Observability for heat equations with time-dependent analytic memory

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.