Null controllability of one dimensional degenerate parabolic equations with first order terms

J. Carmelo Flores; Luz de Teresa

arxiv: 1907.03321 · v1 · pith:FKT5RR2Gnew · submitted 2019-07-07 · 🧮 math.OC · math.AP

Null controllability of one dimensional degenerate parabolic equations with first order terms

J. Carmelo Flores , Luz de Teresa This is my paper

Pith reviewed 2026-05-25 01:11 UTC · model grok-4.3

classification 🧮 math.OC math.AP

keywords null controllabilitydegenerate parabolic equationsCarleman inequalityfirst order termssemilinear parabolic equationsone-dimensional equations

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The pith

Null controllability holds for one-dimensional degenerate semilinear parabolic equations with first-order terms via a new Carleman inequality.

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 one-dimensional degenerate semilinear parabolic equation including first-order terms is null controllable. This result follows from establishing a new Carleman inequality for the corresponding linear degenerate parabolic equation with first-order terms. A reader would care because such controllability allows driving the solution to the zero state in finite time from any initial data using suitable controls. The presence of degeneracy and first-order terms makes standard controllability techniques insufficient, requiring this specialized inequality. The approach extends previous work on degenerate equations by handling the additional first-order terms.

Core claim

We prove a null controllability result for a degenerate semilinear parabolic equation with first order terms in one dimension. The result is obtained by first proving a new Carleman inequality for the associated degenerate linear parabolic equation with first order terms.

What carries the argument

A new Carleman inequality adapted to degenerate linear parabolic equations that include first-order terms, which is used to derive observability and then controllability for the semilinear case.

If this is right

The semilinear system can be driven to the zero state in finite time.
The linear system satisfies an observability inequality derived from the Carleman estimate.
Controllability results extend to cases with first-order drift terms under the given degeneracy.
Suitable controls exist for the nonlinear equation based on the linear theory.

Where Pith is reading between the lines

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

This method might apply to other types of degeneracies or higher-order terms if similar inequalities can be derived.
Connections to stabilization problems where the same estimates could yield exponential decay rates.
Possible extension to boundary control or distributed control in multi-dimensional settings with careful adaptation.

Load-bearing premise

The new Carleman inequality holds under the specific assumptions on the location and type of degeneracy and the coefficients of the first-order terms in the equation.

What would settle it

A counterexample where the Carleman inequality does not hold for a particular choice of degeneracy parameter, leading to lack of controllability for some initial data.

read the original abstract

In this paper we present a null controllability result for a degenerate semilinear parabolic equation with first order terms. The main result is obtained after the proof of a new Carleman inequality for a degenerate linear parabolic equation with first order terms.

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

The paper extends null controllability to degenerate semilinear parabolic equations that include first-order terms by deriving a new Carleman inequality for the linear case, but the abstract leaves the degeneracy type and coefficient bounds unspecified.

read the letter

The paper gives a null controllability result for a degenerate semilinear parabolic equation with first order terms. It obtains this by proving a new Carleman inequality for the linear degenerate parabolic equation with those terms. What stands out is the attempt to include first-order terms in the Carleman estimate for the degenerate case. Many earlier papers on null controllability of degenerate parabolic equations either omit first-order terms or treat them under stronger assumptions on the degeneracy. The authors follow the common path from Carleman to observability to controllability, which is appropriate here. The work is straightforward in its structure and does not show obvious flaws like circular arguments. The result appears to be new for this specific setting. The main limitation is the lack of detail in the abstract about the degeneracy location and the coefficient conditions. It is not clear how the first-order terms affect the construction of the weight functions or if additional restrictions are needed to absorb them. This makes it hard to judge the strength of the new inequality without the full proof. This kind of result is mainly of interest to people working on control problems for parabolic equations with variable coefficients or degeneracy. A reader looking for extensions of Carleman-based controllability methods might find it useful. I would bring this to the next reading group if we are discussing recent papers on degenerate PDE control. I would not cite it in my own work in the next year unless I needed that exact controllability statement. It should go to peer review because the topic is relevant and the approach is standard, so a referee can check the technical details.

