This work considers systems described by the heat equation on the interval [0, π] with L^2 boundary controls and it studies the reachable space at some instant τ > 0. The main results assert that this space is generally sandwiched between two Hilbert spaces of holomorphic functions defined on a square in the complex plane and which has [0, π] as one of the diagonals. More precisely, in the case Dirichlet boundary controls acting at both ends we prove that the reachable space contains the Smirnov space and it is contained in the Bergman space associated to the above mentioned square. The methodology, quite different of the one employed in previous literature, is a direct one. We first represent the input-to-state map as an integral operator whose kernel is a sum of Gaussians and then we study the range of this operator by combining the theory of Riesz bases for Smirnov spaces in polygons and the theory developed by Aikawa, Hayashi and Saitoh on the range of integral transforms, in particular those associated with the heat kernel.

中文翻译：

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从热方程的可达空间到全纯函数的希尔伯特空间

这项工作考虑了由区间 [0, π] 上具有 L^2 边界控制的热方程描述的系统，并研究了某个时刻 τ > 0 的可达空间。主要结果断言，该空间通常夹在两个希尔伯特空间之间定义在复平面中的正方形上并且具有 [0, π] 作为对角线之一的全纯函数。更准确地说，在 Dirichlet 边界控制作用于两端的情况下，我们证明可达空间包含 Smirnov 空间，并且它包含在与上述正方形相关的 Bergman 空间中。这种方法与以前文献中采用的方法大不相同，是一种直接的方法。