Approximate Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Dirichlet Boundary Condition with a Bounded Control
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918056712732672 |
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| author | Fardigola, Larissa Khalina, Kateryna |
| author_facet | Fardigola, Larissa Khalina, Kateryna |
| contents | In the paper, the problems of approximate controllability are studied for the control system $w_t=Δw$, $w(0,x_2,t)=u(x_2,t)$, $x_1\in\mathbb R_+=(0,+\infty)$, $x_2\in\mathbb R$, $t\in(0,T)$, where $u$ is a control belonging to a special subset of $L^\infty(\mathbb R\times (0,T))\cap L^2(\mathbb R\times (0,T))$. It is proved that each initial state belonging to $L^2(\mathbb R_+\times\mathbb R)$ is approximately controllable to an arbitrary end state belonging to $L^2(\mathbb R_+\times\mathbb R)$ by applying these controls. A numerical algorithm of solving the approximate controllability problem for this system is given. The results are illustrated by an example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximate Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Dirichlet Boundary Condition with a Bounded Control Fardigola, Larissa Khalina, Kateryna Optimization and Control Analysis of PDEs In the paper, the problems of approximate controllability are studied for the control system $w_t=Δw$, $w(0,x_2,t)=u(x_2,t)$, $x_1\in\mathbb R_+=(0,+\infty)$, $x_2\in\mathbb R$, $t\in(0,T)$, where $u$ is a control belonging to a special subset of $L^\infty(\mathbb R\times (0,T))\cap L^2(\mathbb R\times (0,T))$. It is proved that each initial state belonging to $L^2(\mathbb R_+\times\mathbb R)$ is approximately controllable to an arbitrary end state belonging to $L^2(\mathbb R_+\times\mathbb R)$ by applying these controls. A numerical algorithm of solving the approximate controllability problem for this system is given. The results are illustrated by an example. |
| title | Approximate Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Dirichlet Boundary Condition with a Bounded Control |
| topic | Optimization and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2506.10466 |