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arxiv: 2604.12862 · v1 · submitted 2026-04-14 · 🧮 math.OC · cs.SY· eess.SY

From Interpolation to mathcal{H}₂ Optimality: Model Reduction for Infinite-Dimensional Linear Control Systems

Cankat Tilki , Tobias Breiten , Serkan Gugercin This is my paper

Pith reviewed 2026-05-10 14:56 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords model reductionH2 optimalityHermite interpolationinfinite-dimensional systemsPetrov-Galerkin projectiondata-driven realizationtransfer functionlinear control systems
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The pith

H2-optimal reduced models for infinite-dimensional linear systems require Hermite interpolation of the transfer function at the mirror images of the reduced poles.

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

The paper shows that the full interpolatory framework for H2-optimal model reduction carries over without change to linear control systems whose state, input, and output spaces are infinite-dimensional. It proves that suitable Petrov-Galerkin projections onto finite-dimensional subspaces enforce interpolation of the original transfer function at chosen points, and it supplies an infinite-dimensional analogue of the Loewner framework that recovers the reduced model directly from input-output data. The central optimality result follows: any locally H2-optimal reduced model must satisfy Hermite interpolation conditions at the reflected poles of the reduced transfer function. If these statements hold, engineers can produce optimal finite-dimensional approximations for distributed-parameter systems such as those arising from PDEs while using only measured data and without assembling the full infinite-dimensional operators.

Core claim

First-order H2 optimality of the reduced-order model requires Hermite interpolation of the full-order transfer function at the mirror images of the reduced model's poles. This characterization remains valid for systems posed as abstract Cauchy problems on Banach spaces. The required interpolation is realized by constructing the trial and test subspaces of the Petrov-Galerkin projection so that the finite-dimensional reduced transfer function matches the original at the prescribed points and their derivatives. A data-driven realization procedure recovers the reduced model from input-output samples alone, without explicit knowledge of the underlying operators.

What carries the argument

Petrov-Galerkin projection onto finite-dimensional trial and test subspaces constructed to enforce Hermite interpolation of the transfer function at selected points.

If this is right

  • Any reduced model obtained by a projection whose subspaces satisfy the interpolation conditions will reproduce the full-order transfer function and its derivatives at the chosen points.
  • The data-driven realization procedure recovers a minimal realization of the reduced system solely from samples of the input-output map.
  • The classical first-order necessary conditions for H2 optimality continue to hold verbatim in the infinite-dimensional setting.
  • Finite-dimensional reduced models can be computed for systems with infinite-dimensional inputs and outputs while preserving the same optimality guarantees that exist in the finite-dimensional theory.

Where Pith is reading between the lines

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

  • The same projection-based construction could be applied after spatial discretization of a PDE, yielding a reduced model whose optimality can be verified directly on the discrete data.
  • Because the data-driven step needs only input-output trajectories, the method offers a route to model reduction for systems where the governing operators are unavailable or too expensive to form explicitly.
  • The persistence of the pole-mirror interpolation condition suggests that existing numerical algorithms for the finite-dimensional H2 problem can be reused once the interpolation points are supplied by the infinite-dimensional data.

Load-bearing premise

The original system can be written as a controlled abstract Cauchy problem on a Banach space so that Petrov-Galerkin projections onto finite subspaces enforce the exact interpolation relations needed for the optimality conditions.

What would settle it

Construct an H2-optimal reduced model for a concrete infinite-dimensional system such as the controlled heat equation on a bounded domain; if the resulting reduced transfer function fails to match the full-order transfer function and its first derivative at the reflected reduced poles, the optimality claim is false.

