Uniform controllability for the wave equation with large potential

This paper investigates the dependence of the control cost for a wave equation with respect to perturbation by a time-independent potential $\text{lmbd}V$ scaled by a large parameter $\text{lmbd}$ 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 $V$. We show that~\eqref{GCC+} is necessary and sufficient for the existence of a uniform \emph{observability cost} with respect to the large parameter $\text{lmbd}$. 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.