodic fractional optimal control problems, arXiv:2304.04454v6 [math.OC],
  9. https://doi.org/10.48550/arXiv.2304.04454.
  10. N.H. Sweilam, T.M. Al–Ajami, R.H.W. Hoppe, Numerical solution of some types of fractional optimal control problems, The Scientific World Journal. Volume 2013, Article ID 306237, https://doi.org/10.1155/2013/306237.
  11. S.M. Shafiof, J. Askari, M. Shams Solary, A numerical solution of fractional optimal control problems using spectral method and hybrid functions, Control and Optimization in Applied Mathematics (COAM). 3(1) Spring–Summer (2018) 1–25.
  12. A. Alizadeh, S. Effati, An iterative approach for solving fractional optimal control problems. J. Vibr. Control. 8 (2016) 1–19.
  13. A. Alizadeh, S. Effati, A. Heydari, Numerical schemes for fractional optimal control problems. J. Dynamic Sys. Measur. Control. 139 (8) (2017) 1–18.
  14. T. Akbarian, M. Keyanpour, A New approach to the numerical solution of fractional order optimal control problems. Applications and Appl. Math. 8 (2013) 523–534.
  15. M. Alipour, D. Rostamy, D. Baleanu, Solving multi–dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. J. Vibr. Control. 19(16) (2012) 2523–2540.
  16. A. Lotfi, S.A. Yousefi, Epsilon-Ritz method for solving a class of fractional constrained optimization problems. J. Optim. Theory Appl. 163(3) (2014) 884–899.
  17. S.S. Zeid, M. Yousefi, Approximated solutions of linear quadratic fractional optimal control problems. JAMSI. 12(2) (2016) 83–94.
  18. Y. Yang, J. Zhang, H. Liu, A.O. Vasilev, An indirect convergent Jacobi spectral collocation method for fractional optimal control problems, Math. Meth. Appl. Sci. 44(4) (2021) 2806–2824.
  19. S. Ghaderi, S. Effati, A. Heydari, A new numerical approach for solving fractional optimal control problems with the Caputo–Fabrizio fractional operator, J. Math. vol. 2022 (2022), Article ID 6680319, https://doi.org/10.1155/2022/6680319.
  20. Z. Pirouzeh, M.H. Noori Skandari, K. Nassiri Pirbazari, A convergent Legendre spectral collocation method for the variable–order fractional–functional optimal control problems, J. Math, vol. 2024 (2024), Article ID 3934093, https://doi.org/10.1155/2024/3934093.
  21. H. Kheiri, M. Jafari, Fractional optimal control of an HIV/AIDS epidemic model with random testing and contact tracing, J. Appl. Math. Comput. 60 (2019) 387–411.
  22. M. Vellappandi, P. Kumar, V. Govindaraj, W. Albalawi, An optimal control problem for Mosaic disease via Caputo fractional derivative, Alexandria Engineering Journal, 61 (2022) 8027–8037.
  23. N.H. Sweilam, S.M. AL-Mekhlafi, Optimal control for a nonlinear mathematical model of tumor under immune suppression: a numerical approach, Optim. Control Appl. Meth. 39(5) (2018), 1581–1596.
  24. K.R. Cheneke,K.P. Rao, G.K. Edessa, Fractional derivative and optimal control analysis of Cholera epidemic model, J. Math. vol. 2022 (2022) Article ID 9075917, https://doi.org/10.1155/2022/9075917.
  25. M. Nadeem, M. Habib, M. Safdar, P.K. Mwanakatwe, Analysis of Climatic model using fractional optimal control, J. Math. vol. 2023 (2023), Article ID 7482381, https://doi.org/10.1155/2023/7482381.
  26. D. Ganeshan, L.J. Darne, Optimal control of hand, foot and mouth disease model using variational iteration method, Int. J. Math. Model. Comput. 8(4) (2018), 239–254.
  27. M. Wameko,. Mathematical model for transmission dynamics of hepatitus C virus with optimal control strategies. Int. J. Math. Model. Comput. 9(3) (2019) 213–237.
  28. F. Fahroo, I.M. Ross, Costate estimation by a Legendre pseudospectral method. J. Guid. Control Dyn. 24(2) (2002) 270–277.
  29. D.A. Benson, G.T. Huntington, T.P. Thorvaldsen, A.V. Rao, Direct trajectory optimization and costate estimation via an orthogonal collocation method. J. Guid. Control Dyn. 29(6) (2006) 1435–1440.
  30. S. Chen, J. Shen, L.-L. Wang, Generalized Jacobi functions and their applications to fractional differential equations, Math. Comput. 85(300) (2016) 1603–1638.
  31. S. Esmaeili, M. Shamsi, Yury Luchko, Numerical solution of fractional differential equations with a collocation method based on M¨untz polynomials, journal Computers and Mathematics with Applications 62 (2011) 918-929.
  32. Bahareh Sadeghi, Mohammad Maleki, Homa Almasieh, Space-time M¨untz spectral collocation approach for parabolic Volterra integro-differential equations with a singular kernel. J. Nonlinear Anal. Appl. (2022) 1-10.
  33. W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, New York, 2004.
  34. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domain, Springer-Verlag, Berlin, (2006).
  35. G.H. Golub, J.H.Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969) 221-230.
  36. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
  37. P. Borwein, T. Erd´elyi, J. Zhang, M¨untz systems and orthogonal M¨untz–Legendre polynomials, Trans. Amer. Math. Soc. 342 (2)(1994) 523–542.
  38. G.V. Milovanovi´c, M¨untz orthogonal polynomials and their numerical evaluation, in: Applications and Computation of Orthogonal Polynomials, in: Internat. Ser. Numer. Math., vol. 131, Birkhauser, Basel, 1999, pp. 179–194.

Most read articles by the same author(s)