Stability under dwell time constraints: Discretization revisited

Thomas Mejstrik, Vladimir Yu. Protasov

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Theorem 5 establishes σh ≤ σ(A) ≤ σh − (1/m) ln(1 − ∥(A − σh I)^2∥ h^2/8) using an extremal multinorm and a geometric short-time bound (Theorem 13) for e^{tA} on [0,h] with constant h^2/8. The candidate solution proves the same double inequality with the same hypotheses and structure: (i) the trivial lower bound via admissible paths of the h-discretization, and (ii) the upper bound via the extremal multinorm, decomposition of dwell intervals, and a short-time bound. The only substantive difference is the derivation of the short-time bound: the paper uses a norm-geometry argument, while the candidate uses a Peano kernel interpolation identity. Both yield the same factor h^2/8 and lead to the same conclusion.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The main theorem provides a computable, quadratically accurate discretization bound for Lyapunov exponents under dwell-time constraints, with clear definitions, a correct proof, and useful algorithmic guidance. The short-time bound in arbitrary norms is elegant and of independent interest. Minor clarifications (explicitly noting the vacuous case of the bound and slightly expanding the lower-bound step) would further improve accessibility.