  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 $D$.
