Geometric control condition for the wave equation with a time-dependent observation domain
type: Item
instance of: scholarly article
title: Geometric control condition for the wave equation with a time-dependent observation domain
zbMATH Open document ID: 1391.35254
author: Gilles Lebeau
author: Peppino Terpolilli
author: Emmanuel Trélat
author: Jérôme Le Rousseau
published in: Analysis \& PDE
publication date: 2017-05-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1607.01527
review text: Investigated is the observability of the classical wave equation \(\square_{g}u=0\), on a \(d\)-dimensional Riemannian manifold \((\Omega,g)\), with homogeneous Dirichlet or Neumann boundary conditions. It is known that the observability property holds if some geometric control conditions, GCC are satisfied. The aim of the article is to extend the GCC to time-dependent observation domains. By using the notion of compressed generalized bicharacteristic, the time-dependent geometric control condition, t-GCC is defined. There is proved that under some special assumptions and if \(Q\) is an open subset of \((\mathbb R \times\overline{\Omega})\) that satisfies the t-GCC and \(T>T_{0}(Q,\Omega)\), the observability inequality holds. Further, as consequences, a result for the control of the wave equation with time-dependent control domain and results on observability with few sensors and stabilization are shown. The main results are then extended to the case of boundary observability. An extensive discussion is dedicated to examples and counterexamples.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 93B07
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 58J45
zbMATH DE Number: 6715609
zbMATH Keywords: geometric control condition
zbMATH Keywords: boundary observability
MaRDI profile type: Publication
OpenAlex ID: W2964225436
arXiv ID: 1607.01527
cites work: Sharp polynomial decay rates for the damped wave equation on the torus
cites work: Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary
cites work: Local energy decay of the wave equation in an exterior problem and without resonance in the neighborhood of the real line
cites work: Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes
cites work: Energy decay for damped wave equations on partially rectangular domains
cites work: Mesures de défaut de compacité, application au système de Lamé
cites work: Controllability of the Linear One-dimensional Wave Equation with Inner Moving Forces
cites work: Microlocal defect measures
cites work: Q4729705
cites work: The analysis of linear partial differential operators. III: Pseudo-differential operators
cites work: Contróle Exact De Léquation De La Chaleur
cites work: Q4040043
cites work: Exact Controllability, Stabilization and Perturbations for Distributed Systems
cites work: Singularities of boundary value problems. I
cites work: Singularities of boundary value problems. II
cites work: Polynomial decay rate for the dissipative wave equation
cites work: Q4767648
cites work: Boundary Value Control of the Higher-Dimensional Wave Equation
cites work: Boundary Value Control Theory of the Higher-Dimensional Wave Equation, Part II
cites work: Exponential stabilization without geometric control
cites work: H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations
DOI: 10.2140/APDE.2017.10.983
description: scientific article