Active Disturbance Rejection for Boundary Control Systems

Jukka-Pekka Humaloja; Lassi Paunonen

REVIEW 3 major objections 4 minor 34 references

Active Disturbance Rejection for Boundary Control Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A single three-block controller estimates unknown boundary disturbances and cancels them without knowing their model, yielding exponential stabilization for a wide class of PDE boundary control systems.

desk verdict A genuinely abstract ADRC framework for boundary control systems, with honest limitations; the main proofs hold up, but the wave example has a controller mismatch and the heat example leans on unstated computations. read the letter →

arxiv 2607.19144 v1 pith:3WBACR6C submitted 2026-07-21 math.OC math.APmath.FA

Jukka-Pekka Humaloja , Lassi Paunonen This is my paper

classification math.OCmath.APmath.FA MSC 93C2593C1093D1535B3593C2047D0647A10

keywords activedisturbancerejectionboundarycontrolsystemswell-posedlinearinfinite-dimensionalwaveequationheatunknowninputexponentialstabilization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that active disturbance rejection—estimate the unknown 'total disturbance' entering a system's boundary and subtract it from the control input—can be made rigorous for a broad class of infinite-dimensional systems governed by PDEs. The authors construct one controller template with three interacting parts: a separator that pre-stabilizes and isolates the known part of the input, an inverter that reconstructs the unknown disturbance from boundary measurements, and a Luenberger observer that tracks the state. Under one key assumption—the inverted system is itself well-posed—the closed-loop system has unique solutions, the state and observer decay exponentially, and the disturbance estimate converges in a weighted L2 sense. The results are applied to one-dimensional wave and heat equations, with the heat case handled by a frequency-domain substitute because its inverse is ill-posed. A sympathetic reader would care because this gives a systematic, verifiable route from an abstract boundary control system to a working disturbance-rejecting stabilizer.

What carries the argument

The central object is the abstract boundary node (B, A, C, Q, B_i), a functional-analytic packaging of a PDE with boundary inputs and outputs, and its partial flow inversion. The controller (3.3) is a three-block structure: the separator (state x_s) pre-stabilizes the plant and removes the known control u from the inversion loop; the inverter (state x_i) solves the inverted boundary condition C_d x_i = y_d - C_d x_s and produces the estimate d̂ = (B_d - L_d C)x_i; the observer (state x̂) is a Luenberger observer in which the estimate d̂ is used both as feedforward cancellation and as a known input. The proof of Theorem 3.2 works by a state transformation z = (x - x_s, x, x - x̂, x - x_s - x_

What would settle it

Take a boundary control system satisfying Assumption 3.1(a)-(c) with TK, TL, and Td exponentially stable, but whose inverted node fails Assumption 3.1(d) and also fails the Theorem 3.6 product bound; simulate the closed loop with a bounded unknown disturbance. If the state does not converge to zero or the disturbance estimate does not converge in the predicted weighted L2 sense, the controller template is refuted for that configuration. The paper itself flags the heat equation as the borderline case where (d) fails and only the frequency-domain substitute rescues it.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.2. For any boundary control system of the form (3.1), with collocated inputs and outputs, if Assumption 3.1 holds—in particular if the inverted boundary node (B_h - L_h C, A_d, B_d - L_d C, 0, 0) has a well-posed output map, so that the boundary measurement can be swapped with the controlled boundary to reconstruct the input—then the closed-loop system formed with the ADRC controller (3.3) is a well-posed boundary node. Every initial state and L2_loc input produces a unique generalised solution; smooth data yield classical solutions. If the three semigroups TK, TL, and Td are exponentially stable, the disturbance estimation error d̂ - d_tot belongs to L2_{ω_d}(

Load-bearing premise

The whole construction leans on the assumption that the measured boundary can be inverted into a well-posed dynamical system whose output is the disturbance estimate; if that inversion is not well-posed (and the parabolic frequency-domain substitute fails), the controller has no proven way to reconstruct the disturbance.

