Optimal actuation for regional approximate controllability of parabolic systems with the fractional Laplacian

This paper addresses the optimal actuation policies for regional approximate controllability of parabolic systems with the fractional Laplacian on a bounded domain. These systems could well model a wide class of physical phenomena, including Lévy flights and stochastic interfaces when traditional approaches appear to fail. To this end, we consider the Sakawa-type controller, which may be possibly unbounded depending on the structure of actuators. An approach on the optimal actuation policies for regional approximate controllability of the studied system in some subregion of its evolution domain is then established via the Hilbert uniqueness method (HUM). It is shown that the optimal control inputs can be explicitly developed with respect to the subregion, the structure of actuators and the spectral theory of fractional Laplacian. Two illustrations are finally included to confirm our results.