Journal of Mathematical Physics, Analysis, Geometry
2022, vol. 18, No 1, pp. 75-104   https://doi.org/10.15407/mag18.01.075     ( to contents , go back )
https://doi.org/10.15407/mag18.01.075

Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Dirichlet Boundary Condition with a Point-Wise Control

Larissa Fardigola

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, 61022, Ukraine
E-mail: fardigola@ilt.kharkov.ua

Kateryna Khalina

B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: khalina@ilt.kharkov.ua

Received December 18, 2021.

Abstract

In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\Delta w$, $w(0,x_2,t)=u(t)\delta(x_2)$, $x_1>0$, $x_2\in\mathbb R$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control. Both necessary and suffcient conditions for controllability and suffcient conditions for approximate controllability in a given time $T$ under a control $u$ bounded by a given constant are obtained in terms of solvability of a Markov power moment problem. Orthogonal bases are constructed in special spaces of Sobolev type. Using these bases, necessary and suffcient conditions for approximate controllability and numerical solutions to the approximate controllability problem are obtained. The results are illustrated by an example.

Mathematics Subject Classification 2010: 93B05, 35K05, 35B30
Key words: heat equation, controllability, approximate controllability, point-wise control, half-plane

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