Backstepping Observer for the Quasilinear Heat Equation with Linear Design Gains: Beyond Local Stability

Kirsten A. Morris; Mohamed Camil Belhadjoudja

arxiv: 2604.25815 · v1 · submitted 2026-04-28 · 📡 eess.SY · cs.SY· math.AP

Backstepping Observer for the Quasilinear Heat Equation with Linear Design Gains: Beyond Local Stability

Mohamed Camil Belhadjoudja , Kirsten A. Morris This is my paper

Pith reviewed 2026-05-07 15:05 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.AP

keywords quasilinear heat equationbackstepping observerexponential stabilityH1 normLyapunov analysisdiffusivity mismatchboundary output

0 comments

The pith

A backstepping observer designed for the linear heat equation achieves exponential convergence of the observation error to zero for the quasilinear version, despite a persistent diffusivity mismatch.

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

The paper applies a boundary-output observer whose gains come from the standard backstepping design for a linear constant-coefficient heat equation to a one-dimensional quasilinear heat equation whose diffusivity depends on the state. It proves that the resulting observation error system is exponentially stable at the origin in the H^1 norm inside an explicitly characterized region of attraction. The proof treats the difference between the nonlinear diffusivity and the constant design value as a state-dependent multiplicative perturbation that need not decay along solutions. A sufficiently refined Lyapunov analysis shows that this perturbation is absorbed without reducing the result to mere boundedness of the error.

Core claim

Viewing the quasilinear system as a perturbation of the linear one, we establish exponential stability of the origin for the observation error dynamics in H^1, with an explicit region of attraction depending on the system parameters, observer gains, and the mismatch between the nonlinear diffusivity and the constant design diffusivity. The observation error converges to zero rather than merely to a neighborhood scaling with this mismatch, even though the mismatch need not decay along trajectories and may remain bounded away from zero.

What carries the argument

Backstepping observer gains computed from the linear heat equation with constant coefficients, together with a Lyapunov function whose analysis absorbs the state-dependent diffusivity mismatch as a multiplicative perturbation.

If this is right

The observation error converges exactly to zero inside the region of attraction even when the diffusivity mismatch stays bounded away from zero.
One of the backstepping gains exhibits a non-monotonic effect on the convergence rate: beyond a certain value further increases degrade performance.
The size of the region of attraction depends explicitly on the mismatch magnitude, the system parameters, and the chosen gains.
The linear design therefore works for this class of quasilinear systems without requiring the mismatch to vanish or the gains to be adapted online.

Where Pith is reading between the lines

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

The same linear-gain observer may remain effective for other quasilinear parabolic equations whose nonlinearity appears only in the diffusion coefficient.
The existence of an optimal gain value suggests that performance tuning in nonlinear PDE observers requires joint consideration of decay rate and region of attraction rather than rate alone.
The explicit dependence of the region of attraction on the mismatch size gives a practical criterion for when the linear design can be used safely on measured nonlinear data.

Load-bearing premise

The mismatch between the nonlinear diffusivity and the constant design diffusivity can be absorbed as a persistent state-dependent perturbation by a sufficiently fine Lyapunov analysis without the analysis collapsing to a mere boundedness result.

What would settle it

Numerical simulation or laboratory experiment in which the observation error fails to approach zero for initial conditions inside the predicted region of attraction, or approaches zero outside that region, would falsify the exponential stability claim.

read the original abstract

We consider the one-dimensional quasilinear heat equation with state-dependent heat capacity and thermal conductivity, and design a boundary-output observer based on the backstepping design for a linear heat equation with constant coefficients. Viewing the quasilinear system as a perturbation of the linear one, we establish exponential stability of the origin for the observation error dynamics in $H^1$, with an explicit region of attraction depending on the system parameters, observer gains, and the mismatch between the nonlinear diffusivity and the constant design diffusivity. Importantly, the observation error converges to zero rather than merely to a neighborhood scaling with this mismatch, even though, in contrast to backstepping-based stabilization of nonlinear PDEs, the mismatch need not decay along trajectories and may remain bounded away from zero, acting as a persistent state-dependent multiplicative perturbation. A technical challenge was to perform a sufficiently-fine Lyapunov analysis that does not yield overly conservative results such as mere boundedness of the observation error. Interestingly, while in the linear case the relationship between one of the backstepping observer gains and the convergence rate is monotonic, we show that in the nonlinear setting this is no longer the case: there may exist an optimal value of that gain, beyond which further increases deteriorate the system's performance. Such behavior cannot be predicted without our analysis: one might expect a priori the decay rate to be freely tunable at the expense of a region of attraction that shrinks to zero as the prescribed rate tends to infinity. However, our Lyapunov analysis (supported by numerical experiments) reveals that this intuition is incorrect.

