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Strong Stability of Linear Functional Equations with Distributed Delays

Yacine Chitour, Felipe Gonçalves Netto, Guilherme Mazanti

TL;DR

The paper investigates robustness of exponential stability for linear functional equations with distributed delays driven by matrix-valued measures, extending the Hale--Silkowski framework from finite delays to general distributed delays. It introduces a perturbation model for delays via pushforwards of the delay measure and defines a spectral-signal quantity $\rho_{HS}(M)$ that generalizes the HS criterion. A key result is the continuity of the spectral abscissa in total variation norm for measures without singular parts, ensuring stability under small perturbations in that regime. The main contribution is a partial counterpart of the HS conjecture in higher dimensions: $\rho_{HS}(M)<1$ is equivalent to strong stability with uniform rate (and local uniform-rate stability) for $M$ in the appropriate class, with a detailed analysis of characteristic functions under perturbations. Together, these results illuminate robust stability under delay perturbations for distributed-delay systems and provide a foundation for further extension of strong stability criteria to matrix-valued distributed delays.

Abstract

This paper considers linear functional equations on $\mathbb R^d$ with distributed delays defined by matrix-valued measures of bounded variation. More precisely, we are interested in providing conditions to ensure that the exponential stability of these systems is preserved under small changes of the parameters which define them. In the special case of difference equations, it is known that exponential stability is preserved under small perturbations of the matrices defining the system, but not of the delays, and an additional condition for preservation of exponential stability under perturbation of the delays is given by the Hale--Silkowski criterion (HSC). In this paper, we extend the treatment of these issues to more general systems. For that purpose, we first put forward an appropriate definition of perturbation on the delays and then propose a conjecture in the spirit of (HSC). We prove several partial results related to that conjecture.

Strong Stability of Linear Functional Equations with Distributed Delays

TL;DR

The paper investigates robustness of exponential stability for linear functional equations with distributed delays driven by matrix-valued measures, extending the Hale--Silkowski framework from finite delays to general distributed delays. It introduces a perturbation model for delays via pushforwards of the delay measure and defines a spectral-signal quantity that generalizes the HS criterion. A key result is the continuity of the spectral abscissa in total variation norm for measures without singular parts, ensuring stability under small perturbations in that regime. The main contribution is a partial counterpart of the HS conjecture in higher dimensions: is equivalent to strong stability with uniform rate (and local uniform-rate stability) for in the appropriate class, with a detailed analysis of characteristic functions under perturbations. Together, these results illuminate robust stability under delay perturbations for distributed-delay systems and provide a foundation for further extension of strong stability criteria to matrix-valued distributed delays.

Abstract

This paper considers linear functional equations on with distributed delays defined by matrix-valued measures of bounded variation. More precisely, we are interested in providing conditions to ensure that the exponential stability of these systems is preserved under small changes of the parameters which define them. In the special case of difference equations, it is known that exponential stability is preserved under small perturbations of the matrices defining the system, but not of the delays, and an additional condition for preservation of exponential stability under perturbation of the delays is given by the Hale--Silkowski criterion (HSC). In this paper, we extend the treatment of these issues to more general systems. For that purpose, we first put forward an appropriate definition of perturbation on the delays and then propose a conjecture in the spirit of (HSC). We prove several partial results related to that conjecture.

Paper Structure

This paper contains 14 sections, 18 theorems, 142 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations, definitions, and basic results
  3. Elementary notations
  4. Main system and its well-posedness
  5. Exponential behavior of solutions
  6. Continuity of the spectral abscissa in TV norm
  7. Strong stability
  8. Strong stability of (\ref{['sys:general']}) and statement of the main result
  9. Properties of rhoHS
  10. Characteristic function of perturbed systems
  11. Proof of Theorem \ref{['GHS-Theorem']} and further comments
  12. Appendix
  13. Results in measure theory
  14. Other technical results

Key Result

Proposition 2.6

Let $M\colon [-1, 0] \to \mathcal{M}_d(\mathbb R)$ be of bounded variation, define $A_M = M(0) - M(0^-)$, and suppose that $\det (\mathop{\mathrm{Id}}\nolimits-A_M) \ne 0$. Then, for every $\phi\in \mathcal{C}_0$, there exists a unique global solution $x\colon [-1,+\infty)\to \mathbb{R}^d$ of sys:ge

Figures (1)

  • Figure 4.1: Diagram of the results of the paper on Conjecture \ref{['GHS-Conjecture']}.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6: GoncalvesNetto2025Strong
  • Definition 2.7
  • Remark 2.8
  • Lemma 3.1
  • proof
  • ...and 41 more