read the original abstract

We develop the interpolatory $\mathcal{H}_2$ optimal model reduction framework for linear control systems posed on infinite dimensional state, input and output spaces. Specifically, we consider linear systems formulated as controlled abstract Cauchy problems on a Banach space and approximate them via Petrov-Galerkin projection onto finite dimensional trial and test subspaces. We show that the resulting reduced order transfer function interpolates the original at prescribed points, and we characterize precisely how the projection subspaces must be constructed to enforce this interpolation. Building on this, we develop a data-driven realization framework -- an infinite dimensional analogue of the Loewner approach -- that recovers the system behavior directly from input-output data without requiring access to the underlying operators. Finally, we derive $\mathcal{H}_2$ optimality conditions for the reduced model and show that the classical interpolatory characterization persists in this infinite dimensional setting: first-order optimality requires Hermite interpolation of the transfer function at the mirror images of the reduced model's poles. Taken together, these results establish that the interpolatory $\mathcal{H}_2$ optimal model reduction theory extends naturally and completely to infinite dimensional linear control systems with infinite dimensional input and output spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the interpolatory H2-optimal model reduction framework to infinite-dimensional linear control systems formulated as controlled abstract Cauchy problems on Banach spaces. It employs Petrov-Galerkin projections onto finite-dimensional trial and test subspaces to enforce transfer-function interpolation at prescribed points, characterizes the required subspace construction, develops a data-driven Loewner-type realization from input-output measurements without access to the underlying operators, and derives that first-order H2 optimality requires Hermite interpolation of the transfer function at the mirror images of the reduced model's poles.

Significance. If the central derivations hold, the work establishes a complete extension of the finite-dimensional interpolatory H2 theory to systems with infinite-dimensional state, input, and output spaces. This is significant for applications in distributed-parameter control, where model reduction of PDE-governed systems is essential. The data-driven recovery framework and the persistence of the classical optimality characterization are notable strengths, as they provide both theoretical insight and practical utility without requiring explicit operator knowledge.

major comments (2)
  1. [Section on Petrov-Galerkin projections and subspace characterization] The characterization of the trial and test subspaces needed to enforce interpolation (detailed after the abstract's description of Petrov-Galerkin projection) does not explicitly verify membership in the domain of the unbounded generator A. For the generic case of unbounded generators, the trial subspace must lie in D(A) for the reduced operator to be well-defined; without this, the projection may not yield a valid finite-dimensional system. This is load-bearing for the claim that the framework extends naturally and completely.
  2. [Section on data-driven realization framework] The data-driven realization framework claims to recover the necessary subspace information from input-output data alone. However, when the generator is unbounded, recovering the required data without implicit access to the resolvent or input operator may require additional regularity or measurement assumptions not stated in the development of the infinite-dimensional Loewner analogue. This underpins the purely data-driven claim.
minor comments (2)
  1. [Introduction and preliminaries] Notation for the infinite-dimensional transfer function and its interpolation conditions could be clarified with an explicit comparison table to the finite-dimensional case to aid readability.
  2. [Abstract] The abstract states the results clearly, but the manuscript would benefit from an early remark on the standing assumptions regarding the well-posedness of the abstract Cauchy problem.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments, which help strengthen the manuscript. We address each major comment below and will incorporate the necessary clarifications and explicit statements in the revised version to ensure the framework is rigorously well-posed for unbounded generators.

read point-by-point responses

  1. Referee: [Section on Petrov-Galerkin projections and subspace characterization] The characterization of the trial and test subspaces needed to enforce interpolation (detailed after the abstract's description of Petrov-Galerkin projection) does not explicitly verify membership in the domain of the unbounded generator A. For the generic case of unbounded generators, the trial subspace must lie in D(A) for the reduced operator to be well-defined; without this, the projection may not yield a valid finite-dimensional system. This is load-bearing for the claim that the framework extends naturally and completely.

    Authors: We agree that explicit verification of domain membership is essential for unbounded generators. In the manuscript, the trial subspace is constructed via resolvent applications of the form (sI - A)^{-1}Bu, which by definition lie in D(A) for s in the resolvent set. The test subspace is chosen accordingly to ensure the Petrov-Galerkin projection yields a well-defined finite-dimensional operator. To address the referee's point directly, we will add an explicit remark in the relevant section stating that the trial subspace is contained in D(A) and verifying that the reduced generator is bounded on the finite-dimensional space. revision: yes

  2. Referee: [Section on data-driven realization framework] The data-driven realization framework claims to recover the necessary subspace information from input-output data alone. However, when the generator is unbounded, recovering the required data without implicit access to the resolvent or input operator may require additional regularity or measurement assumptions not stated in the development of the infinite-dimensional Loewner analogue. This underpins the purely data-driven claim.

