  We consider the control problem of the heat equation on bounded and unbounded
domains, and more generally the corresponding inhomogeneous equation for the
Schr\"odinger semigroup. We show that if the sequence of null-controls
associated to an exhaustion of an unbounded domain converges, then the
solutions do in the same way, and that the control cost estimate carries over
to the limiting problem on the unbounded domain. This allows to infer the
controllability on unbounded domains by studying the control problem on a
sequence of bounded domains.
