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arxiv: 2604.21542 · v1 · submitted 2026-04-23 · 📡 eess.SY · cs.SY

A Characterization of Integral Input-to-state Stability for Hybrid Systems with Memory

Wenbang Wang , Neng Li , Wei Ren This is my paper

Pith reviewed 2026-05-09 20:43 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords integral input-to-state stabilityhybrid systemstime delayKrasovskii approachdissipativitydetectabilitystorage functionalLyapunov characterization
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The pith

Integral input-to-state stability for hybrid systems with memory is equivalent to dissipativity and detectability under regularity assumptions.

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

This paper develops characterizations of integral input-to-state stability for hybrid systems that include memory effects such as time delays. It presents a novel Lyapunov characterization using the Krasovskii approach and introduces dissipativity, detectability, and storage functional notions to describe the same property from different angles. Under mild regularity and convexity assumptions, these descriptions are shown to be equivalent. The equivalences provide a basis for analyzing stability and designing controls in systems with delayed hybrid dynamics, as illustrated by a numerical example.

Core claim

For hybrid systems with memory, the integral input-to-state stability property is equivalent to dissipativity with respect to a storage functional together with a detectability condition, with all notions connected through a Lyapunov functional characterization obtained via the Krasovskii approach, provided the system satisfies mild regularity and convexity assumptions.

What carries the argument

The storage functional, which acts as a Lyapunov-like object to encode the iISS property, together with the dissipativity and detectability conditions adapted to hybrid dynamics that include memory.

If this is right

  • Stability verification can proceed by checking any one of the equivalent descriptions: iISS, dissipativity, or detectability.
  • Control synthesis for hybrid systems with memory can be performed by constructing storage functionals or enforcing dissipativity.
  • The hybrid-system theory for stability properties now extends directly to the time-delay setting.
  • Design methods based on these equivalences apply to any system meeting the regularity and convexity requirements.

Where Pith is reading between the lines

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

  • The same pattern of equivalences may carry over to other stability notions such as ISS for hybrid systems with memory.
  • Engineers could verify the results on concrete delayed systems arising in networked control or process control.
  • The framework suggests a route for extending the characterizations to switched or stochastic hybrid systems that incorporate memory.

Load-bearing premise

The hybrid system dynamics and the candidate functionals must satisfy the stated mild regularity and convexity assumptions.

What would settle it

A concrete hybrid system with memory that satisfies iISS yet fails dissipativity or detectability (or vice versa) while still obeying the regularity and convexity conditions.

Figures

Figures reproduced from arXiv: 2604.21542 by Neng Li, Wei Ren, Wenbang Wang.

Figure 1
Figure 1. Figure 1: Time evolution of the state norm ∥𝑥(𝑡) ∥W and veloc￾ity trajectories under different control inputs. 5 CONCLUSION This paper established a unified iISS framework for hy￾brid systems with memory. By leveraging the Krasovskii approach, we proved the equivalence among Lyapunov cri￾teria, dissipativity, and storage functionals. These findings effectively resolve the analytical challenges posed by the cou￾pling… view at source ↗
read the original abstract

This paper addresses characterizations of Integral Input-to-State Stability (iISS) for hybrid systems with memory. Based on the Krasovskii approach, a novel Lyapunov characterization of iISS is established to extend the hybrid system theory to the time-delay case. In particular, we introduce the notions of dissipativity, detectability and storage functional to describe the iISS property from different perspectives. Under mild regularity and convexity assumptions, the equivalence relations among diverse stability descriptions are established, which lays a solid foundation for the control design. Finally, a numerical example is presented to illustrate the derived results.

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

2 major / 2 minor

Summary. The manuscript develops Lyapunov characterizations of integral input-to-state stability (iISS) for hybrid systems with memory by extending the Krasovskii approach. It introduces notions of dissipativity, detectability, and storage functionals, establishes equivalence relations among these descriptions and iISS under mild regularity and convexity assumptions, and illustrates the results with a numerical example.

Significance. If the equivalences are rigorously established, the work extends hybrid stability theory to the time-delay setting and supplies multiple equivalent perspectives (dissipativity, detectability, storage functionals) that can support control design. The Krasovskii-style construction and the parameter-free character of the equivalences under the stated assumptions are strengths that align with existing results for delay-free hybrid systems.

major comments (2)
  1. [§3.2, Theorem 2] §3.2, Theorem 2: the equivalence between iISS and the storage-functional property is stated to hold under convexity of the functional, yet the proof sketch does not explicitly verify that the hybrid jump map preserves the required integral inequality when the memory kernel is non-trivial; a line-by-line expansion of the jump condition is needed to confirm the claim is load-bearing.
  2. [§4, Example 1] §4, Example 1: the numerical system is asserted to satisfy the mild regularity and convexity hypotheses, but no explicit check (e.g., verification that the chosen delay kernel yields a convex storage functional) is supplied; without this, the example does not fully substantiate the general equivalences.
minor comments (2)
  1. [Preliminaries] The list of standing assumptions (regularity, convexity) is scattered across the preliminaries and main theorems; consolidating them into a single numbered assumption block would improve readability.
  2. [Notation] Notation for the hybrid time domain with memory is introduced after the first theorem; moving the definition to §2 would prevent forward references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive suggestions. We address the major comments point by point below and plan to incorporate the necessary clarifications in the revised version of the manuscript.

read point-by-point responses

  1. Referee: [§3.2, Theorem 2] §3.2, Theorem 2: the equivalence between iISS and the storage-functional property is stated to hold under convexity of the functional, yet the proof sketch does not explicitly verify that the hybrid jump map preserves the required integral inequality when the memory kernel is non-trivial; a line-by-line expansion of the jump condition is needed to confirm the claim is load-bearing.

    Authors: We appreciate this observation. The proof of Theorem 2 uses the convexity of the storage functional to handle the jump dynamics, but we acknowledge that the sketch does not provide an explicit line-by-line verification for the case of non-trivial memory kernels. In the revision, we will expand the proof to include a detailed step-by-step analysis of the jump condition, demonstrating how the integral inequality is preserved under the given assumptions. revision: yes

  2. Referee: [§4, Example 1] §4, Example 1: the numerical system is asserted to satisfy the mild regularity and convexity hypotheses, but no explicit check (e.g., verification that the chosen delay kernel yields a convex storage functional) is supplied; without this, the example does not fully substantiate the general equivalences.

    Authors: We agree that providing an explicit verification would enhance the example. In the revised manuscript, we will add a subsection or paragraph in Section 4 that explicitly checks the regularity conditions and confirms the convexity of the storage functional induced by the chosen delay kernel, thereby substantiating that the example satisfies the hypotheses of the main theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives equivalences among iISS, dissipativity, detectability, and storage-functional characterizations for hybrid systems with memory via a Krasovskii Lyapunov functional under explicit mild regularity and convexity assumptions. All introduced notions are defined independently from standard hybrid Lyapunov theory, with no reduction of any claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The argument structure consists of direct mathematical extensions and equivalence proofs that remain independent of the target claims themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions from stability theory rather than new invented entities or fitted parameters; the equivalences are derived under regularity and convexity conditions that are common but not independently verified here.

axioms (1)
  • domain assumption mild regularity and convexity assumptions
    Invoked to establish equivalence relations among iISS, dissipativity, detectability, and storage functionals.

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Reference graph

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