Editorial extensions

If this is right

For every boundary control system satisfying Assumption 3.1, the same controller template rejects arbitrary L2_loc disturbances and globally Lipschitz boundary nonlinearities without any model of them.
If TK, TL, and Td are exponentially stable, the estimate d̂ converges to the total disturbance with weighted L2 error depending only on initial data, and the state and observer converge exponentially—not merely asymptotically.
The closed-loop system is externally well-posed and incrementally well-posed: it has unique generalised solutions for all inputs, classical solutions for smooth data, and bounded-input bounded-state behavior for L2 or L∞ disturbances.
The design applies to concrete 1D wave and heat equations: the wave example (4.1)-(4.2) and heat example (4.4)-(4.5) both admit unique generalised solutions and exponential stability with the proposed gains.
The frequency-domain condition in Theorem 3.6 gives an explicit verifiable route for parabolic systems whose inverses are ill-posed; verifying that ||H(λ)|| ||(B_d - L_d C)(λ - A_d)^{-1}|| is bounded on C+_β is enough.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The decoupled structure suggests a separation principle: the estimator gains (L, L_i) can be tuned from the three semigroups TK, TL, and Td independently, which may make the ADRC design scale to more complex PDE networks.
The same template will likely apply to systems with multiple inputs and outputs, as long as a block of the transfer function is right-invertible; the paper's framework treats U and Y as Hilbert spaces, so it is ready for MIMO boundary control.
For parabolic systems, the product bound in Theorem 3.6 is a resolvent-decay condition that may be verifiable for other analytic semigroup systems (e.g., higher-dimensional heat or reaction-diffusion equations), where the inverse is similarly ill-posed.
Numerical experiments on the wave example could directly test the weighted L2 bound on d̂ - d_tot and the decay rate max{ω0(TK), ω0(TL), ω0(Td)} against simulations, giving a quantitative falsification check.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a serious paper. It does something genuinely new: it formulates active disturbance rejection for abstract boundary control systems, unifying the PDE-specific ADRC designs for wave, heat, beam, and Schrödinger equations, and it gets stronger guarantees — exponential convergence, external input well-posedness, and output nonlinearities φ_o ≠ 0. The main theorem (3.2) is proven in detail via a coordinate transformation to error variables, cascade well-posedness, and a Gearhart–Prüss–Greiner growth-bound computation. The disturbance-estimation identity is transparent. Theorem 3.6 sensibly extends the framework to parabolic cases where the inverse is ill-posed, exactly the heat-equation situation.

The soft spots are real but mostly fixable. The most concrete issue is the wave controller display. With L = [0 −ℓ; 0 0] as stated in the proof, equation (3.3d) gives w'_s(0) = ℓ(ẇ_s(0) − ẇ(0)), not the ẇ_s(0) − ℓẇ(0) that appears in (4.2); the same mismatch appears in the ˆw equation. This looks like a typo, but as written the concrete controller is not demonstrably covered by Theorem 3.2. The authors need to fix it.

The heat example is the second weak spot. Proposition 4.2 verifies the frequency-domain conditions of Theorem 3.6 by asserting that ∥H(λ)∥ ≲ |λ|^{−3/4} and ∥B_dL(λ−A_d)^{−1}∥ ≲ |λ|^{−1/4} via 'direct computations', without derivation. The Paley–Wiener step is also compressed. These estimates are load-bearing for the heat result; they should be shown or cited properly.

The third issue is the heavy reliance on the same-group, apparently unreviewed preprint [9] for the Section 2 machinery. Readers cannot independently verify the cascade and well-posedness results that Theorem 3.2 builds on. That is a genuine accessibility problem, not a mathematical error, and the authors should either include those results or make [9] publicly verifiable before publication.

Minor points: the introduction's contrast with earlier 'asymptotic only' results sits awkwardly against [33] (Zhou–Weiss), whose title already claims exponential stabilization; and there are several cross-reference typos (e.g., Theorem 2.2/2.3 references, H1(0,1) for H1(0,∞)).

None of this breaks the central argument. The proofs are structurally coherent, the assumptions are honestly stated, and there is no circularity or fitted-parameter recursion. This deserves a serious referee. I would send it to review, with requests to fix the wave controller display, expand the heat computations, and address the [9] dependency.

Referee Report

3 major / 4 minor

Summary. The paper develops an abstract active disturbance rejection control (ADRC) framework for boundary control systems of the form (1.1), with unknown input disturbances and boundary nonlinearities. The proposed controller (3.3) consists of a separator, an inverter, and an observer, and the main result (Theorem 3.2) states well-posedness, exponential stability, and disturbance-estimation error decay under Assumption 3.1. Theorem 3.6 provides a frequency-domain substitute for the inverter well-posedness condition, aimed at parabolic systems. The theory is applied to a one-dimensional wave equation and a one-dimensional heat equation in Section 4.

