Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends

Ro Won Myong; Jo Il Guk; O. Chol Won

doi:doi:10.11648/j.ajam.20251306.12

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Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends

Ro Won Myong^*

, Jo Il Guk

, O. Chol Won

Published in American Journal of Applied Mathematics (Volume 13, Issue 6)

Received: 2 October 2025 Accepted: 15 October 2025 Published: 26 November 2025

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Abstract

In this paper, we studied the necessary conditions for seeking optimal trajectory, optimal control, optimal time and optimal state. Then, we applied these qualitative properties to explore two numerical methods of a variable order fractional optimal control problem until full free ends that the initial time, initial state, terminal time and terminal state are free simultaneously. Based on the relations between variable order fractional calculus operators, i.e., the integral formulas by parts, the necessary conditions of optimality of a variable order fractional optimal control problem until full free ends drove from the Euler-Lagrange equation, applying Hamiltonian and variational principle. To develop the solution methods, we proposed the “optimization first, then discretization” (OFTD) method and the “discretization first, then optimization” (DFTO) method. The OFTD method is to solve the variable order fractional optimal control problem until full free ends by transforming the Euler-Lagrange equation into a nonlinear system using the Grünwald-Letnikov definition and manipulating the transversal conditions as an objective function of error quadratic minimization, i.e., a nonlinear programming type problem with equality constraints. The DFTO method is to solve the problem by transforming the variable order fractional calculus into a classical optimal control problem with integer order using the expansion formulas of the variable order fractional calculus operators. Finally, we demonstrated the validity and accuracy of the proposed methods through various types of numerical test problems.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 6)
DOI	10.11648/j.ajam.20251306.12
Page(s)	393-411
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Variable-Order Fractional Optimal Control Problem, Variable-Order Fractional Calculus, Variable-Order Fractional Differential Equation, Optimality Condition, Approximation Method

References

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[7]	Heydari, M. H., Avazzadeh, Z. New Wavelet Method for Variable Order Fractional Optimal Controls. Asian J. Control 2018; 20(5): 804-817.
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APA Style

Myong, R. W., Guk, J. I., Won, O. C. (2025). Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends. American Journal of Applied Mathematics, 13(6), 393-411. https://doi.org/10.11648/j.ajam.20251306.12

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ACS Style

Myong, R. W.; Guk, J. I.; Won, O. C. Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends. Am. J. Appl. Math. 2025, 13(6), 393-411. doi: 10.11648/j.ajam.20251306.12

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AMA Style

Myong RW, Guk JI, Won OC. Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends. Am J Appl Math. 2025;13(6):393-411. doi: 10.11648/j.ajam.20251306.12

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@article{10.11648/j.ajam.20251306.12,
  author = {Ro Won Myong and Jo Il Guk and O. Chol Won},
  title = {Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends
},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {6},
  pages = {393-411},
  doi = {10.11648/j.ajam.20251306.12},
  url = {https://doi.org/10.11648/j.ajam.20251306.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251306.12},
  abstract = {In this paper, we studied the necessary conditions for seeking optimal trajectory, optimal control, optimal time and optimal state. Then, we applied these qualitative properties to explore two numerical methods of a variable order fractional optimal control problem until full free ends that the initial time, initial state, terminal time and terminal state are free simultaneously. Based on the relations between variable order fractional calculus operators, i.e., the integral formulas by parts, the necessary conditions of optimality of a variable order fractional optimal control problem until full free ends drove from the Euler-Lagrange equation, applying Hamiltonian and variational principle. To develop the solution methods, we proposed the “optimization first, then discretization” (OFTD) method and the “discretization first, then optimization” (DFTO) method. The OFTD method is to solve the variable order fractional optimal control problem until full free ends by transforming the Euler-Lagrange equation into a nonlinear system using the Grünwald-Letnikov definition and manipulating the transversal conditions as an objective function of error quadratic minimization, i.e., a nonlinear programming type problem with equality constraints. The DFTO method is to solve the problem by transforming the variable order fractional calculus into a classical optimal control problem with integer order using the expansion formulas of the variable order fractional calculus operators. Finally, we demonstrated the validity and accuracy of the proposed methods through various types of numerical test problems.
},
 year = {2025}
}

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TY  - JOUR
T1  - Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends

