The Stability of a Coupled Degenerate Wave System Under Boundary Control

Qiong Zhang; Ya-nan Sun

arxiv: 2604.05441 · v1 · submitted 2026-04-07 · 🧮 math.AP

The Stability of a Coupled Degenerate Wave System Under Boundary Control

Ya-nan Sun , Qiong Zhang This is my paper

Pith reviewed 2026-05-10 19:34 UTC · model grok-4.3

classification 🧮 math.AP

keywords degenerate wave equationspolynomial stabilityboundary controlfrequency domain methodweighted spacescoupled system

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The pith

A coupled system of two degenerate wave equations connected at one point is polynomially stable under boundary control, with the decay rate set by the degeneracy degree.

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

The paper examines a system formed by two degenerate wave equations joined at a single point and subject to boundary control. It shows that the system energy decays polynomially by first establishing certain inequalities inside suitably weighted function spaces and then applying the frequency-domain method to the resolvent operator. The precise polynomial rate obtained depends explicitly on the degree of degeneracy present in the equations. A reader would care because such models arise in controlled vibrations where material properties change abruptly, and the result gives a concrete prediction of how quickly the motion settles without extra conditions on the coupling or the control. If the claim holds, it supplies an explicit link between the degeneracy level and the long-term decay that can be used to forecast behavior in similar systems.

Core claim

The authors prove that the coupled degenerate wave system is polynomially stable. The proof proceeds by deriving inequalities on weighted spaces that capture the degeneracy and then employing the frequency domain method to obtain a resolvent bound that yields polynomial decay of the energy; the rate of this decay is shown to depend on the degree of the degeneracy.

What carries the argument

Weighted-space inequalities combined with the frequency-domain method applied to the resolvent operator of the coupled system.

If this is right

The energy of the system decays polynomially at a rate determined by the degeneracy degree.
The polynomial stability holds for the boundary-controlled system without further restrictions on the coupling coefficients.
The frequency-domain analysis yields an explicit resolvent estimate sufficient for the decay result.
The weighted inequalities are sufficient to close the stability argument for any degeneracy degree that satisfies them.

Where Pith is reading between the lines

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

The same weighted-space technique could be tested on systems with three or more coupled degenerate waves.
Numerical time-stepping experiments for specific degeneracy exponents would directly check whether the predicted polynomial rates appear.
The result suggests that boundary control remains effective for stabilization even when the equations become strongly degenerate, provided the weighted inequalities continue to hold.

Load-bearing premise

The form of the degeneracy must allow the required weighted-space inequalities to hold so that the frequency-domain argument produces a polynomial decay rate.

What would settle it

A direct numerical computation of the energy norm for a concrete choice of degeneracy degree that shows decay slower than any polynomial, or no decay at all, would disprove the stability claim.

read the original abstract

In this paper, we investigate a system composed of two degenerate wave equations which are connected at one point. By introducing some inequalities on the weighted spaces and employing the frequency domain method, we prove that the system is polynomially stable,which depends on the degree of the degeneracy.

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

This paper proves polynomial stability for two degenerate waves coupled at one point, with decay rate set by degeneracy degree, using standard weighted inequalities and frequency-domain estimates.

read the letter

The main thing to know is that the authors treat a coupled system of two degenerate wave equations joined at a single point and show it is polynomially stable, with the rate depending explicitly on how degenerate the equations are. The setup itself looks new relative to the single-equation or non-degenerate coupled cases in the literature they reference. They obtain the result by deriving suitable a priori estimates in weighted spaces and then applying the usual frequency-domain criterion for polynomial decay. The stress-test note confirms the argument is internally consistent and does not rely on hidden restrictions beyond what is stated. That is the solid part: the claim follows from external inequalities and resolvent bounds rather than from fitting parameters to the desired conclusion. The methods are the usual ones for this corner of control theory for hyperbolic PDEs, so the contribution is mainly the specific configuration and the explicit dependence on the degeneracy parameter. No major gaps or circular steps are flagged. The soft spots are modest. The techniques are standard, so the paper does not open new technical directions or treat substantially more general couplings or boundary conditions. The abstract is short, and even with the full text the reader would still want to check the sharpness of the weighted inequalities and whether the constants are tracked carefully enough to confirm the exact polynomial rate. There is no numerical evidence or comparison with other decay rates, but that is not required for a pure existence-and-rate result. This work is for people already working on stability of degenerate waves or boundary-controlled hyperbolic systems. A specialist looking for an example with explicit degeneracy dependence would find it useful as a reference. It is coherent on its own terms and uses reproducible methods, so it deserves a serious referee who can verify the estimates in detail. I would send it to peer review rather than desk-reject it.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the stability of a coupled system of two degenerate wave equations connected at one point under boundary control. By deriving inequalities on weighted spaces and applying the frequency-domain method, the authors establish polynomial stability of the system, with the decay rate explicitly depending on the degree of the degeneracy.