Significance. The abstract formulation is a genuine contribution: it unifies and extends several PDE-specific ADRC results, handles boundary nonlinearities and external inputs, and gives exponential convergence rather than mere asymptotic convergence. The proofs of the central well-posedness and stability results are detailed and structurally sound, and the assumptions are not circular. However, the wave controller as displayed in (4.2) does not match the abstract controller (3.3) under the stated parameter choice, so Proposition 4.1 is not currently supported by Theorem 3.2. The heat example verification of Theorem 3.6 also relies on asserted resolvent estimates that are not derived. These issues are local and likely fixable, but they are load-bearing for the applications.

major comments (3)

[§4.1, Eq. (4.2) and proof of Prop. 4.1] The proof of Proposition 4.1 states that the controller (4.2) is represented in the form (3.3) with L = [0 −ℓ; 0 0]. With this L, Eq. (3.3d) gives w'_s(0) = ℓ(˙w_s(0) − ˙w(0)), and Eq. (3.3g) gives ˙hat w'(0) = ℓ(˙hat w(0) + u2_2 − ˙w(0)). The displayed controller (4.2) instead has w'_s(0) = ˙w_s(0) − ℓ ˙w(0) and ˙hat w'(0) = ℓ ˙hat w(0) + ℓu3 − ˙w(0). These are not the same equations. Moreover, Ue = U×Y×U = C×C^2×C is four-dimensional, so the statement ue ∈ L^2_loc(0,∞;C^3) in Proposition 4.1 is inconsistent. As written, Proposition 4.1 does not follow from Theorem 3.2 for the displayed controller. Please correct the display (or the abstract parameter choice) and verify all boundary conditions explicitly.
[§4.2, Prop. 4.2] The verification of Theorem 3.6 for the heat equation rests on the assertions ∥H(λ)∥ ≲ |λ|^{−3/4} and ∥B_dL(λ−A_d)^{−1}∥ ≲ |λ|^{−1/4} on the sector |arg λ| ≤ π/2 + δ, justified only as 'direct computations'. Since Assumption 3.1(d) is explicitly stated to fail for this example, these estimates are the only support for applying Theorem 3.6. The Paley–Wiener reduction of the output-map well-posedness is also only sketched. Please provide the actual calculations (or a precise reference) for both resolvent estimates and for the Paley–Wiener condition.
[§3, proof of Thm. 3.6] In the proof of Theorem 3.6, the uniform boundedness of ∥H_K(λ)∥ ∥B_dL(λ−A_d)^{−1}∥ and of the analogous H_LQ term is established by invoking 'the arguments in the proof of [24, Prop. 2.6]' and 'similar arguments as in the proof of [9, Prop. 2.11]'. This is the crucial step that replaces Assumption 3.1(d). Please state explicitly the resolvent identity and the feedback formula used, and show how the assumed uniform bound on λ ↦ ∥H(λ)∥ ∥(B_d−L_dC)(λ−A_d)^{−1}∥ implies the required bounds.

minor comments (4)

[§4.1] In the controller display (4.2), the symbol u3 is not defined. In the abstract controller, the observer input u2 belongs to Y = C^2; please reconcile notation (e.g., write u2 = (u2_1,u2_2) and identify u3 with u2_2).
[§4.1, proof of Prop. 4.1] When verifying Assumption 3.1(d) via Remark 3.5, the entry Pdd of the matrix P0(λ) should be explicitly identified. From the displayed P0, Pdd appears to be 1/tanh(λ), but this should be stated because the ordering of rows/columns is not self-evident.
[§4.2, proof of Prop. 4.2] The statement 'Since K = −κC, (B1,A1,C1,0,0) has a well-posed output map if (B1,A1,C'_1,0,0) ... has a well-posed output map' is true but should be justified in one sentence, since C1 and C'_1 differ by a bounded invertible transformation.
[General] The paper relies on several key results from the companion preprint [9] (Propositions 2.5, 2.6, 2.7, 2.8, 2.9). Please ensure these results are available or reproduced sufficiently, since they underpin the main proof.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

No new physical entities are postulated: the separator/inverter/observer are algorithmic components, and the total disturbance d_tot = d + phi_s(x) + phi_o(y) is a definition of the lumped unknown input, not a new entity with an independent falsifiable handle. The free parameters are controller gains (theorems hold for a class of them, with stability of TK, TL, Td as verifiable conditions). The main epistemic cost is the un-auditable dependency on the same-group preprint [9] and the example-specific resolvent computations.

