Pith. sign in

REVIEW 1 major objections 70 references

Semi-uniform stability estimates for impedance passive systems with saturated feedback *

T0 review · 1 major / 0 minor · reviewed 2026-07-03 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Saturated output feedback preserves fractional Sobolev regularity in impedance passive systems and establishes semi-uniform input-to-state stability.

desk verdict The paper shows fractional Sobolev regularity transfers from the linear closed-loop to the saturated nonlinear case, letting linear observability give semi-uniform ISS without nonlinear domain work. read the letter →

arxiv 2607.01861 v1 pith:B42ZSAXW submitted 2026-07-02 math.AP

Swann Marx (LS2N , Nantes Univ - ECN) This is my paper · ORCID
classification math.AP
keywords impedancepassivesystemssaturatedfeedbacksemi-uniformstabilityinput-to-statefractionalSobolevregularitywaveequationboundarydampingobservabilityestimates
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that for impedance passive systems, saturated output feedback preserves the fractional Sobolev regularity of both the output and the state whenever the initial condition lies in a suitable interpolation space tied to the linear closed-loop generator. This shared regularity between linear and nonlinear cases allows direct application of linear observability estimates to characterize long-time behavior even with external disturbances. The approach avoids explicit identification of the nonlinear generator domain. Under the impedance passivity assumption together with exact observability and regularity conditions, the result yields a semi-uniform input-to-state stability property, illustrated on a multidimensional wave equation with nonlinear boundary damping.

What carries the argument

The interpolation space associated with the linear closed-loop generator, which transfers fractional regularity to the nonlinear saturated feedback via the impedance passivity framework.

What would settle it

A simulation or calculation for the wave equation example that checks whether the state norm satisfies the semi-uniform decay bound under saturated feedback and a nonzero constant disturbance.

Watch

Extended reading notes

Core claim

Under the assumption that the linear output feedback exponentially stabilizes the system in the absence of saturation and disturbances, the nonlinear closed-loop system preserves fractional Sobolev regularity of the free output in the corresponding interpolation spaces. Combined with exact observability estimates for the linear system, this regularity yields a characterization of the asymptotic behavior and establishes semi-uniform input-to-state stability for the nonlinear system in the presence of disturbances.

Load-bearing premise

The linear output feedback exponentially stabilizes the system in the absence of saturation and disturbances, allowing transfer of regularity and observability to the nonlinear case.

Editorial extensions

If this is right

  • The output and state inherit the fractional regularity of the linear system when initial data are in the interpolation space.
  • Linear observability estimates apply directly to bound the nonlinear closed-loop trajectories.
  • Semi-uniform input-to-state stability holds under impedance passivity and exact observability.
  • The result covers infinite-dimensional examples such as the wave equation with nonlinear boundary damping.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The regularity-transfer technique could simplify analysis for other classes of nonlinear boundary feedback in distributed systems.
  • The framework may extend to time-dependent or state-dependent saturation levels while retaining the same stability conclusion.
  • Similar interpolation-space arguments might apply to other passive structures such as port-Hamiltonian systems.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates the long-time behavior of impedance passive (possibly infinite-dimensional) systems under saturated output feedback in the presence of external disturbances. Assuming exponential stability of the linear closed-loop without saturation or disturbances, it proves that fractional Sobolev regularity of the free output is preserved by the nonlinear saturated feedback when initial data lie in suitable interpolation spaces associated with the linear closed-loop generator. This shared regularity allows direct application of linear observability estimates to nonlinear trajectories, yielding a characterization of asymptotic behavior and, under impedance passivity plus exact observability and regularity assumptions, a semi-uniform input-to-state stability property. The results are illustrated by a multidimensional wave equation with nonlinear boundary damping.

Significance. If the regularity-transfer argument is complete, the work offers a route to semi-uniform ISS estimates for nonlinear infinite-dimensional systems that avoids explicit identification of the nonlinear generator domain, a common technical obstacle in PDE control with nonlinear boundary conditions. The wave-equation example supplies a concrete test case.

major comments (1)
  1. [regularity preservation argument (main theorem on fractional Sobolev regularity)] The central regularity-transfer step (the argument that linear exponential stability of the unsaturated closed-loop implies preservation of interpolation-space regularity and direct applicability of the linear observability inequality once the bounded saturation map and disturbances are present) is load-bearing for the semi-uniform ISS claim. The manuscript must explicitly confirm that the constants and semi-uniform character survive the insertion of the saturated input; without this verification the transfer from linear to nonlinear trajectories remains incomplete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the importance of the regularity-transfer argument. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [regularity preservation argument (main theorem on fractional Sobolev regularity)] The central regularity-transfer step (the argument that linear exponential stability of the unsaturated closed-loop implies preservation of interpolation-space regularity and direct applicability of the linear observability inequality once the bounded saturation map and disturbances are present) is load-bearing for the semi-uniform ISS claim. The manuscript must explicitly confirm that the constants and semi-uniform character survive the insertion of the saturated input; without this verification the transfer from linear to nonlinear trajectories remains incomplete.

