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REVIEW 4 major objections 4 minor 44 references

A Koopman-backstepping approach to data-driven robust output regulation for linear parabolic systems

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A fully data-driven controller can make an unknown linear parabolic system track references and reject unknown disturbances using only finite boundary-output data and two dominant Koopman modes.

desk verdict Novel inverse Sturm-Liouville identification from two Koopman modes is the real contribution; the output regulation claim rests on an unproven exact eigenvalue recovery. read the letter →

arxiv 2506.06451 v1 pith:E3E6MPHO submitted 2025-06-06 eess.SY cs.SY

Joachim Deutscher , Julian Zimmer This is my paper
classification eess.SYcs.SY MSC 93C2093B5234B24
keywords KoopmanoperatorparabolicPDEoutputregulationbacksteppingHankel-DMDinverseSturm-Liouvilleproblemdata-drivencontroldisturbancerejection
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 claims that the robust output regulation problem for a linear parabolic (reaction–diffusion) system with unknown plant and unknown disturbance model can be solved entirely from finite boundary-output measurements. Using the Koopman operator picture of the PDE, the authors extract eigenvalues and modes from sequential output data with Hankel-DMD, then show that two dominant Koopman modes suffice to recover all constant system parameters (diffusivity, reaction coefficient, and the two boundary feedback gains) by solving an inverse Sturm-Liouville problem. Since the disturbance input location vectors are not needed, the recovered parameters plug directly into a backstepping regulator design with an internal model. The paper proves closed-loop stability under small identification errors and robust output regulation for any model uncertainty that does not destabilize the closed loop, and demonstrates the result numerically.

What carries the argument

The engine of the method is the Koopman operator of the extended PDE–ODE system, a linear operator on observables whose eigenvalues are those of the plant plus the disturbance model and whose modes contain the boundary and point-evaluation data. Hankel-DMD approximates these eigenvalues and modes from the sequential measurements through a companion matrix, and SVD-enhanced DMD gives a numerically stable implementation. Two of the resulting Koopman modes and eigenvalues are then fed into the inverse Sturm-Liouville problem, i.e., equations (25)–(27), which recovers $\rho$, $a$, $q_0$, $q_1$ from the mode shapes at the boundaries and at the output point $z_0$. On the control side, the identified parameters enter the backstepping kernel equations (32) and the decoupling equations (36), together with an internal model (29) built from the identified disturbance eigenvalues; the controller is (29), (30).

What would settle it

Take a reaction-diffusion system with known parameters, generate finite-time output data with a persistently acting disturbance, add zero-mean measurement noise of increasing variance, and compare the inverse Sturm-Liouville recovery to the true parameters; if the recovered parameters leave the smallness region used in the stability theorem, the closed-loop tracking error will not converge to zero despite a small Hankel-DMD residual.

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Extended reading notes

Core claim

The central claim is that the unknown coefficients $\rho$, $a$, $q_0$, $q_1$ of the nominal parabolic PDE and the frequencies of the unknown disturbance generator can be recovered from measured output sequences alone, and that this data is enough to build a backstepping regulator that drives the tracking error to zero. The recovery step is the paper's key conceptual move: rather than approximating the PDE by a finite-dimensional model for control, it uses only two Koopman modes and the corresponding eigenvalues, solves the inverse Sturm-Liouville equations (25)–(27), and thereby obtains exact parameter values for the late-lumping design. Theorem 12 states that once the closed loop with the identified nominal model is strongly asymptotically stable, the same controller achieves output regulation for any disturbance input locations and any signal-model matrices, so the regulator is robust to non-destabilizing uncertainties.

Load-bearing premise

The result rests on the assumption that the errors in the recovered parameters $\rho$, $a$, $q_0$, $q_1$ are small enough for the nominal stability proof to apply, and the paper provides no computable bound connecting the size of the Hankel-DMD residual to those parameter errors.

