Optimal regional control for a class of semilinear time-fractional diffusion systems with distributed feedback

Optimal regional control problem for a class of semilinear time-fractional diffusion systems with distributed feedback in a bounded domain is solved in the paper. For this purpose, we first discuss the well-posedness of the considered system and the differentiability of the control-to-state mapping. The existence of optimal controls for the studied optimal regional control problem is then proved. By using fractional-order system's duality theory to generalize the Hilbert uniqueness method, we present an approach on exploring the explicit expression of the optimal control formulae for associated optimal regional control problems. Moreover, to make the controllers implementation simpler and more precise, a particular case when the distributed controller is a kind of Sakawa-type is also investigated. Finally, we present a numerical example to illustrate the efficiency of our proposed approach.