    Authors: We agree that an explicit verification paragraph improves clarity. The boundedness of the saturation map (which maps into a fixed ball) together with the given external disturbance ensures that the effective input to the linear observability inequality remains controlled in the same function space as in the linear case. Because the main regularity theorem already places both the state and output of the nonlinear closed-loop system in the identical interpolation spaces used for the linear observability estimate, the constants and the semi-uniform dependence on the initial-data norm carry over verbatim. In the revised manuscript we will add, immediately after Theorem 3.2, a short remark that records this fact and cites the precise linear observability statement (Assumption 4.1) to make the transfer fully explicit. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation rests on explicit external assumptions without self-referential reductions.

full rationale

The paper states its key assumptions upfront (exponential stability of the linear unsaturated closed-loop, impedance passivity, exact observability and regularity) and derives the semi-uniform ISS property from them via regularity preservation under saturation and application of linear observability estimates. No equations or steps reduce a claimed prediction or result to a fitted input or self-citation by construction; the linear stability assumption is invoked as an independent hypothesis to transfer properties, not derived within the paper. The provided abstract and description contain no self-definitional, fitted-prediction, or load-bearing self-citation patterns, making the chain self-contained against its stated benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on two domain assumptions from control theory for passive systems; no free parameters or invented entities are indicated in the abstract.

assumptions (2)
  • domain assumption The linear output feedback exponentially stabilizes the system in the absence of saturation and disturbances.
    Explicitly stated as the starting assumption in the abstract.
  • domain assumption Exact observability and regularity assumptions hold for the linear system.
    Invoked to obtain the semi-uniform ISS property from the regularity result.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Semi-uniform stability estimates for impedance passive systems with saturated feedback *." pith.science (2026). https://pith.science/paper/B42ZSAXW

@misc{pith2026260701861,
  author       = {Pith},
  title        = {Pith review of: Semi-uniform stability estimates for impedance passive systems with saturated feedback *},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B42ZSAXW}},
  note         = {Machine review of arXiv:2607.01861}
}
read the original abstract

This article investigates the long-time behavior of (possibly infinite-dimensional) impedance passive systems under saturated output feedback and external disturbances. We assume that, in the absence of saturation and disturbances, the underlying linear output feedback exponentially stabilizes the system. Our main contribution is to show that fractional Sobolev regularity of the free output is preserved by the nonlinear feedback. More precisely, if the initial condition belongs to a suitable interpolation space associated with the linear closed-loop generator, then both the output and the state of the nonlinear closed-loop system inherit the corresponding fractional regularity. This regularity is sufficiently weak to be shared by the linear and nonlinear closed-loop systems, thereby avoiding the identification of the nonlinear generator domain. Combining this regularity result with observability estimates for the linear system yields a characterization of the asymptotic behavior of the nonlinear closed-loop system in the presence of disturbances. In particular, under the impedance passivity framework and exact observability and regularity assumptions, we establish a semi-uniform input-tostate stability property. The theory is illustrated by a multidimensional wave equation with nonlinear boundary damping.

Figures

Figures reproduced from arXiv: 2607.01861 by the authors.

Figure 1
Figure 1. Regular linear system in feedback with a saturation [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Regular linear system in feedback with a dead-zone [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages

  1. [1]

    F. Alabau. Stabilisation frontiere indirecte de systemes faiblement coupl´ es. Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics, 328(11):1015–1020, 1999

    work page 1999

  2. [2]

    Alabau-Boussouira

    F. Alabau-Boussouira. Indirect boundary stabilization of weakly coupled hyperbolic systems. SIAM Journal on Control and Optimization , 41(2):511–541, 2002

    work page 2002

  3. [3]

    Alabau-Boussouira

    F. Alabau-Boussouira. Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Applied Mathematics and Optimization , 51(1):61– 105, 2005. 34

    work page 2005

  4. [4]

