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Transcendental Bernstein series for solving nonlinear variable order fractional optimal control problems

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  • Hassani, Hossein
  • Avazzadeh, Zakieh

Abstract

This paper deals with finding an approximate solution of variable order fractional optimal control problems (V-FOCPs) based on Lagrange multiplier optimization technique in combination with the new operational matrix of variable order (VO) fractional derivatives. To carry out the proposed approach, we firstly develop the well-known Bernstein polynomials to the new series of functions namely transcendental Bernstein series (TBS). Then we implement these basis functions to approximate the solutions of V-FOCPs. In fact, the series expansion in TBS with unknown free coefficients and control parameters is the novel idea for solving the fractional systems with less terms of approximation. The convergence analysis of our proposed method, will be guaranteed by proving a new theorem for the TBS. To investigate the practical computational efficiency and accuracy of the presented method, some numerical examples are provided. The experimental results confirm the applicability of the method and a good agreement between the approximate and exact solutions.

Suggested Citation

  • Hassani, Hossein & Avazzadeh, Zakieh, 2019. "Transcendental Bernstein series for solving nonlinear variable order fractional optimal control problems," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    • Handle: RePEc:eee:apmaco:v:362:y:2019:i:c:28
    DOI: 10.1016/j.amc.2019.124563
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    References listed on IDEAS

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    1. Heydari, M.H. & Hooshmandasl, M.R. & Maalek Ghaini, F.M. & Cattani, C., 2016. "Wavelets method for solving fractional optimal control problems," Applied Mathematics and Computation, Elsevier, vol. 286(C), pages 139-154.
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    2. Hosseininia, M. & Heydari, M.H., 2019. "Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 400-407.
    3. Heydari, M.H., 2020. "Chebyshev cardinal functions for a new class of nonlinear optimal control problems generated by Atangana–Baleanu–Caputo variable-order fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    4. Pho, Kim-Hung & Heydari, M.H. & Tuan, Bui Anh & Mahmoudi, Mohammad Reza, 2020. "Numerical study of nonlinear 2D optimal control problems with multi-term variable-order fractional derivatives in the Atangana-Baleanu-Caputo sense," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    5. Soradi-Zeid, Samaneh & Jahanshahi, Hadi & Yousefpour, Amin & Bekiros, Stelios, 2020. "King algorithm: A novel optimization approach based on variable-order fractional calculus with application in chaotic financial systems," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).

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