Moving localized observations and Ces{\`a}ro asymptotic observability for conservative PDEs

TL;DR

This paper introduces a deterministic method for Cesàro asymptotic observability in conservative PDEs using moving localized observations, applicable to various equations and geometries, especially when instantaneous observations are limited.

Contribution

It develops a novel static-to-moving observer design via convexification, switching realization, and spectral tail reduction, enabling asymptotic observability without finite-time GCC.

Findings

Applicable to wave, Klein-Gordon, and Schrödinger equations on homogeneous manifolds.

Works with moving boundary caps and singular boundary models.

Achieves Cesàro asymptotic observability without finite-time GCC.

Abstract

We develop a deterministic large-time mechanism yielding Ces{\`a}ro asymptotic observability inequalities from moving localized observations for conservative evolutions. On each observation interval, exact convexification on a compact measured homogeneous space replaces full observation on the whole observation manifold by a finite convex combination of translates of one prototype subset. A switching realization theorem then turns that static design into a genuinely moving observer, while a Hilbertian tail-reduction proposition shows that interval estimates proved only on growing spectral windows still recover the full conserved energy after Ces{\`a}ro averaging. The resulting design-to-observability chain applies to interior observations for wave, Klein-Gordon, and Schr{\"o}dinger equations on compact measured homogeneous manifolds, to moving boundary caps on the Euclidean ball, and to a singular almost-separated gas-giant boundary model. The framework is especially relevant when each instantaneous observation set is too small for one to expect a finite-time GCC or time-dependent GCC statement.