Editorial extensions

If this is right

  • The controller can be designed without ever identifying the disturbance input location vectors $g_1,\dots,g_4$ and $G_5$; these quantities influence the data but drop out of the regulator design.
  • Because only two dominant modes are used, the parameter recovery is robust in the sense that the modes best represented in the data carry the identification, which keeps the Hankel-DMD errors small.
  • The identified parameters can be inserted directly into existing backstepping kernel solvers, so late-lumping regulation methods apply to plants identified purely from snapshots.
  • If the closed loop with the identified nominal parameters remains strongly asymptotically stable under model uncertainty, output regulation holds for any disturbance model matrices $P_d$ and $p_r^\top$, so the regulator need not be redesigned when the disturbance or reference spectrum changes.
  • Sampling time and data length can be selected from data by minimizing the SVD-enhanced Hankel-DMD residual, giving a systematic data-collection procedure.

Reading between the lines

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

  • The two-mode inverse Sturm-Liouville step suggests that the same identification philosophy could handle spatially varying coefficients, although the paper notes that this would in principle require infinitely many Koopman modes rather than two.
  • If a quantitative error bound were derived from the Hankel-DMD residual to the parameter errors, the 'sufficiently small' condition in the stability theorem would become checkable from data; the current paper leaves that propagation unquantified.
  • The method appears directly testable on Dirichlet boundary conditions and measurement-noise studies; the paper states the Dirichlet extension but leaves noise analysis to future work.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a fully data-driven regulator for robust output regulation of a class of 1-D linear parabolic PDEs with boundary control, unknown plant parameters, and an unknown exosystem generating disturbances. The method uses Hankel-DMD to extract Koopman modes and eigenvalues from boundary output data, solves an inverse Sturm-Liouville problem with two dominant modes to identify the plant parameters ρ, a, q0, q1 and the disturbance eigenvalues, and then applies a backstepping-based robust regulator from prior work. The central theoretical claim (Theorem 12) is that the controller achieves asymptotic tracking and disturbance rejection for any non-destabilizing model uncertainties, provided the nominal closed loop is strongly asymptotically stable. A numerical example demonstrates the approach on an unstable parabolic system with a sinusoidal disturbance and a ramp reference.

Significance. If the claims are valid, this would be a meaningful advance: it appears to be the first fully data-driven output regulation result for distributed-parameter systems in which neither the plant nor the disturbance model is known and in which the disturbance input locations need not be identified. The paper also contributes an extension of Hankel-DMD to PDE-ODE systems and gives conditions for exact Hankel-DMD in Theorem 6. The inverse Sturm-Liouville recovery from only two modes is elegant and could be of independent interest. However, the significance is tempered by the fact that the main regulation theorem rests on an exact-eigenvalue assumption that is stated heuristically, and the stability theorem is proved only for unquantified small identification errors. The numerical validation is noiseless and tunes its hyperparameters on the same data, so the robustness claims are not yet convincingly supported.