    Alabau-Boussouira

    F. Alabau-Boussouira. On some recent advances on stabilization for hyperbolic equations. In Control of partial differential equations , volume 2048 of Lecture Notes in Math., pages 1–100. Springer, Heidelberg, 2012

    work page

  5. [5]

    Balogoun, S

    I. Balogoun, S. Marx, and F. Plestan. Sliding mode control for a class of linear infinite- dimensional systems. arXiv preprint arXiv:2210.13465 , 2022

    work page arXiv 2022

  6. [6]

    V. Barbu. Nonlinear differential equations of monotone types in Banach spaces . Springer Science & Business Media, 2010

    work page 2010

  7. [7]

    Bardos, G

    C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM journal on control and optimization , 30(5):1024–1065, 1992

    work page 1992

  8. [8]

    Bergh and J

    J. Bergh and J. L¨ ofstr¨ om.Interpolation spaces: an introduction, volume 223. Springer Science & Business Media, 2012

    work page 2012

Show all 70 references
  1. [9]

    Borichev and Y

    A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Mathematische Annalen, 347(2):455–478, 2010

    2010

  2. [10]

    H. Brezis. Op´ erateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, 1973

    1973

  3. [11]

    Chill, L

    R. Chill, L. Paunonen, D. Seifert, R. Stahn, and Y. Tomilov. Nonuniform stability of damped contraction semigroups. Analysis & PDE , 16(5):1089–1132, 2023

    2023

  4. [12]

    Chitour, S

    Y. Chitour, S. Marx, and G. Mazanti. One-dimensional wave equation with set-valued bound- ary damping: well-posedness, asymptotic stability, and decay rates. ESAIM: COCV , 27:84, 2021

    2021

  5. [13]

    Chitour, S

    Y. Chitour, S. Marx, and C. Prieur. Lp-asymptotic stability analysis of a 1d wave equation with a nonlinear damping. Journal of Differential Equations , 269(10):8107–8131, 2020

    2020

  6. [14]

    Curtain and H

    R. Curtain and H. Zwart. Stabilization of collocated systems by nonlinear boundary control. Systems & control letters , 96:11–14, 2016

    2016

  7. [15]

    Curtain and G

    R.F. Curtain and G. Weiss. Exponential stabilization of well-posed systems by colocated feedback. SIAM Journal on Control and Optimization , 45(1):273–297, 2006

    2006

  8. [16]

    Curtain and G

    R.F. Curtain and G. Weiss. Strong stabilization of (almost) impedance passive systems by static output feedback. Mathematical Control & Related Fields, 9(4), 2019

    2019

  9. [17]

    Di Nezza, G

    E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional sobolev spaces. Bulletin des sciences math´ ematiques, 136(5):521–573, 2012. 35

    2012

  10. [18]

    Engel and R

    K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations , volume

  11. [19]

    Guiver and H

    C. Guiver and H. Logemann. A circle criterion for strong integral input-to-state stability. Automatica, 111:108641, 2020

    2020

  12. [20]

    Guiver, H

    C. Guiver, H. Logemann, and M. R. Opmeer. Transfer functions of infinite-dimensional systems: positive realness and stabilization. Mathematics of Control, Signals, and Systems , 29:1–61, 2017

    2017

  13. [21]

    Guiver, H

    C. Guiver, H. Logemann, and M. R. Opmeer. A mixed passivity/small-gain theorem for sobolev input-output stability. SIAM Journal on Control and Optimization , 62(6):3042–3075, 2024

    2024

  14. [22]

    Guiver, H

    C. Guiver, H. Logemann, and M. R. Opmeer. Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability. Mathematics of Control, Signals, and Systems , 36(4):729–773, 2024

    2024

  15. [23]

    Hante and M

    F.M. Hante and M. Sigalotti. Converse Lyapunov theorems for switched systems in Banach and Hilbert spaces. SIAM Journal on Control and Optimization , 49(2):752–770, 2011

    2011

  16. [24]

    Hastir and L

    A. Hastir and L. Paunonen. Well-posedness and stability of infinite-dimensional systems under monotone feedback. arXiv preprint arXiv:2503.16092 , 2025

    2025

  17. [25]

    A. Isidori. Nonlinear control systems: an introduction . Springer, 1985

    1985

  18. [26]

