Reachable space of the Hermite heat equation with boundary control

TL;DR

This paper investigates the set of states reachable via boundary controls for the Hermite heat equation, revealing that these states include certain holomorphic functions in a complex domain related to the spatial segment.

Contribution

It characterizes the reachable states for the Hermite heat equation with boundary control, linking them to functions in a Bergman space and holomorphic functions near a specific square.

Findings

Reachable states extend to functions in a Bergman space.

Holomorphic functions in a neighborhood of the square are reachable.

Results apply to both boundary points at the origin and symmetric segments.

Abstract

We discuss reachable states for the Hermite heat equation on a segment with boundary $L^2$-controls. The Hermite heat equation corresponds to the heat equation to which a quadratic potential is added. We will discuss two situations: when one endpoint of the segment is the origin and when the segment is symmetric with respect to the origin. One of the main results is that reachable states extend to functions in a Bergman space on a square one diagonal of which is the segment under consideration, and that functions holomorphic in a neighborhood of this square are reachable.