  We make two remarks about the null-controllability of the heat equation with
Dirichlet condition in unbounded domains. Firstly, we give a geometric
necessary condition (for interior null-controllability in the Euclidean
setting)which implies that one can not go infinitely far away from the control
region without tending to the boundary (if any), but also applies when the
distance to the control region is bounded. The proof builds on heat kernel
estimates. Secondly, we describe a class of null-controllable heat equations on
unbounded product domains. Elementary examples include an infinite strip in the
plane controlled from one boundary and an infinite rod controlled from an
internal infinite rod. The proof combines earlier results on compact manifolds
with a new lemma saying that the null-controllability of an abstract control
system and its null-controllability cost are not changed by taking its tensor
product with a system generated by a non-positive self-adjoint operator.
