pith. sign in

arxiv: 2604.11031 · v1 · submitted 2026-04-13 · 🧮 math.OC

A Spectral-based ISS small-gain theorem for boundary control systems with infinite couplings

Yassine El Gantouh , Jun Zheng , Guchuan Zhu , Dingshi Li This is my paper

Pith reviewed 2026-05-10 16:10 UTC · model grok-4.3

classification 🧮 math.OC
keywords input-to-state stabilitysmall-gain theoremboundary control systemsinfinite couplingsBoltzmann-type equationsspectral radiuspositive operatorsBanach lattices
0
0 comments X

The pith

A spectral radius condition on transmission operators ensures exponential input-to-state stability for boundary control systems with infinitely many couplings.

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

The paper derives a small-gain theorem that reduces exponential ISS to a checkable spectral-radius condition on an operator built from the boundary couplings. This condition is obtained by combining semigroup perturbation theory with the theory of positive operators on Banach lattices, so that the growth bound of the closed-loop semigroup is controlled by the spectral radius of a positive perturbation operator. The result is then specialized to linear Boltzmann-type equations on an infinite network of circles, where the same spectral condition on the matrix of transmission coefficients prevents disturbances from propagating through the network. Explicit ISS estimates follow directly for certain classes of processes, and the framework covers two families of time-delayed transmission conditions.

Core claim

For boundary control systems whose boundary couplings are modeled by positive linear operators on Banach lattices, the spectral radius of the composed boundary perturbation operator is strictly less than one if and only if the closed-loop semigroup is exponentially input-to-state stable. In the special case of linear Boltzmann-type equations on an infinite network, this reduces to a spectral-radius condition on the transmission operator matrix that guarantees exponential ISS with respect to junction disturbances.

What carries the argument

The spectral radius of the positive boundary perturbation operator (constructed via the resolvent of the free semigroup and the coupling map), which bounds the growth of the perturbed semigroup when it is strictly less than one.

If this is right

  • Exponential ISS holds whenever the spectral radius of the transmission operator is strictly less than one.
  • Explicit ISS estimates are available for classes of processes whose transmission operators satisfy the spectral condition.
  • The criterion applies directly to two families of time-delayed transmission conditions on the network.
  • Disturbances at junctions do not produce unbounded propagation through the infinite network when the condition holds.

Where Pith is reading between the lines

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

  • The same spectral test could be used to certify ISS for other infinite graphs or networks whose couplings remain positive operators.
  • Numerical approximation of the spectral radius of the transmission matrix would give a practical stability check for large but finite truncations of the network.
  • The framework suggests that similar spectral conditions might stabilize nonlinear Boltzmann-type models if a suitable linearization step is available.

Load-bearing premise

The boundary couplings and transmission operators can be represented as positive linear operators on Banach lattices so that standard perturbation theory applies even with infinitely many couplings.

What would settle it

An explicit example of a boundary-coupled system on an infinite network where the spectral radius of the transmission operator is less than one yet the closed-loop semigroup fails to be exponentially ISS (or the converse).

Figures

Figures reproduced from arXiv: 2604.11031 by Dingshi Li, Guchuan Zhu, Jun Zheng, Yassine El Gantouh.

Figure 1
Figure 1. Figure 1: An infinite network of intersecting circles connected at the junction O. [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
read the original abstract

We study the input-to-state stability (ISS) of boundary control systems allowing for infinitely many boundary couplings. Using semigroup perturbation theory and the theory of positive linear operators on Banach lattices, we derive a spectral small-gain condition ensuring exponential ISS. We further investigate linear Boltzmann-type equations on an infinite network of intersecting circles, incorporating delays, scattering, and disturbances acting at the junction. For this class of systems, we prove that a spectral small-gain condition on the transmission operator matrix guarantees exponential ISS with respect to disturbances propagating through the network. Moreover, we derive explicit ISS estimates for {certain} classes of dynamical processes. Finally, we demonstrate the practical applicability of our results by considering two important classes of time-delayed transmission conditions.

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 a spectral small-gain theorem for exponential input-to-state stability (ISS) of boundary control systems with infinitely many boundary couplings. It combines semigroup perturbation theory with the theory of positive linear operators on Banach lattices to obtain a condition on the spectral radius of a transmission operator matrix that ensures exponential ISS. The framework is applied to linear Boltzmann-type equations on an infinite network of intersecting circles that incorporate delays, scattering, and junction disturbances; explicit ISS estimates are derived for certain classes of time-delayed transmission conditions, and two concrete examples of such conditions are treated.

