Controllability problems for partial differential equations aim to understand how one can influence the evolution of a system through localized actions. In this talk we will focus on the controllability of the wave equation perturbed by a potential $\mathsf{V}$. According to the Hilbert Uniqueness Method (HUM), controllability is equivalent to an observability inequality, whose validity is characterized by the Geometric Control Condition (GCC) of Bardos–Lebeau–Rauch (1992). More precisely, this equivalence relates the existence of an observability cost $\mathfrak{C}_{\mathrm{obs}} = \mathfrak{C}_{\mathrm{obs}}(\|\mathsf{V}\|_{L^\infty})$ depending on the size of the potential, to the geometric condition (GCC).

For a fixed potential $V_0$ and a scaling parameter $\rho \geq 1$, we consider the wave equation perturbed by the potential $\rho V_0$. 
We introduce a potential-dependent geometric condition ($\mathrm{GCC}_{V_0}$), which contain the influence of the potential on the dynamics, and show that this condition is equivalent to the existence of a uniform observability cost $\mathfrak{C}_{\mathrm{obs}}(V_0)$ with respect to $\rho$.