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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.10169 |
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| _version_ | 1866909476842373120 |
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| author | Fardigola, Larissa Khalina, Kateryna |
| author_facet | Fardigola, Larissa Khalina, Kateryna |
| contents | In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=Δw$, $w_{x_1}(0,x_2,t)=u(t)δ(x_2)$, $x_1>0$, $x_2\in\mathbb R$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control. To this aid, it is investigated the set $\mathcal{R}_T(0)\subset L^2((0,+\infty)\times\mathbb R)$ of its end states which are reachable from $0$. It is established that a function $f\in\mathcal{R}_T(0)$ can be represented in the form $f(x)=g\big(|x|^2\big)$ a.e. in $(0,+\infty)\times\mathbb R$ where $g\in L^2(0,+\infty)$. In fact, we reduce the problem dealing with functions from $L^2((0,+\infty)\times\mathbb R)$ to a problem dealing with functions from $L^2(0,+\infty)$. Both a necessary and sufficient condition for controllability and a sufficient condition 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. Using the Laguerre functions (forming an orthonormal basis of $L^2(0,+\infty)$), necessary and sufficient conditions for approximate controllability and numerical solutions to the approximate controllability problem are obtained. It is also shown that there is no initial state that is null-controllable in a given time $T$. The results are illustrated by an example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10169 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Neumann Boundary Condition with a Point-Wise Control Fardigola, Larissa Khalina, Kateryna Analysis of PDEs Optimization and Control In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=Δw$, $w_{x_1}(0,x_2,t)=u(t)δ(x_2)$, $x_1>0$, $x_2\in\mathbb R$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control. To this aid, it is investigated the set $\mathcal{R}_T(0)\subset L^2((0,+\infty)\times\mathbb R)$ of its end states which are reachable from $0$. It is established that a function $f\in\mathcal{R}_T(0)$ can be represented in the form $f(x)=g\big(|x|^2\big)$ a.e. in $(0,+\infty)\times\mathbb R$ where $g\in L^2(0,+\infty)$. In fact, we reduce the problem dealing with functions from $L^2((0,+\infty)\times\mathbb R)$ to a problem dealing with functions from $L^2(0,+\infty)$. Both a necessary and sufficient condition for controllability and a sufficient condition 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. Using the Laguerre functions (forming an orthonormal basis of $L^2(0,+\infty)$), necessary and sufficient conditions for approximate controllability and numerical solutions to the approximate controllability problem are obtained. It is also shown that there is no initial state that is null-controllable in a given time $T$. The results are illustrated by an example. |
| title | Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Neumann Boundary Condition with a Point-Wise Control |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2409.10169 |