New results on biorthogonal families in cylindrical domains and controllability consequences

TL;DR

This paper develops new methods to construct biorthogonal families for moment problems related to null controllability of heat equations in cylindrical domains, providing novel insights into minimal control time.

Contribution

It introduces an abstract construction of biorthogonal families under partial spectral assumptions, enabling analysis of controllability in higher-dimensional cylindrical domains.

Findings

Existence of biorthogonal families with norm estimates

Characterization of minimal control time for heat equations

Application to controllability in cylindrical domains

Abstract

In this article we consider moment problems equivalent to null controllability of some linear parabolic partial differential equations in space dimension higher than one. For these moment problems, we prove existence of an associated biorthogonal family and estimate its norm. The considered setting requires the space domain to be a cylinder and the evolution operator to be tensorized. Roughly speaking, we assume that the so-called Lebeau-Robbiano spectral inequality holds but only for the eigenvectors of the transverse operator. In the one dimensional tangent variable we assume the solvability of block moment problem as introduced in [Benabdallah, Boyer and Morancey - \textit{Ann. H. Lebesgue.} 3 (2020)]. We apply this abstract construction of biorthogonal families to the characterization of the minimal time for simultaneous null controllability of two heat-like equations in a cylindrical domain. To the best of our knowledge, this result is unattainable with other known techniques.