2024/07/02 by Nicolas Burq, Burq, Nicolas, Belhassen Dehman +3 · 1 citation
Computer Science · Engineering · Mathematics · #34A99 #35L05 #35L20 #35Q49 #35R05 #35S05 #93B07 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2407.02255
openalex publication_date 2024/07/02 · openalex created_date 2024/07/06 · openalex updated_date 2026/07/28
The celebrated geometric control condition of Bardos, Lebeau, and Rauch is necessary and sufficient for wave observability [1,7] and exact controllability. It requires that any point in phase-space be transported by the generalized geodesic flow to the region of observation in some finite time. The initial smoothness (\Cinf) required on the coefficients of the metric to prove that exact control and geometric control are essentially equivalent was subsequently relaxed to \Con2-metrics/coefficients and \Con3-domains [2] which is close to the optimal smoothness required to preserve a generalized geodesic flow. In this article, we investigate a natural generalization of the geometric control condition that makes sense for \Con1-metrics and we prove that wave observability holds under this condition. Moreover, we establish that the observability property is stable under rougher (Lipschitz) perturbation of the metric. We also provide a geometric necessary condition for wave observability to hold. Transport equations that describe the propagation of semi-classical measures are at the heart of the arguments. They are natural extensions to geometries with boundaries of usual transport equations. This article is mainly dedicated to the proof of such propagation equations in this very rough context.