From the reachable space of the heat equation to Hilbert spaces of holomorphic functions

Abstract

This work considers systems described by the heat equation on the interval $[0,\pi ]$ with $L^2$ boundary controls and it studies the reachable space at some instant $\tau >0$. 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 $[0,\pi ]$ as one of the diagonals. More precisely, in the case of Dirichlet boundary controls acting at both ends we prove that the reachable space contains the Hardy–Smirnov space and is contained in the Bergman space associated to the square. The methodology, quite different from the one employed in the 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 Hardy–Smirnov spaces in polygons and a result of Aikawa, Hayashi and Saitoh on the range of integral transforms associated with the heat kernel.