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

Cai Ruiyang; Ge Fudong; Chen Yangquan; Kou Chunhai

doi:10.3934/mcrf.2019033

Article Contents

2020, Volume 10, Issue 1: 141-156. Doi: 10.3934/mcrf.2019033

This issue Previous Article A moment approach for entropy solutions to nonlinear hyperbolic PDEs Next Article Controllability properties of degenerate pseudo-parabolic boundary control problems

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
^* Corresponding author

Received: June 30, 2018

Revised: January 31, 2019

Early access: April 2019

Published: February 2020

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)

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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.

Keywords:

Ultra-slow diffusion processes,
regional gradient controllability,
Hadamard-Caputo time fractional derivative,
Hilbert Uniqueness Method.

Mathematics Subject Classification: Primary: 93B05, 35R11; Secondary: 60J60.

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