Propagation of smallness and control for heat equations

TL;DR

This paper extends propagation of smallness and spectral projector estimates for heat equations on manifolds, enabling null controllability with controls on sets of positive Lebesgue measure, even with Lipschitz coefficients and zero measure sets.

Contribution

It introduces a new approach based on Logunov-Malinnikova's method, extending spectral projector estimates to arbitrary sets and applying these to control problems with minimal regularity assumptions.

Findings

Extended spectral projector estimates to arbitrary sets of positive measure.

Achieved null controllability with controls supported on zero measure sets.

Dropped constant coefficient and analyticity assumptions for Laplace operators.

Abstract

In this note we investigate propagation of smallness properties for solutions to heat equations. We consider spectral projector estimates for the Laplace operator with Dirichlet or Neumann boundary conditions on a Riemanian manifold with or without boundary. We show that using the new approach for the propagation of smallness from Logunov-Malinnikova [7, 6, 8] allows to extend the spectral projector type estimates from Jerison-Lebeau [3] from localisation on open set to localisation on arbitrary sets of non zero Lebesgue measure; we can actually go beyond and consider sets of non vanishing d -- $\delta$ ($\delta$ > 0 small enough) Hausdorf measure. We show that these new spectral projector estimates allow to extend the Logunov-Malinnikova's propagation of smallness results to solutions to heat equations. Finally we apply these results to the null controlability of heat equations with controls localised on sets of positive Lebesgue measure. A main novelty here with respect to previous results is that we can drop the constant coefficient assumptions (see [1, 2]) of the Laplace operator (or analyticity assumption, see [4]) and deal with Lipschitz coefficients. Another important novelty is that we get the first (non one dimensional) exact controlability results with controls supported on zero measure sets.