Complete controllability for a class of fractional evolution equations with uncertainty

Nguyen Thi Kim Son; Nguyen Phuong Dong; Le Hoang Son; Alireza Khastan; Hoang Viet Long

doi:10.3934/eect.2020104

Article Contents

2022, Volume 11, Issue 1: 95-124. Doi: 10.3934/eect.2020104

This issue Previous Article Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces Next Article Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"

Complete controllability for a class of fractional evolution equations with uncertainty

1.
Faculty of Natural Science, Hanoi Metropolitan University, Hanoi, Vietnam
2.
Department of Mathematics, Hanoi Pedagogical University 2, Hanoi, Vietnam
3.
Vietnam National University, Hanoi, Vietnam
4.
Department of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
5.
Faculty of Information Technology, People's Police University of Technology and Logistics, Bac Ninh, Vietnam

^* Corresponding author: Hoang Viet Long
^* Corresponding author: Hoang Viet Long

Received: May 1, 2019

Revised: August 1, 2020

Early access: December 2, 2020

Published online: December 2, 2020

This work is supported by NAFOSTED - Vietnam under grant contract 101.02-2018.311.

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Abstract

In this paper, we study the complete controllability for a class of fractional evolution equations with a common type of fuzzy uncertainty. By using Hausdorff measure of noncompactness and Krasnoselskii's fixed point theorem in complete semilinear metric space, we give some sufficient conditions of the controllability for the fuzzy fractional evolution equations without involving the compactness of strongly continuous semigroup and the perturbation function. In addition, the controllable problem is considered in a subspace of fuzzy numbers in which the gH-differences always exist, that guarantees the satisfaction of hypotheses of the problem. An application example related to electrical circuit is given to illustrate the effectiveness of theoretical results.

Keywords:

Mathematics Subject Classification: 39A26, 93C42, 34A07.

Citation:

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Figure 1. The level sets $ [u]^\alpha $ of a triangular fuzzy number $ u $

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Figure 2. The gH-differences $ w = u\ominus_{gH}v $ and $ z = v\ominus_{gH}u $ of fuzzy numbers $ u = (3, 6, 9) $ and $ v = (0, 1, 2)$

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Figure 3. The electrical circuit diagram

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Figure 4. The fuzzy solutions of the electrical circuit model without control input solved by $ \mathtt{fde12.m} $ and $\mathtt{flmm2.m} $

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Table 1. The parameters of the electrical circuit

Parameters	Description	The value
$ R_1 $	The first resistance	$ 1 $ $ \Omega $
$ R_2 $	The second resistance	$ 2 $ $ \Omega $
$ R_3 $	The third resistance	$ 1 $ $ \Omega $
$ L_1 $	The inductance of the wire 1	$ 0.5 $ H
$ L_2 $	The inductance of the wire 2	$ 1 $ H
$ b $	The amplitude ("approximately $ 0.2 $")	$ (0.15, 0.2, 0.25) $
$ \beta $	The fractional order	$ \frac{1}{2} $
$ [0, T] $	Time	$ [0, 1] $

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