Measure and continuous vector field at a boundary I: propagation equations and wave observability

TL;DR

This paper extends the geometric control condition for wave observability to less smooth metrics, proving observability under these conditions and analyzing stability with rougher metrics using propagation equations for semi-classical measures.

Contribution

It generalizes wave observability conditions to ext{C}^1-metrics and establishes stability under Lipschitz perturbations, advancing understanding of wave control in rough geometries.

Findings

Wave observability holds under generalized geometric control condition for ext{C}^1-metrics.

Observability is stable under Lipschitz perturbations of the metric.

Propagation equations for semi-classical measures are developed for rough geometries.

Abstract

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.