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RESEARCH ARTICLE

Variational analysis and numerical methods for modified fractional optimal control problems with applications

Amin Jajarmi1,2* ,  Dumitru Baleanu3,4
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1 Department of Electrical Engineering, University of Bojnord, Bojnord , Iran
2 Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai, Tamil Nadu , India
3 Department of Computer Science and Mathematics, Lebanese American University, Beirut , Lebanon
4 Institute of Space Sciences, Magurele-Bucharest , Romania
IJOCTA, 026360254 https://doi.org/10.36922/IJOCTA026360254
Received: 5 September 2026 | Revised: 3 October 2026 | Accepted: 7 October 2026 | Published online: 9 November 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

This study presents a new modified fractional optimal control framework for dynamical systems governed by the left-sided Modified Atangana-Baleanu-Caputo (MABC) state equation , 0 < α < 1, including theoretical foundations, optimality conditions, a numerical solution approach, and two practical applications. This formulation explicitly incorporates nonlocal memory and hereditary effects that are important in epidemiological and biomedical systems. We begin by developing the core analytical properties of the α-th-order fractional derivatives and integrals with a generalized Mittag-Leffler kernel, including a fractional integration-by-parts formula that is fundamental to the variational analysis. Using these tools, we formulate our fractional optimal control problem and rigorously prove the existence of an optimal control under standard convexity and regularity assumptions. The integration-by-parts identity then enables the derivation of the necessary optimality conditions, yielding the left-sided state system, the right-sided adjoint system, and a pointwise characterization of the optimal control, which naturally reduces to the classical Pontryagin Maximum Principle as α → 1−. To solve the resulting coupled left-right fractional optimality system, we combine a forward-backward sweep method with a high-order predictor-corrector numerical algorithm based on fractional Adams-Bashforth and Adams-Moulton steps, supported by a global error convergence rate of , where h is the time step size. The practical relevance and advantages of the extended framework are demonstrated through two detailed case studies: (i) an SEIR model of human Respiratory Syncytial Virus (RSV) transmission with seasonal forcing and treatment control, and (ii) a three-component cancer-immune interaction model with multiple therapeutic controls. Numerical experiments for fractional orders α ∈ {0.995, 0.997, 0.999} demonstrate that the new formulation more effectively captures memory effects compared with the classical Caputo approach, resulting in a greater reduction in infection peaks for the RSV system and a consistently lower tumor burden in the cancer model. These results show that the modified Mittag-Leffler-kernel-based fractional derivative is an effective tool for modeling and optimally controlling complex systems that exhibit hereditary dynamics.

Keywords
Modified fractional derivative
Fractional optimal control
Predictor-corrector scheme
Epidemiological and biomedical applications
Funding
None.
Conflict of interest
The authors declare they have no competing interests.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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