  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.
