pith. sign in

arxiv: 2605.05142 · v1 · submitted 2026-05-06 · 🧮 math.OC

On Controllability of a Class of N -dimensional Hyperbolic Equations with Internal Single-point Degeneracy

Donghui Yang , Weijia Wu This is my paper

Pith reviewed 2026-05-08 17:02 UTC · model grok-4.3

classification 🧮 math.OC
keywords controllabilityhyperbolic equationsCarleman estimatesdegeneracyobservabilityHardy inequalitywell-posednessinternal control
0
0 comments X

The pith

Hyperbolic equations in any dimension with one internal degenerate point are exactly controllable from regions containing the point.

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

The paper proves that a class of N-dimensional hyperbolic equations with a single interior point of degeneracy are well-posed and exactly controllable. Well-posedness follows from the Hardy inequality. The main step is deriving an observability inequality by means of a Carleman estimate whose weight function is specially built to neutralize the degeneracy without creating new singularities or breaking the needed geometric conditions. Exact controllability then follows by the standard duality between observability and controllability.

Core claim

By constructing a weight function in the Carleman estimate that cancels the influence of the degenerate null point, the authors obtain the observability inequality for the system, which in turn yields exact controllability even when the control region includes the degenerate point.

What carries the argument

The custom weight function used in the Carleman estimate, designed to negate the effect of the single-point degeneracy while maintaining pseudoconvexity.

If this is right

  • The equation admits a unique weak solution via the Hardy inequality.
  • Observability holds from any open set containing the degenerate point.
  • Exact controllability is achieved in the natural energy space for the hyperbolic system.
  • The result covers arbitrary dimension N.

Where Pith is reading between the lines

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

  • The technique may extend to other types of degeneracy or to systems with multiple singular points.
  • Similar weight constructions could be tested on related controllability problems for degenerate parabolic equations.
  • Explicit bounds on the control time or norm might follow from refining the Carleman constants.

Load-bearing premise

A weight function exists for the Carleman estimate that removes the degeneracy's impact without violating pseudoconvexity conditions or introducing new singularities.

What would settle it

A specific N-dimensional degenerate hyperbolic equation for which no such Carleman weight function can be found, or for which the observability inequality fails when the control region contains the degeneracy.

read the original abstract

This paper explores the controllability of a class of N-dimensional hyperbolic equations featuring a single interior degenerate point. Firstly, we establish the well-posedness of the equation through the application of the Hardy inequality. Following this, we primarily utilize the Carleman estimate method to derive the observability inequality. By leveraging the equivalence between observability and controllability, we deduce the exact controllability of the equation. It is worth noting that our selected control region includes the degenerate null point. In the Carleman estimate, we adopt a unique approach to construct the weight function, effectively negating the influence of the degenerate region.

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

1 major / 0 minor

Summary. The paper claims to establish exact controllability for a class of N-dimensional hyperbolic equations with a single interior point of degeneracy. It proceeds by proving well-posedness via the Hardy inequality, deriving an observability inequality from a Carleman estimate that employs a custom weight function designed to neutralize the effect of the degeneracy, and then invoking the standard observability-controllability equivalence, with the control region permitted to contain the degenerate point.

Significance. If the central claims are substantiated, the result would extend controllability theory for hyperbolic PDEs to higher-dimensional degenerate cases while allowing controls that include the degeneracy, which is a non-standard feature. The custom weight-function construction for the Carleman estimate, if shown to preserve pseudoconvexity, could serve as a reusable technique. The use of the Hardy inequality for well-posedness is a standard and appropriate choice that strengthens the foundation.

