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

This paper investigates linear time-invariant dissipative Hamiltonian differential-algebraic systems by characterizing the conditions for robust asymptotic stability and deriving exact criteria and bounds for the loss of this stability under structure-preserving perturbations.

Original authors: Peter Benner, Volker Mehrmann, Anshul Prajapati, Punit Sharma

Published 2026-05-15

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CC BY 4.0

Original authors: Peter Benner, Volker Mehrmann, Anshul Prajapati, Punit Sharma

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a complex machine, like a giant, intricate clockwork system or a network of water pipes, that is designed to settle down and stop moving eventually. In the world of mathematics and engineering, this is called a dissipative Hamiltonian system. "Dissipative" means it loses energy (like friction slowing a spinning top), and "Hamiltonian" refers to the specific way its energy is stored and organized.

Sometimes, these systems are "differential-algebraic" (DAE). Think of this as a machine where some parts move freely (differential), but other parts are locked in place by rigid rules or constraints (algebraic).

The Problem: Is the Machine Sturdy?

The authors of this paper are asking a very practical question: How much can we shake this machine before it breaks or starts behaving wildly?

In the real world, we never know the exact numbers for our machines. There are always tiny errors in how we build them (modeling errors) or how we measure them (uncertainty).

Robustly Asymptotically Stable: This is a fancy way of saying, "The machine is so well-built that even if we nudge it slightly, it will still settle down safely and stop."
The Danger Zone: If we push it too hard, three bad things can happen:
1. Singularity: The machine gets stuck in a way where the rules break down completely (it becomes impossible to solve).
2. High Index: The machine becomes "confused." Instead of just following simple rules, it starts needing to know its future to determine its present, which makes it unstable.
3. Imaginary Eigenvalues: The machine stops settling down and starts oscillating forever (like a pendulum that never stops swinging) or explodes.

The Paper's Goal: Measuring the "Safety Margin"

The authors want to calculate the exact distance to the edge of the safety zone. They call this the "stability radius."

Think of the system as a ball sitting in a deep bowl. The bottom of the bowl is the perfect, stable state. The rim of the bowl is the point where the ball falls out and becomes unstable.

The paper asks: How far is the ball from the rim?
Crucially, they only allow "structured perturbations." This means they only push the ball in ways that respect the machine's internal rules. For example, if the machine is a car, you can't magically turn the engine into a toaster (that would be an unstructured change). You can only change the weight of the tires, the friction of the brakes, or the stiffness of the springs.

The Three Ways the Machine Can Fail

The paper breaks down the "distance to disaster" into three specific scenarios:

The "Singularity" Distance: How much can we change the parts before the system becomes mathematically impossible to solve?
- Analogy: Imagine a bridge. If you remove too many support beams, the bridge collapses into a pile of rubble where no one can walk across. This is the distance to singularity.
The "High Index" Distance: How much can we change the parts before the system becomes "confused" and requires impossible calculations to predict its behavior?
- Analogy: Imagine a driver who suddenly needs to know what the traffic light will be next week to decide whether to brake now. The system becomes overly complex and unstable.
The "Imaginary Eigenvalue" Distance: How much can we change the parts before the system starts oscillating forever instead of stopping?
- Analogy: Imagine a swing. If you push it just right, it keeps going back and forth forever. If you push it too hard, it might fly off the chains. This is the distance to the point where the system refuses to stop.

What They Found

The authors developed a set of mathematical formulas (like a ruler) to measure these distances.

They looked at systems where everything (mass, damping, stiffness) could be slightly wrong.
They also looked at systems where only some parts (like just the damping or stiffness) could be wrong, while the main structure remained fixed.

They found that:

If you allow errors in all parts of the system, the "safety margin" is usually smaller (it's easier to break).
If you only allow errors in specific parts, the system is often more robust (harder to break).
They provided exact formulas to calculate these distances, rather than just guessing.

Real-World Examples Used

To prove their math works, they tested it on three types of systems:

Mechanical Systems: Like a car suspension or a building frame, where heavy and light masses interact.
Fluid Dynamics: Simulating water flow (like in a pipe or the ocean), where pressure and velocity are linked.
Porous Media: Like water moving through wet sand or soil, which is very complex to model.

The Bottom Line

This paper gives engineers and mathematicians a precise tool to say, "Our system is safe, and here is exactly how much error we can tolerate before it becomes dangerous." It tells us not just if a system is stable, but how close it is to the edge of instability, ensuring that the complex machines we rely on (from bridges to power grids) remain safe even when our measurements aren't perfect.

Technical Summary: Characterization of Stability Radii for Robustly Asymptotically Stable Dissipative Hamiltonian Differential-Algebraic Systems

Problem Statement
This paper addresses the robust asymptotic stability of linear time-invariant dissipative Hamiltonian differential-algebraic equation (dHDAE) systems of the form $E \dot{x} = (J - R)Qx$ . While it is well-established that the finite eigenvalues of such systems lie in the closed left half-plane and that purely imaginary eigenvalues (excluding zero) are semisimple, these systems are not necessarily robustly asymptotically stable. Small perturbations in the coefficient matrices can destroy regularity, increase the system index, or move eigenvalues to the imaginary axis, leading to instability.

