Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

Fardigola Larissa; Khalina Kateryna

doi:10.3934/mcrf.2020034

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2021, Volume 11, Issue 1: 211-236. Doi: 10.3934/mcrf.2020034

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Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

Fardigola Larissa^1, , and
Khalina Kateryna^2,

1.
B. Verkin Institute for Low Temperature Physics and Engineering, of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine, V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61077, Ukraine
2.
B. Verkin Institute for Low Temperature Physics and Engineering, of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine

^* Corresponding author: Larissa Fardigola
^* Corresponding author: Larissa Fardigola

Received: September 2019

Revised: June 2020

Early access: August 2020

Published: March 2021

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Abstract

In the paper, the problems of controllability and approximate controllability are studied for the control system $ w_t = w_{xx} $, $ w_x(0,\cdot) = u $, $ x>0 $, $ t\in(0,T) $, where $ u\in L^\infty(0,T) $ is a control. It is proved that each initial state of the system is approximately controllable to each target state in a given time $ T $. A necessary and sufficient condition for controllability in a given time $ T $ is obtained in terms of solvability of a Markov power moment problem. It is also shown that there is no initial state which is null-controllable in a given time $ T $. Orthogonal bases are constructed in $ H^1 $ and $ H_1 $. Using these bases, numerical solutions to the approximate controllability problem are obtained. The results are illustrated by examples.

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Mathematics Subject Classification: Primary:93B05, 35K05;Secondary:35B30.

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Figure 1. The functions $ u_l^n $

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Figure 2. (A)–(D): The controls $ u_N $ defined by (70). (E), (F): The influence of these controls on the end state of the solution to (6), (7) with $ W^T(x) = \frac{1}{\sqrt\pi} \int_0^T e^{-\frac{x^2}{4\xi}}\frac{2\xi-1}{\sqrt\xi}d\xi $ and $ u = u_N $.

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Figure 3. (A)–(D): The controls $ u_N $ defined by (70). (E), (F): The influence of these controls on the end state of the solution to (6), (7) with $ W^T(x) = \frac{1}{\sqrt\pi} \int_0^T e^{-\frac{x^2}{4\xi}}\frac{2(\xi-1)^2-1}{\sqrt\xi}d\xi $ and $ u = u_N $.

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Figure 4. (A), (B): The controls $ u_{N,l} $ defined by (70). (C), (D): The influence of these controls on the end state $ W_N^l $ of the solution to (6), (7) with $ W^T(x) = \cosh xe^{-\frac{x^2}{2}-\frac{1}{4}} $ and $ u = u_{N,l} $.

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Table 1. The estimates for $ \left\| W^T-W_N^l\right\|^1 $

	$ \varepsilon_N^1 $	$ \varepsilon_{N,l}^2 $	$ \varepsilon_N^1+\varepsilon_{N,l}^2 $
$ N=3 $, $ l=100 $	0.18666	0.12756	0.31422
$ N=3 $, $ l=200 $	0.18666	0.05927	0.24593
$ N=4 $, $ l=150 $	0.01535	0.08648	0.10183
$ N=4 $, $ l=400 $	0.01535	0.03038	0.04573

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