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

Andreas Hartmann (1) , Marcu-Antone Orsoni (1)

1 IMB - Institut de Mathématiques de Bordeaux

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].

Domaines: Variables complexes [math.CV]; Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Andreas Hartmann, Marcu-Antone Orsoni. Separation of singularities for the Bergman space and application to control theory. Journal de Mathématiques Pures et Appliquées, 2021, 150, pp.181-201. ⟨10.1016/j.matpur.2021.04.009⟩. ⟨hal-03194639⟩