The central problem is to quantify the "distance" from a given robustly asymptotically stable dHDAE system to the boundary of the set of such systems. Specifically, the authors seek to characterize the smallest structured perturbations required to cause:

Singularity (loss of regularity).
An increase in the differential-algebraic index to greater than one.
The appearance of purely imaginary eigenvalues (loss of asymptotic stability).

The analysis considers two types of structured perturbations:

$S_d$ (Dissipative): Perturbations that preserve the sign-definiteness of $E$ and $R$ (i.e., $\Delta E \le 0, \Delta R \le 0$ ) and the skew-Hermitian structure of $J$ .
$S_i$ (Indefinite): Perturbations that preserve the Hermitian/skew-Hermitian structures but do not necessarily preserve the sign-definiteness constraints on the perturbations themselves, only that the resulting matrices remain positive semidefinite.

Methodology
The authors employ a combination of structural matrix theory, spectral analysis, and optimization techniques:

Structural Characterization: The paper utilizes the Kronecker canonical form and a specific "staircase form" (Lemma 8) derived via unitary transformations to analyze the regularity and index of the dHDAE pencil. Robust asymptotic stability is characterized by three conditions: regularity, index $\le 1$ , and a negative spectral abscissa (Theorem 11).
Reduction to Standard Forms: The analysis often assumes $Q$ is invertible, allowing the system to be transformed into the form $E \dot{x} = (J - R)x$ with $E=E^* \ge 0$ , $R=R^* \ge 0$ , and $J=-J^*$ .
Mapping Problems: To compute the distance to instability, the authors formulate the problem as finding the minimal norm perturbation $(\Delta E, \Delta J, \Delta R)$ that satisfies specific algebraic constraints (e.g., making a matrix singular or creating an eigenvalue on the imaginary axis). They leverage minimal norm solutions to Hermitian and skew-Hermitian mapping problems (Theorems 15 and 16).
Optimization Formulations: The distances are expressed as optimization problems involving generalized Rayleigh quotients. For cases where $E$ $E$ is fixed (partial perturbations), exact characterizations are derived. For full perturbations (where $E$ $E$ varies), the authors derive bounds.
- The distance to singularity and higher index involves minimizing sums of squared norms of perturbations subject to rank conditions.
- The distance to imaginary eigenvalues involves minimizing over a frequency parameter $\omega$ and a vector $x$ .
Numerical Implementation: The derived optimization problems are solved using a two-level strategy: an inner loop solving a nonlinear eigenvalue problem (NEPv) via level-shifted self-consistent field (SCF) iterations, and an outer loop minimizing over the frequency parameter $\omega$ using standard optimization routines (e.g., fminsearch, fmincon).

Key Contributions and Results

Exact Characterizations for Partial Perturbations: When the matrix $E$ $E$ is fixed and only $J$ $J$ and $R$ $R$ are perturbed, the paper provides exact formulas for the structured distances to singularity, higher index, and imaginary eigenvalues (Theorems 24–26).
- For $S_i$ perturbations with $R > 0$ , the distance to singularity is exactly $\sqrt{\lambda_{\min}(N^*(R^2 - J^2)N)}$ , where $N$ spans the null space of $E$ .
- The distance to imaginary eigenvalues for $S_i$ is given by $\inf_{\omega} \sigma_{\min}([R, i\omega E - J])$ .
Bounds for Full Perturbations: When perturbations are allowed in all matrices ( $E, J, R$ $E, J, R$ ), the authors derive explicit lower and upper bounds:
- Distance to Singularity: Exact formulas are provided for $S_d$ (Theorem 20) and bounds for $S_i$ (Theorem 21).
- Distance to Higher Index: Upper bounds are derived (Theorem 22) by employing a two-stage perturbation approach (first perturbing $E$ to increase the null space dimension, then perturbing $J$ and $R$ ). The authors note that due to second-order perturbation terms, these are currently upper bounds rather than exact characterizations.
- Distance to Imaginary Eigenvalues: Lower bounds are derived for both $S_d$ and $S_i$ (Theorem 18). These involve complex optimization over generalized Rayleigh quotients and eigenvalues of specific matrix pencils.
Algorithmic Framework: A step-by-step algorithm (Section 2.7) is presented to check robust asymptotic stability and compute the associated distance bounds using standard linear algebra decompositions (SVD, spectral decomposition).

Significance and Claims
The paper claims to provide the first comprehensive characterization of stability radii for dHDAE systems under structure-preserving perturbations. While the computation of stability radii for general unstructured DAEs is known to be difficult, the authors demonstrate that the specific geometric and algebraic properties of dissipative Hamiltonian systems allow for the derivation of explicit formulas and computable bounds.

The significance lies in the ability to quantify the tolerance of physical models (such as those in structural mechanics, fluid dynamics, and porous media flow) to modeling and discretization errors. By determining the distance to the nearest singular, high-index, or unstable system, engineers and scientists can assess the reliability of their dHDAE models.

The authors are modest regarding the tightness of the bounds for full perturbations (specifically for the distance to higher index), acknowledging that the current results are upper bounds due to the complexity of second-order perturbation terms. They identify the determination of exact formulas for these cases and the development of more efficient numerical optimization methods as necessary directions for future research.