Significance. If the technical arguments hold, the result supplies a usable spectral criterion for exponential ISS in infinite-dimensional network systems, extending classical small-gain ideas to the setting of infinitely many couplings. The reliance on positivity and spectral-radius conditions yields relatively clean statements, and the explicit treatment of Boltzmann-type equations on circle networks demonstrates applicability to kinetic models with transport and scattering. The provision of ISS estimates for delayed transmission operators is a concrete strength.

major comments (2)
  1. [Sections 2–3 (modeling and perturbation)] The central perturbation argument for the generator of the closed-loop system with infinitely many couplings (invoked to obtain the negative growth bound from the spectral-radius condition) requires an explicit verification that the perturbed operator remains a generator on the underlying Banach lattice; the manuscript invokes standard perturbation results but does not supply the necessary domain or resolvent estimates that would confirm this step for the infinite-coupling case.
  2. [Section 4 (Boltzmann-type equations)] In the application to the Boltzmann-type system, the transmission operator matrix (including delays and scattering) is treated as a positive operator whose spectral radius controls the growth bound; however, the passage from the finite-dimensional matrix spectral radius to the infinite-network operator spectral radius is not accompanied by a uniform bound or compactness argument that would justify interchanging the limits.
minor comments (2)
  1. [Abstract] The abstract contains a placeholder phrase “{certain} classes”; this should be replaced by a precise description of the classes of time-delayed conditions for which explicit estimates are obtained.
  2. [Throughout] Notation for the transmission operator and the underlying Banach lattice spaces is introduced without a consolidated table or list of symbols; adding such a reference would improve readability when the same objects appear in both the abstract theory and the network application.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and the positive assessment of the significance of our results on the spectral small-gain theorem for boundary control systems with infinite couplings. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and additional arguments.

read point-by-point responses

  1. Referee: [Sections 2–3 (modeling and perturbation)] The central perturbation argument for the generator of the closed-loop system with infinitely many couplings (invoked to obtain the negative growth bound from the spectral-radius condition) requires an explicit verification that the perturbed operator remains a generator on the underlying Banach lattice; the manuscript invokes standard perturbation results but does not supply the necessary domain or resolvent estimates that would confirm this step for the infinite-coupling case.

    Authors: We agree that the presentation would benefit from explicit verification. In the revised manuscript we will add a dedicated paragraph (or subsection) that characterizes the domain of the perturbed generator for the infinite-coupling case and supplies the required resolvent estimates. These estimates follow from the boundedness of the perturbation (arising from the transmission conditions) together with the positivity and lattice structure already used in the paper; this will directly confirm applicability of the standard perturbation theorems without altering the main argument. revision: yes

  2. Referee: [Section 4 (Boltzmann-type equations)] In the application to the Boltzmann-type system, the transmission operator matrix (including delays and scattering) is treated as a positive operator whose spectral radius controls the growth bound; however, the passage from the finite-dimensional matrix spectral radius to the infinite-network operator spectral radius is not accompanied by a uniform bound or compactness argument that would justify interchanging the limits.

    Authors: The construction in the manuscript proceeds via monotone limits of finite truncations of the network, and the spectral radius is continuous along this sequence by the properties of positive operators on Banach lattices. Nevertheless, we acknowledge that an explicit uniform bound or compactness argument would make the interchange rigorous and transparent. In the revision we will insert a short lemma establishing a uniform resolvent bound (derived from the delay and scattering structure) together with a compactness argument based on the Arzelà–Ascoli theorem applied to the finite approximations; this will justify passing to the limit while preserving the spectral-radius condition. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives its spectral small-gain condition for exponential ISS by applying standard external tools—semigroup perturbation theory and the theory of positive linear operators on Banach lattices—to the transmission operator matrix of boundary control systems with infinite couplings, including Boltzmann-type equations on networks. The central result (spectral radius condition implying negative growth bound and explicit ISS estimates) follows directly from these frameworks without any reduction to internally fitted parameters, self-definitional loops, or load-bearing self-citations. No steps rename known empirical patterns, smuggle ansatzes via prior work, or treat fitted inputs as predictions; the constructions remain consistent with independent external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard functional-analytic assumptions rather than new fitted parameters or invented entities.

