ON THE REACHABLE SET FOR THE ONE-DIMENSIONAL HEAT EQUATION

摘要

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 is an element of (-L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L-0 L, any function which can be extended analytically on the square {x + iy, vertical bar x vertical bar + vertical bar y vertical bar L-0} 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, vertical bar x vertical bar + vertical bar y vertical bar L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions.