Local controllability around a regular solution and null-controllability of scattering solutions for semilinear wave equations
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911221455781888 |
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| author | Perrin, Thomas |
| author_facet | Perrin, Thomas |
| contents | On a Riemannian manifold with or without boundary, and whether bounded or unbounded, we consider a semilinear wave (or Klein-Gordon) equation with a subcritical nonlinearity (either defocusing or focusing). We establish local controllability around a partially analytic solution, under the Geometric Control Condition. Specifically, some blow-up solutions can be controlled. In the case of a Klein-Gordon equation on a non-trapping exterior domain of small dimension, we prove the null-controllability of scattering solutions. The proof is based on local energy decay and global-in-time Strichartz estimates. Several consequences are presented, including the null-controllability of a solution initiated near the ground state in some focusing cases, and exact controllability in some defocusing cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06373 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local controllability around a regular solution and null-controllability of scattering solutions for semilinear wave equations Perrin, Thomas Analysis of PDEs Optimization and Control On a Riemannian manifold with or without boundary, and whether bounded or unbounded, we consider a semilinear wave (or Klein-Gordon) equation with a subcritical nonlinearity (either defocusing or focusing). We establish local controllability around a partially analytic solution, under the Geometric Control Condition. Specifically, some blow-up solutions can be controlled. In the case of a Klein-Gordon equation on a non-trapping exterior domain of small dimension, we prove the null-controllability of scattering solutions. The proof is based on local energy decay and global-in-time Strichartz estimates. Several consequences are presented, including the null-controllability of a solution initiated near the ground state in some focusing cases, and exact controllability in some defocusing cases. |
| title | Local controllability around a regular solution and null-controllability of scattering solutions for semilinear wave equations |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2312.06373 |