Non null-controllability properties of the Grushin-like heat equation on 2D-manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914231673159680 |
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| author | Vanlaere, Roman |
| author_facet | Vanlaere, Roman |
| contents | We study the internal non null-controllability properties of the heat equation on 2-dimensional almost-Riemannian manifolds with an interior singularity, and under the assumption that the closure of the control zone does not contain the whole singularity. We show that if locally, around the singularity, the sub-Riemannian metric can be written in a Grushin form, or equivalently the sub-Laplacian writes as a generalized Grushin operator, then, achieving null-controllability requires at least a minimal amount of time. As locally the manifold looks like a rectangular domain, we consequently focus ourselves on the non null-controllability properties of the generalized Grushin-like heat equation on various Euclidean domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_00997 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non null-controllability properties of the Grushin-like heat equation on 2D-manifolds Vanlaere, Roman Optimization and Control Analysis of PDEs 93C05, 93C20, 93B05, 35K65 We study the internal non null-controllability properties of the heat equation on 2-dimensional almost-Riemannian manifolds with an interior singularity, and under the assumption that the closure of the control zone does not contain the whole singularity. We show that if locally, around the singularity, the sub-Riemannian metric can be written in a Grushin form, or equivalently the sub-Laplacian writes as a generalized Grushin operator, then, achieving null-controllability requires at least a minimal amount of time. As locally the manifold looks like a rectangular domain, we consequently focus ourselves on the non null-controllability properties of the generalized Grushin-like heat equation on various Euclidean domains. |
| title | Non null-controllability properties of the Grushin-like heat equation on 2D-manifolds |
| topic | Optimization and Control Analysis of PDEs 93C05, 93C20, 93B05, 35K65 |
| url | https://arxiv.org/abs/2503.00997 |