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 central claim of exponential convergence to zero for the observer error despite non-vanishing mismatch runs into a basic problem: the origin is not an equilibrium of the error dynamics.

read the letter

The main takeaway is that this paper tries to push a linear backstepping observer to a quasilinear heat equation and get exponential stability of the error to zero in H1, with an explicit region of attraction, even when the diffusivity mismatch stays bounded away from zero. That is the headline result, and it is presented as going beyond local stability or mere boundedness results in prior work on nonlinear PDE observers. They also report that one of the gains shows non-monotonic behavior with respect to convergence rate, which is backed by numerics and not obvious from the linear case. Those elements are the genuinely new pieces worth noting. The numerics appear to illustrate the gain insight in a concrete way, and the attempt to keep the design gains linear while handling the quasilinear plant is a reasonable direction for applications like thermal control. The explicit dependence of the region of attraction on parameters and mismatch is also a concrete step forward from purely local arguments. The soft spot is the equilibrium question. Substituting zero error into the error PDE leaves a nonzero term from the mismatch operator acting on the plant state, since the observer is built from the linear design and the mismatch is allowed to persist. This means the origin is not invariant, so asymptotic stability to zero cannot hold in the stated form. The Lyapunov analysis is described as fine enough to absorb the perturbation without conservative boundedness, but that cannot overcome the fact that zero is not a solution of the error equation. The abstract is clear that the mismatch need not decay, which makes the zero-convergence claim difficult to reconcile with the dynamics. This work is for researchers extending backstepping to nonlinear distributed-parameter systems or designing observers for diffusion processes. A reader focused on practical gain tuning or explicit estimates could extract value from the numerics and the non-monotonic observation, even if the main theorem needs scrutiny. It deserves peer review to check the full error equation derivation and Lyapunov estimates rather than a desk rejection, because the topic is relevant and the simulations add something tangible.

Referee Report

1 major / 2 minor

Summary. The manuscript designs a boundary-output observer for the one-dimensional quasilinear heat equation with state-dependent heat capacity and thermal conductivity, using backstepping gains computed from the corresponding linear constant-coefficient problem. Treating the quasilinear plant as a perturbation of the linear design, the authors claim to prove exponential stability of the zero observation error in the H^1 norm, with an explicit region of attraction that depends on system parameters, observer gains, and the (possibly persistent and non-vanishing) diffusivity mismatch. The error is asserted to converge exactly to zero rather than to a mismatch-scaled neighborhood, and the analysis reveals a non-monotonic dependence of the convergence rate on one design gain, contrary to the linear case.

Significance. If the central claims are valid, the result would be a notable contribution to nonlinear PDE observer design. It would show that linear backstepping can be applied to quasilinear systems while still obtaining asymptotic (not merely ultimate bounded) error convergence despite a persistent state-dependent multiplicative perturbation, which exceeds the conclusions of most perturbation analyses. The explicit region-of-attraction estimate and the identification of an optimal gain value (supported by numerics) offer practical design guidance. The technical feature of a sufficiently refined Lyapunov argument that avoids overly conservative boundedness conclusions is a strength of the approach.

major comments (1)

[Abstract and error-system derivation (presumably §3–4)] Abstract and error-system derivation (presumably §3–4): the origin is not an equilibrium of the observation-error dynamics when the diffusivity mismatch remains bounded away from zero. Substituting e ≡ 0 into the error PDE yields a nonzero residual equal to the difference between the quasilinear operator applied to the plant trajectory u(t) and the linear design operator; the right-hand side does not vanish. This contradicts the claim of exponential stability of the origin in H^1 and invalidates any subsequent Lyapunov argument that treats the mismatch as an absorbable perturbation while asserting convergence to zero rather than a neighborhood.