major comments (4)
  1. [Section 5, Remark 10 and Theorem 12] The internal model (29) must contain the exact eigenvalues of the disturbance model (3), since output regulation for a sinusoidal exosystem requires exact internal-model replication; a small eigenvalue error δ produces a steady-state tracking error whose amplitude scales with δ and the plant gain at that frequency. Remark 10 merely asserts that persistently exciting disturbances allow these eigenvalues to be determined accurately, but the Hankel-DMD of Theorem 4 provides only approximate eigenvalues λ̂_i = ln μ_i/t_s, with no bound relating ∥σ(S_d) − σ(Ŝ_d)∥ to the residual in (15) or (17). Consequently, the proof invoked from [10,14] does not apply, and the conclusion of Theorem 12 is not established under the assumptions actually stated. The manuscript needs either a provable condition under which the identified eigenvalues coincide with σ(S_d), or an explicit error-propagation analysis that would downgrade the claim to practical regulation with quantified steady-state error.
  2. [Section 5, Theorem 11 and Remark 5] Theorem 11 guarantees nominal closed-loop stability only for 'sufficiently small' identification errors Δρ̂, Δâ, Δq̂0, Δq̂1, but no quantitative bound is derived from Hankel-DMD residuals to these parameter errors. Remark 5 selects ts and n by minimizing the Frobenius residual on the same dataset, so the reported accurate parameter recovery in the example does not provide independent validation that the required smallness holds. For noisy, short, or poorly excited data the hypothesis of Theorem 11 may fail without any way to detect it from the available residual. The authors should either provide an explicit continuity bound or clearly formulate the smallness condition as an assumption and discuss how it could be verified.
  3. [Section 4.2, eqs. (25)–(27)] The inverse Sturm-Liouville problem is solved from two Koopman modes and eigenvalues, and the manuscript acknowledges that the resulting transcendental equation 'may have multiple solutions' and that a valid solution is verified by re-solving the eigenvalue problem. With exact algebraic data this verification may select the correct branch, but with noisy or approximate Hankel-DMD data multiple candidate parameter sets may pass the eigenvalue verification within the noise level. The paper provides no uniqueness criterion, no continuity result for the map from measured modes/eigenvalues to (ρ, a, q0, q1), and no procedure to select the correct solution in the approximate case. This gap affects the well-definedness of the identified controller, which is load-bearing for the nominal design in Section 5.
  4. [Section 7] The numerical example uses noiseless simulated data and chooses ts = 0.104 and n = 6 by minimizing the Hankel-DMD residual on the same dataset, which masks potential overfitting and does not test robustness to measurement noise or to data sets not perfectly aligned with the dominant subspace. Since the paper's abstract and introduction emphasize a data-driven method, a validation with noisy data or with an independent test set would substantially strengthen the demonstration. As written, the example verifies the theory only in an idealized scenario that already satisfies the exactness assumptions of Theorem 6.
minor comments (4)
  1. [Section 1] In the introduction, 'By extended the Hankel-DMD' should read 'By extending the Hankel-DMD'.
  2. [Remark 5] Remark 5 refers to 'Theorem 6', but Theorem 6 is stated after the remark and concerns exact Hankel-DMD, not the SVD-enhanced variant described in the remark. The cross-reference should be corrected or the remark should refer to the SVD-enhanced implementation directly.
  3. [Theorem 4] The symbol MΦ̂_i is used in (17), but the matrix M defining the Koopman modes is introduced in (13); the dependence is clear from context but a short reminder would improve readability.
  4. [Throughout] There are several minor typographical issues, e.g., 'disturbance model is determined from measurement data' could be phrased more precisely, and 'steal industry' should be 'steel industry' in Section 2.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity (score 2): the output-regulation claim is conditionally derived from published backstepping-regulator theorems and is tested against the true plant; the only flagged dependence is Remark 10, which asserts exact recovery of the disturbance spectrum from the Hankel-DMD fit without an error bound, a robustness gap rather than a by-construction reduction.

  1. other [Remark 10 (Sec. 5, p. 7) bridging Theorem 4 (Hankel-DMD) to Theorem 12 (Sec. 6, p. 8)]
    "In (28) it is assumed that the spectrum σ(Sd) can be determined exactly from the data. This is justified, because the disturbances are assumed to be persistently exciting and thus are well represented in the sequential data. Hence, they can be determined accurately using the Hankel-DMD."

    The internal model (29) is built directly from the Hankel-DMD-fitted spectrum σ( S̃d ), and Theorem 12's conclusion e_y→0 holds only if that fitted spectrum equals the true exosystem spectrum σ(Sd). Theorem 4 supplies only approximate eigenvalues λ̂_i = ln(μ_i)/t_s with the residual bound (17), and no theorem in the paper bounds ∥σ(Sd) − σ(S̃d)∥ in terms of that residual or of Remark 5's in-sample residual minimization. Remark 10 bridges the data fit to the theorem premise by assertion ('it is assumed ... can be determined exactly'), not by a derived error bound.

