Fractional optimal control of dark matter dynamics in cosmological models

This paper develops a fractional optimal control framework for dark matter dynamics in cosmological models. The proposed model describes the coupled evolution of dark matter and dark energy in an expanding universe governed by the Caputo fractional derivative of order 0 < α ≤ 1 . The fractional formulation incorporates memory effects into the evolution of the cosmic density components and extends the classical integer-order dark sector model. A phenomenological interaction term is used to describe the exchange between dark matter and dark energy, while a bounded time-dependent control variable is introduced to regulate the deviation between the controlled densities and prescribed reference profiles. The control is interpreted as an effective calibration or interaction-modulation mechanism rather than a direct physical intervention on dark matter. The admissible controls are taken as bounded measurable functions satisfying box constraints. The existence of an optimal control is established using compactness of the admissible set, continuity of the control-to-state map, convexity arguments, and weak lower semicontinuity of the cost functional. Necessary optimality conditions are derived by applying a fractional Pontryagin maximum principle. The resulting optimality system consists of the Caputo fractional state equations, right-sided fractional adjoint equations, transversality conditions, and a projection formula for the optimal control. A numerical scheme based on the fractional Adams–Bashforth–Moulton predictor–corrector method combined with the forward–backward sweep algorithm is proposed to solve the coupled state–adjoint system. Numerical simulations are performed to examine the effects of the fractional order, the dark sector interaction coefficient, and the control cost on the evolution of dark matter and dark energy densities. The sensitivity analysis shows that the fractional order and the control weight have significant influence on the controlled dynamics. The results indicate that fractional memory and optimal control provide a flexible mathematical framework for analyzing controlled dark sector evolution in phenomenological cosmological models.