Time-fractional optimal control in the Bergman model for type 1 diabetes

Essamy Amina; Karim Marouane; Ferjouchia Hanane; M. Z. Youssef; Mahmoud A. Zaky; Essamy Amina; Karim Marouane; Ferjouchia Hanane; M. Z. Youssef; Mahmoud A. Zaky

doi:10.3934/math.2026546

AIMS Mathematics

2026, Volume 11, Issue 5: 13233-13256. doi: 10.3934/math.2026546

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Time-fractional optimal control in the Bergman model for type 1 diabetes

1.
LAMS, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'Sik, Hassan Ⅱ University of Casablanca, Morocco
2.
Multidisciplinary Research and Innovation Laboratory (LPRI), Moroccan School of Engineering Sciences (EMSI), Casablanca 20250, Morocco
3.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia

Received: 04 February 2026 Revised: 06 April 2026 Accepted: 20 April 2026 Published: 12 May 2026
MSC : 49J15, 93B05, 92B05, 34A08

This study investigates a time-fractional optimal control problem for the Bergman minimal model of type 1 diabetes by employing the Caputo derivative of order $q\in(0, 1)$ to incorporate memory effects into the glucose–insulin dynamics. First, the well-posedness of the fractional-order model is established by proving the existence, uniqueness, boundedness, and positivity of its solutions. Subsequently, an optimal control problem describing a normalized therapeutic intervention is formulated, and the existence of an optimal control is rigorously proved. By applying the fractional Pontryagin maximum principle, the corresponding optimality system is derived, and the optimal control law is explicitly characterized. The obtained results demonstrate that the proposed control strategy effectively stabilizes the glucose dynamics around the desired equilibrium while reducing excessive insulin administration. In comparison with the classical integer-order model, the fractional formulation provides a more realistic representation of delayed physiological responses. Finally, numerical simulations, carried out using a predictor–corrector scheme together with a forward–backward sweep algorithm, confirm the effectiveness and reliability of the proposed approach for blood glucose regulation.
- time fractional derivative,
- Bergman's minimal model,
- optimal control,
- type 1 diabetes
Citation: Essamy Amina, Karim Marouane, Ferjouchia Hanane, M. Z. Youssef, Mahmoud A. Zaky. Time-fractional optimal control in the Bergman model for type 1 diabetes[J]. AIMS Mathematics, 2026, 11(5): 13233-13256. doi: 10.3934/math.2026546

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Abstract

This study investigates a time-fractional optimal control problem for the Bergman minimal model of type 1 diabetes by employing the Caputo derivative of order $q\in(0, 1)$ to incorporate memory effects into the glucose–insulin dynamics. First, the well-posedness of the fractional-order model is established by proving the existence, uniqueness, boundedness, and positivity of its solutions. Subsequently, an optimal control problem describing a normalized therapeutic intervention is formulated, and the existence of an optimal control is rigorously proved. By applying the fractional Pontryagin maximum principle, the corresponding optimality system is derived, and the optimal control law is explicitly characterized. The obtained results demonstrate that the proposed control strategy effectively stabilizes the glucose dynamics around the desired equilibrium while reducing excessive insulin administration. In comparison with the classical integer-order model, the fractional formulation provides a more realistic representation of delayed physiological responses. Finally, numerical simulations, carried out using a predictor–corrector scheme together with a forward–backward sweep algorithm, confirm the effectiveness and reliability of the proposed approach for blood glucose regulation.

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