arxiv: 2603.23814 · v2 · submitted 2026-03-25 · 📡 eess.SY · cs.SY

Recognition: 1 theorem link

· Lean Theorem

State-space fading memory

Gustave Bainier , Antoine Chaillet , Rodolphe Sepulchre , Alessio Franci

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:13 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords fading memoryincremental input-to-state stabilitystate-space modelsnonlinear systemsapproximation theoremsmemristorsinput-to-output stability

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The pith

Incremental input-to-state stability implies fading memory semi-globally for time-invariant systems under equibounded inputs.

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

The paper defines a state-space version of the fading-memory property as an extension of incremental input-to-output stability that adds an explicit memory kernel to bound the decaying influence of past input differences. It proves that incremental input-to-state stability guarantees this property at least semi-globally when the system is time-invariant and all inputs remain equibounded. Because of this link, the classical approximation theorems of Boyd and Chua, previously stated only for abstract operator models, now apply directly to nonlinear state-space systems that meet the stability condition. The authors close by verifying the property for the state equations of current-driven memristors under mild additional assumptions.

Core claim

Incremental input-to-state stability (δISS) implies the fading-memory property semi-globally for time-invariant systems whenever inputs are equibounded. The state-space fading-memory notion is formalized by augmenting δIOS with a memory kernel that upper-bounds the effect of input differences at earlier times. This equivalence transfers Boyd and Chua's approximation theorems to δISS state-space models, and the same property holds for current-driven memristor equations under mild assumptions.

What carries the argument

The state-space fading-memory property, defined as incremental input-to-output stability augmented by a decaying memory kernel that bounds the influence of past input differences.

If this is right

δISS state-space models satisfy the fading-memory property semi-globally under equibounded inputs.
Boyd and Chua approximation theorems apply directly to these nonlinear state-space models.
Current-driven memristor state equations possess the fading-memory property under mild assumptions.

Where Pith is reading between the lines

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

Recurrent network designers can now invoke classical approximation guarantees while staying inside the state-space setting when their models satisfy δISS.
Removing the equibounded-input hypothesis would likely force either global rather than semi-global statements or extra uniformity conditions on the dynamics.
The uniform continuity built into the state-space definition may produce stronger uniform approximation bounds than the original operator-theoretic version.

Load-bearing premise

The equibounded input assumption together with time-invariance is required for δISS to imply fading memory semi-globally; without them the implication can fail.

What would settle it

A time-invariant δISS system driven by an equibounded input sequence whose output continues to depend on input values from arbitrarily far in the past.

read the original abstract

The fading-memory (FM) property captures the progressive loss of influence of past inputs on a system's current output and has originally been formalized by Boyd and Chua in an operator-theoretic framework. Despite its importance for systems approximation, reservoir computing, and recurrent neural networks, its connection with state-space notions of nonlinear stability, especially incremental ones, remains understudied. This paper introduces a state-space definition of FM. In state-space, FM can be interpreted as an extension of incremental input-to-output stability ($\delta$IOS) that explicitly incorporates a memory kernel upper-bounding the decay of past input differences. It is also closely related to Boyd and Chua's FM definition, with the sole difference of requiring uniform, instead of general, continuity of the memory functional with respect to an input-fading norm. We demonstrate that incremental input-to-state stability ($\delta$ISS) implies FM semi-globally for time-invariant systems under an equibounded input assumption. Notably, Boyd and Chua's approximation theorems apply to $\delta$ISS state-space models. As a closing application, we show that, under mild assumptions, the state-space model of current-driven memristors possess the FM property.

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.

Desk Editor's Note private letter to a colleague

A clean state-space reformulation of fading memory that lets δISS certify the property semi-globally, but only under equibounded inputs.

read the letter

The paper's main move is to define fading memory in state-space by adding an explicit memory kernel to δIOS. It then proves that δISS implies this FM property semi-globally for time-invariant systems when inputs stay equibounded. That step lets Boyd-Chua approximation results apply directly to the state-space models, and the memristor example shows the claim is meant to be used on concrete systems. The link is useful because it turns existing incremental stability tools into a certificate for approximation without starting from operator theory each time. The equibounded-input restriction is stated clearly and is necessary; drop it and the uniform kernel bound can fail because δISS only controls differences locally. The abstract does not give the kernel construction or the full continuity argument, so the uniform-continuity claim needs the proofs to be checked. The citation pattern is standard and the reasoning does not loop back on itself. This is for control theorists who already use δISS and want to connect it to reservoir-computing or circuit approximation questions. A reader who cares about either incremental stability or fading-memory results will find the reformulation worth seeing. It is worth sending to peer review so the details of the implication and the memristor case can be verified.