full rationale

I walked the derivation chain: data (4) → Hankel matrix (14) → companion matrix F (Lemma 3) → DMD eigenvalue/mode estimates (Theorem 4; exactness conditions in Theorem 6) → two slowest modes solve the inverse Sturm-Liouville problem (25)–(27) for ρ, a, q0, q1 (Sec. 4.2) → backstepping regulator (29), (30) designed along [10, 14] → Theorem 11 proves exponential stability for sufficiently small identification errors Δρ̂, Δâ, Δq̂0, Δq̂1 with a self-contained appendix proof → Theorem 12 imports the regulation conclusion from [10, 14] under a strong-stability assumption. At no point does an equation reduce to its own input: the ISLP genuinely recovers parameters from measured mode values, and the controller is then simulated against the true plant with 70% parameter perturbations (Figs. 2–3), i.e., an external benchmark that would fail if the identified parameters were wrong. The regulator-design theorems [10, 12, 14] are self-citations and are load-bearing for Theorem 12's proof, but they are published, parameter-free theorems that do not contain the present data-driven result, so under the review rules they count as real evidence rather than circularity. The genuine weaknesses are: (i) Remark 10 asserts exact recovery of σ(Sd) with no quantitative bound from DMD residuals — the skeptic's attack identifies a real correctness gap, not a circular reduction; (ii) Remark 5 selects t_s and n by minimizing the DMD residual on the same dataset, so the 'accurate' parameter claim in Remark 9 is not validated independently of the training data. Both concerns affect the robustness of the proof, not the logical independence of the conclusion from the fitted inputs, so the appropriate finding is no significant circularity (score 2).

Assumptions & free parameters 9 free parameters · 8 assumptions · 0 invented entities

The central claim rests on the system class assumptions, the rank condition on the disturbance transfer function, the richness of the data, and the qualitative small-error assumption in Theorem 11. The estimated parameters are fitted from data; the regulator design additionally depends on prior backstepping results.

free parameters (9)
  • diffusion coefficient rho_hat = 1.510 (example)
    Recovered from data via Hankel-DMD and inverse Sturm-Liouville (Section 4.2, Section 7).
  • reaction coefficient a_hat = 8.020 (example)
    Recovered from data via Hankel-DMD and inverse Sturm-Liouville (Section 4.2, Section 7).
  • boundary parameter q0_hat = 2.491 (example)
    Recovered from data via Hankel-DMD and inverse Sturm-Liouville (Section 4.2, Section 7).
  • boundary parameter q1_hat = -1.992 (example)
    Recovered from data via Hankel-DMD and inverse Sturm-Liouville (Section 4.2, Section 7).
  • disturbance frequency omega_d_hat = 3.1416 (example)
    Recovered as a Koopman eigenvalue from the Hankel-DMD data (Section 7).
  • sampling time t_s = 0.104 (example)
    Chosen by minimizing the Hankel residual norm over a grid (Remark 5).
  • Hankel order n = 6 (example)
    Chosen together with t_s by residual minimization (Remark 5).
  • backstepping design parameter mu_c = 5 (chosen)
    Design parameter set by the user to set the decay rate (Section 5).
  • internal model eigenvalue placement = -4.5 +/- j pi, -4, -5 (chosen)
    Placed by the feedback gain k_ϖ^T during design (Section 7).
assumptions (8)
  • domain assumption The plant is a 1D linear parabolic PDE with constant nominal coefficients, boundary control at z=1, and interior point measurement (Equations (1)-(2)).
    The entire method is specialized to this system class; the backstepping and inverse Sturm-Liouville steps rely on this form.
  • domain assumption The disturbance model (3) is an observable ODE with simple eigenvalues on the imaginary axis.
    Needed to ensure the internal model captures the disturbance and that the data reveal its eigenvalues.
  • domain assumption rank F_d(lambda_i) = q for the disturbance eigenvalues, which implies q <= 3 (Theorem 6, Remark 7).
    Required for exact Hankel-DMD recovery of all disturbance-related Koopman modes.
  • domain assumption The initial condition has nonzero components in the n selected modes, and the dominant modes are well represented in the data (Equation (16), Remark 9).
    Otherwise the Hankel matrix loses rank and the Koopman modes cannot be recovered.
  • domain assumption The Hankel-DMD identification errors are sufficiently small so that the nominal closed-loop remains exponentially stable (Theorem 11).
    The theorem is qualitative; no quantitative bound is given for the allowable error in terms of data length, sampling time, or noise.
  • domain assumption The measurement data are noise-free or noise is negligible.
    The paper does not model noise, and Section 8 lists measurement noise as future work. This is an implicit assumption in the derivation and example.
  • standard math Standard Sturm-Liouville and Riesz basis results used in Lemma 1 and Section 4.2.
    Background from [9,20,27] used without proof.
  • standard math The backstepping regulator results from the authors' prior work [10,14] are correct and applicable.
    Theorem 12 is stated to follow directly from [10,14]; the paper relies on these results.

