Controllability of a fractional output linear system with constraints

The primary objective of this research is to generalize the concept of controllability with constraints to cases where the output function is a Riemann–Liouville fractional derivative of order More precisely, if , we obtain the enlarged controllability of the system state and enlarged controllability of the gradient state of the system is obtained with Moreover, we aim to characterize the optimal control that enables us to guide a linear parabolic system to a fractional final state within the evolution system's domain. We adopt two approaches to solve the aforementioned problem: The first is based on the Lagrangian technique, and the second one employs the subdifferential theory. Subsequently, we develop an algorithm to compute the minimum energy control. Finally, we provide numerical simulations to illustrate the validity of the obtained theoretical results.