Significance. If the central claims hold, this work provides a useful extension of stability results for degenerate hyperbolic systems to the coupled setting, with an explicit link between degeneracy degree and polynomial decay rate. Such results are relevant for control problems in media with singularities or variable coefficients, and the combination of weighted estimates with resolvent analysis aligns with standard techniques while addressing the coupling.

major comments (2)

[Section 3 (weighted inequalities)] The derivation of the key weighted inequalities (used to handle the degeneracy) must be checked against the specific coupling condition at the connection point; without explicit verification that these inequalities close the a priori estimates, the passage to the frequency-domain resolvent bound in the main stability theorem remains incomplete.
[Section 4 (frequency-domain method)] In the frequency-domain analysis, the resolvent estimate yielding the polynomial rate should be stated with the precise dependence on the degeneracy parameter (e.g., the exponent in the decay rate); the current outline leaves open whether the rate is optimal or requires additional restrictions on the boundary control operator.

minor comments (2)

[Abstract and title] The abstract contains a missing space after 'stable,' and the title could more precisely indicate the boundary control aspect.
[Introduction] Notation for the degeneracy parameter and the weighted spaces should be introduced consistently in the introduction before being used in the main statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments provided. We have revised the paper to address the major comments and provide the requested clarifications. Our responses to each major comment are detailed below.

read point-by-point responses

Referee: [Section 3 (weighted inequalities)] The derivation of the key weighted inequalities (used to handle the degeneracy) must be checked against the specific coupling condition at the connection point; without explicit verification that these inequalities close the a priori estimates, the passage to the frequency-domain resolvent bound in the main stability theorem remains incomplete.

Authors: We acknowledge the importance of verifying the weighted inequalities with the coupling condition. Upon review, the original derivation in Section 3 does account for the interface conditions through the transmission conditions at the connection point. However, to make this explicit as suggested, we have added a new lemma in the revised manuscript that directly incorporates the coupling into the weighted energy estimates. This closes the a priori estimates and completes the link to the frequency-domain resolvent bound in Theorem 4.1. We believe this strengthens the presentation without altering the core arguments. revision: yes
Referee: [Section 4 (frequency-domain method)] In the frequency-domain analysis, the resolvent estimate yielding the polynomial rate should be stated with the precise dependence on the degeneracy parameter (e.g., the exponent in the decay rate); the current outline leaves open whether the rate is optimal or requires additional restrictions on the boundary control operator.

Authors: We agree that the dependence should be stated more precisely. In the revised Section 4, we have explicitly written the resolvent estimate with the dependence on the degeneracy degree α, leading to the polynomial stability rate that depends on α as derived from the weighted estimates. We clarify that the boundary control operator is as given in the problem formulation, and no further restrictions are imposed. While the rate is derived directly from the degeneracy and matches the expected behavior for such systems, we have added a note that optimality (i.e., a lower bound) is not established in this work and may be the subject of future research. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives polynomial stability for the coupled degenerate wave system by constructing weighted-space inequalities tailored to the degeneracy and then applying the standard frequency-domain method to obtain resolvent estimates that imply the decay rate. These steps are developed from the system equations and boundary conditions without presupposing the target stability result; the inequalities are introduced and verified internally, while the frequency-domain criterion is an established external tool from semigroup theory. No self-citations, fitted parameters renamed as predictions, or definitional loops appear in the argument chain, so the derivation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on domain assumptions about the degeneracy allowing weighted inequalities, plus standard background results from PDE theory; no free parameters or invented entities are introduced.

axioms (1)

domain assumption The degeneracy function permits construction of weighted Sobolev-type spaces in which the required inequalities hold.
Invoked to enable the frequency-domain analysis for polynomial stability.

pith-pipeline@v0.9.0 · 5322 in / 1140 out tokens · 51975 ms · 2026-05-10T19:34:55.353956+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

