Boundary controllability of the Korteweg-de Vries equation: The Neumann case
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911224703221760 |
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| author | Capistrano-Filho, R. de A. da Silva, J. S. |
| author_facet | Capistrano-Filho, R. de A. da Silva, J. S. |
| contents | This article gives a necessary first step to understanding the critical set phenomenon for the Korteweg-de Vries (KdV) equation posed on interval $[0,L]$ considering the Neumann boundary conditions with only one control input. We showed that the KdV equation is controllable in the critical case, i.e., when the spatial domain $L$ belongs to the set $\mathcal{R}_c$, where $c\neq-1$ and $$
\mathcal{R}_c:=\left\{\frac{2π}{\sqrt{3(c+1)}}\sqrt{m^2+ml+m^2};\ m,l\in \mathbb{N}^*\right\}\cup\left\{\frac{mπ}{\sqrt{c+1}};\ m\in \mathbb{N}^*\right\}, $$ the KdV equation is exactly controllable in $L^2(0,L)$. The result is achieved using the return method together with a fixed point argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04977 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boundary controllability of the Korteweg-de Vries equation: The Neumann case Capistrano-Filho, R. de A. da Silva, J. S. Analysis of PDEs Optimization and Control This article gives a necessary first step to understanding the critical set phenomenon for the Korteweg-de Vries (KdV) equation posed on interval $[0,L]$ considering the Neumann boundary conditions with only one control input. We showed that the KdV equation is controllable in the critical case, i.e., when the spatial domain $L$ belongs to the set $\mathcal{R}_c$, where $c\neq-1$ and $$ \mathcal{R}_c:=\left\{\frac{2π}{\sqrt{3(c+1)}}\sqrt{m^2+ml+m^2};\ m,l\in \mathbb{N}^*\right\}\cup\left\{\frac{mπ}{\sqrt{c+1}};\ m\in \mathbb{N}^*\right\}, $$ the KdV equation is exactly controllable in $L^2(0,L)$. The result is achieved using the return method together with a fixed point argument. |
| title | Boundary controllability of the Korteweg-de Vries equation: The Neumann case |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2310.04977 |