minor comments (2)

[Abstract] The abstract refers to 'numerical experiments' that illustrate the non-monotonic gain behavior, but the manuscript provides no details on the discretization scheme, parameter values, or initial conditions used in those simulations, which limits reproducibility.
[Notation and §3] Notation for the mismatch term (k(u) − κ) and the precise definition of the error PDE should be introduced earlier and kept consistent across the stability theorem statement and the Lyapunov analysis to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the detailed and constructive feedback on our manuscript. The major comment raises a fundamental point about the error dynamics, which we address below.

read point-by-point responses

Referee: Abstract and error-system derivation (presumably §3–4): the origin is not an equilibrium of the observation-error dynamics when the diffusivity mismatch remains bounded away from zero. Substituting e ≡ 0 into the error PDE yields a nonzero residual equal to the difference between the quasilinear operator applied to the plant trajectory u(t) and the linear design operator; the right-hand side does not vanish. This contradicts the claim of exponential stability of the origin in H^1 and invalidates any subsequent Lyapunov argument that treats the mismatch as an absorbable perturbation while asserting convergence to zero rather than a neighborhood.

Authors: We thank the referee for this observation. After careful reconsideration of the error system derivation, we agree that the origin is not an equilibrium of the observation error dynamics in the presence of a non-vanishing diffusivity mismatch. Substituting the zero error into the error PDE indeed produces a residual term corresponding to the operator mismatch applied to the plant state. Consequently, our original claim of exponential convergence of the error to zero is not accurate. We will revise the manuscript to establish ultimate boundedness of the observation error, with an explicit bound that depends on the mismatch, rather than asymptotic stability to the origin. The abstract, introduction, and stability theorems will be updated to reflect this. The Lyapunov analysis will be modified to demonstrate that the error enters and remains in a neighborhood whose radius can be made small if the mismatch is small. While this weakens the result compared to the original claim, we believe the explicit region of attraction and the non-monotonic gain behavior still offer useful insights for observer design in quasilinear systems. We will also include a discussion of this limitation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs the observer via standard linear backstepping applied to a constant-coefficient heat equation, then performs a direct Lyapunov analysis on the nonlinear observation-error PDE obtained by subtracting the observer from the quasilinear plant. The central claim of exponential stability of the origin in H^1 (with explicit region of attraction) is obtained by bounding the persistent mismatch term through sufficiently refined estimates that absorb it without reducing the result to a tautology or to any fitted quantity defined by the authors. No load-bearing step equates the target stability statement to an input by construction, and the analysis does not rely on self-citations whose content is itself unverified or circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard well-posedness assumptions for quasilinear parabolic equations and on the existence of a backstepping transformation for the linear reference system; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The quasilinear coefficients are positive, sufficiently smooth, and bounded away from zero so that the system is well-posed in H^1.
Required for the perturbation view and for the Lyapunov analysis to close.
standard math A backstepping observer exists and is exponentially stable for the linear constant-coefficient heat equation.
This is the design foundation reused for the nonlinear case.

pith-pipeline@v0.9.0 · 5590 in / 1439 out tokens · 44222 ms · 2026-05-07T15:05:26.109646+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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Belhadjoudja, M

M.C. Belhadjoudja, M. Maghenem, E. Witrant, State Estimation for the Kuramoto–Sivashinsky Equation Using Scanning Outputs,IEEE Transactions on Automatic Control, vol. 70, no. 12, pp. 8398-8405, 2025

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Si and C

Y. Si and C. Xie, Local stabilization of semilinear parabolic system by boundary control,Asian Journal of Control, 23(1), 591-599, 2021

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Zeldovich and Y.P

Y.B. Zeldovich and Y.P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (1st ed.), Academic Press, 1966

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Zhang and K

M. Zhang and K. Morris, Sensor Choice for Minimum Error Variance Estimation,IEEE Transactions on Automatic Control, vol. 63, no. 2, pp. 315-330, 2018. A Definition of constants and functions LetL α(s) (resp.,L α′(s)), for somes≥0, be the Lipschitz constant ofα(resp., ofα ′) on [−s, s]. We recall thatαand α′ are Lipschitz on bounded sets becauseαis of clas...