free parameters (1)

controller gain operators K, L, Li (kappa, ell in examples) = kappa>0, ell>0 arbitrary
Designer-chosen operators, not fitted to data. Theorems are conditional on K, L, Li satisfying Assumption 3.1 (feedback well-posedness, H-infinity conditions) and on exponential stability of TK, TL, Td; for the examples any positive kappa, ell work. No data-fitting loop exists.

assumptions (7)

standard math Well-posed linear system and boundary node framework (Staffans; Malinen-Staffans; Tucsnak-Weiss): input/output/input-output maps, transfer functions, growth bounds, Gearhart-Pruess-Greiner and Paley-Wiener admissibility tools.
Background operator theory invoked throughout Sections 2-4 (refs [18,21,27,28]); treated as unproved standard tools.
domain assumption Section 2 machinery from unreviewed same-group preprint [9] (Fkirine-Paunonen, arXiv:2605.28189): Propositions 2.5, 2.7, 2.8, 2.9, Lemma 2.12 (classical-solution regularity, well-posedness criteria, feedback and cascade well-posedness, partial flow inversion).
Load-bearing for Theorem 3.2; cited, not proved here, and not machine-checked or yet peer-reviewed. A reader cannot independently verify these foundations.
domain assumption The plant is a well-posed boundary node with collocated structure (3.2) and augmented well-posed node (Assumption 3.1(a)).
Defines the class of systems the controller applies to; Section 3, equations (3.1)-(3.2).
domain assumption Feedback well-posedness: (I - PK(.))^{-1} in H-infinity(C+_beta; L(U)) and (I - PL(.))^{-1} in H-infinity(C+_beta; L(Y)) (Assumption 3.1(c)).
Ensures the state-feedback and output-injection loops are well-defined; verified in the examples via dissipativity (wave) and Theorem 2.7(c) (heat).
domain assumption Partial flow inversion well-posedness: (B_h - L_h C, A_d, B_d - L_d C, 0, 0) has a well-posed output map (Assumption 3.1(d)), or the Theorem 3.6 substitute: lambda maps to ||H(lambda)|| ||(B_d - L_d C)(lambda - A_d)^(-1)|| uniformly bounded on C+_beta.
The invertibility premise of ADRC - the measured output must reconstruct the input asymptotically. Fails for naive heat configurations; the paper discloses this and handles it with Theorem 3.6.
domain assumption Nonlinearities: phi_s : X -> U locally Lipschitz with linear growth, phi_o : Y -> U globally Lipschitz (Assumption 3.1(b)).
Standard well-posedness conditions for the nonlinear closed loop; used in Propositions 2.6 and 2.10.
domain assumption Exponential stability of the auxiliary semigroups TK, TL, Td (hypotheses of Theorem 3.2(b)-(c) and Theorem 3.6).
Gives the exponential rates in the conclusions. For the wave example verified via [1, Thm 3.5], [2, Thm 10.1]; for the heat example via analytic semigroup theory. Not an automatic consequence of the controller design.

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Cite this review

Pith. "Pith review of Active Disturbance Rejection for Boundary Control Systems." pith.science (2026). https://pith.science/paper/3WBACR6C

@misc{pith2026260719144,
  author       = {Pith},
  title        = {Pith review of: Active Disturbance Rejection for Boundary Control Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3WBACR6C}},
  note         = {Machine review of arXiv:2607.19144}
}

read the original abstract

We consider stabilisation of abstract boundary control systems and controlled partial differential equations with general unknown input disturbances and unmodeled nonlinearities at the input. We utilise the active disturbance rejection control approach to design a controller which rejects the input disturbance and achieves stability and external well-posedness of the closed-loop system for a class of boundary control systems with collocated inputs and outputs. We apply our main results to design controllers for one-dimensional wave and heat equations.

Figures

Figures reproduced from arXiv: 2607.19144 by the authors.

