We use some results from the theory of reproducing kernel Hilbert spaces to show that the reachable space of the heat equation for a finite rod with either one or two Dirichlet boundary controls is a RKHS of analytic functions on a square, and we compute its reproducing kernel as an infinite double series. We also show that the null reachable space of the heat equation for the half line with Dirichlet boundary data is a RKHS of analytic functions on a sector, whose reproducing kernel is (essentially) the sum of pullbacks of the Bergman and Hardy kernels on the half plane \(\mathbb {C}^+\).

中文翻译：

---

作为再生核希尔伯特空间的有限棒热方程的可达空间

我们使用再生核 Hilbert 空间理论的一些结果来证明具有一个或两个 Dirichlet 边界控制的有限杆的热方程的可达空间是正方形上解析函数的 RKHS，我们计算了它的再生核作为无限双级数。我们还表明，具有狄利克雷边界数据的半线的热方程的零可达空间是扇区上解析函数的 RKHS，其再现内核（本质上）是一半上的 Bergman 和 Hardy 内核的回调之和平面\(\mathbb {C}^+\)。