Observability of wave equations with time- and space-dependent potentials: a counterexample under GCC and a positive criterion on the disk
Abstract
We show that the geometric control condition (GCC) does not guarantee internal observability for a wave equation with a smooth time-dependent potential. On a bounded connected domain with analytic boundary, we construct a nonempty open observation set such that the pair satisfies GCC and the free Dirichlet wave is observable. For this same domain and observation set, we construct a smooth 2π-periodic potential for which there exists a nonzero smooth Dirichlet wave that vanishes on the observed cylinder for all time. The wave is the unique mild solution in the space XΩ=L2(Ω)× H−1(Ω) on every finite time interval, so no finite observation horizon yields an observability inequality for this fixed potential. The construction starts from a real-valued local counterexample to unique continuation across a noncharacteristic hyperplane. This local statement is of independent interest, since earlier constructions for the wave operator use complex-valued waves and potentials. The global counterexample is then obtained by joining flat oscillatory solutions along the edges of a polygonal observation set, repairing their sum at the vertices without changing its zero set, and using a spatial transition to a common time-independent pair near the boundary. The planar construction also yields counterexamples in every space dimension n≥3. For comparison, and to identify a geometry that supplies both propagation and continuation, we show on the unit disk that an observation region containing a central disk and a neighborhood of an embedded cross-cut yields observability for every weak wave at sufficiently large observation time, with constants uniform over bounded measurable potentials. The proof uses a cut, logarithmic polar coordinates, and a Carleman weight whose favorable flux is confined to the observed artificial boundary.
Related
- Observability for wave equations with critically singular potentials
- Observability from Measurable Sets for One-Dimensional Heat Equations with Bounded Spacetime Potentials
- Robust Observability for Schrödinger Equations with Singular Potentials on 2D Compact Hyperbolic Surfaces
- Algebraic Criterion for Nonlinear Unknown Input Observability in Systems Without Control Inputs
- Regularity under Integral Closure: A General Criterion and a Sharp Counterexample
BibTeX
@misc{liao2026observability,
title = {{Observability of wave equations with time- and space-dependent potentials: a counterexample under GCC and a positive criterion on the disk}},
author = {Zhonghua Liao and Qi Lü and Xu Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25327},
eprint = {2610.25327},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25327},
}
RIS
TY - UNPB
TI - Observability of wave equations with time- and space-dependent potentials: a counterexample under GCC and a positive criterion on the disk
AU - Zhonghua Liao
AU - Qi Lü
AU - Xu Zhang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.25327
UR - https://arxiv.org/abs/2610.25327
AB - We show that the geometric control condition (GCC) does not guarantee internal observability for a wave equation with a smooth time-dependent potential. On a bounded connected domain with analytic boundary, we construct a nonempty open observation set such that the pair satisfies GCC and the free Dirichlet wave is observable. For this same domain and observation set, we construct a smooth 2π-periodic potential for which there exists a nonzero smooth Dirichlet wave that vanishes on the observed cylinder for all time. The wave is the unique mild solution in the space X_Ω=L^2(Ω)× H^−1(Ω) on every finite time interval, so no finite observation horizon yields an observability inequality for this fixed potential. The construction starts from a real-valued local counterexample to unique continuation across a noncharacteristic hyperplane. This local statement is of independent interest, since earlier constructions for the wave operator use complex-valued waves and potentials. The global counterexample is then obtained by joining flat oscillatory solutions along the edges of a polygonal observation set, repairing their sum at the vertices without changing its zero set, and using a spatial transition to a common time-independent pair near the boundary. The planar construction also yields counterexamples in every space dimension n≥3. For comparison, and to identify a geometry that supplies both propagation and continuation, we show on the unit disk that an observation region containing a central disk and a neighborhood of an embedded cross-cut yields observability for every weak wave at sufficiently large observation time, with constants uniform over bounded measurable potentials. The proof uses a cut, logarithmic polar coordinates, and a Carleman weight whose favorable flux is confined to the observed artificial boundary.
ER -