The reachable space of the heat equation for a finite rod as a Reproducing Kernel Hilbert Space

TL;DR

This paper characterizes the reachable space of the heat equation on a finite rod and a half-line as Reproducing Kernel Hilbert Spaces of analytic functions, computing their kernels and exploring boundary control effects.

Contribution

It demonstrates that the reachable spaces for the heat equation with boundary controls are RKHSs of analytic functions and explicitly computes their reproducing kernels.

Findings

Reachable space for finite rod with boundary controls is an RKHS of analytic functions.

Null reachable space for half-line with boundary data is an RKHS with kernels related to Bergman and Hardy spaces.

Results extend to cases with Neumann boundary conditions.

Abstract

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. 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 $\mathbb{C}^+$. We also consider the case with Neumann boundary data.