Separation of singularities for the Bergman space and application to control theory

TL;DR

This paper proves a decomposition property of the Bergman space for convex polygons, and applies it to characterize the reachable space in a boundary control problem for the heat equation, confirming a previous conjecture.

Contribution

It establishes a new decomposition result for Bergman spaces on convex polygons and applies it to solve a conjecture in control theory related to the heat equation.

Findings

Bergman space on convex polygons decomposes into sums of spaces on half planes.

The reachable space of the heat equation on an interval is a specific Bergman space.

Confirmed a conjecture linking Bergman spaces and control theory.

Abstract

In this paper, we solve a separation of singularities problem in the Bergman space. More precisely, we show that if $P\subset \mathbb{C}$ is a convex polygon which is the intersection of $n$ half planes, then the Bergman space on $P$ decomposes into the sum of the Bergman spaces on these half planes. The result applies to the characterization of the reachable space of the one-dimensional heat equation on a finite interval with boundary controls. We prove that this space is a Bergman space of the square which has the given interval as a diagonal. This gives an affirmative answer to a conjecture raised in [HKT20].