AU  - Ro Won Myong
AU  - Jo Il Guk
AU  - O. Chol Won
Y1  - 2025/11/26
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251306.12
DO  - 10.11648/j.ajam.20251306.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 393
EP  - 411
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251306.12
AB  - In this paper, we studied the necessary conditions for seeking optimal trajectory, optimal control, optimal time and optimal state. Then, we applied these qualitative properties to explore two numerical methods of a variable order fractional optimal control problem until full free ends that the initial time, initial state, terminal time and terminal state are free simultaneously. Based on the relations between variable order fractional calculus operators, i.e., the integral formulas by parts, the necessary conditions of optimality of a variable order fractional optimal control problem until full free ends drove from the Euler-Lagrange equation, applying Hamiltonian and variational principle. To develop the solution methods, we proposed the “optimization first, then discretization” (OFTD) method and the “discretization first, then optimization” (DFTO) method. The OFTD method is to solve the variable order fractional optimal control problem until full free ends by transforming the Euler-Lagrange equation into a nonlinear system using the Grünwald-Letnikov definition and manipulating the transversal conditions as an objective function of error quadratic minimization, i.e., a nonlinear programming type problem with equality constraints. The DFTO method is to solve the problem by transforming the variable order fractional calculus into a classical optimal control problem with integer order using the expansion formulas of the variable order fractional calculus operators. Finally, we demonstrated the validity and accuracy of the proposed methods through various types of numerical test problems.

VL  - 13
IS  - 6
ER  -

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Author Information

Ro Won Myong

Department of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0000-0001-5559-0384
Jo Il Guk

Department of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

http://orcid.org/0009-0003-6532-0531
O. Chol Won

Department of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

http://orcid.org/0000-0002-4329-215X

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APA Style

Myong, R. W., Guk, J. I., Won, O. C. (2025). Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends. American Journal of Applied Mathematics, 13(6), 393-411. https://doi.org/10.11648/j.ajam.20251306.12

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ACS Style

Myong, R. W.; Guk, J. I.; Won, O. C. Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends. Am. J. Appl. Math. 2025, 13(6), 393-411. doi: 10.11648/j.ajam.20251306.12

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AMA Style

Myong RW, Guk JI, Won OC. Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends. Am J Appl Math. 2025;13(6):393-411. doi: 10.11648/j.ajam.20251306.12

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@article{10.11648/j.ajam.20251306.12,
  author = {Ro Won Myong and Jo Il Guk and O. Chol Won},
  title = {Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends
},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {6},
  pages = {393-411},
  doi = {10.11648/j.ajam.20251306.12},
  url = {https://doi.org/10.11648/j.ajam.20251306.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251306.12},
  abstract = {In this paper, we studied the necessary conditions for seeking optimal trajectory, optimal control, optimal time and optimal state. Then, we applied these qualitative properties to explore two numerical methods of a variable order fractional optimal control problem until full free ends that the initial time, initial state, terminal time and terminal state are free simultaneously. Based on the relations between variable order fractional calculus operators, i.e., the integral formulas by parts, the necessary conditions of optimality of a variable order fractional optimal control problem until full free ends drove from the Euler-Lagrange equation, applying Hamiltonian and variational principle. To develop the solution methods, we proposed the “optimization first, then discretization” (OFTD) method and the “discretization first, then optimization” (DFTO) method. The OFTD method is to solve the variable order fractional optimal control problem until full free ends by transforming the Euler-Lagrange equation into a nonlinear system using the Grünwald-Letnikov definition and manipulating the transversal conditions as an objective function of error quadratic minimization, i.e., a nonlinear programming type problem with equality constraints. The DFTO method is to solve the problem by transforming the variable order fractional calculus into a classical optimal control problem with integer order using the expansion formulas of the variable order fractional calculus operators. Finally, we demonstrated the validity and accuracy of the proposed methods through various types of numerical test problems.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Optimality Condition and Numerical Methods of Variable-Order Fractional Optimal Control Problem with Full Free Ends

AU  - Ro Won Myong
AU  - Jo Il Guk
AU  - O. Chol Won
Y1  - 2025/11/26
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251306.12
DO  - 10.11648/j.ajam.20251306.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 393
EP  - 411
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251306.12
AB  - In this paper, we studied the necessary conditions for seeking optimal trajectory, optimal control, optimal time and optimal state. Then, we applied these qualitative properties to explore two numerical methods of a variable order fractional optimal control problem until full free ends that the initial time, initial state, terminal time and terminal state are free simultaneously. Based on the relations between variable order fractional calculus operators, i.e., the integral formulas by parts, the necessary conditions of optimality of a variable order fractional optimal control problem until full free ends drove from the Euler-Lagrange equation, applying Hamiltonian and variational principle. To develop the solution methods, we proposed the “optimization first, then discretization” (OFTD) method and the “discretization first, then optimization” (DFTO) method. The OFTD method is to solve the variable order fractional optimal control problem until full free ends by transforming the Euler-Lagrange equation into a nonlinear system using the Grünwald-Letnikov definition and manipulating the transversal conditions as an objective function of error quadratic minimization, i.e., a nonlinear programming type problem with equality constraints. The DFTO method is to solve the problem by transforming the variable order fractional calculus into a classical optimal control problem with integer order using the expansion formulas of the variable order fractional calculus operators. Finally, we demonstrated the validity and accuracy of the proposed methods through various types of numerical test problems.

VL  - 13
IS  - 6
ER  -

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