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Cite this review

Pith. "Pith review of A Koopman-backstepping approach to data-driven robust output regulation for linear parabolic systems." pith.science (2026). https://pith.science/paper/E3E6MPHO

@misc{pith2026250606451,
  author       = {Pith},
  title        = {Pith review of: A Koopman-backstepping approach to data-driven robust output regulation for linear parabolic systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E3E6MPHO}},
  note         = {Machine review of arXiv:2506.06451}
}
read the original abstract

In this paper a solution of the data-driven robust output regulation problem for linear parabolic systems is presented. Both the system as well as the ODE, i.e., the disturbance model, describing the disturbances are unknown, but finite-time sequential data obtained from measurements of the output to be controlled and additional boundary outputs are available. The data-driven controller is designed in the Koopman operator framework for PDEs, where the Koopman modes and eigenvalues are obtained from data using Hankel-DMD. It is shown that all system parameters and the eigenvalues of the disturbance model can be recovered from the available measurements by solving an inverse Sturm-Liouville problem. This allows to directly apply backstepping methods for the robust regulator design. For this, closed-loop stability in the presence of small errors in the Hankel-DMD is verified in the nominal case. Robust output regulation is shown for non-destabilizing model uncertainties. A numerical example demonstrates the results of the paper.

Figures

Figures reproduced from arXiv: 2506.06451 by the authors.

Figure 1
Figure 1. Eigenvalues of the nominal system ( ), the distur￾bance model ( ) and the eigenvalues resulting from the Han￾kel-DMD ( ). x(z, −0.1) = 0, where s(t) is the step function. The dom￾inant modes are then well contained in the output data (4) for t ≥ 0. Afterwards, the SVD-enhanced Hankel￾DMD is used, because of numerically well-posedness. By calculating the norm of the residuum Rsvd over dif￾ferent sampling times ts and… view at source ↗
Figure 3
Figure 3. Closed-loop disturbance behavior of y(t) ( ) for d(t) = sin(πt) and r(t) ≡ 0. Upper plot shows the nominal case and lower plot with model uncertainty. lation. 8 Concluding remarks It is straightforward to extend the considered state feedback regulator by a backstepping observer using the available data. Further research considers systems with spatially varying coefficients and the extension to second-order systems l… view at source ↗

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Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    Ackermann

    J. Ackermann. Sampled-Data Control Systems . Springer, Berlin, 1985

    work page 1985

  2. [2]

    Arbabi and I

    H. Arbabi and I. Mezi´ c. Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator. SIAM J. Appl. Dyn. Syst., 16:2096–2126, 2017

    work page 2017

  3. [3]

    Aulisa and D

    E. Aulisa and D. Gilliam. A Practical Guide to Geometric Regulation for Distributed Parameter Systems . CRC Press, Boca Raton, 2016

    work page 2016

  4. [4]

    Banks and K

    H. Banks and K. Kunisch. Estimation Techniques for Distributed Parameter Systems . Birkh¨ auser, Boston, 1998

    work page 1998

  5. [5]