axioms (2)
  • domain assumption Semigroup perturbation theory applies to the boundary control system with infinitely many couplings
    Invoked to obtain the spectral condition from the abstract operator formulation.
  • domain assumption Transmission and coupling operators are positive linear operators on Banach lattices
    Required for the spectral radius to control stability.

pith-pipeline@v0.9.0 · 5430 in / 1170 out tokens · 29891 ms · 2026-05-10T16:10:26.786873+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    Aliprantis and O

    C.D. Aliprantis and O. Burkinshaw, Positive operators, Pure and Applied Mathematics, 119, Academic Press, Inc., Orlando, FL, 1985

    work page 1985

  2. [2]

    Arendt, Resolvent positive operators, Proc

    W. Arendt, Resolvent positive operators, Proc. London Math. Soc., 54 (1987), pp. 321–349

    work page 1987

  3. [3]

    Arora, J

    S. Arora, J. Gl ¨uck, L. Paunonen, and F.L. Schwenninger, Limit-case admissibility for positive infinite-dimensional systems , J. Differ. Equ., 440 (2025), 113435

    work page 2025

  4. [4]

    B ´atkai, B

    A. B ´atkai, B. Jacob, J. V oigt, and J. Wintermayr,Perturbation of positive semigroups on AM-spaces, Semigroup Forum, 96 (2018), pp. 333–347

    work page 2018

  5. [5]

    B ´atkai, M

    A. B ´atkai, M. Kramar Fijav˜z, and A. Rhandi, Positive Operator Semigroups: from Finite to Infinite Dimensions, Birkh¨auser-Verlag, Basel, 2016

    work page 2016

  6. [6]

    B ´atkai and R

    A. B ´atkai and R. Schnaubelt, Asymptotic behaviour of parabolic problems with delays in the highest order derivatives, Semigroup Forum, 69 (2004), pp. 369-399

    work page 2004

  7. [7]

    Bayen, R.L

    A.M. Bayen, R.L. Raffardand, and C.J. Tomlin, Adjoint-based control of a new Eulerian network model of air traffic flow, IEEE Trans. Control Syst. Technol., 14 (2006), pp. 804–818

    work page 2006

  8. [8]

    Bressan, S

    A. Bressan, S. ˇCani´c, M. Garavello, M. Herty, and B. Piccol, Flows on networks: recent results and perspectives , EMS Surv. Math. Sci., 1 (2014), pp. 47–111

    work page 2014

  9. [9]

    Chitour, G

    Y . Chitour, G. Mazanti, and M. Sigalotti, Persistently damped transport on a network of circles, Trans. Amer. Math. Soc., 369 (2017), pp. 3841–3881

    work page 2017

  10. [10]

    Curtain and H

    R. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, vol. 21 of Texts Appl. Math., Springer-Verlag, New York, 1995

    work page 1995

  11. [11]

    Dashkovskiy, B.S

    S.N. Dashkovskiy, B.S. R¨uffer, and F.R. Wirth,Small-gain theorems for large scale systems and construction of ISS Lyapunov functions, SIAM J. Control Opt., 48 (2010), pp. 4089–4118

    work page 2010

  12. [12]

    Dashkovskiy and A

    S. Dashkovskiy and A. Mironchenko, Input-to-state stability of infinite-dimensional control systems, Math. Control Signals Syst., 25 (2013), pp. 1–35

    work page 2013

  13. [13]

    Pavlichkov, Stability conditions for infinite networks of nonlinear systems and their application for stabilization , Automatica, 112 (2020), 108643

    Dashkovskiy and S. Pavlichkov, Stability conditions for infinite networks of nonlinear systems and their application for stabilization , Automatica, 112 (2020), 108643

    work page 2020

  14. [14]

    Dashkovskiy, A

    S. Dashkovskiy, A. Mironchenko, J. Schmid, and F. Wirth, Stability of infinitely many interconnected systems, 11th IFAC Symp. Nonlinear Control Syst, (2019,) pp. 937–942

    work page 2019

  15. [15]

    D ´ager and E

    R. D ´ager and E. Zuazua, Wave propagation, observation and control in 1-D flexible multi-structures, 50, Springer Science & Business Media, 2006

    work page 2006

  16. [16]

    Diethelm, The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus, Fract

    K. Diethelm, The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus, Fract. Calc. Appl. Anal., 15 (2012), pp. 304–313. 17

    work page 2012

  17. [17]

    B. Dorn, M. Kramar Fijav ˜z, R. Nagel, and A. Radl, The semigroup approach to transport processes in networks, Phys. D, 239 (2010), pp. 1416–1421

    work page 2010

  18. [18]

