  This paper investigates the link between the null controllability property
for some abstract parabolic problems and an inequality that can be seen as a
quantified Fattorini-Hautus test. Depending on the hypotheses made on the
abstract setting considered we prove that this inequality either gives the
exact minimal null control time or at least gives the qualitative property of
existence of such a minimal time. We also prove that for many known examples of
minimal time in the parabolic setting, this inequality recovers the value of
this minimal time.
