  The controllability cost for the heat equation as the control time $T$ goes
to 0 is well-known of the order $e^{C/T}$ for some positive constant $C$,
depending on the controlled domain and for all initial datum. In this paper, we
prove that the constant $C$ can be chosen to be arbitrarily small if the
support of the initial data is sufficiently close to the controlled domain, but
not necessary inside the controlled domain. The proof is in the spirit on
Lebeau and Robbiano's approach in which a new spectral inequality is
established. The main ingredient of the proof of the new spectral inequality is
three-sphere inequalities with partial data.
