Admissibility theory in abstract Sobolev scales and transfer function growth at high frequencies

TL;DR

This paper explores the admissibility of control and observation operators in abstract Sobolev scales for semigroups, establishing equivalences with resolvent bounds and applying results to wave equations to derive energy decay rates.

Contribution

It introduces a novel framework linking admissibility in Sobolev scales with resolvent estimates and transfer function growth, with applications to wave equation boundary control.

Findings

Equivalence of admissibility in fractional domain and interpolation spaces.

Quantified resolvent bounds imply frequency-domain estimates.

Derived optimal asymptotics for wave transfer functions and energy decay rates.

Abstract

For strongly continous semigroups on Hilbert spaces, we investigate admissibility properties of control and observation operators shifted along continuous scales of spaces built by means of either interpolation and extrapolation or functional calculus. Our results show equivalence of admissibility in, on the one hand, a fractional domain of the generator and, on the other hand, a (different, in general) quadratic interpolation space of the same "Sobolev order". Furthermore, such properties imply quantified resolvent bounds in the original state space topology. When the semigroup is a group, the resulting frequency-domain estimates are in fact equivalent to the aforementioned time-domain properties. In the case of systems with both control and observation, we are able to translate input-output regularity properties into high-frequency growth rates of operator-valued transfer functions. As an application, based on results by Lasiecka, Triggiani and Tataru on interior and boundary regularity of the wave equation under Neumann control, we derive optimal asymptotics for the Neumann-to-Dirichlet wave transfer function. With that in hand, we establish non-uniform energy decay rates for the wave equation posed in a rectangle and subject to Neumann damping on an arbitrary open subset of the boundary.