Sharp geometric condition for null-controllability of the heat equation on $\mathbb{R}^d$ and consistent estimates on the control cost

TL;DR

This paper establishes a precise geometric condition called thickness for null-controllability of the heat equation on  and provides explicit estimates of control costs, including for bounded domains with various boundary conditions.

Contribution

It introduces the necessary and sufficient thickness condition for null-controllability of the heat equation on  and derives explicit control cost estimates that align with the unbounded case.

Findings

Thickness condition characterizes null-controllability.

Control cost estimates depend explicitly on geometric parameters.

Results extend to bounded domains with various boundary conditions.

Abstract

In this note we study the control problem for the heat equation on $\mathbb{R}^d$, $d\geq 1$, with control set $\omega\subset\mathbb{R}^d$. We provide a necessary and sufficient condition (called $(\gamma, a)$-\emph{thickness}) on $\omega$ such that the heat equation is null-controllable in any positive time. We give an estimate of the control cost with explicit dependency on the characteristic geometric parameters of the control set. Finally, we derive a control cost estimate for the heat equation on cubes with periodic, Dirichlet, or Neumann boundary conditions, where the control sets are again assumed to be thick. We show that the control cost estimate is consistent with the $\mathbb{R}^d$ case.