    Authors: We thank the referee for highlighting this. The infinite-dimensional Loewner analogue is formulated under the standing assumption that input-output measurements are generated by sufficiently regular inputs (e.g., in the domain of the input operator and yielding outputs in the observation space) so that transfer-function evaluations at the interpolation points can be recovered directly from the data. This ensures no implicit access to the resolvent or operators is needed. We will revise the section to explicitly list these regularity and measurement assumptions, thereby clarifying the conditions under which the purely data-driven recovery holds. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension of finite-dimensional theory is self-contained.

full rationale

The paper poses infinite-dimensional systems as abstract Cauchy problems on Banach spaces, applies Petrov-Galerkin projections to enforce transfer-function interpolation at prescribed points, develops a data-driven Loewner-type realization from input-output data, and derives that H2 first-order optimality requires Hermite interpolation at the mirror images of the reduced poles. These steps rely on standard functional-analytic arguments and projection properties that are stated directly rather than defined in terms of the target result or recovered only via self-citation chains. No equation or claim reduces by construction to a fitted parameter, a renamed known pattern, or an unverified self-citation; the infinite-dimensional extension is presented as a direct generalization whose validity can be checked against the stated assumptions on the generator and subspaces.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard functional-analysis assumptions for abstract Cauchy problems and transfer-function interpolation; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Linear control systems can be formulated as controlled abstract Cauchy problems on a Banach space.
    This is the foundational setup stated in the abstract for the infinite-dimensional systems.

pith-pipeline@v0.9.0 · 5518 in / 1217 out tokens · 68390 ms · 2026-05-10T14:56:23.238155+00:00 · methodology

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Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    A. C. Antoulas.Approximation of Large-Scale Dynamical Systems. Advances in Design and Control. Society for Industrial and Applied Mathematics, 2009

    work page 2009

  2. [2]

    A. C. Antoulas, C. A. Beattie, and S. G¨ u˘ gercin.Interpolatory Methods for Model Reduction. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2020

    work page 2020

  3. [3]

    Arendt, C.J.K

    W. Arendt, C.J.K. Batty, and F. Neubrander.Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics. Springer Basel, 2013

    work page 2013

  4. [4]

    Beattie and S

    C. Beattie and S. G¨ u˘ gercin. Realization-independentH 2-approximation. In2012 IEEE 51st IEEE Conference on Decision and Control (CDC), pages 4953–4958, 2012

    work page 2012

  5. [5]

    Benner and T

    P. Benner and T. Breiten. Model order reduction based on system balancing. In Model reduction and approximation: theory and algorithms, chapter 6, pages 261–

    work page

  6. [6]

    C. Bonnet. Approximation of infinite dimensional discrete time linear systems via bal- anced realizations and an application to fractional filters. InAnalysis and Optimiza- tion of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems, 1993

    work page 1993

  7. [7]

    Breiten and B

    T. Breiten and B. H¨ oveler. On the approximability of Koopman-based operator Lyapunov equations.SIAM Journal on Control and Optimization, 61(5):3131–3155, 2023

    work page 2023

  8. [8]

    Breiten and T

    T. Breiten and T. Stykel. Balancing-related model reduction methods.System-and Data-Driven Methods and Algorithms, 1:15–56, 2021

    work page 2021

  9. [9]

    Bunse-Gerstner, D

    A. Bunse-Gerstner, D. Kubalinska, G. Vossen, and D. Wilczek.H 2-norm optimal model reduction for large scale discrete dynamical MIMO systems.Journal of Com- putational and Applied Mathematics, 233(5):1202–1216, 2010. Special Issue Dedicated to William B. Gragg on the Occasion of His 70th Birthday

    work page 2010

  10. [10]

    Curtain and K

    R. Curtain and K. Morris. Transfer functions of distributed parameter systems: A tutorial.Automatica, 45(5):1101–1116, 2009

    work page 2009

  11. [11]

    K. J. Engel and R. Nagel.One-Parameter Semigroups for Linear Evolution Equations, volume 63. 06 2001

    work page 2001

  12. [12]

    Gallivan, A

    K. Gallivan, A. Vandendorpe, and P. Van Dooren. Sylvester equations and projection-based model reduction.Journal of Computational and Applied Mathe- matics, 162(1):213–229, 2004

    work page 2004

  13. [13]

    K. Glover. All optimal Hankel-norm approximations of linear multivariable systems and theirL ∞-error bounds.International Journal of Control, 39(6):1115–1193, 1984

    work page 1984

  14. [14]

    Glover, R

    K. Glover, R. F. Curtain, and J. R. Partington. Realisation and approximation of linear infinite-dimensional systems with error bounds.SIAM Journal on Control and Optimization, 26(4):863–898, 1988

    work page 1988

  15. [15]

