Observability for heat equations with time-dependent analytic memory

TL;DR

This paper analyzes the observability of heat equations with time-dependent analytic memory kernels, providing geometric conditions for observation sets and introducing new methods applicable to memory-involved models.

Contribution

It introduces a novel methodology for observability of heat equations with memory, characterizing geometric conditions for observation sets, which was not addressed by existing methods.

Findings

Characterization of geometric conditions for observation sets

Development of a new methodology based on flow decomposition and analyticity

Application to control problems and open questions in the field

Abstract

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.