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Numerical Computation of Optimal Control Problems with Atangana–Baleanu Fractional Derivatives

Author

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  • Chongyang Liu

    (Shandong Technology and Business University)

  • Changjun Yu

    (Shanghai University)

  • Zhaohua Gong

    (Shandong Technology and Business University)

  • Huey Tyng Cheong

    (Sunway University)

  • Kok Lay Teo

    (Sunway University)

Abstract

In this paper, a computational method is proposed for solving a class of fractional optimal control problems subject to canonical constraints of equality and inequality. Fractional derivatives are described in the Atangana–Baleanu-Caputo sense, and their fractional orders can be different. To solve this problem, we present a discretization scheme based on the trapezoidal rule and a novel numerical integration technique. Then, the gradient formulas of the cost and constraint functions with respect to the decision variables are derived. Furthermore, a gradient-based optimization algorithm for solving the discretized optimal control problem is developed. Finally, the applicability and effectiveness of the proposed algorithm are verified through three non-trivial example problems.

Suggested Citation

  • Chongyang Liu & Changjun Yu & Zhaohua Gong & Huey Tyng Cheong & Kok Lay Teo, 2023. "Numerical Computation of Optimal Control Problems with Atangana–Baleanu Fractional Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 798-816, May.
    • Handle: RePEc:spr:joptap:v:197:y:2023:i:2:d:10.1007_s10957-023-02212-5
    DOI: 10.1007/s10957-023-02212-5
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    References listed on IDEAS

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    1. Chongyang Liu & Zhaohua Gong & Changjun Yu & Song Wang & Kok Lay Teo, 2021. "Optimal Control Computation for Nonlinear Fractional Time-Delay Systems with State Inequality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 83-117, October.
    2. Dumitru Baleanu & Amin Jajarmi & Mojtaba Hajipour, 2017. "A New Formulation of the Fractional Optimal Control Problems Involving Mittag–Leffler Nonsingular Kernel," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 718-737, December.
    3. Chongyang Liu & Zhaohua Gong & Kok Lay Teo & Song Wang, 2022. "Optimal Control of Nonlinear Fractional-Order Systems with Multiple Time-Varying Delays," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 856-876, June.
    4. Gong, Zhaohua & Liu, Chongyang & Teo, Kok Lay & Yi, Xiaopeng, 2022. "Optimal control of nonlinear fractional systems with multiple pantograph‐delays," Applied Mathematics and Computation, Elsevier, vol. 425(C).
    5. Wen Li & Song Wang & Volker Rehbock, 2019. "Numerical Solution of Fractional Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 556-573, February.
    6. M. H. Heydari & M. Razzaghi, 2022. "A new class of orthonormal basis functions: application for fractional optimal control problems," International Journal of Systems Science, Taylor & Francis Journals, vol. 53(2), pages 240-252, January.
    7. Bin Li & Chang Jun Yu & Kok Lay Teo & Guang Ren Duan, 2011. "An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem," Journal of Optimization Theory and Applications, Springer, vol. 151(2), pages 260-291, November.
    8. Gong, Zhaohua & Liu, Chongyang & Teo, Kok Lay & Wang, Song & Wu, Yonghong, 2021. "Numerical solution of free final time fractional optimal control problems," Applied Mathematics and Computation, Elsevier, vol. 405(C).
    9. Zafar, Zain Ul Abadin & Zaib, Sumera & Hussain, Muhammad Tanveer & Tunç, Cemil & Javeed, Shumaila, 2022. "Analysis and numerical simulation of tuberculosis model using different fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    10. Kok Lay Teo & Bin Li & Changjun Yu & Volker Rehbock, 2021. "Applied and Computational Optimal Control," Springer Optimization and Its Applications, Springer, number 978-3-030-69913-0, December.
    11. Pan Mu & Lei Wang & Chongyang Liu, 2020. "A Control Parameterization Method to Solve the Fractional-Order Optimal Control Problem," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 234-247, October.
    12. M. H. Heydari & R. Tavakoli & M. Razzaghi, 2022. "Application of the extended Chebyshev cardinal wavelets in solving fractional optimal control problems with ABC fractional derivative," International Journal of Systems Science, Taylor & Francis Journals, vol. 53(12), pages 2694-2708, September.
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    Cited by:

    1. Set, Erhan & Akdemi̇r, Ahmet Ocak & Karaoğlan, Ali̇, 2024. "New integral inequalities for synchronous functions via Atangana–Baleanu fractional integral operators," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).
    2. Liu, Chongyang & Zhou, Tuo & Gong, Zhaohua & Yi, Xiaopeng & Teo, Kok Lay & Wang, Song, 2023. "Robust optimal control of nonlinear fractional systems," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).

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