    Jacob, A

    B. Jacob, A. Mironchenko, J. R. Partington, and F. Wirth. Noncoercive Lyapunov functions for input-to-state stability of infinite-dimensional systems. SIAM Journal on Control and Optimization, 58(5):2952–2978, 2020

    2020

  19. [27]

    Jacob, F.L

    B. Jacob, F.L. Schwenninger, and L. A Vorberg. Remarks on input-to-state stability of collocated systems with saturated feedback. Mathematics of Control, Signals, and Systems , 32(3):293–307, 2020

    2020

  20. [28]

    H.K. Khalil. Nonlinear Systems Second Edition . Prentice Hall, Inc., 1996

    1996

  21. [29]

    Komornik

    V. Komornik. Exact Controllability and Stabilization. The Multiplier Method . Masson and John Wiley, 1994

    1994

  22. [30]

    Komornik and E

    V. Komornik and E. Zuazua. A direct method for the boundary stabilization of the wave equation. Journal de Math´ ematiques Pures et Appliqu´ ees, 69:33–54, 1990

    1990

  23. [31]

    Kreyszig

    E. Kreyszig. Introductory functional analysis. Wiley Classics Library, 2007. 36

    2007

  24. [32]

    Lax and R

    P.D. Lax and R. S. Phillips. Scattering theory, volume 26. Academic press, 1990

    1990

  25. [33]

    J. L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications: Vol. 1, volume 1. Springer Science & Business Media, 2012

    2012

  26. [34]

    K. Liu. Locally distributed control and damping for the conservative systems. SIAM Journal on Control and Optimization , 35(5):1574–1590, 1997

    1997

  27. [35]

    W. Liu, Y. Chitour, and E. Sontag. On finite-gain stabilizability of linear systems subject to input saturation. SIAM Journal on Control and Optimization , 34(4):1190–1219, 1996

    1996

  28. [36]

    S. Marx. Coercive ISS-Lyapunov functionals for regular infinite-dimensional systems and applications. to appear in Mathematics of Control, Signals, and Systems , 2025

    2025

  29. [37]

    S. Marx, E. Cerpa, C. Prieur, and V. Andrieu. Global stabilization of a Korteweg-de Vries equation with a saturating distributed control. SIAM Journal on Control and Optimization , 55(3):1452–1480, 2017

    2017

  30. [38]

    S. Marx, Y. Chitour, and C. Prieur. Stability analysis of dissipative systems subject to nonlinear damping via Lyapunov techniques. IEEE Transactions on Automatic Control, 2020

    2020

  31. [39]

    Marx and G

    S. Marx and G. Weiss. Well-posed impedance passive nonlinear systems with monotone feedback. submitted, 2025

    2025

  32. [40]

    Mironchenko and C

    A. Mironchenko and C. Prieur. Input-to-State Stability of Infinite-Dimensional Systems: Recent Results and Open Questions. SIAM Rev., 62(3):529–614, 2020

    2020

  33. [41]

    Mironchenko and F

    A. Mironchenko and F. Wirth. Characterizations of input-to-state stability for infinite- dimensional systems. IEEE Trans. Automat. Control, 63(6):1602–1617, 2018

    2018

  34. [42]

    Miyadera

    I. Miyadera. Nonlinear semigroups. Translations of mathematical monographs, 1992

    1992

  35. [43]

    Parada, E

    H. Parada, E. Cr´ epeau, and C. Prieur. Global well-posedness of the KdV equation on a star-shaped network and stabilization by saturated controllers. SIAM Journal on Control and Optimization, 60(4):2268–2296, 2022

    2022

  36. [44]

    Paunonen, D

    L. Paunonen, D. Seifert, and N. Vanspranghe. Admissibility theory in abstract sobolev scales and transfer function growth at high frequencies. arXiv preprint arXiv:2412.14786 , 2024

    2024

  37. [45]

    Pierre and J

    M. Pierre and J. Vancostenoble. Strong decay for one-dimensional wave equations with non- monotone boundary damping. Control and Cybernetics, 29(2):473–484, 2000

    2000

  38. [46]

    Singh, S

    S. Singh, S. Fueyo, and G. Weiss. Projected incrementally scattering passive systems on closed convex sets. Systems & Control Letters, 197:106033, 2025. 37

    2025

  39. [47]

    Singh and G

    S. Singh and G. Weiss. Second order systems on Hilbert spaces with nonlinear damping. SIAM Journal on Control and Optimization , 61(4):2630–2654, 2023

    2023

  40. [48]