major comments (1)
  1. [Carleman estimate construction (referenced in abstract)] The abstract states that a unique construction of the weight function is used in the Carleman estimate to negate the influence of the degenerate region. The manuscript must explicitly exhibit this weight function (presumably in the section detailing the Carleman estimate) and verify that the associated quadratic form involving its Hessian remains positive definite with respect to the principal symbol of the hyperbolic operator in every neighborhood of the interior degeneracy point. Without this verification, the global Carleman inequality cannot be guaranteed, undermining the subsequent observability inequality and controllability conclusion. This is the load-bearing step identified in the proof chain.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for identifying the critical step in our proof. We provide a point-by-point response to the major comment below.

read point-by-point responses

  1. Referee: The abstract states that a unique construction of the weight function is used in the Carleman estimate to negate the influence of the degenerate region. The manuscript must explicitly exhibit this weight function (presumably in the section detailing the Carleman estimate) and verify that the associated quadratic form involving its Hessian remains positive definite with respect to the principal symbol of the hyperbolic operator in every neighborhood of the interior degeneracy point. Without this verification, the global Carleman inequality cannot be guaranteed, undermining the subsequent observability inequality and controllability conclusion. This is the load-bearing step identified in the proof chain.

    Authors: We agree with the referee that the explicit construction and verification are essential for the rigor of the argument. The weight function is constructed in Section 3 of the manuscript using a specific form that incorporates a correction term to cancel the degeneracy at the interior point. We will revise the manuscript to include an explicit display of this weight function and a detailed computation verifying that the quadratic form defined by the Hessian of the weight function is positive definite relative to the principal symbol of the operator, uniformly in a neighborhood of the degeneracy point. This verification will be presented as a separate lemma to highlight its role in establishing the Carleman estimate. revision: yes

Circularity Check

0 steps flagged

No circularity: standard PDE controllability chain with independent weight construction

full rationale

The derivation proceeds from well-posedness (via Hardy inequality) to a Carleman estimate with a custom weight function chosen to handle the degeneracy, yielding an observability inequality, followed by the standard duality equivalence to exact controllability. None of these steps reduce the target result to a fitted parameter, self-definition, or load-bearing self-citation; the weight construction is presented as an original ansatz satisfying the required pseudoconvexity conditions, and the equivalence is a general theorem independent of the specific equation. The provided text contains no equations or citations that collapse the controllability conclusion back onto its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the Hardy inequality (standard in analysis) and the existence of a suitable weight function for the Carleman estimate. No free parameters or invented entities are introduced.

axioms (2)
  • standard math Hardy inequality holds for the degenerate coefficients
    Invoked to establish well-posedness of the degenerate hyperbolic equation.
  • domain assumption A weight function exists that cancels the degeneracy effect while preserving Carleman-estimate pseudoconvexity
    Central technical step stated in the abstract but not derived.

pith-pipeline@v0.9.0 · 5400 in / 1289 out tokens · 52286 ms · 2026-05-08T17:02:15.555669+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

24 extracted references · 24 canonical work pages

  1. [1]

    Alabau-Boussouira, P

    F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability,J. Evol. Equ., 6(2006), 161–204

    work page 2006

  2. [2]

    Alabau-Boussouira, P

    F. Alabau-Boussouira, P. Cannarsa, and G. Leugering, Control and stabilization of degenerate wave equations,SIAM J. Control Optim., 55(2017), 2052–2087

    work page 2017

  3. [3]

    Allal, A

    B. Allal, A. Moumni and J. Salhi, Boundary controllability for a degenerate and singular wave equation,Math. Methods Appl. Sci., 45(2022), no. 17, 11526–11544

    work page 2022

  4. [4]

    Bai and S

    J. Bai and S. Chai, Exact controllability for some degenerate wave equations,Math. Methods Appl. Sci., 43(2020), 7292–7302

    work page 2020

  5. [5]

    Bai and S

    J. Bai and S. Chai, Exact controllability of wave equations with interior degeneracy and one- sided boundary control,J. Syst. Sci. Complex., 36(2023), 656–671

    work page 2023

  6. [6]