    Grubi˘ si´ c and D

    L. Grubi˘ si´ c and D. Kressner. On the eigenvalue decay of solutions to operator lya- punov equations.Systems & Control Letters, 73:42–47, 2014. Preprint. 2026-04-15 C. Tilki, T. Breiten, S. G¨ u˘ gercin: Model Reduction for Infinite-Dimensional Systems31

    work page 2014

  16. [16]

    G¨ u˘ gercin, A

    S. G¨ u˘ gercin, A. C. Antoulas, and C. Beattie.H2 model reduction for large-scale linear dynamical systems.SIAM Journal on Matrix Analysis and Applications, 30(2):609– 638, 2008

    work page 2008

  17. [17]

    Guiver and M

    C. Guiver and M. R. Opmeer. Model reduction by balanced truncation for sys- tems with nuclear hankel operators.SIAM Journal on Control and Optimization, 52(2):1366–1401, 2014

    work page 2014

  18. [18]

    Harkort and J

    C. Harkort and J. Deutscher. Krylov subspace methods for linear infinite-dimensional systems.IEEE Transactions on Automatic Control, 56(2):441–447, 2011

    work page 2011

  19. [19]

    Harkort and J

    C. Harkort and J. Deutscher. Stability and passivity preserving Petrov–Galerkin approximation of linear infinite-dimensional systems.Automatica, 48(7):1347–1352, 2012

    work page 2012

  20. [20]

    Hille and R

    E. Hille and R. S. Phillips.Functional analysis and semi-groups, volume 31. American Mathematical Soc., 1996

    work page 1996

  21. [21]

    T. C. Ionescu and O. V. Iftime. Moment matching with prescribed poles and zeros for infinite-dimensional systems. In2012 American Control Conference (ACC), pages 1412–1417, 2012

    work page 2012

  22. [22]

    Kailath.Linear Systems

    T. Kailath.Linear Systems. Information and System Sciences Series. Prentice-Hall, 1980

    work page 1980

  23. [23]

    A. J. Mayo and A. C. Antoulas. A framework for the solution of the generalized real- ization problem.Linear Algebra and its Applications, 425(2):634–662, 2007. Special Issue in honor of Paul Fuhrmann

    work page 2007

  24. [24]

    Megginson.An Introduction to Banach Space Theory

    R.E. Megginson.An Introduction to Banach Space Theory. Graduate Texts in Math- ematics. Springer New York, 1998

    work page 1998

  25. [25]

    Meier and D

    L. Meier and D. Luenberger. Approximation of linear constant systems.IEEE Trans- actions on Automatic Control, 12(5):585–588, 1967

    work page 1967

  26. [26]

    B. Moore. Principal component analysis in linear systems: Controllability, observ- ability, and model reduction.IEEE Transactions on Automatic Control, 26(1):17–32, 1981

    work page 1981

  27. [27]

    Mullis and R

    C. Mullis and R. A. Roberts. Synthesis of minimum roundoff noise fixed point digital filters.IEEE Transactions on Circuits and Systems, 23(9):551–562, 1976

    work page 1976

  28. [28]

    M. R. Opmeer, T. Reis, and W. Wollner. Finite-rank ADI iteration for operator Lyapunov equations.SIAM Journal on Control and Optimization, 51(5):4084–4117, 2013

    work page 2013

  29. [29]

    Reis and T

    T. Reis and T. Selig. Balancing transformations for infinite-dimensional systems with nuclear Hankel operator.Integral Equations and Operator Theory, 79:67–105, 2014

    work page 2014

  30. [30]

    Van Dooren, K

    P. Van Dooren, K. A. Gallivan, and P-A. Absil.H2-optimal model reduction of MIMO systems.Applied Mathematics Letters, 21(12):1267–1273, 2008

    work page 2008

  31. [31]

    Yosida.Functional Analysis, volume 123 ofGrundlehren der Mathematischen Wissenschaften

    K. Yosida.Functional Analysis, volume 123 ofGrundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, fifth edition, 1978. Preprint. 2026-04-15 C. Tilki, T. Breiten, S. G¨ u˘ gercin: Model Reduction for Infinite-Dimensional Systems32

    work page 1978

  32. [32]

    K. Zhou, J. C. Doyle, and K. Glover.Robust and Optimal Control. Feher/Prentice Hall Digital. Prentice Hall, 1996. Preprint. 2026-04-15

    work page 1996