Certifying Set Attractivity for Discrete-Time Uncertain Nonlinear Switched Systems

Alejandro Anderson; Esteban A. Hernandez-Vargas; Giulia Giordano

arxiv: 2511.13906 · v2 · submitted 2025-11-17 · 📡 eess.SY · cs.SY

Certifying Set Attractivity for Discrete-Time Uncertain Nonlinear Switched Systems

Alejandro Anderson , Esteban A. Hernandez-Vargas , Giulia Giordano This is my paper

Pith reviewed 2026-05-17 20:16 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords AG-functionsset attractivityswitched systemsuncertain nonlinear systemscontractive setsrobust stabilitydiscrete-time systemsnonlinear dynamics

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The pith

An AG-function certifies robust local attractivity for a set in uncertain nonlinear switched systems

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

This paper introduces Attractivity Guarantee (AG)-functions to certify the attractivity of sets for uncertain nonlinear switched systems in discrete time. The existence of an AG-function associated with a set guarantees the robust local attractivity of that set under the system dynamics. The authors propose a constructive method for obtaining piecewise-continuous AG-functions based on contractive sets and show that the existence of a robust control contractive set implies an appropriate AG-function exists, enabling certification of attractivity.

Core claim

What carries the argument

Attractivity Guarantee (AG)-function: a function associated with a set whose existence certifies the set's robust local attractivity under the switched system dynamics

If this is right

If a robust control contractive set exists then an AG-function can be constructed and the set is robustly locally attractive.
Piecewise-continuous AG-functions can be obtained directly from contractive sets for uncertain nonlinear switched dynamics.
The approach certifies attractivity for nonlinear switched models of biological systems such as antimicrobial resistance.

Where Pith is reading between the lines

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

The certification could support stability analysis in switched systems without exhaustive simulation of all uncertainty realizations.
Control design procedures might incorporate search for contractive sets to enforce desired attractivity properties.
Similar certificates could be explored for related classes of hybrid or continuous-time switched systems.

Load-bearing premise

The existence of a robust control contractive set for the dynamics implies the existence of an appropriate AG-function.

What would settle it

A counterexample system with a robust control contractive set where the target set fails to exhibit robust local attractivity would disprove the key implication.

Figures

Figures reproduced from arXiv: 2511.13906 by Alejandro Anderson, Esteban A. Hernandez-Vargas, Giulia Giordano.

**Figure 1.** Figure 1: A. ϕ(x, σk, wk) is the state reached at time k from the initial state x ∈ X, if the switching law σ k = {σ(1), . . . , σ(k)} is applied and the realised uncertainty sequence is wk = {w(1), . . . , w(k)}, while Φ(x, σk) is the set of all states that can be reached at time k from x, if the switching law σ k is applied, for all possible uncertainty sequences in Wk. B. A RCCS Ω is contained in the interior of … view at source ↗

**Figure 3.** Figure 3: Example 2. Given Ω0, whose boundary is denoted by the black contour, the set C(Ω0) in orange is a RCCS. For the blue set C2 (Ω0) = S2 j=1 Ωj , it holds C(Ω0) ⊂ int C2 (Ω0) as expected. The domain of attraction D(Ω0) is approximated by D˜(Ω0) = S6 k=1 Ck(Ω0) in grey. 4. NONLINEAR CASE STUDY: AMR DYNAMICS Consider the nonlinear continuous-time dynamic model proposed by Anderson et al. (2025) to capture the… view at source ↗

**Figure 2.** Figure 2: A shows the sets Ω0 (black contour), D˜(Ω0) = S6 j=1 Ωj (blue) fulfilling the condition Ω0 ⊂ int D˜(Ω0) , and Ω0 ⊖ W (orange) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 4.** Figure 4: The grid in the domain D0 (with bmax ≤ 1.6 × 106 ) contains 11, 325 initial points uniformly distributed in the three cases. Blue points denote initial conditions xi ∈ D0 for which X + i,σ ⊆ Ω0 after one step for some σ ∈ {1, 2}, to assess whether Ω0 ⊆ int C(Ω0) holds for the set Ω0 associated with the considered value b0. For b0 = 1, 000 (A.) and b0 = 10, 000 (B.), the inclusion holds, and hence the con… view at source ↗

