MEASURE PROPAGATION ALONG C^0-VECTOR FIELD AND WAVE CONTROLLABILITY ON A ROUGH COMPACT MANIFOLD - HAL UNIV-PARIS8 - open access Access content directly
Preprints, Working Papers, ... (Preprint) Year : 2022

MEASURE PROPAGATION ALONG C^0-VECTOR FIELD AND WAVE CONTROLLABILITY ON A ROUGH COMPACT MANIFOLD

Abstract

The celebrated Rauch-Taylor/Bardos-Lebeau-Rauch geometric control condition is central in the study of the observability of the wave equation linking this propery to highfrequency 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 C 1-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 stabililty of the observability property and the sensitivity of the control process under a perturbation of the metric of regularity as low as Lipschitz.
Fichier principal
Vignette du fichier
BDLR-part0.pdf (488.65 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03866679 , version 1 (22-11-2022)
hal-03866679 , version 2 (01-06-2023)

Identifiers

  • HAL Id : hal-03866679 , version 1

Cite

Nicolas Burq, Belhassen Dehman, Jérôme Le Rousseau. MEASURE PROPAGATION ALONG C^0-VECTOR FIELD AND WAVE CONTROLLABILITY ON A ROUGH COMPACT MANIFOLD. 2022. ⟨hal-03866679v1⟩
167 View
126 Download

Share

Gmail Mastodon Facebook X LinkedIn More