Null-controllability and control cost estimates for the heat equation on unbounded and large bounded domains

TL;DR

This paper reviews recent advances in controlling the heat equation on large or unbounded domains, introducing new spectral inequalities and control cost estimates that improve existing bounds and apply to generalized heat equations.

Contribution

It presents new spectral inequalities and an improved abstract control cost estimate, enhancing understanding of heat equation controllability on large and unbounded domains.

Findings

Sharp control cost bounds in various regimes

Control on unbounded domains approximated by bounded domain problems

Applicability to generalized heat equations with Schrödinger semigroups

Abstract

We survey recent results on the control problem for the heat equation on unbounded and large bounded domains. First we formulate new uncertainty relations, respectively spectral inequalities. Then we present an abstract control cost estimate which improves upon earlier results. It is particularly interesting when combined with the earlier mentioned spectral inequalities since it yields sharp control cost bounds in several asymptotic regimes. We also show that control problems on unbounded domains can be approximated by corresponding problems on a sequence of bounded domains forming an exhaustion. Our results apply also for the generalized heat equation associated with a Schr\"odinger semigroup.