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For time-invariant delay systems, global asymptotic stability does not imply uniform global attractivity

Antoine Chaillet, Fabian Wirth, Andrii Mironchenko, Lucas Brivadis

TL;DR

This work investigates whether global asymptotic stability ($GAS$) implies uniform global attractivity ($UGA$) for time-delay systems (TDS). By adapting a recent counterexample, the authors construct a four-state, two-delay system that is forward complete ($FC$), locally exponentially stable (LES), and globally attractive, yet fails to be forward bound-reach (not $BRS$), not $UGA$, and not weakly $WUGA$; artificially large transients can occur from bounded initial sets. The key insight is that forward completeness does not guarantee bounded reachability in infinite-dimensional state spaces, distinguishing TDS behavior from finite-dimensional ODEs. This negative result highlights the need for additional conditions (e.g., $BRS$ or alternative function spaces) when asserting uniform convergence properties in TDS and has practical implications for stability analysis and controller design in systems with delays.

Abstract

Adapting a counterexample recently proposed by J.L. Mancilla-Aguilar and H. Haimovich, we show here that, for time-delay systems, global asymptotic stability does not ensure that solutions converge uniformly to zero over bounded sets of initial states. Hence, the convergence might be arbitrarily slow even if initial states are confined to a bounded set.

For time-invariant delay systems, global asymptotic stability does not imply uniform global attractivity

TL;DR

This work investigates whether global asymptotic stability () implies uniform global attractivity () for time-delay systems (TDS). By adapting a recent counterexample, the authors construct a four-state, two-delay system that is forward complete (), locally exponentially stable (LES), and globally attractive, yet fails to be forward bound-reach (not ), not , and not weakly ; artificially large transients can occur from bounded initial sets. The key insight is that forward completeness does not guarantee bounded reachability in infinite-dimensional state spaces, distinguishing TDS behavior from finite-dimensional ODEs. This negative result highlights the need for additional conditions (e.g., or alternative function spaces) when asserting uniform convergence properties in TDS and has practical implications for stability analysis and controller design in systems with delays.

Abstract

Adapting a counterexample recently proposed by J.L. Mancilla-Aguilar and H. Haimovich, we show here that, for time-delay systems, global asymptotic stability does not ensure that solutions converge uniformly to zero over bounded sets of initial states. Hence, the convergence might be arbitrarily slow even if initial states are confined to a bounded set.
Paper Structure (16 sections, 2 theorems, 40 equations, 1 figure)

This paper contains 16 sections, 2 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Definitions
  5. The counter-example
  6. Main statement
  7. Rationale behind this example
  8. Proofs
  9. Proof of Lemma \ref{['lemma1']}
  10. Proof of \ref{['item0']}): forward completeness
  11. Proof of \ref{['item2']}): local exponential stability
  12. Proof of \ref{['item4']}): global asymptotic stability
  13. Proof of \ref{['item1']}): no boundedness of reachability sets
  14. Proof of \ref{['item5']}): no uniform global attractivity
  15. Proof of \ref{['item6']}): no weak uniform global attractivity
  16. ...and 1 more sections

Key Result

Proposition 1

Given any $c>0$, the time-delay system eq-10 However, there exists $c>0$ such that the TDS eq-10

Figures (1)

  • Figure 1: Graphical representation of $z_{10}$.

Theorems & Definitions (7)

  • Definition 1: FC, BRS
  • Definition 2: GA, UGA
  • Definition 3: WGA, WUGA
  • Definition 4: GAS, UGAS
  • Definition 5: LES
  • Proposition 1: GAS & LES $\nRightarrow$ WUGA
  • Lemma 1: Arbitrarily large transients