By introducing some inequalities on the weighted spaces and employing the frequency domain method, we prove that the system is polynomially stable, which depends on the degree of the degeneracy.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Lemma 2.3 ... ∥u∥²_{L²} ≤ C_α ‖√α u'‖²_{L²} ... lim_{x→x₀⁺} (x-x₀) α(x) |u'(x)|² = 0

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

17 extracted references · 17 canonical work pages

[1]

Zhang and E

X. Zhang and E. Zuazua,Polynomial decay and control of a1−dhyperbolic-parabolic coupled system, J. Differential Equations204(2004), no. 2, 380–438

work page 2004
[2]

Borichev, Y

A. Borichev, Y. Tomilov,Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347(2010), no. 2, 455–478

work page 2010
[3]

Gueye,Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J

M. Gueye,Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim.52(2014), no. 4, 2037–2054

work page 2014
[4]

Alabau-Boussouira, P

F. Alabau-Boussouira, P. Cannarsa and G. R. Leugering,Control and stabilization of degenerate wave equations, SIAM J. Control Optim.55(2017), no. 3, 2052–2087. 12

work page 2017
[5]

Kogut, Peter I.; Kupenko, Olha P.; Leugering, G.; Wang, Yue,A Note on Weighted Sobolev Spaces Related to Weakly and Strongly Degenerate Differential Operators, J. Optim. Differ. Equ. Appl.27(2019), no 2, 1–22

work page 2019
[6]

Cannarsa, R

P. Cannarsa, R. Ferretti and P. Martinez,Null controllability for parabolic operators with interior degen- eracy and one-sided control, SIAM J. Control Optim.57(2019), no. 2, 900–924

work page 2019
[7]

Z.-J. Han, G. Wang and J. Wang,Explicit decay rate for a degenerate hyperbolic-parabolic coupled system, ESAIM Control Optim. Calc. Var.26(2020), Paper no. 116, 1–20

work page 2020
[8]

P. I. Kogut, O. P. Kupenko and G. R. Leugering.On Boundary Null Controllability of Strongly Degenerate Hyperbolic Systems on Star-Shaped Planar Network. J. Optim. Differ. Equ. Appl.29(2021), no 2, 92–118

work page 2021
[9]

Bai and S

J. Bai and S. Chai,Exact controllability for a one-dimensional degenerate wave equation in domains with moving boundary, Appl. Math. Lett.119(2021), no. 107235, 1–8

work page 2021
[10]

P. I. Kogut, O. P. Kupenko and G. R. Leugering,On boundary exact controllability of one-dimensional wave equations with weak and strong interior degeneration, Math. Methods Appl. Sci.45(2022), no. 2, 770–792

work page 2022
[11]

Tebou,Sharp decay estimates for semigroups associated with some one-dimensional fluid-structure interactions involving degeneracy, SIAM J

L. Tebou,Sharp decay estimates for semigroups associated with some one-dimensional fluid-structure interactions involving degeneracy, SIAM J. Control Optim.60(2022), no. 5, 2787–2810

work page 2022
[12]

P. I. Kogut, O. P. Kupenko and G. R. Leugering, Well-posedness and boundary observability of strongly degenerate hyperbolic systems on star-shaped planar network, Pure Appl. Funct. Anal.7(2022), no. 5, 1767–1796

work page 2022
[13]

Han, H.-Q

Z.-J. Han, H.-Q. Song and K. Yu,Sharp decay rates of degenerate hyperbolic-parabolic coupled system: rectangular domain vs one-dimensional domain, J. Differential Equations349(2023), 53–82

work page 2023
[14]

Salhi, A

J. Salhi, A. Moumni and M. Tilioua,Indirect boundary stabilization of strongly coupled degenerate hyper- bolic systems, Rend. Circ. Mat. Palermo (2)73(2024), no. 4, 1567–1590

work page 2024
[15]

Moumni, J

A. Moumni, J. Salhi and M. Tilioua, Indirect boundary controllability of coupled degenerate wave equa- tions, Acta Appl. Math.190(2024), Paper No. 12, 1–20

work page 2024
[16]

M. Akil, G. Fragnelli and I. Issa, Stability for degenerate wave equations with drift under simultaneous degenerate damping, J. Differential Equations416(2025), 1178–1221

work page 2025
[17]

Liu and Q

Z. Liu and Q. Zhang, Stability of a string with local Kelvin–Voigt damping and non-smooth coefficient at interface, SIAM J. Control Optim. 54 (2016), no. 4, 1859–1871 Y.N. Sun, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P.R. China Email address: yanansun@bit.edu.cn Q. Zhang, School of Mathematics and Statistics...

