This new research aims to extend the topic of the enlarged controllability of a fractional output linear system. Thus, we characterize the optimal control by two methods, ensuring that the Riemann-Liouville fractional derivative of the final state of the considered system lies between two given functions on a subregion of the evolution domain. Firstly, we transform the considered problem into the saddle point using the Lagrangian multiplier approach. Then, in the second one, we provide the technique of the subdifferential, which allows us to present the cost-explicit formula of the minimum energy control. Moreover, we construct an algorithm of Uzawa type to illustrate the theoretical results obtained through numerical simulations.