A New Approach to Stability of Delay Differential Equations with Time-Varying Delays via Isospectral Reduction

Quinlan Leishman, Benjamin Webb

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:57 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves delay-independent exponential convergence (Theorem 1.2) under intrinsic stability α(M)<0, with M built from abs*(D_x f) and |D_{y_i} f|, for uniformly bounded, uniformly continuous delays, using an isospectral-reduction/operator-joint-spectral-radius route (Sections 3–6) . The candidate solution establishes the same conclusion via a direct Dini-derivative/Halanay-type comparison in a diagonally weighted ∞-norm, leveraging Hurwitz–Metzler structure of M (via d ≫ 0 with Md ≪ 0). Assumptions align with the paper’s setup, and the technical steps are standard for monotone/Metzler bounds. Hence both are correct, with substantially different proofs.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript presents a new, operator-theoretic pathway to delay-independent exponential stability for a broad nonlinear DDE class with time-varying delays. The technique (isospectral reduction + generalized spectral radius control) is of genuine interest to the DDE community and complements classical comparison approaches. The technical chain (comparison inequality, compactness, generalized Berger–Wang) is plausible and well-motivated, though some intermediate regularity/compactness arguments could be streamlined or clarified. The contribution is specialized but solid and likely to influence subsequent work on delay-independent stability.