An observability estimate for the wave equation and applications to the Neumann boundary controllability for semi-linear wave equations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909311652855808 |
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| author | Claret, Sue |
| author_facet | Claret, Sue |
| contents | We give a boundary observability result for a $1$d wave equation with a potential. We then deduce with a Schauder fixed-point argument the existence of a Neumann boundary control for a semi-linear wave equation $\partial_{tt}y - \partial_{xx}y + f(y) = 0$ under an optimal growth assumption at infinity on $f$ of the type $s\ln^2s$. Moreover, assuming additional assumption on $f'$, we construct a minimizing sequence which converges to a control. Numerical experiments illustrate the results. This work extends to the Neumann boundary control case the work of Zuazua in $1993$ and the work of Münch and Trélat in $2022$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An observability estimate for the wave equation and applications to the Neumann boundary controllability for semi-linear wave equations Claret, Sue Optimization and Control Analysis of PDEs We give a boundary observability result for a $1$d wave equation with a potential. We then deduce with a Schauder fixed-point argument the existence of a Neumann boundary control for a semi-linear wave equation $\partial_{tt}y - \partial_{xx}y + f(y) = 0$ under an optimal growth assumption at infinity on $f$ of the type $s\ln^2s$. Moreover, assuming additional assumption on $f'$, we construct a minimizing sequence which converges to a control. Numerical experiments illustrate the results. This work extends to the Neumann boundary control case the work of Zuazua in $1993$ and the work of Münch and Trélat in $2022$. |
| title | An observability estimate for the wave equation and applications to the Neumann boundary controllability for semi-linear wave equations |
| topic | Optimization and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2409.07214 |