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arxiv: 2605.13891 · v1 · pith:T7LLG5YGnew · submitted 2026-05-12 · 🧮 math.DS · cs.NA· math.NA· math.OC

Characterization of stability radii for robustly asymptotically stable dissipative Hamiltonian differential-algebraic systems

Peter Benner , Volker Mehrmann , Anshul Prajapati , Punit Sharma This is my paper

Pith reviewed 2026-05-15 05:44 UTC · model grok-4.3

classification 🧮 math.DS cs.NAmath.NAmath.OC
keywords stability radiusdissipative Hamiltonian systemsdifferential-algebraic equationsrobust asymptotic stabilitystructure-preserving perturbationslinear systems
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The pith

Dissipative Hamiltonian differential-algebraic systems remain robustly asymptotically stable exactly when the smallest structure-preserving perturbation that destroys stability has positive size.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies linear time-invariant dissipative Hamiltonian differential-algebraic systems and determines when they stay asymptotically stable under perturbations that keep the same structure. It supplies exact conditions that must hold for this robust stability and gives precise bounds on how large a perturbation can grow before stability fails. These results matter for models of constrained mechanical and physical systems, where knowing the safe range of parameter changes helps predict when the model will remain reliable. The characterization supplies both the threshold at which stability is lost and the explicit form of the critical perturbations.

Core claim

For linear time-invariant dissipative Hamiltonian differential-algebraic systems that are asymptotically stable, the stability radius with respect to structure-preserving perturbations is characterized exactly, giving the precise bounds on perturbations that preserve asymptotic stability.

What carries the argument

The stability radius of the system under structure-preserving perturbations, which measures the minimal size of a perturbation that keeps the dissipative Hamiltonian structure while driving the system to instability.

If this is right

  • A positive stability radius guarantees that all sufficiently small structure-preserving perturbations leave the system asymptotically stable.
  • The radius supplies an explicit, computable distance to the nearest destabilizing perturbation within the allowed structure class.
  • Algebraic constraints in the DAE are incorporated directly into the radius calculation.
  • Loss of robust stability occurs precisely when a perturbation reaches the derived bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The radius could be used to certify numerical integrators that respect the Hamiltonian structure during simulation.
  • Similar radius formulas might extend to time-varying or mildly nonlinear Hamiltonian DAEs.
  • In control design the radius offers a quantitative guideline for choosing feedback that enlarges the stability margin.

Load-bearing premise

Perturbations must preserve the dissipative Hamiltonian structure of the linear time-invariant differential-algebraic system.

What would settle it

A concrete structure-preserving perturbation whose size is strictly smaller than the computed stability radius yet makes the system asymptotically unstable would disprove the claimed bound.

Figures

Figures reproduced from arXiv: 2605.13891 by Anshul Prajapati, Peter Benner, Punit Sharma, Volker Mehrmann.

Figure 1
Figure 1. Figure 1: Eigenvalue perturbation curves of dHDAE system with respect to structured complex pertur [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
read the original abstract

We study linear time-invariant dissipative Hamiltonian differential-algebraic systems. We characterize when the systems are robustly asymptotically stable and derive exact conditions and bounds when this property is lost under structure-preserving perturbations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies linear time-invariant dissipative Hamiltonian differential-algebraic systems. It characterizes when these systems are robustly asymptotically stable and derives exact conditions together with explicit bounds for the loss of this property under structure-preserving perturbations.

Significance. If the central derivations hold, the work supplies a precise, structure-exploiting characterization of stability radii for a practically relevant class of DAEs. This is valuable for robust analysis in constrained mechanical systems, circuit models, and other applications where the Hamiltonian structure must be respected. The approach rests on standard negative-definiteness conditions for the dissipation matrix after accounting for the skew-symmetric Hamiltonian part, combined with a perturbation analysis that preserves the block structure; when the derivations are fully verified this yields non-conservative bounds that can be computed directly from the system matrices.

minor comments (2)
  1. The abstract states the main claim but does not indicate the explicit form of the stability radius or the key matrix condition used; adding one sentence summarizing the central theorem would improve readability for readers scanning the front matter.
  2. Notation for the dissipation matrix and the perturbation class is introduced without a dedicated preliminary section; a short table collecting the standing assumptions on the Hamiltonian, dissipation, and constraint blocks would help readers track the structure-preserving conditions throughout the proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on stability radii for dissipative Hamiltonian DAEs and for recommending minor revision. The referee's assessment accurately reflects the paper's focus on exact conditions and structure-preserving perturbation bounds. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central derivation characterizes stability radii for dissipative Hamiltonian DAEs via standard matrix-theoretic conditions (negative definiteness of the dissipation matrix after accounting for the Hamiltonian structure) and structure-preserving perturbation analysis. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the exact conditions and bounds follow directly from the given linear time-invariant system form without renaming known results or smuggling ansatzes. The analysis is self-contained against external matrix theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available, so ledger is minimal and based on stated setup.

axioms (1)
  • domain assumption Systems are linear time-invariant dissipative Hamiltonian differential-algebraic systems
    Core modeling assumption stated in the abstract.

pith-pipeline@v0.9.0 · 5334 in / 1012 out tokens · 33502 ms · 2026-05-15T05:44:41.091011+00:00 · methodology

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Reference graph

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