# American Institute of Mathematical Sciences

March  2021, 11(1): 211-236. doi: 10.3934/mcrf.2020034

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

 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

Received  September 2019 Revised  June 2020 Published  March 2021 Early access  August 2020

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.

Citation: Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034
##### References:

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##### References:
The functions $u_l^n$
(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 $u = u_N$.
(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 $u = u_N$.
(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 $u = u_{N,l}$.
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
 $\varepsilon_N^1$ $\varepsilon_{N,l}^2$ $\varepsilon_N^1+\varepsilon_{N,l}^2$ $N=3$, $l=100$ 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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