A universal example for quantitative semi-uniform stability

TL;DR

This work characterizes quantitative semi-uniform stability for 1-D port-Hamiltonian $C_0$-semigroups by relating resolvent growth on the imaginary axis to a derived boundary-matrix quantity. It introduces a universal port-Hamiltonian example whose decay rates, ranging from near $t^{-1/2}$ to slower polynomial forms, are governed by Diophantine approximation properties of an irrational parameter $\alpha$, via the fundamental matrix and odd/odd rational approximations. The authors develop a rigorous framework linking resolvent bounds $M(\eta)$ to $m(\eta)=\sup_{t\in[-\eta,\eta]}\|T_t^{-1}\|$, and prove both upper and lower rate results, including a.a. results, badly approximable cases, and gamma-controlled rates, with optimality statements. Overall, the paper reveals a deep connection between energy decay in 1-D hyperbolic port-Hamiltonian systems and number-theoretic properties, enabling explicit and near-optimal decay rates across broad regimes.

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.

Theorems & Definitions (66)

• Lemma 2.1
• proof
• Theorem 2.2: WaurickZwart2023
• Theorem 2.3
• Remark 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• proof : Proof of Theorem \ref{['thm:MforPhs']}