    Singh and G

    S. Singh and G. Weiss. The local representation of incrementally scattering passive nonlinear systems. Mathematics of Control, Signals, and Systems , pages 1–32, 2024

    2024

  41. [49]

    Singh, G

    S. Singh, G. Weiss, and M. Tucsnak. Non-linear damping for scattering-passive systems in the maxwell class. IFAC-PapersOnLine, 53(2):7458–7465, 2020

    2020

  42. [50]

    Singh, G

    S. Singh, G. Weiss, and M. Tucsnak. Abstract nonlinear control systems. In 2021 60th IEEE Conference on Decision and Control (CDC) , pages 6181–6187. IEEE, 2021

    2021

  43. [51]

    M. Slemrod. Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control. Mathematics of Control, Signals and Systems , 2(3):847–857, 1989

    1989

  44. [52]

    E. D. Sontag and Y. Wang. On characterizations of the input-to-state stability property. Systems & Control Letters , 24(5):351–359, 1995

    1995

  45. [53]

    O. J. Staffans. Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems. Mathematics of Control, Signals and Systems , 15:291–315, 2002

    2002

  46. [54]

    Staffans

    O.J. Staffans. Well-posed linear systems, volume 103. Cambridge university press, 2005

    2005

  47. [55]

    Sussmann, Eduardo Sontag, and Y

    H.J. Sussmann, Eduardo Sontag, and Y. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Transactions on Automatic Control , 39(12), 1994

    1994

  48. [56]

    Tarbouriech, G

    S. Tarbouriech, G. Garcia, J.M. Gomes da Silva Jr, and I. Queinnec.Stability and Stabilization of Linear Systems with Saturating Actuators . Springer-Verlag, 2011

    2011

  49. [57]

    D. Tataru. On the regularity of boundary traces for the wave equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze , 26(1):185–206, 1998

    1998

  50. [58]

    A. R. Teel. Global stabilization and restricted tracking for multiple integrators with bounded controls. Systems & Control Letters , 18:165–171, 1992

    1992

  51. [59]

    Tucsnak and G

    M. Tucsnak and G. Weiss. Observation and control for operator semigroups . Springer, 2009

    2009

  52. [60]

    Tucsnak and G

    M. Tucsnak and G. Weiss. Well-posed systems—the LTI case and beyond. Automatica, 50(7):1757–1779, 2014

    2014

  53. [61]

    Vanspranghe, F

    N. Vanspranghe, F. Ferrante, and C. Prieur. Velocity stabilization of a wave equation with a nonlinear dynamic boundary condition. IEEE Transactions on Automatic Control , 67(12):6786–6793, 2022. 38

    2022

  54. [62]

    M. Wakaiki. Semi-uniform input-to-state stability of infinite-dimensional systems. Mathemat- ics of Control, Signals, and Systems , 34(4):789–817, 2022

    2022

  55. [63]

    G. Weiss. Admissibility of unbounded control operators. SIAM Journal on Control and Optimization, 27(3):527–545, 1989

    1989

  56. [64]

    G. Weiss. Regular linear systems with feedback. Mathematics of Control, Signals and Systems, 7:23–57, 1994

    1994

  57. [65]

    G. Weiss. Transfer functions of regular linear systems (I): Characterizations of regularity. Transactions of the American Mathematical Society , 342(2):827–854, 1994

    1994

  58. [66]

    Weiss and R

    G. Weiss and R. Rebarber. Optimizability and estimatability for infinite-dimensional linear systems. SIAM Journal on Control and Optimization , 39(4):1204–1232, 2000

    2000

  59. [67]

    Weiss and M

    G. Weiss and M. Tucsnak. How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance. ESAIM: Control, Optimisation and Calculus of Variations, 9:247–273, 2003

    2003

  60. [68]

    Xu and G

    C.-Z. Xu and G. Q. Xu. Saturated boundary feedback stabilization of a linear wave equation. SIAM J. Control Optim. , 57(1):290–309, 2019

    2019

  61. [69]

    Zuazua, H Ammari, Y

    E. Zuazua, H Ammari, Y. Capdeboscq, and H. Kang. A remark on the observability of conser- vative linear systems. Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion , 577:47–59, 2012

    2012

  62. [70]

    Uniform stabilization of the wave equation by nonlinear boundary feedback

    Enrike Zuazua. Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim. , 28(2):466–477, 1990. 39

    1990

Pith tools

Reviewed July 3, 2026 · model on record in the stance chip above.