Referee Report

0 major / 2 minor

Summary. The paper establishes a null controllability result for a one-dimensional degenerate semilinear parabolic equation with first-order terms. The controllability is obtained from a new Carleman inequality proved for the associated linear degenerate parabolic equation with first-order terms under suitable degeneracy and coefficient assumptions.

Significance. If the Carleman inequality holds, the result meaningfully extends the literature on controllability of degenerate parabolic equations by accommodating first-order terms, which appear in many applied models. The standard Carleman-to-observability pipeline is followed, and the provision of an explicit new estimate for this setting strengthens the contribution.

minor comments (2)

The abstract states that the controllability follows from the Carleman inequality but supplies no explicit statement of the degeneracy location, type, or coefficient assumptions; these should be summarized in the abstract for clarity.
Notation for the weight functions and the precise form of the degeneracy (e.g., the function a(x) or its vanishing order) should be introduced earlier in the introduction to help readers track the assumptions through the Carleman proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds by first establishing a new Carleman inequality for the linear degenerate parabolic operator that includes first-order terms, then applying the resulting observability estimate to obtain null controllability for the semilinear problem. This follows the standard independent Carleman-to-observability pipeline; the abstract and description supply no evidence that the Carleman estimate is defined in terms of the controllability result, that parameters are fitted and then relabeled as predictions, or that load-bearing uniqueness claims rest on self-citations. The central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, background axioms, or new entities; all such elements would be contained in the full proof.

pith-pipeline@v0.9.0 · 5549 in / 924 out tokens · 28375 ms · 2026-05-25T01:11:57.358466+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

and Fragnelli, G

Alabau-Boussouira, F., Cannarsa, P. and Fragnelli, G. , Carleman estimates for de- generate parabolic operators with applications to null con trollability, J. Evol. Equ. 6 (2006), no. 2, 161-204

work page 2006
[2]

and Barbu, V

Anita, S. and Barbu, V. , Null controllability of nonlinear convective heat equatio ns, ESAIM: Control, Optim. Calc. Var., 5 (2000), 157-173

work page 2000
[3]

D; Ara ´ujo, B

Araruna, F. D; Ara ´ujo, B. S. V. ; Fern ´andez-Cara, E.; Stackelberg-Nash null controlla- bility for some linear and semilinear degenerate parabolic equations. Math. Control Signals Systems 30 (2018), no. 3 https://doi.org/10.1007/s00498- 018-0220-6

work page doi:10.1007/s00498- 2018
[4]

, Partial Diﬀerential Equations and Boundary Value Problems , Vol

Barbu, V. , Partial Diﬀerential Equations and Boundary Value Problems , Vol. 441, Kluwer Academic Publishers, London, 1997

work page 1997
[5]

, On local controllability of Navier-Stokes equations , Adv

Barbu, V. , On local controllability of Navier-Stokes equations , Adv. Diﬀerential Equations 8 (12) (2003), 1481-1498

work page 2003
[6]

, An´ alisis Funcional

Brezis, H. , An´ alisis Funcional. theoremr ´ ıa y aplicaciones, Alianza Editorial, Madrid, 1984

work page 1984
[7]

R., De Menezes, S

Cabanilla, V. R., De Menezes, S. B. and Zuazua, E. , Null controllability in unbounded domains for semilinear heat equations with nonlinearities involving gradient terms , Journal of Optimization Theory and Applications, 110 (2) (2001), 245-264

work page 2001
[8]

and Pallara, D

Campiti, M., Metafune, G. and Pallara, D. , Degenerate self-adjoint evolution equations on the unit interval , Semigroup Forum, Vol. 57 (1998), 1-36

work page 1998
[9]

and De Teresa, L

Cannarsa, P. and De Teresa, L. , Insensitizing controls for one dimensional degenerate parabolic equations, EJDE., Vol. 2009(2009), No. 73, pp. 1-21

work page 2009
[10]

and Fragnelli, G

Cannarsa, P. and Fragnelli, G. ,Null controllability of semilinear degenerate parabolic equations in bounded domains , EJDE (2006), N.136, pp.1-20

work page 2006
[11]

and Rocchetti, D., Null controllability of degenerate para- bolic operators with drift , Netw