work page 2018

[1] [1]

Afshar, K

S. Afshar, K. Morris, A. Khajepour, State-of-charge estimation using an EKF-based adaptive observer,IEEE Transactions on Control Systems Technology, 27(5), 1907- 1923, 2018

work page 1907

[2] [2]

Al Jamal and K

R. Al Jamal and K. Morris, Linearized stability of partial differential equations with application to stabilization of the Kuramoto–Sivashinsky equation,SIAM Journal on Control and Optimization, 56(1), 120-147, 2018

work page 2018

[3] [3]

Arnesano, G.M

M. Arnesano, G.M. Revel, F. Seri, A tool for the optimal sensor placement to optimize temperature monitoring in large sports spaces,Automation in Construction, 68, 223-234, 2016

work page 2016

[4] [4]

Aronson, The porous medium equation,In Nonlinear Diffusion Problems, Berlin, Heidelberg: Springer Berlin Heidelberg, pp

D.G. Aronson, The porous medium equation,In Nonlinear Diffusion Problems, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 1-46, 2006

work page 2006

[5] [5]

Belhadjoudja, M.A

M.C. Belhadjoudja, M.A. Maghenem, E. Witrant, C. Prieur, Adaptive boundary control of the Kuramoto–Sivashinsky equation under intermittent sensing,Automatica, 178, 112379, 2025

work page 2025

[6] [6]

Belhadjoudja, M

M.C. Belhadjoudja, M. Maghenem, E. Witrant, State Estimation for the Kuramoto–Sivashinsky Equation Using Scanning Outputs,IEEE Transactions on Automatic Control, vol. 70, no. 12, pp. 8398-8405, 2025

work page 2025

[7] [7]

M. C. Belhadjoudja, M. Maghenem, E. Witrant, M. Krstic, Boundary Stabilization of Quasilinear Parabolic PDEs that Blow Up in Open Loop for Arbitrarily Small Initial Conditions,2025 IEEE 64th Conference on Decision and Control (CDC), Rio de Janeiro, Brazil, pp. 1086-1091, 2025

work page 2025

[8] [8]

M. C. Belhadjoudja, M. Maghenem, E. Witrant, M. Krstic, Stabilization of Quasilinear Parabolic Equations by Cubic Feedback at Boundary with Estimated Region of Attraction, Under review

work page

[9] [9]

Boussinesq, Recherches th´ eoriques sur l’´ ecoulement des nappes d’eau infiltr´ ees dans le sol et sur le d´ ebit des sources, Journal de Math

J. Boussinesq, Recherches th´ eoriques sur l’´ ecoulement des nappes d’eau infiltr´ ees dans le sol et sur le d´ ebit des sources, Journal de Math. Pures et Appliqu´ ees, 10: 5–78, 1904

work page 1904

[10] [10]

Castillo, E

F. Castillo, E. Witrant, C. Prieur, and L. Dugard, Boundary observers for linear and quasi-linear hyperbolic systems with application to flow control,Automatica, 49(11), 3180-3188, 2013

work page 2013

[11] [11]

Coron, R

J.M. Coron, R. Vazquez, M. Krstic, G. Bastin, Local exponentialH 2 stabilization of a 2×2 quasilinear hyperbolic system using backstepping,SIAM Journal on Control and Optimization, 51(3), 2005-2035, 2013

work page 2005

[12] [12]

Demetriou, Guidance of Mobile Actuator-Plus-Sensor Networks for Improved Control and Estimation of Distributed Parameter Systems,IEEE Transactions on Automatic Control, vol

M.A. Demetriou, Guidance of Mobile Actuator-Plus-Sensor Networks for Improved Control and Estimation of Distributed Parameter Systems,IEEE Transactions on Automatic Control, vol. 55, no. 7, pp. 1570-1584, 2010

work page 2010

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Geshkovski, Null-controllability of perturbed porous medium gas flow,ESAIM: Control, Optimisation and Calculus of Variations, 26, 85., 2020

