A dependence of the cost of fast controls for the heat equation on the support of initial datum

TL;DR

This paper shows that the controllability cost for the heat equation can be made arbitrarily small if the initial data's support is close to the controlled domain, using a new spectral inequality based on three-sphere inequalities.

Contribution

It introduces a novel spectral inequality that allows the controllability cost constant to be arbitrarily small under specific support conditions of initial data.

Findings

Controllability cost can be minimized by positioning initial data near the controlled domain.

A new spectral inequality is established using three-sphere inequalities with partial data.

The approach extends Lebeau and Robbiano's method to broader initial data supports.

Abstract

The controllability cost for the heat equation as the control time $T$ goes to 0 is well-known of the order $e^{C/T}$ for some positive constant $C$, depending on the controlled domain and for all initial datum. In this paper, we prove that the constant $C$ can be chosen to be arbitrarily small if the support of the initial data is sufficiently close to the controlled domain, but not necessary inside the controlled domain. The proof is in the spirit on Lebeau and Robbiano's approach in which a new spectral inequality is established. The main ingredient of the proof of the new spectral inequality is three-sphere inequalities with partial data.