A Remark on Null Exact Controllability of the Heat Equation

It is well known that the heat equation $u_t-\Delta u=f\chi_{\omega}$ in $(0,T)\times\Omega$ with homogeneous Dirichlet boundary conditions is null exactly controllable for any T>0 and any open nonempty subset $\omega$ of $\Omega$. In this note we show that this property may be obtained as a singular limit of the exact controllability properties of singularly perturbed damped wave equations with a changing controller.