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.