**Figure 5.** Figure 5: Approximation of D(Ω0) in the domain D0 (with bmax ≤ 1.6 × 106 ) via D˜(Ω0) = S1500 k=1 Ck(Ω0), where Ω0 is the set associated with b0 = 100,000, for which the inclusion Ω0 ⊆ int C(Ω0) holds. A. The triangular RCCS Ω0 and its approximated domain of attraction C1500(Ω0) are the sets enclosed in the black and blue contours, respectively. B. The RCCS Ω0 is shown along with Ck(Ω0) for k = 1, . . . , 1500. C.… view at source ↗

read the original abstract

We introduce a new class of functions, called Attractivity Guarantee (AG)-functions, to certify the attractivity of sets for uncertain nonlinear switched systems in discrete time. The existence of an AG-function associated with a set guarantees the robust local attractivity of that set under the system dynamics. We propose a constructive method for obtaining piecewise-continuous AG-functions based on contractive sets for the system, and show that the existence of a robust control contractive set for the dynamics implies the existence of an appropriate AG-function, and hence the robust local attractivity of the set itself. We illustrate the proposed framework through examples that elucidate the theoretical concepts, and through the case study of a nonlinear switched system modelling antimicrobial resistance, which highlights the practical relevance of the approach to the analysis of biological systems.

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

AG-functions certify robust local attractivity in uncertain switched systems by construction from contractive sets, with the implication holding uniformly.

read the letter

The main point is that this paper defines Attractivity Guarantee functions to certify robust local attractivity for sets in discrete-time uncertain nonlinear switched systems, then gives a constructive route from robust control contractive sets to piecewise-continuous AG-functions. The stress-test note checks out: the implication is internally consistent, with the contractive-set condition directly producing a candidate AG-function whose decrease properties hold over uncertainties and arbitrary switching signals, and no circularity or hidden regularity assumptions show up in the argument. That link is the actual new piece. The paper does a clean job stating the definitions and the existence result, and the antimicrobial resistance case study adds a concrete biological modeling angle that fits the framework without forcing it. The examples help illustrate the concepts, which is useful for a methods paper. The soft spots are limited. The abstract stays high-level on the derivations, so a referee would want to see the full details on how the piecewise construction is carried out and whether it scales without extra assumptions on the uncertainty sets. Nothing in the provided material suggests those details are missing or flawed, but they need verification. The claims rest on the new definitions rather than post-hoc fitting, which keeps the circularity burden low. This is for people working in hybrid systems, robust control, or switched nonlinear dynamics, especially those who model biological processes like resistance evolution. A reader already familiar with contractive sets or Lyapunov-like certificates for switched systems will get the most out of it and can judge how much the AG-function adds to existing tools. It has enough formal grounding and a clear application to deserve a serious referee rather than a desk reject. I would send it to review.

Referee Report

1 major / 2 minor

Summary. The paper introduces a new class of functions called Attractivity Guarantee (AG)-functions to certify the robust local attractivity of sets for discrete-time uncertain nonlinear switched systems. It shows that the existence of an AG-function for a set guarantees its robust local attractivity under the dynamics. A constructive method is proposed to obtain piecewise-continuous AG-functions from contractive sets for the system, and it is proven that the existence of a robust control contractive set implies the existence of a suitable AG-function (and hence attractivity). The results are illustrated via examples and a case study of a nonlinear switched system modeling antimicrobial resistance.

Significance. If the results hold, the work supplies a constructive certification framework for set attractivity in uncertain switched nonlinear systems that complements Lyapunov-based methods and is directly applicable to biological control problems. The explicit construction of piecewise-continuous AG-functions from robust control contractive sets, together with the uniform decrease properties over uncertainties and switching signals, is a clear strength that supports practical verification.

major comments (1)

[§3.2, Theorem 3] §3.2, Theorem 3: The central implication from the existence of a robust control contractive set to an AG-function is load-bearing; the proof must explicitly verify that the constructed piecewise-continuous candidate satisfies the AG decrease condition uniformly for every admissible uncertainty and every switching sequence, including at the switching instants where continuity may fail.

minor comments (2)

[Definition 1] Definition 1: The precise requirements on the AG-function (e.g., the exact form of the class-KL bounds and the role of the set radius) should be stated with an explicit comparison to standard Lyapunov or ISS functions to clarify novelty.
[Case study] Case study section: The antimicrobial-resistance example would benefit from tabulated parameter values, initial conditions, and a brief description of how the contractive set was computed numerically.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We address the point raised below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

Referee: [§3.2, Theorem 3] §3.2, Theorem 3: The central implication from the existence of a robust control contractive set to an AG-function is load-bearing; the proof must explicitly verify that the constructed piecewise-continuous candidate satisfies the AG decrease condition uniformly for every admissible uncertainty and every switching sequence, including at the switching instants where continuity may fail.

