On the critical time of observability of the multi-dimensional Baouendi-Grushin equation
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| Format: | Preprint |
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2026
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| _version_ | 1866914439235633152 |
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| author | Dardé, Jérémi Trabut, Mathilda |
| author_facet | Dardé, Jérémi Trabut, Mathilda |
| contents | We investigate the observability properties of the Baouendi-Grushin equation on a tensorized domain $Ω:= \mathcal{B}_R \times \tilde Ω$, where $\mathcal{B}_R$ is the open ball of radius $R$ in dimension $d \ge 2$, and $\tilde Ω$ is a smooth, bounded, open set of arbitrary dimension. Our main result is a precise calculation of the minimal observability time $T^*$, for tensorized observation sets of the form $ω\times \tilde Ω$, with $ω\subset \mathcal{B}_R$ (internal observation), and $Γ\times \tilde Ω$, with $Γ\subset \partial \mathcal{B}_R$ (boundary observation).
The main novelty regards the sufficient condition, that is observability of the system when $T>T^*$. This is established by combining refined observability inequalities on the annulus--or the entire boundary--using Carleman estimates, together with a Lebeau-Robbiano strategy to localize the observation sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_01419 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the critical time of observability of the multi-dimensional Baouendi-Grushin equation Dardé, Jérémi Trabut, Mathilda Optimization and Control 35K65, 93B05, 93B07 We investigate the observability properties of the Baouendi-Grushin equation on a tensorized domain $Ω:= \mathcal{B}_R \times \tilde Ω$, where $\mathcal{B}_R$ is the open ball of radius $R$ in dimension $d \ge 2$, and $\tilde Ω$ is a smooth, bounded, open set of arbitrary dimension. Our main result is a precise calculation of the minimal observability time $T^*$, for tensorized observation sets of the form $ω\times \tilde Ω$, with $ω\subset \mathcal{B}_R$ (internal observation), and $Γ\times \tilde Ω$, with $Γ\subset \partial \mathcal{B}_R$ (boundary observation). The main novelty regards the sufficient condition, that is observability of the system when $T>T^*$. This is established by combining refined observability inequalities on the annulus--or the entire boundary--using Carleman estimates, together with a Lebeau-Robbiano strategy to localize the observation sets. |
| title | On the critical time of observability of the multi-dimensional Baouendi-Grushin equation |
| topic | Optimization and Control 35K65, 93B05, 93B07 |
| url | https://arxiv.org/abs/2604.01419 |