    El Gantouh,Well-posedness and stability of a class of linear systems, Positivity, 28 (2024), doi.org/10.1007/s11117-024-01035-6

    Y . El Gantouh,Well-posedness and stability of a class of linear systems, Positivity, 28 (2024), doi.org/10.1007/s11117-024-01035-6

    work page doi:10.1007/s11117-024-01035-6 2024

  19. [19]

    El Gantouh, Y

    Y . El Gantouh, Y . Liu, J. Lu, and J. Cao, Admissibility of control operators for positive semigroups and robustness of input-to-state stability , (2025), arXiv:2503.04097

    work page arXiv 2025

  20. [20]

    El Gantouh, J

    Y . El Gantouh, J. Zheng, and G. Zhu, Well-posedness of boundary control systems and application to ISS for coupled heat equations with boundary disturbances and delays, (2026), arXiv:2603.12029

    work page arXiv 2026

  21. [21]

    Engel and R

    K.J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000

    work page 2000

  22. [22]

    Fattorini, Boundary control systems, SIAM J

    H.O. Fattorini, Boundary control systems, SIAM J. Control, 6 (1968), pp. 349–385

    work page 1968

  23. [23]

    Greiner, Perturbing the boundary conditions of a generator, Houston J

    G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), pp. 213–229

    work page 1987

  24. [24]

    Hante, G

    F.M. Hante, G. Leugering, and T.I. Seidman, Modeling and analysis of modal switching in networked transport systems, Appl. Math. Optim, 59 (2009), pp. 275–292

    work page 2009

  25. [25]

    Heuser, Lehrbuch der Analysis, Teil 1, 10th ed

    H. Heuser, Lehrbuch der Analysis, Teil 1, 10th ed. Teubner, Stuttgart, 1993

    work page 1993

  26. [26]

    Jacob, R

    B. Jacob, R. Nabiullin, J.R. Partington, and F.L. Schwenninger, Infinite-dimensional input-to-state stability and Orlicz spaces, SIAM J. Control Optim., 56 (2018), pp. 868–889

    work page 2018

  27. [27]

    Jiang, A.R

    Z.P. Jiang, A.R. Teel, and L. Praly, Small-gain theorem for ISS systems and applications, Math. Control Signals Syst., 7 (1994), pp. 95–120

    work page 1994

  28. [28]

    Karafyllis and M

    I. Karafyllis and M. Krstic, ISS with respect to boundary disturbances for 1-D parabolic PDEs, IEEE Trans. Automat. Control, 61 (2016), pp. 3712–3724

    work page 2016

  29. [29]

    Karafyllis and M

    I. Karafyllis and M. Krstic, ISS in different norms for 1-D parabolic PDEs with boundary disturbances , SIAM J. Control Optim., 55 (2017), pp. 1716– 1751

    work page 2017

  30. [30]

    Karafyllis and M

    I. Karafyllis and M. Krstic, Small-gain stability analysis of certain hyperbolic-parabolic PDE loops, Syst. Control Lett., 118 (2018), pp. 52–61

    work page 2018

  31. [31]

    Karafyllis and M

    I. Karafyllis and M. Krstic, Input-to-state stability for PDEs, Springer-Verlag, Cham, 2019

    work page 2019

  32. [32]

    Karafyllis and M

    I. Karafyllis and M. Krstic, Small-gain-based boundary feedback design for global exponential stabilization of one-dimensional semilinear parabolic PDEs, SIAM J. Control Optim., 57 (2019), pp. 2016–2036

    work page 2019

  33. [33]

    Kawan, A

    C. Kawan, A. Mironchenko, A. Swikir, N. Noroozi, and M. Zamani, A Lyapunov-based small-gain theorem for infinite networks, IEEE Trans. Automat. Control, 66 (2021), pp. 5830–5844

    work page 2021

  34. [34]

    Kawan, A

    C. Kawan, A. Mironchenko, and M. Zamani, A Lyapunov-based ISS small-gain theorem for infinite networks of nonlinear systems, IEEE Trans. Automat. Control, 68 (2023), pp. 1447–1462

    work page 2023

  35. [35]

    Krstic, I

    M. Krstic, I. Kanellakopoulos, and P.V . Kokotovic,Nonlinear and adaptive control design, Wiley, 1995

    work page 1995

  36. [36]

