A universal example for quantitative semi-uniform stability

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.

💡 Research Summary

The paper investigates quantitative semi‑uniform stability for C₀‑semigroups generated by one‑dimensional port‑Hamiltonian systems. After a brief historical overview of stability concepts—uniform exponential stability, strong stability, and the more recent notion of semi‑uniform stability (also known as polynomial stability)—the authors focus on the resolvent growth function
M(η)=sup_{|t|≤η}‖R(i t,−A)‖,
where A is the generator of the contraction semigroup T(t)=e^{-tA}.

Section 2 establishes a precise relationship between M(η) and the norm of the inverse of a boundary matrix Tₜ defined by the fundamental solution matrix Φₜ of the underlying differential system. Under the standing assumptions (H(t) symmetric positive definite, P₁ invertible, W full rank, and iℝ⊂ρ(A)), the fundamental matrix is uniformly bounded, i.e. supₜ‖Φₜ‖<∞. Theorem 2.3 shows that there exist constants c,C>0 such that for all η≥0
c m(η) ≤ M(η) ≤ C m(η),
where m(η)=sup_{|t|≤η}‖Tₜ⁻¹‖. Lemma 2.5 provides the upper bound M(η) ≤ C(‖Tₜ⁻¹‖+1) and Lemma 2.6 the converse estimate. Consequently, the growth of the resolvent on the imaginary axis is completely governed by the behaviour of the boundary matrix Tₜ. Moreover, if m is a function of positive increase (a technical condition satisfied by all regularly varying functions with positive index), then M inherits the same property (Remark 2.4). This observation allows the authors to apply the abstract semi‑uniform stability theorem from Appendix A, which translates resolvent growth into decay rates for the semigroup via the estimate
‖T(t)A^{-1}‖ = O(γ^{-1}(t))
when M(η) behaves like γ(η).

Section 3 presents a universal example that realizes a wide spectrum of decay rates. The authors consider the constant Hamiltonian matrix H = diag(1,α)^{-1} with α>0, the identity P₁, zero P₀, and a fixed boundary matrix W. The fundamental matrix reduces to Φₜ = e^{i t H^{-1}} and the boundary matrix becomes Tₜ = f_W e^{i t H^{-1}} with determinant
det Tₜ = 1 + ½(e^{i t}+e^{i α t}).
From this explicit formula they deduce:

• Strong stability holds iff α is irrational (Theorem 3.1).
• Exponential stability never holds for any α (Proposition 3.1).

The key observation is that the norm of Tₜ⁻¹ is dictated by how closely α can be approximated by rational numbers. If α is badly approximable (i.e., there exists c>0 such that |α−p/q| ≥ c/q² for all integers p,q), then ‖Tₜ⁻¹‖ grows at most like √|t|, leading to a decay rate of order t^{-1/2} for the semigroup. More generally, by selecting α whose continued‑fraction convergents p_n/q_n have prescribed growth, one can force ‖Tₜ⁻¹‖ to behave like |t|^{β} for any β∈(0,½). This yields semigroups whose energy decays like t^{-β}, i.e. any algebraic rate slower than t^{-1/2}. Theorem 3.2 formalises this statement, showing that for each β<½ there exists an α (irrational) such that the associated semigroup satisfies
‖T(t)A^{-1}‖ = O(t^{-β})
but not faster. The construction relies on classical results from Diophantine approximation (Appendix B), in particular metric properties of continued fractions and the measure‑theoretic prevalence of numbers with prescribed approximation exponents.

The remainder of the paper (Sections 4–7) supplies the technical proofs of the auxiliary lemmas, the resolvent estimates, and the number‑theoretic lemmas needed to control the growth of the convergents. The authors also discuss the role of the “positive increase” condition, showing that the resolvent bound M(η) automatically inherits this property from m(η).

In summary, the article provides a complete quantitative description of semi‑uniform stability for a broad class of port‑Hamiltonian C₀‑semigroups. By linking the resolvent growth to a simple matrix inverse and exploiting Diophantine approximation, the authors construct a universal family of examples that can realize any polynomial decay rate slower than t^{-1/2}. This bridges operator‑theoretic stability analysis with number‑theoretic approximation theory, offering a new perspective on how the arithmetic nature of system parameters influences the long‑time behaviour of PDE‑based control systems.