    Bernstein

    D. Bernstein. Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas . Princeton University Press, Princeton, New Jersey, 2018

    work page 2018

  6. [6]

    Brunton and N

    S. Brunton and N. Kutz. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, 2022

    work page 2022

  7. [7]

    K. Chen, J. Tu, and C. Rowley. Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlin. Science , 22:887–915, 2012

    work page 2012

  8. [8]

    Curtain and H

    R. Curtain and H. Zwart. Introduction to Infinite- Dimensional Systems Theory . Springer-Verlag, New York, 2020

    work page 2020

Show all 44 references
  1. [9]

    Delattre, D

    C. Delattre, D. Dochain, and J. Winkin. Sturm-Liouville systems are Riesz-spectral systems. Int. J. Appl. Math. Comput. Sci. , 13:481–484, 2003

    2003

  2. [10]

    Deutscher

    J. Deutscher. Backstepping design of robust output feedback regulators for boundary controlled parabolic PDEs. IEEE Trans. Autom. Control, 61:2288–2294, 2016

    2016

  3. [11]

    Deutscher

    J. Deutscher. Data-driven robust output regulation of continuous-time LTI systems using Koopman operators. IEEE Trans. Autom. Control, 69:8774–8781, 2024

    2024

  4. [12]

    Deutscher

    J. Deutscher. Data-driven control of parabolic systems using Koopman eigenstructure assignment. IEEE Trans. Autom. Control, 70:665–672, 2025

    2025

  5. [13]

    Deutscher and S

    J. Deutscher and S. Kerschbaum. Output regulation for coupled linear parabolic PIDEs. Automatica, 100:360–370, 2019

    2019

  6. [14]

    Deutscher and S

    J. Deutscher and S. Kerschbaum. Robust output regulation by state feedback control for coupled linear parabolic PIDEs. IEEE Trans. Autom. Control, 65:2207–2214, 2020

    2020

  7. [15]

    Drmaˇ c, I

    Z. Drmaˇ c, I. Mezi´ c, and R. Mohr. Data driven modal decompositions: Analysis and enhancements. SIAM J. Sci. Comp., 40:A2253–A2285, 2018

    2018

  8. [16]

    Engel and R

    K.-J. Engel and R. Nagel. One-parameter Semigroups for Linear Evolution Equations . Springer Verlag, New York, 2000

    2000

  9. [17]

    Gehring and J

    N. Gehring and J. Rudolph. An algebraic algorithm for parameter identification in a class of systems described by linear partial differential equations. PAMM, 16:39–42, 2016

    2016

  10. [18]

    E.G. Gilbert. Controllability and observability in multivariable control systems. SIAM Journal Control, Ser. A, 1:128–151, 1963

    1963

  11. [19]

    Goswami and D

    D. Goswami and D. Paley. Bilinearization, reachability, and optimal control of control-affine nonlinear systems: a Koopman spectral approach. IEEE Trans. Autom. Control , 67:2715–2728, 2022

    2022

  12. [20]

    Guo and J.-M

    B.-Z. Guo and J.-M. Wang. Control of Wave and Beam PDEs: The Riesz Basis Approach . Springer, Cham, 2019. 10

    2019

  13. [21]

    Guo and R.-X

    B.-Z. Guo and R.-X. Zhao. Output regulation for a heat equation with unknown exosystem. Automatica, 138:110159, 2022

    2022

  14. [22]

    F. Huang. Strong asymptotic stability of linear dynamical systems in Banach spaces. J. Diff. Equns, 104:307–324, 1993

    1993

  15. [23]

    Kaiser, J

    E. Kaiser, J. Kutz, and S. Brunton. Data-driven discovery of Koopman eigenfunctions for control. Machine Learning: Science and Technology, 2, 2021. Art. no. 035023

    2021

  16. [24]

    Kerschbaum

    S. Kerschbaum. Backstepping Control of Coupled Parabolic Systems with Varying Parameters . F AU University Press, Doctoral dissertation, University Erlangen, 2021

