Reachable states and holomorphic function spaces for the 1-D heat equation

TL;DR

This paper characterizes the set of states reachable by the 1-D heat equation with boundary control, showing they are exactly the sum of two Bergman spaces on sectors, refining previous analytic descriptions.

Contribution

It introduces a new precise description of the reachable states as a sum of two Bergman spaces, improving understanding of the heat equation's controllability in complex function spaces.

Findings

Reachable states are exactly the sum of two Bergman spaces on sectors.

Includes the Smirnov-Zygmund space and a weighted Bergman space in the description.

Refines previous bounds between Smirnov and Bergman spaces for the heat equation.

Abstract

The description of the reachable states of the heat equation is one of the central questions in control theory. The aim of this work is to present new results for the 1-D heat equation with boundary control on the segment $[0, \pi]$. In this situation it is known that the reachable states are holomorphic in a square $D$ the diagonal of which is given by $[0,\pi]$. The most precise results obtained recently say that the reachable space is contained between two well known spaces of analytic function: the Smirnov space $E^2(D)$ and the Bergman space $A^2(D)$. We show that the reachable states are exactly the sum of two Bergman spaces on sectors the intersection of which is $D$. In order to get a more precise information on this sum of Bergman spaces, we also prove that it includes the Smirnov-Zygmund space $E_{L\log^+\!L}(D)$ as well as a certain weighted Bergman space on $D$.