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

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.