    2021

  17. [25]

    Kn¨ uppel and F

    T. Kn¨ uppel and F. Woittennek. A contribution to parameter identification in infinite-dimensional systems. In European Control Conference (ECC), pages 1591–1596, 2013

    2013

  18. [26]

    Kong and A

    Q. Kong and A. Zettl. Eigenvalues of regular Sturm-Liouville problems. J. Diff. Eqns. , 131:1–19, 1996

    1996

  19. [27]

    Kravchenko

    V. Kravchenko. Direct and Inverse Sturm-Liouville Problems. Birk¨ auser, Cham, 2020

    2020

  20. [28]

    Krstic, L

    M. Krstic, L. Bhan, and Y. Shi. Neural operators of backstepping controller and observer gain functions for reaction-diffusion PDEs. Automatica, 164:111649, 2024

    2024

  21. [29]

    Krstic and A

    M. Krstic and A. Smyshlyaev. Boundary Control of PDEs — A Course on Backstepping Designs . SIAM, Philadelphia, 2008

    2008

  22. [30]

    Kubrulsy

    C. Kubrulsy. Distributed parameter system indentification – A survey. Int. J. Control , 26:509–535, 1977

    1977

  23. [31]

    Lancaster and M

    P. Lancaster and M. Tismenetsky. The Theory of Matrices . Academic Press, San Diego, 1985

    1985

  24. [32]

    L. Ljung. System Identification. Prentice Hall, Upper Saddle River, 1998

    1998

  25. [33]

    A. Mauroy. Koopman operator framework for spectral analysis and identification of infinite-dimensional systems. Mathematics, 9:2495, 2021

    2021

  26. [34]

    Mauroy and J

    A. Mauroy and J. Goncalves. Parameter estimation and identification of nonlinear systems with the Koopman operator. In A. Mauroy, I. Mezi´ c, and Y. Susuki, editors, The Koopman Operator in Systems and Control: Concepts, Methodologies, and Applications , pages 335–357. Springe...

    2020

  27. [35]

    Mauroy, I

    A. Mauroy, I. Mezi´ c, and Y. Susuki.The Koopman Operator in Systems and Control . Springer Cham, London, 2020

    2020

  28. [36]

    I. Mezi´ c. On numerical approximations of the Koopman operator. Mathematics, 10:1180, 2022

    2022

  29. [37]

    Nakao and I

    H. Nakao and I. Mezi´ c. Spectral analysis of the Koopman operator for partial differential equations. Chaos, 30:113131, 2020

    2020

  30. [38]

    Y. Orlov. On general properties of eigenvalues and eigenfunctions of a Sturm-Liouville operator: comments on ”ISS with respect to boundary disturbances for 1-d parabolic PDEs”. IEEE Trans. Autom. Control, 62:5970–5973, 2017

    2017

  31. [39]

    Pandolfi and H

    L. Pandolfi and H. Zwart. Stability of perturbed linear distributed parameter systems. Syst. Cont. Lett., 17:257–264, 1991

    1991

  32. [40]

    Polis and R

    M. Polis and R. Goodson. Parameter identification in distributed systems: A synthesizing overview. Proc. IEEE, 64:45–61, 1976

    1976

  33. [41]

    Rudolph and F

    J. Rudolph and F. Woittennek. An algebraic approach to parameter identification in linear infinite dimensional systems. In Medit. Conf. Control Aut. (MED) , pages 332– 337, 2008

    2008

  34. [42]

    P. Schmid. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mechanics , 656:5–28, 2010

    2010

  35. [43]

    Smyshlyaev and M

    A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Trans. Autom. Control, 49:2185–2202, 2004

    2004

  36. [44]

    H. Zwart. Transfer functions for infinite-dimensional systems. Syst. Control Lett. , 52:247–255, 2004. 11

    2004

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Reviewed August 7, 2026 · model on record in the stance chip above.