On the Regional Controllability for Hadamard-Caputo Fractional Ultra-Slow Diffusion Processes

In this article, we address the regional controllability problem for ultra-slow diffusion processes governed by Hadamard-Caputo time fractional diffusion systems. Some necessary and sufficient conditions on both regional exact controllability and regional approximate controllability of the considered systems are first obtained. Secondly, we provide a necessary and sufficient condition on the minimum number of the actuators to achieve the regional approximate controllability. Moreover, by applying the Hilbert Uniqueness Method (HUM), the optimal actuators with minimum energy can be chosen among the whole admissible ones. Finally, an example is given to illustrate our results.