**Figure 2.** The structure of the active disturbance rejection controller. The closed-loop system consisting of (3.1) and the controller (3.3) has state xe(t) = (x(t), xs(t), xi(t), xˆ(t)) ∈ Xe := X4 , input ue(t) = (u1(t), u2(t), d(t)) ∈ Ue := U × Y × U, and output ye(t) = (y(t), yˆ(t), ˆd(t), u0(t)) ∈ Ye := [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

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Works this paper leans on

34 extracted references · 2 linked inside Pith

[9]

Dynamic Stabilisat ion of Boundary Control Systems

Mohamed Fkirine and Lassi Paunonen. Dynamic Stabilisat ion of Boundary Control Systems. arXiv e-prints , page arXiv:2605.28189, May 2026

arXiv 2026
[33]

Output feedback expon ential stabilization for one-dimensional unstable wave equations with boundary con trol matched distur- bance

Hua-Cheng Zhou and George Weiss. Output feedback expon ential stabilization for one-dimensional unstable wave equations with boundary con trol matched distur- bance. SIAM J. Control Optim. , 56(6):4098–4129, 2018

2018
[1]

Non- uniform stability of damped contraction semigroups

Ralph Chill, Lassi Paunonen, David Seifert, Reinhard St ahn, and Yuri Tomilov. Non- uniform stability of damped contraction semigroups. Anal. PDE , 16(5):1089–1132, 2023

2023
[2]

The rate at which energy de cays in a string damped at one end

Steven Cox and Enrique Zuazua. The rate at which energy de cays in a string damped at one end. Indiana Univ. Math. J. , 44(2):545–573, 1995. ACTIVE DISTURBANCE REJECTION CONTROL FOR BCS 29

1995
[3]

Stabiliza- tion of a Timoshenko beam system with boundary matched distu rbances by designing novel disturbance estimator

Xueru Fan, Cheng-Zhong Xu, Xi Xu, Chunhai Kou, and Hua-Ch eng Zhou. Stabiliza- tion of a Timoshenko beam system with boundary matched distu rbances by designing novel disturbance estimator. Math. Control Relat. Fields , 15(2):690–711, 2025

2025
[4]

Distributed disturba nce estimator and appli- cation to stabilization for multi-dimensional wave equati on with corrupted boundary observation

Hongyinping Feng and Bao-Zhu Guo. Distributed disturba nce estimator and appli- cation to stabilization for multi-dimensional wave equati on with corrupted boundary observation. Automatica J. IF AC, 66:25–33, 2016

2016
[5]

Active disturbance re jection control: Old and new results

Hongyinping Feng and Bao-Zhu Guo. Active disturbance re jection control: Old and new results. Annu. Rev. Control , 44:238–248, 2017

2017
[6]

A new active disturban ce rejection control to output feedback stabilization for a one-dimensional anti- stable wave equation with disturbance

Hongyinping Feng and Bao-Zhu Guo. A new active disturban ce rejection control to output feedback stabilization for a one-dimensional anti- stable wave equation with disturbance. IEEE Trans. Automat. Control , 62(8):3774–3787, 2017

2017

Show all 34 references

[7]

New unknown input obse rver and output feed- back stabilization for uncertain heat equation

Hongyinping Feng and Bao-Zhu Guo. New unknown input obse rver and output feed- back stabilization for uncertain heat equation. Automatica J. IF AC, 86:1–10, 2017

2017
[8]

Obs ervers and disturbance rejection control for a heat equation

Hongyinping Feng, Cheng-Zhong Xu, and Peng-Fei Yao. Obs ervers and disturbance rejection control for a heat equation. IEEE Trans. Automat. Control , 65(11):4957– 4964, 2020

2020
[10]

Chris Guiver, Hartmut Logemann, and Mark R. Opmeer. Inﬁ nite-dimensional Lur’e systems: input-to-state stability and convergence proper ties. SIAM J. Control Op- tim., 57(1):334–365, 2019

2019
[11]

Sliding mode and active di sturbance rejection control to stabilization of one-dimensional anti-stable wave equa tions subject to disturbance in boundary input

Bao-Zhu Guo and Feng-Fei Jin. Sliding mode and active di sturbance rejection control to stabilization of one-dimensional anti-stable wave equa tions subject to disturbance in boundary input. IEEE Trans. Automat. Control , 58(5):1269–1274, 2013

2013
[12]

Output feedback stabiliz ation for one-dimensional wave equation subject to boundary disturbance

Bao-Zhu Guo and Feng-Fei Jin. Output feedback stabiliz ation for one-dimensional wave equation subject to boundary disturbance. IEEE Trans. Automat. Control , 60(3):824–830, 2015

2015
[13]

Active Disturbance Rejection Control for Nonlinear Systems: An Introduction

Bao-Zhu Guo and Zhi-Liang Zhao. Active Disturbance Rejection Control for Nonlinear Systems: An Introduction . John Wiley & Sons Inc., 2016

