On the reachable set for the one-dimensional heat equation

TL;DR

This paper characterizes the set of all states reachable by boundary controls for the one-dimensional heat equation, showing it consists of functions extendable to certain complex squares, using Carleman estimates and complex analysis.

Contribution

It provides a nearly sharp description of the reachable set for the heat equation with boundary controls, linking real controllability to complex analyticity properties.

Findings

Reachable functions extend analytically to a square of size L0 > L.

The reachable set is characterized by complex analyticity up to a square of size L.

Method employs Carleman estimates and Cauchy's formula for holomorphic functions.

Abstract

The goal of this article is to provide a description of the reachable set of the one-dimensional heat equation, set on the spatial domain x $\in$ (--L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L0 \textgreater{} L any function which can be extended analytically on the square {x + iy, |x| + |y| $\le$ L0} belongs to the reachable set. This result is nearly sharp as one can prove that any function which belongs to the reachable set can be extended analytically on the square {x + iy, |x| + |y| \textless{} L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions.