This research aims to advance the concept of fractional controllability with constraints in the context of fractional systems in the Caputo sense, where the output function is described by a fractional Riemann-Liouville derivative of order \(\gamma \in [0,1]\) . Consequently, the objective is to characterize optimal control by two distinct methods, ensuring that the fractional Riemann-Liouville derivative of the final state remains between two prescribed functions p(.) and q(.). In particular, if \(\gamma = 0\) , we obtain global enlarged controllability over the evolution domain. On the other hand, with \(\gamma =1\) , we obtain the enlarged controllability of the gradient of the output system. What’s more, when \(p(.) = q(.)\) , we’re talking about exact controllability. The problem is approached in two ways: the first uses the Lagrangian method, and the second the subdifferential theory. To validate the theoretical results, we develop an Uzawa-type algorithm and demonstrate its application through numerical simulations.
