  In this paper we focus on the null controllability problem for the heat
equation with the so-called inverse square potential and a memory term. To this
aim, we first establish the null controllability for a nonhomogeneous singular
heat equation by a new Carleman inequality with weights which do not blow up at
t=0. Then the null controllability property is proved for the singular heat
equation with memory under a condition on the kernel, by means of Kakutani's
fixed-point Theorem.
