The propagation of smallness property and its utility in controllability problems

Abstract

In this article, wewould like to transmit the origins and the importance of the propagation of smallness property, delving into its origins and analyzing its utility in parabolic controllability problems. Generally speaking, the propagation of smallness property analyzes and tries to quantify the rate of growth of a function in one domain knowing its values in two other domains related to the first one. We will mention the first historical ideas and results related to this property (harmonic measure, two-constants theorem, Hadamard three-circles theorem), and, then, we will see how we have applied them in order to solve parabolic evolutions’ interior and boundary controllability problems.