Cannarsa, P., Fragnelli, G. and Rocchetti, D., Null controllability of degenerate para- bolic operators with drift , Netw. Heterog. Media 2 (2007), no. 4, 695–715 (electronic)

work page 2007
[12]

and V ancostenoble, J

Cannarsa, P., Fragnelli, G. and V ancostenoble, J. , Regional controllability of semilin- ear degenerate parabolic equations in bounded domains , J. Math. Anal. Appl. 320 (2006), no. 2, 804-818

work page 2006
[13]

and V ancostenoble, J

Cannarsa, P., Martinez, P. and V ancostenoble, J. , Carleman estimates for degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), no. 1, 1-19. SIAM J. Control Optim. 47 (2008), no. 1, 1–19

work page 2008
[14]

and V ancostenoble, J

Cannarsa, P., Martinez, P. and V ancostenoble, J. , Null controllability of degenerate heat equations, Adv. Diﬀerential Equations, 10 (2) (2005), 153-190

work page 2005
[15]

and V ancostenoble, J

Cannarsa, P., Martinez, P. and V ancostenoble, J. , Persistent regional null controlla- bility for a class of degenerate parabolic equations , Communications on Pure and Applied Analysis, 3 (2004), 607-635

work page 2004
[16]

F abre, C., Puel, J. P. and Zuazua, E., Approximate controllability of the semilinear heat equation, Proceedings of the Royal Society of Edinburgh, 125 (1995) 31-61

work page 1995
[17]

F attorini, H. O. and Russell, D. L., Exact controllability theorems for linear parabolic equations in one space dimension , Arch. Rat. Mech. Anal. 4 (1971) 272-292

work page 1971
[18]

Fern´andez-Cara, E., Guerrero, S., Imanuvilov, Yu. O. and Puel, J. P., Local exact controllability of the Navier-Stokes system , J. Math Pures Appl. (9) 83 (12) (2004) 1501-1542

work page 2004
[19]

Flores, C., De Teresa, L., Carleman estimates for degenerate parabolic equations wit h ﬁrst order terms and applications . C.R. Acad. Sci. Paris. I 348 (2010) 391-396

work page 2010
[20]

Fursikov, A. V. and Imanuvilov, Yu. O., Controllability of evolution equations , Lectures Notes Series 34, Seoul National University, Seoul, Korea, 1996

work page 1996
[21]

Imanuvilov,Yu. O. and Yamamoto, M. , Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations , Publ. Res. Math. Sci. 39 (2003), no. 2,227-274

work page 2003
[22]

and Magenes, E

Lions, J.-L. and Magenes, E. , Non-homogeneous boundary value problems and applications , Vol. I, Die Grundlehren der mathematischen Wissenschaften , Band 181, Springer-Verlag, New York-Heidelberg, 1972

work page 1972
[23]

and V ancostenoble, J

Martinez, P. and V ancostenoble, J. ,Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ. 6 (2006), no. 2, 325–362

work page 2006
[24]

Control Optim., 42 (2) (2004) 709-728

Martinez, P., Raymond, J.-P and V ancostenoble, J., Regional null controllability of Crocco type linearized equation , SIAM J. Control Optim., 42 (2) (2004) 709-728. 20 J. CARMELO FLORES AND LUZ DE TERESA

work page 2004
[25]

Oleinkik, O. A. and Samokhin, V. N., Mathematical Models in Boundary Layer Theory , Applied Mathematics and Mathematical Computation, 15, 1999, Chapman and Hall/CRC, Boca Raton, London, New York

work page 1999
[26]

and Kufner, A

Opic, B. and Kufner, A. , Hardy-Type Inequalities, Longman Scientiﬁc and Technical, Har- low, UK, 1990

work page 1990
[27]

Russell, D. L. ,A unifed boundary controllability theory for hyperbolic an d parabolic partial diﬀerential equations, Stud. Appl. Math. 52 (1973), 189-221

work page 1973
[28]

, Compact sets in the spaces Lp(0, T ; B), Annali di Matematica Pura ed Aplicata, IV, Vol

Simon, J. , Compact sets in the spaces Lp(0, T ; B), Annali di Matematica Pura ed Aplicata, IV, Vol. CXL VI (1987), pp 65-96

work page 1987
[29]

Mathematical Control Theory: An Introduction Birkh¨ auser, 2008

Zabczyk, J. Mathematical Control Theory: An Introduction Birkh¨ auser, 2008

work page 2008
[30]

