Large-time optimal observation domain for linear parabolic systems

Abstract

Given a well-posed linear evolution system settled on a domain $\Omega$ of ${\mathbb{R}}^d$, an observation subset $\omega \subset \Omega$ and a time horizon $T$, the observability constant is defined as the largest possible nonnegative constant such that the observability inequality holds for the pair $(\omega ,T)$. In this article we investigate the large-time behavior of the observation domain that maximizes the observability constant over all possible measurable subsets of a given Lebesgue measure. We prove that it converges exponentially, as the time horizon goes to infinity, to a limit set that we characterize. The mathematical technique is new and relies on a quantitative version of the bathtub principle.