2016
[14]

The active disturbance rejection control to sta- bilization for multi-dimensional wave equation with bound ary control matched dis- turbance

Bao-Zhu Guo and Hua-Cheng Zhou. The active disturbance rejection control to sta- bilization for multi-dimensional wave equation with bound ary control matched dis- turbance. IEEE Trans. Automat. Control , 60(1):143–157, 2015

2015
[15]

From PID to active disturbance rejection control

Jingqing Han. From PID to active disturbance rejection control. IEEE Trans. Ind. Electron., 56(3):900–906, March 2009

2009
[16]

Wel l-posedness of inﬁnite- dimensional linear systems with nonlinear feedback

Anthony Hastir, Federico Califano, and Hans Zwart. Wel l-posedness of inﬁnite- dimensional linear systems with nonlinear feedback. Systems Control Lett., 128:19–25, 2019

2019
[17]

Well-posedness and stability of inﬁnite-dimensional systems under monotone feedback

Anthony Hastir and Lassi Paunonen. Well-posedness and stability of inﬁnite-dimensional systems under monotone feedback. arXiv e-prints , page arXiv:2503.16092, March 2025

2025 arXiv
[18]

Linear Port-Hamiltonian Systems on Inﬁnite- Dimensional Spaces , volume 223 of Operator Theory: Advances and Applications

Birgit Jacob and Hans Zwart. Linear Port-Hamiltonian Systems on Inﬁnite- Dimensional Spaces , volume 223 of Operator Theory: Advances and Applications . Birkh¨ auser, Basel, 2012

2012
[19]

Output feedback stabilizat ion of an ODE-Schr¨ odinger cascade system subject to boundary control matched unknown disturbance

Yan-Na Jia and Jun-Jun Liu. Output feedback stabilizat ion of an ODE-Schr¨ odinger cascade system subject to boundary control matched unknown disturbance. J. Dyn. Control Syst. , 26(2):393–405, 2020

2020
[20]

On stabilization of an axially moving string with a tip mass subject to an unbou nded disturbance

Abdelkarim Kelleche, Fardin Saedpanah, and Athmane Ab dallaoui. On stabilization of an axially moving string with a tip mass subject to an unbou nded disturbance. Math. Methods Appl. Sci. , 46(14):15564–15580, 2023

2023
[21]

Jarmo Malinen and Olof J. Staﬀans. Conservative bounda ry control systems. J. Dif- ferential Equations , 231(1):290–312, 2006

2006
[22]

Output feedback exponential stabiliza tion for a 1-d wave PDE with dynamic boundary

Zhan-Dong Mei. Output feedback exponential stabiliza tion for a 1-d wave PDE with dynamic boundary. J. Math. Anal. Appl. , 508(1):Paper No. 125860, 23, 2022

2022
[23]

Output feedback expo nential stabilization for a one-dimensional wave equation with control matched nonli near disturbance

Zhan-Dong Mei and Hua-Cheng Zhou. Output feedback expo nential stabilization for a one-dimensional wave equation with control matched nonli near disturbance. IEEE Trans. Automat. Control , 66(5):2273–2280, 2021. 30 J.-P. HUMALOJA AND L. PAUNONEN

2021
[24]

Stab ility of abstract coupled sys- tems

Serge Nicaise, Lassi Paunonen, and David Seifert. Stab ility of abstract coupled sys- tems. J. Funct. Anal. , page 110909, 2025

2025
[25]

Stability and robust regulation of pas sive linear systems

Lassi Paunonen. Stability and robust regulation of pas sive linear systems. SIAM J. Control Optim. , 57(6):3827–3856, 2019

2019
[26]

Inﬁnite-dimensional linear systems with unbounded control and observation: A functional analytic approach

Dietmar Salamon. Inﬁnite-dimensional linear systems with unbounded control and observation: A functional analytic approach. Trans. Amer. Math. Soc. , 300(2):383– 431, 1987

1987
[27]

Well-Posed Linear Systems

Olof Staﬀans. Well-Posed Linear Systems . Cambridge University Press, 2005

2005
[28]

Well-posed systems—T he LTI case and beyond

Marius Tucsnak and George Weiss. Well-posed systems—T he LTI case and beyond. Automatica J. IF AC, 50(7):1757–1779, 2014

2014
[29]