    Banerjee, N

    A. Banerjee, N. Garofalo, and R. Manna, Carleman estimates for Baouendi-Grushin opera- tors with applications to quantitative uniqueness and strong unique continuation,Appl. Anal., 101(2022), 3667–3688

    work page 2022

  7. [7]

    Baudouin, A

    L. Baudouin, A. Mercado-Saucedo and A. Osses, A global Carleman estimate in a transmis- sion wave equation and application to a one-measurement inverse problem,Inverse Problems 23(2007), 257–278

    work page 2007

  8. [8]

    Beauchard, P

    K. Beauchard, P. Cannarsa, and R. Guglielmi, Null controllability of Grushin-type operators in dimension two,J. Eur. Math. Soc., 16(2014), 67–101

    work page 2014

  9. [9]

    Beauchard, L

    K. Beauchard, L. Miller, and M. Morancey, 2D Grushin-type equations: minimal time and null controllable data,J. Differential Equations, 259(2015), 5813–5845

    work page 2015

  10. [10]

    Bellassoued and M

    M. Bellassoued and M. Yamamoto,Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, Tokyo, 2017. 27

    work page 2017

  11. [11]

    Cannarsa, P

    P. Cannarsa, P. Martinezk, and J. Vancostenoble,Global Carleman Estimates for Degenerate Parabolic Operators with Applications, Mem. Amer. Math. Soc., 239(2016), no. 1133

    work page 2016

  12. [12]

    Catrina and Z

    F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions,Comm. Pure Appl. Math. 54(2001), 229–258

    work page 2001

  13. [13]

    Evans,Partial Differential Equations, second edition, Amer

    L. Evans,Partial Differential Equations, second edition, Amer. Math. Soc., Providence, RI, 2010

    work page 2010

  14. [14]

    Flores and L

    C. Flores and L. de Teresa, Carleman estimates for degenerate parabolic equations with first order terms and applications,C. R. Math. Acad. Sci. Paris, 348(2010), 391–396

    work page 2010

  15. [15]

    X. Fu, J. Yong, and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations,SIAM J. Control Optim., 46(2007), 1578–1614

    work page 2007

  16. [16]

    Fragnelli and D

    G. Fragnelli and D. Mugnai,Control of Degenerate and Singular Parabolic Equations-Carleman Estimates and Observability, Springer, Cham, 2021

    work page 2021

  17. [17]

    Lions, Remarks on approximate controllability,J

    J.L. Lions, Remarks on approximate controllability,J. Anal. Math., 59(1992), 103–116

    work page 1992

  18. [18]

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

    M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equa- tions,SIAM J. Control Optim., 52(2014), 2037–2054

    work page 2014

  19. [19]

    Martinez, J

    P. Martinez, J. Raymond, and J. Vancostenoble, Regional null controllability of a linearized Crocco-type equation,SIAM J. Control Optim., 42(2003), 709–728

    work page 2003

  20. [20]

    R. T. Rockafellar, Duality and stability in extremum problems involving convex functions, Pacific J. Math.21(1967), 167–187

    work page 1967

  21. [21]

    Sakthivel, K

    K. Sakthivel, K. Balachandran, R. Sowrirajan, and J.H. Kim, On exact null controllability of Black-Scholes equation,Kybernetika (Prague), 44(2008), 685–704

    work page 2008

  22. [22]

    Zhang and H

    M. Zhang and H. Gao, Persistent regional null controllability of some degenerate wave equa- tions,Math. Methods Appl. Sci., 40(2017), 5821–5830

    work page 2017

  23. [23]

    Zhang and H

    M. Zhang and H. Gao, Interior controllability of semi-linear degenerate wave equations,J. Math. Anal. Appl., 457(2018), 10–22

    work page 2018

  24. [24]

    Zuazua,Exact controllability and stabilization of the wave equation, Springer, Cham, 2024

    E. Zuazua,Exact controllability and stabilization of the wave equation, Springer, Cham, 2024. 28

    work page 2024