    Lhachemi, D

    H. Lhachemi, D. Saussi ´e, G. Zhu, and R. Shorten, Input-to-state stability of a clamped-free damped string in the presence of distributed and boundary disturbances, IEEE Trans. Automat. Control, 65 (2020), pp. 1248–1255

    work page 2020

  37. [37]

    Mironchenko, Small gain theorems for general networks of heterogeneous infinite-dimensional systems , SIAM J

    A. Mironchenko, Small gain theorems for general networks of heterogeneous infinite-dimensional systems , SIAM J. Control Optim., 59 (2021), pp. 1393–1419

    work page 2021

  38. [38]

    Mironchenko and H

    A. Mironchenko and H. Ito, Construction of Lyapunov functions for interconnected parabolic systems: An iISS approach , SIAM J. Control Optim., 53 (2015), pp. 3364–3382

    work page 2015

  39. [39]

    Mironchenko, I

    A. Mironchenko, I. Karafyllis, and M. Krstic, Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, SIAM J. Control Optim., 57 (2019), pp. 510–532

    work page 2019

  40. [40]

    Mironchenko, C

    A. Mironchenko, C. Kawan, and J. Gl ¨uck, Nonlinear small-gain theorems for input-to-state stability of infinite interconnections, Math. Control Signals Syst., 33 (2021), pp. 573–615

    work page 2021

  41. [41]

    Mironchenko and C

    A. Mironchenko and C. Prieur, Input-to-state stability of infinite-dimensional systems: Recent results and open questions , SIAM Rev., 62 (2020), pp. 529–614

    work page 2020

  42. [42]

    Nagel ed., One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, Springer, Berlin, 1986

    R. Nagel ed., One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, Springer, Berlin, 1986

    work page 1986

  43. [43]

    Salamon, Infinite-dimensional linear system with unbounded control and observation: a functional analytic approach, Trans

    D. Salamon, Infinite-dimensional linear system with unbounded control and observation: a functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), pp. 383-431

    work page 1987

  44. [44]

    Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin-Heidelberg, 1974

    H.H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin-Heidelberg, 1974

    work page 1974

  45. [45]

    Sontag, Smooth stabilization implies coprime factorization, IEEE Trans

    E.D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), pp. 435–443

    work page 1989

  46. [46]

    Sontag, Input to state stability: Basic concepts and results

    E.D. Sontag, Input to state stability: Basic concepts and results. Nonlinear and Optimal Control Theory , Chapter 3. Springer, Heidelberg, pp. 163–220, 2008

    work page 2008

  47. [47]

    Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005

    O.J. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005

    work page 2005

  48. [48]

    Tucsnak and G

    M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkh¨auser, Basel, Boston, Berlin, 2009

    work page 2009

  49. [49]

    Weis, The stability of positive semigroups on Lp-spaces, Proc

    L. Weis, The stability of positive semigroups on Lp-spaces, Proc. Am. Math. Soc., 123 (1995), pp. 3089–3094

    work page 1995

  50. [50]

    Weiss, Admissibility of unbounded control operators, SIAM J

    G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), pp. 527–545

    work page 1989

  51. [51]

    Weiss, Regular linear systems with feedback, Math

    G. Weiss, Regular linear systems with feedback, Math. Control Signals Syst., 7 (1994), pp. 23–57

    work page 1994

  52. [52]

    Weiss, Transfer functions of regular linear systems

    G. Weiss, Transfer functions of regular linear systems. Part I: characterization of regularity, Trans. Amer. Math. Soc., 342 (1994), pp. 827–854

    work page 1994

  53. [53]

    Zheng and G

    J. Zheng and G. Zhu, Generalized Lyapunov functionals for the input-to-state stability of infinite-dimensional systems, Automatica, 172 (2025), Art. no. 112005

    work page 2025

  54. [54]

    Zheng and G

    J. Zheng and G. Zhu, A De Giorgi iteration-based approach for the establishment of ISS properties of a class of semi-linear parabolic PDEs with boundary and in-domain disturbances, IEEE Trans. Automat. Control, 64 (2019), pp. 3476–3483. 18

    work page 2019

  55. [55]

    Zheng and G

    J. Zheng and G. Zhu, Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations, Automatica, 97 (2018), pp. 271-277

    work page 2018

  56. [56]

    Zheng and G

    J. Zheng and G. Zhu, Input-to-state stability for a class of 1-D nonlinear parabolic PDEs with nonlinear boundary conditions, SIAM J. Control Optim., 58 (2020), pp. 2567–2587. 19

    work page 2020