Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller

Abstract

This article is concerned in the first place with the short-time observability constant of the heat equation from a subdomain $\omega$ of a bounded domain $M$. The constant is of the form $e^{\frac{K}{T}}$, where $K$ depends only on the geometry of $M$ and $\omega$. Luc Miller (JDE, 2004) conjectured that $K$ is (universally) proportional to the square of the maximal distance from $\omega$ to a point of $M$. We show in particular geometries that $K$ may blow up like $|\log (r){|}^2$ when $\omega$ is a ball of radius $r$, hence disproving the conjecture. We then prove in the general case the associated upper bound on this blowup. We also show that the conjecture is true for positive solutions of the heat equation. The proofs rely on the study of the maximal vanishing rate of (sums of) eigenfunctions. They also yield lower and upper bounds for other geometric constants appearing as tunneling constants or approximate control costs. As an intermediate step in the proofs, we provide a uniform Carleman estimate for Lipschitz metrics. The latter also implies uniform spectral inequalities and observability estimates for the heat equation in a bounded class of Lipschitz metrics, which are of independent interest.