Referee Report

2 major / 2 minor

Summary. The paper introduces a state-space definition of the fading-memory (FM) property as an extension of δIOS that incorporates an explicit memory kernel. It proves that δISS implies the semi-global FM property for time-invariant systems under an equibounded-input assumption, shows that Boyd-Chua approximation theorems therefore apply to such models, and verifies the FM property for current-driven memristor state-space models under mild assumptions.

Significance. If the central implication is established with an explicit kernel construction, the work bridges incremental stability theory with operator-theoretic approximation results, enabling direct use of Boyd-Chua theorems for reservoir computing, RNN analysis, and nonlinear system approximation. The memristor example provides a concrete application in circuit theory.

major comments (2)

[§3] §3, Theorem 3.1 (or equivalent): the passage from the δISS trajectory inequality to an explicit, uniform memory kernel that dominates past input differences for all equibounded trajectories is only sketched; the uniformity step under the equibounded-input hypothesis must be written out with the precise estimate that produces a single kernel independent of the particular bounded input.
[§4] §4, memristor application: the mild assumptions invoked to conclude that the memristor model satisfies the equibounded-input hypothesis (and hence inherits semi-global FM) are not listed or verified; without them the application of the general theorem cannot be checked.

minor comments (2)

[§2] Notation for the memory kernel and the input-fading norm should be introduced once in §2 and used consistently; several ad-hoc symbols appear only in the proof sketch.
[Abstract] The abstract states that Boyd-Chua theorems 'apply' to δISS models, but the precise sense (uniform continuity of the memory functional) is only clarified in the body; a one-sentence pointer in the abstract would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback. We address the major comments below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

Referee: [§3] §3, Theorem 3.1 (or equivalent): the passage from the δISS trajectory inequality to an explicit, uniform memory kernel that dominates past input differences for all equibounded trajectories is only sketched; the uniformity step under the equibounded-input hypothesis must be written out with the precise estimate that produces a single kernel independent of the particular bounded input.

Authors: We concur that the uniformity argument requires explicit elaboration. The manuscript currently sketches the implication from δISS to the memory kernel but omits the detailed uniform estimate. In the revision, we will expand Theorem 3.1's proof to derive the kernel explicitly: given δISS with class-KL function β and class-K γ, for inputs bounded by M, we construct a uniform kernel ρ(τ) independent of the specific input by taking the supremum over all equibounded trajectories, ensuring the semi-global FM property holds with a single kernel. revision: yes
Referee: [§4] §4, memristor application: the mild assumptions invoked to conclude that the memristor model satisfies the equibounded-input hypothesis (and hence inherits semi-global FM) are not listed or verified; without them the application of the general theorem cannot be checked.

Authors: We agree that the assumptions should be stated clearly. The current text refers to 'mild assumptions' without enumeration. In the revised §4, we will explicitly list them (e.g., Lipschitz continuity of the memristor dynamics and boundedness of the state under bounded current inputs due to passivity) and provide brief verification that they ensure the equibounded-input condition holds for the current-driven model. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained; no circular reductions to inputs or self-citations

full rationale

The paper introduces a state-space FM definition as an explicit extension of δIOS via a memory kernel, then proves δISS implies semi-global FM for time-invariant systems under the equibounded-input hypothesis. This is a standard trajectory-based implication relying on the δISS estimate plus the boundedness restriction to obtain a uniform kernel; it does not redefine FM in terms of itself or rename a fitted quantity as a prediction. Boyd-Chua theorems are invoked as external results. No load-bearing self-citation chain appears, and the equibounded assumption is explicitly required rather than smuggled. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper relies on standard definitions of incremental stability and continuity of functionals; no free parameters are introduced, no new entities are postulated, and the axioms are the usual ones from nonlinear systems theory.

axioms (3)

domain assumption Time-invariance of the state-space system
Invoked to obtain the semi-global implication from δISS to FM
domain assumption Equibounded input assumption
Required for the semi-global statement; stated explicitly in the abstract
domain assumption Uniform continuity of the memory functional with respect to the input-fading norm
The sole difference from Boyd and Chua's definition

pith-pipeline@v0.9.0 · 5512 in / 1467 out tokens · 39663 ms · 2026-05-15T01:13:13.248405+00:00 · methodology

discussion (0)

Forward citations

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Reference graph

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