<p>We present a general framework for time-fractional optimal control problems posed in non-normable state spaces, formulated as projective limits of Hilbert scales. The system dynamics are governed by a Caputo time-fractional evolution equation with a generator <i>A</i> satisfying level-wise monotonicity and resolvent estimates. Controls may act in the domain or on the boundary via operators that are only well-defined in the distributional sense. We establish well-posedness of the state equation using level-wise Galerkin approximations, prove a fractional integration-by-parts identity adapted to the projective-limit setting, and derive the associated adjoint equation. The optimal control is characterized by a projection formula that remains valid for both distributed and boundary control scenarios. A consistent discretization strategy is proposed, ensuring the discrete state and adjoint operators satisfy the same duality relations as in the continuous theory. Numerical experiments—including scalar fractional ODEs and PDEs with both distributed and boundary control—validate the theoretical results, demonstrating convergence and computational efficiency. Potential extensions include time-dependent operators, measure-valued controls, and free-final-time problems.</p>