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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| Datum: | 2023 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2023
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| Online Zugang: | https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1024 |
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| Назва журналу: | Journal of Mathematical Physics, Analysis, Geometry |
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Journal of Mathematical Physics, Analysis, Geometry| Zusammenfassung: | 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. |
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