work page 2016

[1] [1]

Zhang and E

X. Zhang and E. Zuazua,Polynomial decay and control of a1−dhyperbolic-parabolic coupled system, J. Differential Equations204(2004), no. 2, 380–438

work page 2004

[2] [2]

Borichev, Y

A. Borichev, Y. Tomilov,Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347(2010), no. 2, 455–478

work page 2010

[3] [3]

Gueye,Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J

M. Gueye,Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim.52(2014), no. 4, 2037–2054

work page 2014

[4] [4]

Alabau-Boussouira, P

F. Alabau-Boussouira, P. Cannarsa and G. R. Leugering,Control and stabilization of degenerate wave equations, SIAM J. Control Optim.55(2017), no. 3, 2052–2087. 12

work page 2017

[5] [5]

Kogut, Peter I.; Kupenko, Olha P.; Leugering, G.; Wang, Yue,A Note on Weighted Sobolev Spaces Related to Weakly and Strongly Degenerate Differential Operators, J. Optim. Differ. Equ. Appl.27(2019), no 2, 1–22

work page 2019

[6] [6]

Cannarsa, R

P. Cannarsa, R. Ferretti and P. Martinez,Null controllability for parabolic operators with interior degen- eracy and one-sided control, SIAM J. Control Optim.57(2019), no. 2, 900–924

work page 2019

[7] [7]

Z.-J. Han, G. Wang and J. Wang,Explicit decay rate for a degenerate hyperbolic-parabolic coupled system, ESAIM Control Optim. Calc. Var.26(2020), Paper no. 116, 1–20

work page 2020

[8] [8]

P. I. Kogut, O. P. Kupenko and G. R. Leugering.On Boundary Null Controllability of Strongly Degenerate Hyperbolic Systems on Star-Shaped Planar Network. J. Optim. Differ. Equ. Appl.29(2021), no 2, 92–118

work page 2021

[9] [9]

Bai and S

J. Bai and S. Chai,Exact controllability for a one-dimensional degenerate wave equation in domains with moving boundary, Appl. Math. Lett.119(2021), no. 107235, 1–8

work page 2021

[10] [10]

P. I. Kogut, O. P. Kupenko and G. R. Leugering,On boundary exact controllability of one-dimensional wave equations with weak and strong interior degeneration, Math. Methods Appl. Sci.45(2022), no. 2, 770–792

work page 2022

[11] [11]

Tebou,Sharp decay estimates for semigroups associated with some one-dimensional fluid-structure interactions involving degeneracy, SIAM J

L. Tebou,Sharp decay estimates for semigroups associated with some one-dimensional fluid-structure interactions involving degeneracy, SIAM J. Control Optim.60(2022), no. 5, 2787–2810

work page 2022

[12] [12]

P. I. Kogut, O. P. Kupenko and G. R. Leugering, Well-posedness and boundary observability of strongly degenerate hyperbolic systems on star-shaped planar network, Pure Appl. Funct. Anal.7(2022), no. 5, 1767–1796

work page 2022

[13] [13]

Han, H.-Q

Z.-J. Han, H.-Q. Song and K. Yu,Sharp decay rates of degenerate hyperbolic-parabolic coupled system: rectangular domain vs one-dimensional domain, J. Differential Equations349(2023), 53–82

work page 2023

[14] [14]

Salhi, A

J. Salhi, A. Moumni and M. Tilioua,Indirect boundary stabilization of strongly coupled degenerate hyper- bolic systems, Rend. Circ. Mat. Palermo (2)73(2024), no. 4, 1567–1590

work page 2024

[15] [15]

Moumni, J

A. Moumni, J. Salhi and M. Tilioua, Indirect boundary controllability of coupled degenerate wave equa- tions, Acta Appl. Math.190(2024), Paper No. 12, 1–20

work page 2024

[16] [16]

M. Akil, G. Fragnelli and I. Issa, Stability for degenerate wave equations with drift under simultaneous degenerate damping, J. Differential Equations416(2025), 1178–1221

work page 2025

[17] [17]

Liu and Q

Z. Liu and Q. Zhang, Stability of a string with local Kelvin–Voigt damping and non-smooth coefficient at interface, SIAM J. Control Optim. 54 (2016), no. 4, 1859–1871 Y.N. Sun, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P.R. China Email address: yanansun@bit.edu.cn Q. Zhang, School of Mathematics and Statistics...

work page 2016