B. Geshkovski, Null-controllability of perturbed porous medium gas flow,ESAIM: Control, Optimisation and Calculus of Variations, 26, 85., 2020. 14

work page 2020

[14] [14]

Herbinger, M

F. Herbinger, M. Bhouri, D. Groulx, Investigation of heat transfer inside a PCM-air heat exchanger: a numerical parametric study,Heat and Mass Transfer, 54(8), 2433-2442, 2018

work page 2018

[15] [15]

Lhachemi, Local Output Feedback Stabilization of a Nonlinear Kuramoto–Sivashinsky Equation,in IEEE Transactions on Automatic Control, vol

H. Lhachemi, Local Output Feedback Stabilization of a Nonlinear Kuramoto–Sivashinsky Equation,in IEEE Transactions on Automatic Control, vol. 68, no. 12, pp. 8165- 8170, 2023

work page 2023

[16] [16]

Jadachowski, T

L. Jadachowski, T. Meurer, A. Kugi, Backstepping observers for periodic quasi-linear parabolic PDEs,IFAC Proceedings Volumes, 47(3), 7761-7766, 2014

work page 2014

[17] [17]

K. Kant, A. Shukla, A. Sharma, Performance evaluation of fatty acids as phase change material for thermal energy storage,Journal of Energy Storage, 6, 153-162, 2016

work page 2016

[18] [18]

Krstic and A

M. Krstic and A. Smyshlyaev, Boundary control of PDEs: A course on backstepping designs,SIAM, 2008

work page 2008

[19] [19]

O. A. Ladyzhenskaia, V. A. Solonnikov, and N. N. Ural’tseva, Linear and quasi-linear equations of parabolic type,(Vol. 23). American Mathematical Soc., 1968

work page 1968

[20] [20]

Lhachemi, I

H. Lhachemi, I. Munteanu, C. Prieur, Boundary Output Feedback Stabilization of Nonlinear Diffusion Equations, IEEE Transactions on Automatic Control, to appear

work page

[21] [21]

Liu and X

X. Liu and X. Zhang, On the local controllability of a class of multidimensional quasilinear parabolic equations,Comptes Rendus Mathematique, 347(23-24), 1379-1384, 2009

work page 2009

[22] [22]

Privat, E

Y. Privat, E. Tr´ elat, E. Zuazua, Optimal location of sensors and actuators for wave and heat processes,In PDE’s, Dispersion, Scattering theory and Control theory, No. 30, pp. 123-139, 2013

work page 2013

[23] [23]

Roub´ ıˇ cek, Nonlinear partial differential equations with applications,Basel: Birkh¨ auser Basel, 2005

T. Roub´ ıˇ cek, Nonlinear partial differential equations with applications,Basel: Birkh¨ auser Basel, 2005

work page 2005

[24] [24]

Schaum et al., Observer design for a nonlinear heat equation: Application to semiconductor wafer processing, Journal of Process Control, 119, 34-43, 2022

A. Schaum et al., Observer design for a nonlinear heat equation: Application to semiconductor wafer processing, Journal of Process Control, 119, 34-43, 2022

work page 2022

[25] [25]

Si and C

Y. Si and C. Xie, Local stabilization of semilinear parabolic system by boundary control,Asian Journal of Control, 23(1), 591-599, 2021

work page 2021

[26] [26]

V´ azquez, The porous medium equation: mathematical theory,Clarendon Press, 2006

J.L. V´ azquez, The porous medium equation: mathematical theory,Clarendon Press, 2006

work page 2006

[27] [27]

Zeldovich and Y.P

Y.B. Zeldovich and Y.P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (1st ed.), Academic Press, 1966

work page 1966

[28] [28]

Zhang and K

M. Zhang and K. Morris, Sensor Choice for Minimum Error Variance Estimation,IEEE Transactions on Automatic Control, vol. 63, no. 2, pp. 315-330, 2018. A Definition of constants and functions LetL α(s) (resp.,L α′(s)), for somes≥0, be the Lipschitz constant ofα(resp., ofα ′) on [−s, s]. We recall thatαand α′ are Lipschitz on bounded sets becauseαis of clas...

work page 2018