On the reachable states for the boundary control of the heat equation

TL;DR

This paper characterizes the set of states reachable through boundary control of the 1D heat equation, linking controllability to analyticity and Gevrey class functions, and introduces new interpolation techniques.

Contribution

It establishes a novel connection between boundary controllability and analytic or Gevrey class functions, using a new Borel interpolation theorem and the flatness approach.

Findings

Reachable states with square integrable controls are analytic functions.

Analytic functions on a disk can be reached with Gevrey class boundary controls.

The method combines flatness and a new Borel interpolation theorem.

Abstract

We are interested in the determination of the reachable states for the boundary control of the one-dimensional heat equation. We consider either one or two boundary controls. We show that reachable states associated with square integrable controls can be extended to analytic functions onsome square of C, and conversely, that analytic functions defined on a certain disk can be reached by using boundary controlsthat are Gevrey functions of order 2. The method of proof combines the flatness approach with some new Borel interpolation theorem in some Gevrey class witha specified value of the loss in the uniform estimates of the successive derivatives of the interpolating function.