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Wavelets method for solving fractional optimal control problems

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  • Heydari, M.H.
  • Hooshmandasl, M.R.
  • Maalek Ghaini, F.M.
  • Cattani, C.

Abstract

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) is proposed for solving a class of fractional optimal control problems (FOCPs). In the proposed method, the FOCP under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved. To this end, the fractional derivative of the state variable and the control variable are expanded by the LWs with unknown coefficients. Then, the operational matrix of the Riemann–Liouville fractional integration with some properties of the LWs are employed to achieve a nonlinear algebraic equation, in place of the performance index and a linear system of algebraic equations, in place of the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extrema, which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers is applied. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:286:y:2016:i:c:p:139-154
    DOI: 10.1016/j.amc.2016.04.009
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    1. 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).
    2. Fakhrodin Mohammadi & Hossein Hassani, 2019. "Numerical Solution of Two-Dimensional Variable-Order Fractional Optimal Control Problem by Generalized Polynomial Basis," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 536-555, February.
    3. Habibirad, Ali & Azin, Hadis & Hesameddini, Esmail, 2023. "A capable numerical meshless scheme for solving distributed order time-fractional reaction–diffusion equation," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    4. Ghanbari, Behzad & Atangana, Abdon, 2020. "A new application of fractional Atangana–Baleanu derivatives: Designing ABC-fractional masks in image processing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 542(C).
    5. Baghani, Omid, 2022. "SCW-iterative-computational method for solving a wide class of nonlinear fractional optimal control problems with Caputo derivatives," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 540-558.
    6. 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.
    7. Araz Noori Dalawi & Mehrdad Lakestani & Elmira Ashpazzadeh, 2022. "An Efficient Algorithm for the Multi-Scale Solution of Nonlinear Fractional Optimal Control Problems," Mathematics, MDPI, vol. 10(20), pages 1-16, October.
    8. Heydari, M.H. & Razzaghi, M., 2021. "Piecewise Chebyshev cardinal functions: Application for constrained fractional optimal control problems," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    9. Baghani, Omid, 2021. "Second Chebyshev wavelets (SCWs) method for solving finite-time fractional linear quadratic optimal control problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 343-361.
    10. Heydari, M.H. & Avazzadeh, Z. & Mahmoudi, M.R., 2019. "Chebyshev cardinal wavelets for nonlinear stochastic differential equations driven with variable-order fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 124(C), pages 105-124.
    11. Heydari, Mohammad Hossein & Avazzadeh, Zakieh & Haromi, Malih Farzi, 2019. "A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 215-228.
    12. Tirumalasetty Chiranjeevi & Raj Kumar Biswas, 2017. "Discrete-Time Fractional Optimal Control," Mathematics, MDPI, vol. 5(2), pages 1-12, April.
    13. 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.
    14. Heydari, Mohammad Hossein & Avazzadeh, Zakieh, 2018. "Legendre wavelets optimization method for variable-order fractional Poisson equation," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 180-190.

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