Large-time optimal observation domain for linear parabolic systems

Given a well-posed linear evolution system settled on a domain Ω of ℝ^d, an observation subset ω⊂Ω 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 (ω,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.