A universal example for quantitative semi-uniform stability

TL;DR

This paper characterizes quantitative semi-uniform stability for port-Hamiltonian $C_0$-semigroups and provides a universal example class with decay rates slower than $t^{-1/2}$, linked to Diophantine approximation.

Contribution

It introduces a characterization of semi-uniform stability and constructs a universal example class with arbitrary decay rates based on number theory.

Findings

Characterization of semi-uniform stability for port-Hamiltonian semigroups

Construction of examples with decay rates slower than $t^{-1/2}$

Connection between decay rates and Diophantine approximation

Abstract

We characterise quantitative semi-uniform stability for $C_0$-semigroups arising from port-Hamiltonian systems, complementing recent works on exponential and strong stability. With the result, we present a simple universal example class of port-Hamiltonian $C_0$-semigroups exhibiting arbitrary decay rates slower than $t^{-1/2}$. The latter is based on results from the theory of Diophantine approximation, as the decay rates will be strongly related to the approximation properties of irrational numbers by rationals obtained from cut-offs of continued fraction expansions.