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arxiv: 1907.11673 · v1 · pith:KPXHIKLDnew · submitted 2019-07-26 · 📡 eess.SY · cs.SY· math.OC

A characterization of strong iISS for time-varying impulsive systems

Hernan Haimovich , Jose L. Mancilla-Aguilar , Paula Cardone This is my paper

Pith reviewed 2026-05-24 15:13 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords iISSimpulsive systemstime-varying systemsstrong stability0-GUASUBEBSintegral input-to-state stability
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The pith

Integral input-to-state stability equals strong zero-input stability plus bounded-energy state boundedness for time-varying impulsive systems.

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

The paper establishes that iISS for time-varying impulsive systems is equivalent to the combination of strong 0-GUAS under zero input and the UBEBS property. This extends the known characterization from general time-varying systems, but requires stability to be strong, meaning the asymptotic decay depends on both elapsed time and the number of impulses. Because converse Lyapunov theorems are unavailable for these systems, the result gives a direct way to confirm iISS without constructing a Lyapunov function. A reader cares because it supplies a concrete test for stability in systems that experience jumps at irregular times.

Core claim

For time-varying impulsive systems the iISS property holds if and only if the system is strongly globally uniformly asymptotically stable under zero input and satisfies the uniformly bounded energy input-bounded state property.

What carries the argument

Strong asymptotic stability, in which the rate of decay depends on both elapsed time and the cumulative number of impulses.

If this is right

  • iISS can be verified by checking strong zero-input stability and the UBEBS property separately.
  • No iISS-Lyapunov function is needed to establish the property.
  • The same equivalence applies to switched systems when impulses are treated as switches.
  • Results cover general nonlinear dynamics, not only linear or time-invariant cases.

Where Pith is reading between the lines

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

  • If only weak stability is assumed the equivalence is expected to break for some impulsive systems.
  • The distinction between strong and weak stability may be useful for analyzing hybrid or networked control systems that reset at discrete events.
  • Similar characterizations could be tested in discrete-time impulsive models by replacing continuous time with step count.

Load-bearing premise

Stability must be interpreted in the strong sense so that decay accounts for the number of impulses as well as elapsed time.

What would settle it

A concrete time-varying impulsive system that meets iISS yet fails either strong 0-GUAS or UBEBS, or satisfies both properties without being iISS.

read the original abstract

For general time-varying or switched (nonlinear) systems, converse Lyapunov theorems for stability are not available. In these cases, the integral input-to-state stability (iISS) property is not equivalent to the existence of an iISS-Lyapunov function but can still be characterized as the combination of global uniform asymptotic stability under zero input (0-GUAS) and uniformly bounded energy input-bounded state (UBEBS). For impulsive systems, asymptotic stability can be weak (when the asymptotic decay depends only on elapsed time) or strong (when such a decay depends also on the number of impulses that occurred). This paper shows that the mentioned characterization of iISS remains valid for time-varying impulsive systems, provided that stability is understood in the strong sense.

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 paper claims that the standard characterization of integral input-to-state stability (iISS) as the combination of global uniform asymptotic stability under zero input (0-GUAS) and uniformly bounded energy input-bounded state (UBEBS) extends to time-varying impulsive systems, but only when asymptotic stability is understood in the strong sense (decay depending on both elapsed time and the number of impulses) rather than the weak sense.

Significance. If the result holds, it supplies a Lyapunov-free test for iISS in a class of systems where converse Lyapunov theorems are unavailable, clarifying why the strong/weak distinction is necessary for the equivalence. This is a modest but useful technical extension of existing iISS characterizations to impulsive time-varying dynamics.

minor comments (2)
  1. [Abstract and §1] The abstract and introduction should explicitly state the precise definitions of strong vs. weak 0-GUAS (including the role of the impulse counting function) before claiming the characterization; this would make the necessity of the distinction clearer without requiring the reader to consult prior references.
  2. [§2] Notation for the impulse times and the hybrid time domain should be introduced with a short table or diagram in §2 to avoid ambiguity when the strong stability definition is applied in the proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The report contains no specific major comments requiring point-by-point replies.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes an equivalence between strong iISS and the combination of 0-GUAS plus UBEBS for time-varying impulsive systems by direct appeal to the definitions of these stability notions (strong vs. weak asymptotic stability). No parameter fitting, self-referential construction, or load-bearing self-citation is present; the result is a standard converse-style characterization proved from the given definitions without reducing any claimed prediction or theorem to its own inputs by construction. The distinction between strong and weak stability is explicitly required for the equivalence and is not smuggled in via prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard definitions from stability theory for impulsive systems; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Standard definitions of 0-GUAS, UBEBS, and iISS extended to impulsive systems with strong/weak variants
    The paper invokes these as background from prior literature on iISS and impulsive stability.

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Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

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