  In this paper, we analyze the null controllability property for a degenerate
parabolic equation involving memory terms with a locally distributed control.
We first derive a null controllability result for a nonhomogeneous degenerate
heat equation via new Carleman estimates with weighted time functions that do
not blow up at t= 0. Then this result is successfully used with a classical
fixed point to obtain null controllability for the initial memory system.
