Uniform controllability for the wave equation with large potential
arXiv:2607.11702
The paper studies how the cost of controlling a wave equation on a compact Riemannian manifold depends on a large, time‑independent potential, introducing a modified geometric control condition that characterizes when the observability cost remains uniformly bounded.
Abstract
This paper investigates the dependence of the control cost for a wave equation with respect to perturbation by a time-independent potential $\lmbd V$ scaled by a large parameter $\lmbd$ on a compact Riemannian manifold. We introduce the geometric control condition~\eqref{GCC+}, a variant of the geometric control condition of Bardos--Lebeau--Rauch--Taylor, tailored to accommodate the influence of the potential . We show that~\eqref{GCC+} is necessary and sufficient for the existence of a uniform \emph{observability cost} with respect to the large parameter $\lmbd$. We provide geometric examples satisfying~\eqref{GCC+} and estimate the blow-up rate of the \emph{observability cost} in situations where it fails. The proofs rely on semiclassical and second microlocal defect measures.