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, π]. In this situation it is known that the reachable states are holomorphic in a square D the diagonal of which is given by [0, π]. 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_{LlogL}(D) as well as a certain weighted Bergman space on D.

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一维热方程的可达状态和全纯函数空间

热方程可达状态的描述是控制理论的核心问题之一。这项工作的目的是为段 [0, π] 上具有边界控制的一维热方程提供新结果。在这种情况下，已知可达状态在正方形 D 中是全纯的，其对角线由 [0, π] 给出。最近获得的最精确的结果表明，可达空间包含在两个众所周知的解析函数空间之间：Smirnov 空间 E^2(D) 和 Bergman 空间 A^2(D)。我们证明可达状态正好是扇区上的两个伯格曼空间的总和，扇区的交集是 D。为了获得关于这个伯格曼空间总和的更精确的信息，