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Measure propagation
along a $\mathscr{C}^0$-vector field and wave controllability
on a rough compact manifold
Nicolas Burq, Belhassen Dehman and JĂ©rĂ´me Le Rousseau |
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Vol. 17 (2024), No. 8, 2683–2717
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Abstract |
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The celebrated Rauch–Taylor/Bardos–Lebeau–Rauch geometric control condition is central in the study of the observability of the wave equation linking this property to high-frequency propagation along geodesics that are the rays of geometric optics. This connection is best understood through the propagation properties of microlocal defect measures that appear as solutions to the wave equation concentrate. For a sufficiently smooth metric this propagation occurs along the bicharacteristic flow. If one considers a merely -metric, this bicharacteristic flow may however not exist. The Hamiltonian vector field is only continuous; bicharacteristics do exist (as integral curves of this continuous vector field) but uniqueness is lost. Here, on a compact manifold without boundary, we consider this low-regularity setting, revisit the geometric control condition, and address the question of support propagation for a measure solution to an ODE with continuous coefficients. This leads to a sufficient condition for the observability and equivalently the exact controllability of the wave equation. Moreover, we investigate the stability of the observability property and the sensitivity of the control process under a perturbation of the metric of regularity as low as Lipschitz. |
Keywords
wave equation, observability, measure, transport,
low-regularity
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Mathematical Subject Classification
Primary: 35L05, 35Q49, 35Q93, 59J40, 58J45, 58J47
Secondary: 93B05, 93B07
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Milestones
Received: 25 January 2022
Revised: 5 January 2023
Accepted: 10 February 2023
Published: 12 October 2024
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Authors |
| Nicolas Burq | |
| Laboratoire de Mathématiques
d’Orsay Université Paris-Saclay Orsay France CNRS, UMR 8628 & Institut Universitaire de France |
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| Belhassen Dehman | |
| Faculté des Sciences Université de Tunis El Manar Tunis Tunisia |
ENIT-LAMSIN Ecole Nationale d’Ingénieurs de Tunis Tunis Tunisia |
| JĂ©rĂ´me Le Rousseau | |
| Laboratoire d’Analyse, Géomérie et
applications (LAGA) CNRS, UMR 7539 Université Sorbonne Paris Nord Villetaneuse France |
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| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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