Summary: We consider the linear heat equation with potentials in an \(n\)-dimensional domain of the form \(\Omega_\varepsilon=\Omega\times(0, \varepsilon)\), where \(\Omega\) is a bounded, smooth open set of \(\mathbb{R}^{n-1}\), with \(n\geq 2\) and \(\varepsilon\) a small parameter. We study the null controllability problem when the control acts in a cylindrical subdomain \(\omega_\varepsilon=\omega\times(0,\varepsilon)\), where \(\omega\subset\Omega\) is an open and non empty subset of \(\Omega\). We prove that, under appropriate boundary conditions and for suitable classes of potentials, the heat equation is uniformly null controllable as \(\varepsilon\to 0\). We also prove the convergence of the controls to a control for the null controllability of \(a(n-1)\)-dimensional heat equation in \(\Omega\).