Existence, Optimal Control, and Numerical Analysis of a Caputo Fractional Model for Oxygen Saturation Regulation

Fractional-order models are widely recognized for their ability to capture memory and hereditary effects in biological and physiological systems. In this paper, we develop and analyze a Caputo fractional-order dynamical model for the regulation of blood oxygen saturation (SpO2) under bounded control inputs. The model incorporates nonlinear saturation mechanisms and auxiliary state variables to represent delayed oxygen transport and adaptation effects. By reformulating the system as an operator equation in a suitable Banach space, sufficient conditions for existence and uniqueness of solutions are established using fixed-point theory. An optimal control problem is then formulated to steer oxygen saturation toward a prescribed safe target level, and the existence of an optimal control is proved via compactness arguments and the direct method of the calculus of variations. Numerical simulations are provided to illustrate the theoretical findings and to demonstrate the impact of the fractional order on transient oxygen saturation dynamics, including comparison with the classical integer-order case. The results show that fractional modeling offers a mathematically rigorous and physiologically interpretable framework for describing delayed oxygenation responses and achieving stable regulation under bounded control constraints.