# American Institute of Mathematical Sciences

March  2020, 10(1): 141-156. doi: 10.3934/mcrf.2019033

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

 1 College of Information Science and Technology, Donghua University, Shanghai 201620, China 2 School of Computer Science, China University of Geosciences, Wuhan 430074, China 3 School of Engineering (MESA-Lab), University of California, Merced, CA 95343, USA 4 Department of Applied Mathematics, Donghua University, Shanghai 201620, China

* Corresponding author

Received  July 2018 Revised  February 2019 Published  April 2019

Fund Project: The second author is supported by the Fundamental Research Funds for the Central Universities, China University of Geosciences, Wuhan (No.CUG170627), the Natural Science Foundation of China (NSFC, No.KZ18W30084) and the Hubei NSFC (No.2017CFB279).

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.

Citation: Ruiyang Cai, Fudong Ge, Yangquan Chen, Chunhai Kou. Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative. Mathematical Control & Related Fields, 2020, 10 (1) : 141-156. doi: 10.3934/mcrf.2019033
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