Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition
In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}{\rho}\left(kw_x\right)_x+\gamma w$, $\left.\left(\sqrt{\frac{k}{\rho}}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2023
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oai:jmag.ilt.kharkiv.ua:article-10242023-11-29T18:04:11Z Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition Fardigola, Larissa Khalina, Kateryna рiвняння теплопровiдностi керованiсть наближена керованiсть heat equation controllability approximate controllability In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}{\rho}\left(kw_x\right)_x+\gamma w$, $\left.\left(\sqrt{\frac{k}{\rho}}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that any initial state of the control system is not controllable to the origin except the zero initial state in a given time $T>0$. However, each initial state of the control system is approximately controllable to any target state in a given time $T>0$. Due to transformation operator generated by the equation data $\rho$, $k$, $\gamma$, the main results are obtained from their analogues obtained earlier in the case of constant coefficients ($\rho=k=1$, $\gamma=0$). Applying this operator is a focal point of the paper. The results are illustrated by examples. Mathematical Subject Classification 2020: 93B05, 35K05, 35B30. У роботi дослiджено проблеми керованостi та наближеної керованостi для керованої системи $w_t=\frac{1}{\rho}\left(kw_x\right)_x+\gamma w$, $\left.\left(\sqrt{\frac{k}{\rho}}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, де $u$ є керуванням, $u\in L^\infty(0,T)$. Доведено, що жодний початковий стан (крiм нульового) не може бути скерованим до нуля за заданий час $T>0$. Проте, кожний початковий стан керованої системи є наближено керованим у будь-який цiльовий стан за заданий час $T>0$. Завдяки оператору перетворення, породженому параметрами рiвняння $\rho$, $k$, $\gamma$, основнi результати роботи одержано з їх аналогiв для випадку сталих коефiцiєнтiв ($\rho=k=1$, $\gamma=0$). Застосування цього оператора є ключовим моментом роботи. Результати проiлюстровано прикладами. Mathematical Subject Classification 2020: 93B05, 35K05, 35B30. Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України 2023-11-29 Article Article application/pdf https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1024 10.15407/mag19.03.616 Journal of Mathematical Physics, Analysis, Geometry; Vol. 19 No. 3 (2023); 616-641 Журнал математической физики, анализа, геометрии; Том 19 № 3 (2023); 616-641 Журнал математичної фізики, аналізу, геометрії; Том 19 № 3 (2023); 616-641 1817-5805 1812-9471 en https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1024/jm19-0616e |
| institution |
Journal of Mathematical Physics, Analysis, Geometry |
| baseUrl_str |
|
| datestamp_date |
2023-11-29T18:04:11Z |
| collection |
OJS |
| language |
English |
| topic |
рiвняння теплопровiдностi керованiсть наближена керованiсть |
| spellingShingle |
рiвняння теплопровiдностi керованiсть наближена керованiсть Fardigola, Larissa Khalina, Kateryna Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition |
| topic_facet |
рiвняння теплопровiдностi керованiсть наближена керованiсть heat equation controllability approximate controllability |
| format |
Article |
| author |
Fardigola, Larissa Khalina, Kateryna |
| author_facet |
Fardigola, Larissa Khalina, Kateryna |
| author_sort |
Fardigola, Larissa |
| title |
Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition |
| title_short |
Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition |
| title_full |
Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition |
| title_fullStr |
Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition |
| title_full_unstemmed |
Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition |
| title_sort |
controllability problems for the heat equation with variable coefficients on a half-axis controlled by the neumann boundary condition |
| title_alt |
Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition |
| description |
In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}{\rho}\left(kw_x\right)_x+\gamma w$, $\left.\left(\sqrt{\frac{k}{\rho}}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that any initial state of the control system is not controllable to the origin except the zero initial state in a given time $T>0$. However, each initial state of the control system is approximately controllable to any target state in a given time $T>0$. Due to transformation operator generated by the equation data $\rho$, $k$, $\gamma$, the main results are obtained from their analogues obtained earlier in the case of constant coefficients ($\rho=k=1$, $\gamma=0$). Applying this operator is a focal point of the paper. The results are illustrated by examples.
Mathematical Subject Classification 2020: 93B05, 35K05, 35B30. |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України |
| publishDate |
2023 |
| url |
https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1024 |
| work_keys_str_mv |
AT fardigolalarissa controllabilityproblemsfortheheatequationwithvariablecoefficientsonahalfaxiscontrolledbytheneumannboundarycondition AT khalinakateryna controllabilityproblemsfortheheatequationwithvariablecoefficientsonahalfaxiscontrolledbytheneumannboundarycondition |
| first_indexed |
2025-09-26T01:40:38Z |
| last_indexed |
2025-09-26T01:40:38Z |
| _version_ |
1850836704333660160 |