# Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative

## TL;DR

This paper studies how to control ultra-slow diffusion processes with Hadamard-Caputo fractional derivatives, providing conditions for controllability, methods to determine actuator placement, and explicit optimal controllers.

## Contribution

It introduces new controllability criteria and an approach for optimal control design for systems governed by Hadamard-Caputo fractional derivatives.

## Key findings

- Derived necessary and sufficient conditions for regional gradient controllability.
- Proposed a method to determine the minimum number of strategic actuators.
- Established the existence and uniqueness of the optimal controller using HUM.

## Abstract

This paper investigates the regional gradient controllability for ultra-slow diffusion processes governed by the time fractional diffusion systems with a Hadamard-Caputo time fractional derivative. Some necessary and sufficient conditions on regional gradient exact and approximate controllability are first given and proved in detail. Secondly, we propose an approach on how to calculate the minimum number of $\omega-$strategic actuators. Moreover, the existence, uniqueness and the concrete form of the optimal controller for the system under consideration are presented by employing the Hilbert Uniqueness Method (HUM) among all the admissible ones. Finally, we illustrate our results by an interesting example.