English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We characterize the observability property (and, by duality, thecontrollability and the stabilization) of the wave equation on a Riemannianmanifold $\Omega,$ with or without boundary, where the observation (or control)domain is time-varying. We provide a condition ensuring observability, in termsof propagating bicharacteristics. This condition extends the well-knowngeometric control condition established for fixed observation domains. As oneof the consequences, we prove that it is always possible to find atime-dependent observation domain of arbitrarily small measure for which theobservability property holds. From a practical point of view, this means thatit is possible to reconstruct the solutions of the wave equation with only fewsensors (in the Lebesgue measure sense), at the price of moving the sensors inthe domain in an adequate way.We provide several illustrating examples, inwhich the observationdomain is the rigid displacement in $\Omega$ of a fixeddomain, withspeed $v,$ showing that the observability property depends both on$v$and on the wave speed. Despite the apparent simplicity of some ofourexamples, the observability property can depend on nontrivialarithmeticconsiderations.

Geometric control condition for the wave equation with a time-dependent observation domain