# 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  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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