Sampling Observability for Heat Equations with Memory

TL;DR

This paper investigates the conditions under which the heat equation with memory can be observed through finite-time sampling on small regions, providing criteria for selecting observation times and locations based on the memory kernel.

Contribution

It establishes a two-sided sampling observability inequality for heat equations with memory and offers a method to choose observation times and regions tailored to the memory kernel.

Findings

Derived sharp sufficient conditions for sampling observability.

Provided a method to select observation times and regions based on the memory kernel.

Showed the dependence of observation timing on the memory kernel.

Abstract

This paper studies the sampling observability for the heat equations with memory in the lower-order term, where the observation is conducted at a finite number of time instants and on a small open subset at each time instant. We present a two-sided sampling observability inequality and give a sharp sufficient condition to ensure the aforementioned inequality. We also provide a method to select the time instants and then to design the observation regions, based on a given memory kernel, such that the above-mentioned inequality holds for these time instants and observation regions. Additionally, we demonstrate that the positions of these time instants depend significantly on the memory kernel.