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

摘要

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.