, Nonlinear Functional Analysis and its Applications , Applications to Mathe- matical Physics, IV, Springer, 1985

Zeidler, E. , Nonlinear Functional Analysis and its Applications , Applications to Mathe- matical Physics, IV, Springer, 1985. J. Carmelo Flores Academia de Matem ´aticas, Universidad Aut ´onoma de la Ciudad de M ´exico, A venida la Corona 320, Col. Loma la Palma,, Del. Gustavo A. Madero, M ´exico D.F., C.P. 07160., Mexico E-mail address : cflores@matem.u...

work page 1985

[1] [1]

and Fragnelli, G

Alabau-Boussouira, F., Cannarsa, P. and Fragnelli, G. , Carleman estimates for de- generate parabolic operators with applications to null con trollability, J. Evol. Equ. 6 (2006), no. 2, 161-204

work page 2006

[2] [2]

and Barbu, V

Anita, S. and Barbu, V. , Null controllability of nonlinear convective heat equatio ns, ESAIM: Control, Optim. Calc. Var., 5 (2000), 157-173

work page 2000

[3] [3]

D; Ara ´ujo, B

Araruna, F. D; Ara ´ujo, B. S. V. ; Fern ´andez-Cara, E.; Stackelberg-Nash null controlla- bility for some linear and semilinear degenerate parabolic equations. Math. Control Signals Systems 30 (2018), no. 3 https://doi.org/10.1007/s00498- 018-0220-6

work page doi:10.1007/s00498- 2018

[4] [4]

, Partial Diﬀerential Equations and Boundary Value Problems , Vol

Barbu, V. , Partial Diﬀerential Equations and Boundary Value Problems , Vol. 441, Kluwer Academic Publishers, London, 1997

work page 1997

[5] [5]

, On local controllability of Navier-Stokes equations , Adv

Barbu, V. , On local controllability of Navier-Stokes equations , Adv. Diﬀerential Equations 8 (12) (2003), 1481-1498

work page 2003

[6] [6]

, An´ alisis Funcional

Brezis, H. , An´ alisis Funcional. theoremr ´ ıa y aplicaciones, Alianza Editorial, Madrid, 1984

work page 1984

[7] [7]

R., De Menezes, S

Cabanilla, V. R., De Menezes, S. B. and Zuazua, E. , Null controllability in unbounded domains for semilinear heat equations with nonlinearities involving gradient terms , Journal of Optimization Theory and Applications, 110 (2) (2001), 245-264

work page 2001

[8] [8]

and Pallara, D

Campiti, M., Metafune, G. and Pallara, D. , Degenerate self-adjoint evolution equations on the unit interval , Semigroup Forum, Vol. 57 (1998), 1-36

work page 1998

[9] [9]

and De Teresa, L

Cannarsa, P. and De Teresa, L. , Insensitizing controls for one dimensional degenerate parabolic equations, EJDE., Vol. 2009(2009), No. 73, pp. 1-21

work page 2009

[10] [10]

and Fragnelli, G

Cannarsa, P. and Fragnelli, G. ,Null controllability of semilinear degenerate parabolic equations in bounded domains , EJDE (2006), N.136, pp.1-20

work page 2006

[11] [11]

and Rocchetti, D., Null controllability of degenerate para- bolic operators with drift , Netw

Cannarsa, P., Fragnelli, G. and Rocchetti, D., Null controllability of degenerate para- bolic operators with drift , Netw. Heterog. Media 2 (2007), no. 4, 695–715 (electronic)

work page 2007

[12] [12]

and V ancostenoble, J

Cannarsa, P., Fragnelli, G. and V ancostenoble, J. , Regional controllability of semilin- ear degenerate parabolic equations in bounded domains , J. Math. Anal. Appl. 320 (2006), no. 2, 804-818

work page 2006

[13] [13]

and V ancostenoble, J

Cannarsa, P., Martinez, P. and V ancostenoble, J. , Carleman estimates for degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), no. 1, 1-19. SIAM J. Control Optim. 47 (2008), no. 1, 1–19

work page 2008

[14] [14]

and V ancostenoble, J

Cannarsa, P., Martinez, P. and V ancostenoble, J. , Null controllability of degenerate heat equations, Adv. Diﬀerential Equations, 10 (2) (2005), 153-190

work page 2005

[15] [15]

and V ancostenoble, J

Cannarsa, P., Martinez, P. and V ancostenoble, J. , Persistent regional null controlla- bility for a class of degenerate parabolic equations , Communications on Pure and Applied Analysis, 3 (2004), 607-635

work page 2004

[16] [16]

