  This paper presents a complete analysis of the observability property of heat
equations with time-dependent real analytic memory kernels. More precisely, we
characterize the geometry of the space-time measurable observation sets
ensuring sharp observability inequalities, which are relevant both for control
and inverse problems purposes.
  Despite the abundant literature on the observation of heat-like equations,
existing methods do not apply to models involving memory terms.
  We present a new methodology and observation strategy, relying on the
decomposition of the flow, the time-analyticity of solutions and the
propagation of singularities. This allows us to obtain a sufficient and
necessary geometric condition on the measurable observation sets for sharp
two-sided observability inequalities. In addition, some applications to control
and relevant open problems are presented.