George Weiss and Ruth F. Curtain. Dynamic stabilizatio n of regular linear systems. IEEE Trans. Automat. Control , 42(1):4–21, 1997

1997
[30]

Disturbance esti mator based output feed- back exponential stabilization for Euler-Bernoulli beam e quation with boundary con- trol

Hua-Cheng Zhou and Hongyinping Feng. Disturbance esti mator based output feed- back exponential stabilization for Euler-Bernoulli beam e quation with boundary con- trol. Automatica J. IF AC, 91:79–88, 2018

2018
[31]

Stabilization fo r Euler-Bernoulli beam equa- tion with boundary moment control and disturbance via a new d isturbance estimator

Hua-Cheng Zhou and Hongyinping Feng. Stabilization fo r Euler-Bernoulli beam equa- tion with boundary moment control and disturbance via a new d isturbance estimator. J. Dyn. Control Syst. , 27(2):247–259, 2021

2021
[32]

Unknown input observer design and output feedback stabilization for multi-dimensional wave equati on with boundary control matched uncertainty

Hua-Cheng Zhou and Bao-Zhu Guo. Unknown input observer design and output feedback stabilization for multi-dimensional wave equati on with boundary control matched uncertainty. J. Diﬀerential Equations , 263(4):2213–2246, 2017

2017
[34]

Output-based disturbance rejection co ntrol for 1-D anti-stable Schr¨ odinger equation with boundary input matched unknown disturbance

Huacheng Zhou. Output-based disturbance rejection co ntrol for 1-D anti-stable Schr¨ odinger equation with boundary input matched unknown disturbance. Internat. J. Robust Nonlinear Control , 27(18):4686–4705, 2017. (J.-P. Humaloja) Department of Electrical and Computer Enginee...

2017

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [9]

Dynamic Stabilisat ion of Boundary Control Systems

Mohamed Fkirine and Lassi Paunonen. Dynamic Stabilisat ion of Boundary Control Systems. arXiv e-prints , page arXiv:2605.28189, May 2026

arXiv 2026

[2] [33]

Output feedback expon ential stabilization for one-dimensional unstable wave equations with boundary con trol matched distur- bance

Hua-Cheng Zhou and George Weiss. Output feedback expon ential stabilization for one-dimensional unstable wave equations with boundary con trol matched distur- bance. SIAM J. Control Optim. , 56(6):4098–4129, 2018

2018

[3] [1]

Non- uniform stability of damped contraction semigroups

Ralph Chill, Lassi Paunonen, David Seifert, Reinhard St ahn, and Yuri Tomilov. Non- uniform stability of damped contraction semigroups. Anal. PDE , 16(5):1089–1132, 2023

2023

[4] [2]

The rate at which energy de cays in a string damped at one end

Steven Cox and Enrique Zuazua. The rate at which energy de cays in a string damped at one end. Indiana Univ. Math. J. , 44(2):545–573, 1995. ACTIVE DISTURBANCE REJECTION CONTROL FOR BCS 29

1995

[5] [3]

Stabiliza- tion of a Timoshenko beam system with boundary matched distu rbances by designing novel disturbance estimator

Xueru Fan, Cheng-Zhong Xu, Xi Xu, Chunhai Kou, and Hua-Ch eng Zhou. Stabiliza- tion of a Timoshenko beam system with boundary matched distu rbances by designing novel disturbance estimator. Math. Control Relat. Fields , 15(2):690–711, 2025

2025

[6] [4]

Distributed disturba nce estimator and appli- cation to stabilization for multi-dimensional wave equati on with corrupted boundary observation

Hongyinping Feng and Bao-Zhu Guo. Distributed disturba nce estimator and appli- cation to stabilization for multi-dimensional wave equati on with corrupted boundary observation. Automatica J. IF AC, 66:25–33, 2016

2016

[7] [5]

Active disturbance re jection control: Old and new results

Hongyinping Feng and Bao-Zhu Guo. Active disturbance re jection control: Old and new results. Annu. Rev. Control , 44:238–248, 2017

2017

[8] [6]

A new active disturban ce rejection control to output feedback stabilization for a one-dimensional anti- stable wave equation with disturbance

Hongyinping Feng and Bao-Zhu Guo. A new active disturban ce rejection control to output feedback stabilization for a one-dimensional anti- stable wave equation with disturbance. IEEE Trans. Automat. Control , 62(8):3774–3787, 2017

2017