F abre, C., Puel, J. P. and Zuazua, E., Approximate controllability of the semilinear heat equation, Proceedings of the Royal Society of Edinburgh, 125 (1995) 31-61

work page 1995

[17] [17]

F attorini, H. O. and Russell, D. L., Exact controllability theorems for linear parabolic equations in one space dimension , Arch. Rat. Mech. Anal. 4 (1971) 272-292

work page 1971

[18] [18]

Fern´andez-Cara, E., Guerrero, S., Imanuvilov, Yu. O. and Puel, J. P., Local exact controllability of the Navier-Stokes system , J. Math Pures Appl. (9) 83 (12) (2004) 1501-1542

work page 2004

[19] [19]

Flores, C., De Teresa, L., Carleman estimates for degenerate parabolic equations wit h ﬁrst order terms and applications . C.R. Acad. Sci. Paris. I 348 (2010) 391-396

work page 2010

[20] [20]

Fursikov, A. V. and Imanuvilov, Yu. O., Controllability of evolution equations , Lectures Notes Series 34, Seoul National University, Seoul, Korea, 1996

work page 1996

[21] [21]

Imanuvilov,Yu. O. and Yamamoto, M. , Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations , Publ. Res. Math. Sci. 39 (2003), no. 2,227-274

work page 2003

[22] [22]

and Magenes, E

Lions, J.-L. and Magenes, E. , Non-homogeneous boundary value problems and applications , Vol. I, Die Grundlehren der mathematischen Wissenschaften , Band 181, Springer-Verlag, New York-Heidelberg, 1972

work page 1972

[23] [23]

and V ancostenoble, J

Martinez, P. and V ancostenoble, J. ,Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ. 6 (2006), no. 2, 325–362

work page 2006

[24] [24]

Control Optim., 42 (2) (2004) 709-728

Martinez, P., Raymond, J.-P and V ancostenoble, J., Regional null controllability of Crocco type linearized equation , SIAM J. Control Optim., 42 (2) (2004) 709-728. 20 J. CARMELO FLORES AND LUZ DE TERESA

work page 2004

[25] [25]

Oleinkik, O. A. and Samokhin, V. N., Mathematical Models in Boundary Layer Theory , Applied Mathematics and Mathematical Computation, 15, 1999, Chapman and Hall/CRC, Boca Raton, London, New York

work page 1999

[26] [26]

and Kufner, A

Opic, B. and Kufner, A. , Hardy-Type Inequalities, Longman Scientiﬁc and Technical, Har- low, UK, 1990

work page 1990

[27] [27]

Russell, D. L. ,A unifed boundary controllability theory for hyperbolic an d parabolic partial diﬀerential equations, Stud. Appl. Math. 52 (1973), 189-221

work page 1973

[28] [28]

, Compact sets in the spaces Lp(0, T ; B), Annali di Matematica Pura ed Aplicata, IV, Vol

Simon, J. , Compact sets in the spaces Lp(0, T ; B), Annali di Matematica Pura ed Aplicata, IV, Vol. CXL VI (1987), pp 65-96

work page 1987

[29] [29]

Mathematical Control Theory: An Introduction Birkh¨ auser, 2008

Zabczyk, J. Mathematical Control Theory: An Introduction Birkh¨ auser, 2008

work page 2008

[30] [30]

, Nonlinear Functional Analysis and its Applications , Applications to Mathe- matical Physics, IV, Springer, 1985

Zeidler, E. , Nonlinear Functional Analysis and its Applications , Applications to Mathe- matical Physics, IV, Springer, 1985. J. Carmelo Flores Academia de Matem ´aticas, Universidad Aut ´onoma de la Ciudad de M ´exico, A venida la Corona 320, Col. Loma la Palma,, Del. Gustavo A. Madero, M ´exico D.F., C.P. 07160., Mexico E-mail address : cflores@matem.u...

work page 1985