Laplace–Carleson embeddings and weighted infinite-time admissibility

Abstract

In this paper, we will establish necessary and sufficient conditions for a Laplace-Carleson embedding to be bounded for certain spaces of functions on the positive half-line. We will use these results to characterise weighted (infinite-time) admissibility of control and observation operators. We present examples of weighted admissibility criterion for one-dimensional heat equation with Neumann boundary conditions, and a cetrain parabolic diagonal system which was previously known to be not admissible in the unweighted setting.

Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane

“…then we say that µ is a Carleson measure for A p ν . The notion of a Carleson measure has been introduced by Lennart Carleson in [2] to solve the corona problem, and since then has found many other applications (for example, in the context of spaces of analytic functions on the half-plane, it is used to describe the admissibility criterion for control and observation operators, see [10], [11], [14], [15]). Lemma 1.…”

“…The classical Hardy and weighted Bergman spaces defined on the open unit disk of the complex plane and the open right complex half-plane may be viewed as discrete and continuous counterparts (this is discussed for example in [13] and [14]). The continuous case is sometimes more appropriate when we consider applications of these spaces (see [10], [11] and [15]).…”