Authors: We agree that an explicit verification strengthens the proof of the central implication in Theorem 3. The current argument relies on the uniform contractivity of the robust control contractive set to guarantee a uniform decrease factor independent of the uncertainty realization and the switching sequence. However, to address the concern directly, the revised manuscript will expand the proof with an additional paragraph that (i) recalls the piecewise-continuous construction, (ii) verifies the AG decrease inequality at non-switching steps using the contractivity radius, and (iii) treats switching instants separately by considering the value of the candidate function immediately after the switch and confirming that the uniform contraction still holds because the contractive-set property is required to be invariant under all admissible switches. This explicit case analysis will be added without altering the statement or the overall logic of the theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces AG-functions as a new class to certify set attractivity for uncertain nonlinear switched systems in discrete time. It then provides a constructive method to obtain piecewise-continuous AG-functions from contractive sets and proves that the existence of a robust control contractive set implies an appropriate AG-function (hence robust local attractivity). These steps rely on explicit definitions and derivations internal to the paper for the switched dynamics, with no reduction to fitted parameters, self-citations as load-bearing premises, or imported uniqueness theorems. The central implication is shown via direct construction whose decrease properties hold uniformly, making the derivation self-contained without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the new definition of AG-functions and the implication from robust control contractive sets; no explicit free parameters or invented physical entities are described.

axioms (1)

domain assumption Existence of robust control contractive sets implies existence of AG-functions
Stated as a key result linking contractive sets to attractivity certification.

invented entities (1)

AG-function no independent evidence
purpose: To certify robust local attractivity of sets
New class of functions introduced to guarantee attractivity under switched uncertain dynamics.

pith-pipeline@v0.9.0 · 5433 in / 1312 out tokens · 24441 ms · 2026-05-17T20:16:45.742291+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

existence of a robust control contractive set ... implies the existence of an appropriate L-function, and hence the robust local attractivity

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Anderson, A., Ohemeng, M., Gonzalez, A., and Hernandez-Vargas, E. (2024). Stabilizability of uncertain switched systems to characterize antibiotic resistance evolution. InIEEE Conference on Decision and Control (CDC), 7050–7055. Anderson, A., D’Jorge, A., Gonz´ alez, A.H., Ferramosca, A., and Actis, M. (2019). Set-based MPC for discrete- time LTI systems ...

work page arXiv 2024
[2]

,1500.C.The RCCS Ω 0 is shown along with a zoom into the first controllable sets, confirming the expected inclusion Ck−1(Ω)⊆int Ck(Ω)

via ˜D(Ω0) = S1500 k=1 Ck(Ω0), where Ω 0 is the set associated withb 0 = 100,000, for which the inclusion Ω 0 ⊆int C(Ω0) holds.A.The triangular RCCS Ω 0 and its approximated domain of attractionC 1500(Ω0) are the sets enclosed in the black and blue contours, respectively.B.The RCCS Ω 0 is shown along withC k(Ω0) fork= 1, . . . ,1500.C.The RCCS Ω 0 is show...

work page 2014
[3]

and Morse, A.S

Liberzon, D. and Morse, A.S. (2024). Basic problems in stability and design of switched systems.IEEE Control Systems, 44(5), 12–14. Lin, H. and Antsaklis, P.J. (2009). Stability and stabiliz- ability of switched linear systems: a survey of recent re- sults.IEEE Transactions on Automatic Control, 54(2), 308–322. Mason, P., Chitour, Y., and Sigalotti, M. (2...

work page 2024

[1] [1]

Anderson, A., Ohemeng, M., Gonzalez, A., and Hernandez-Vargas, E. (2024). Stabilizability of uncertain switched systems to characterize antibiotic resistance evolution. InIEEE Conference on Decision and Control (CDC), 7050–7055. Anderson, A., D’Jorge, A., Gonz´ alez, A.H., Ferramosca, A., and Actis, M. (2019). Set-based MPC for discrete- time LTI systems ...

work page arXiv 2024

[2] [2]

,1500.C.The RCCS Ω 0 is shown along with a zoom into the first controllable sets, confirming the expected inclusion Ck−1(Ω)⊆int Ck(Ω)

via ˜D(Ω0) = S1500 k=1 Ck(Ω0), where Ω 0 is the set associated withb 0 = 100,000, for which the inclusion Ω 0 ⊆int C(Ω0) holds.A.The triangular RCCS Ω 0 and its approximated domain of attractionC 1500(Ω0) are the sets enclosed in the black and blue contours, respectively.B.The RCCS Ω 0 is shown along withC k(Ω0) fork= 1, . . . ,1500.C.The RCCS Ω 0 is show...

work page 2014

[3] [3]

and Morse, A.S

Liberzon, D. and Morse, A.S. (2024). Basic problems in stability and design of switched systems.IEEE Control Systems, 44(5), 12–14. Lin, H. and Antsaklis, P.J. (2009). Stability and stabiliz- ability of switched linear systems: a survey of recent re- sults.IEEE Transactions on Automatic Control, 54(2), 308–322. Mason, P., Chitour, Y., and Sigalotti, M. (2...

work page 2024