Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908542561157120 |
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| author | Coron, Jean-Michel Krieger, Joachim Xiang, Shengquan |
| author_facet | Coron, Jean-Michel Krieger, Joachim Xiang, Shengquan |
| contents | We study wave maps from the circle to a general compact Riemannian manifold. We prove that the global controllability of this geometric equation is characterized precisely by the homotopy class of the data. As a remarkable intermediate result, we establish uniform-time global controllability between steady states, providing a partial answer to an open problem raised by Dehman, Lebeau and Zuazua (2003). Finally, we obtain quantitative exponential stability around closed geodesics with negative sectional curvature. This work highlights the rich interplay between partial differential equations, differential geometry, and control theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy Coron, Jean-Michel Krieger, Joachim Xiang, Shengquan Analysis of PDEs Differential Geometry Optimization and Control We study wave maps from the circle to a general compact Riemannian manifold. We prove that the global controllability of this geometric equation is characterized precisely by the homotopy class of the data. As a remarkable intermediate result, we establish uniform-time global controllability between steady states, providing a partial answer to an open problem raised by Dehman, Lebeau and Zuazua (2003). Finally, we obtain quantitative exponential stability around closed geodesics with negative sectional curvature. This work highlights the rich interplay between partial differential equations, differential geometry, and control theory. |
| title | Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy |
| topic | Analysis of PDEs Differential Geometry Optimization and Control |
| url | https://arxiv.org/abs/2509.12779 |