  We explore further controllability problems through a standard least square
approach. By setting up a suitable error functional $E$, and putting $m(\ge0)$
for the infimum, we interpret approximate controllability by asking $m=0$,
while exact controllability corresponds, in addition, to demanding that $m$ is
attained. We also provide a condition, formulated entirely in terms of the
error $E$, which turns out to be equivalent to the unique continuation
property, and to approximate controllability. Though we restrict attention here
to the 1D, homogeneous heat equation to explain the main ideas, they can be
extended in a similar way to many other scenarios some of which have already
been explored numerically, due to the flexibility of the procedure for the
numerical approximation.
