Exact output tracking for the one-dimensional heat equation and applications to the interpolation problem in Gevrey classes of order 2

TL;DR

This paper characterizes exactly which boundary outputs can be tracked in the 1D heat equation with Neumann control, linking them to Gevrey regularity, and applies this to solve an interpolation problem in Gevrey classes of order 2.

Contribution

It provides a complete functional characterization of trackable boundary outputs for the heat equation and introduces new analytical tools for Gevrey class interpolation problems.

Findings

Trackable outputs form a Gevrey class of order 2.

Finite and infinite time horizon cases are fully characterized.

Improves the classical interpolation results in Gevrey-$2$ classes.

Abstract

This paper provides a complete characterization of the Dirichlet boundary outputs that can be exactly tracked in the one-dimensional heat equation with Neumann boundary control. The problem consists in describing the set of boundary traces generated by square-integrable controls over a finite or infinite time horizon. We show that these outputs form a precise functional space related to Gevrey regularity of order 2. In the infinite-time case, the trackable outputs are precisely those functions whose successive derivatives satisfy a weighted summability condition, which corresponds to specific Gevrey classes. For finite-time horizons, an additional compatibility condition involving the reachable space of the system provides a full characterization. The analysis relies on Fourier-Laplace transform, properties of Hardy spaces, the flatness method, and a new Plancherel-type theorem for Hilbert spaces of Gevrey functions. Beyond control theory, our results yield an optimal solution to the classical interpolation problem in Gevrey-$2$ classes, which improves results of Mitjagin on the optimal loss factor. The techniques developed here also extend to variants of the heat system with different boundary conditions or observation points.