  We study the controlled heat equations with analytic memory from two
perspectives: reachable subspaces and control regions. Due to the hybrid
parabolic-hyperbolic phenomenon of the equations, the support of a control
needs to move in time to efficiently control the dynamics.We show that under a
sharp sufficient geometric condition imposed to the control regions, the
difference between reachable subspaces of the controlled heat equations, with
and without memory, is precisely given by a Sobolev space. The appearance of
this Sobolev space is attributed to the memory which makes the equation having
the wave-like nature. The main ingredients in the proofs of our main results
are as: first, the decomposition of the flow (generated by the equation with
the null control) given in [31], second, an observability inequality built up
in this paper.
