Energy function approximations for differential algebraic polynomial systems of Stokes-type

Hamza Adjerid; Jeff Borggaard

arxiv: 2507.18909 · v2 · submitted 2025-07-25 · 🧮 math.OC

Energy function approximations for differential algebraic polynomial systems of Stokes-type

Hamza Adjerid , Jeff Borggaard This is my paper

Pith reviewed 2026-05-19 03:13 UTC · model grok-4.3

classification 🧮 math.OC

keywords energy functionsHJB equationsdifferential algebraic equationsKronecker productspolynomial approximationsStokes systemsstrangeness indexnonlinear control

0 comments

The pith

Stokes-type differential-algebraic polynomial systems admit energy function approximations through Kronecker-product methods after separating variables via the strangeness framework.

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

The paper extends polynomial approximations to Hamilton-Jacobi-Bellman equations for energy functions to systems that include linear drift terms with a Stokes-type differential-algebraic equation structure. By using the strangeness framework, the approach creates distinct sets of differential and algebraic variables where the differential equations depend only on the differential variables. This separation preserves the polynomial structure, allowing Kronecker product techniques to generate approximations for the energy functions. The method is illustrated on polynomial feedback control examples and includes a sparsity-preserving variant to maintain efficiency in the original system representation.

Core claim

For polynomial systems whose linear drift exhibits Stokes-type DAE structure, the strangeness framework partitions the variables into differential and algebraic components such that the differential equations involve solely the differential variables. Existing Kronecker-product polynomial approximations to the HJB equations can then be applied to approximate the energy functions. A formulation is also given that preserves the sparsity of the original system matrices. This extension is demonstrated for two polynomial feedback control problems.

What carries the argument

The strangeness framework for differential-algebraic equations, which separates algebraic and differential variables to enable direct application of Kronecker-product based polynomial approximations to the associated Hamilton-Jacobi-Bellman equations.

If this is right

Energy functions can be approximated for nonlinear systems with Stokes-type DAE constraints using polynomial methods.
Nonlinear balanced truncation becomes applicable to these DAE systems via the approximated energy functions.
Feedback controllers can be designed using the polynomial energy function approximations for such systems.
Sparsity structure of the original DAE can be maintained in the approximation process for computational efficiency.
The technique extends prior Kronecker-product HJB approximations to include linear drift terms of DAE form.

Where Pith is reading between the lines

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

The separation method may extend to other DAE classes that possess a well-defined strangeness index.
Preserving sparsity could significantly reduce computational costs for high-dimensional control problems.
This approach opens the door to applying similar approximations in model reduction techniques for constrained nonlinear systems.

Load-bearing premise

The DAE must admit a strangeness index that cleanly partitions the variables so the differential equations involve only differential variables and the algebraic equations relate both without destroying the polynomial structure required for Kronecker approximations.

What would settle it

Observing a Stokes-type DAE where the strangeness-based separation results in differential equations that still couple to algebraic variables or break the polynomial form would show that the method does not generally enable the existing Kronecker approximations.

read the original abstract

Energy functions are generalizations of controllability and observability Gramians to nonlinear systems and as such find applications in both nonlinear balanced truncation and feedback control. These energy functions are solutions to the Hamilton-Jacobi-Bellman (HJB) equations partial differential equations defined over spatial dimensions determined by the number of state variables in the nonlinear system. Thus, they cannot be resolved with local basis functions for problems of even modest dimension. In this paper, we extend recent results that utilize Kronecker products to generate polynomial approximations to HJB equations. Specifically, we consider the addition of linear drift terms that exhibit a Stokes-type differential-algebraic equation (DAE) structure. This extension leverages the so-called strangeness framework for DAEs to create separate sets of algebraic and differential variables with differential equations that only involve the differential variables and algebraic equations that relate them both. At this point, the existing polynomial approximations using Kronecker products can be applied to find approximations to the energy functions. This approach is demonstrated for two polynomial feedback control problems. The standard transformation destroys the sparsity in the original system. Thus, we also present a formulation that preserves the original sparsity structure.

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

This extends Kronecker-product HJB approximations to Stokes-type DAEs via strangeness splitting and adds a sparsity-preserving variant, with demonstrations on two polynomial examples.

read the letter

This paper takes the recent Kronecker-product method for polynomial approximations to Hamilton-Jacobi-Bellman equations and adapts it to systems with Stokes-type differential-algebraic structure. They use the strangeness index to separate differential and algebraic variables so that the differential equations depend only on the differential ones, letting the prior approximation carry over directly. They also supply a version that avoids destroying the original sparsity during the transformation. The two numerical examples on polynomial feedback control problems show the approach in action and confirm that the separation works as intended for these cases. The explicit transformations and the sparsity fix are the concrete advances here. The main limitation is that the method stays tied to polynomial systems where the strangeness index produces a clean partition, and the error behavior still rests on the earlier Kronecker results without fresh bounds for the DAE setting. The examples illustrate feasibility but do not compare against other DAE approximation techniques. The citations track the prior work properly without circularity. This is the sort of targeted methodological note that would interest people doing nonlinear model reduction or control of constrained systems. A reader already working with energy functions or HJB approximations in modest dimensions would get direct value from the splitting procedure and the sparsity option. It deserves peer review because the extension is carried through with explicit steps and supporting calculations.

Referee Report

1 major / 0 minor

Summary. The manuscript extends Kronecker-product polynomial approximations to Hamilton-Jacobi-Bellman equations for nonlinear systems by incorporating Stokes-type differential-algebraic equation (DAE) structure. It applies the strangeness index framework to partition variables into differential and algebraic sets such that differential equations depend only on differential variables, then directly applies prior approximation techniques; a sparsity-preserving formulation is also derived to avoid destroying the original sparsity. The method is illustrated via numerical demonstrations on two polynomial feedback-control examples.

Significance. If the approximations are shown to satisfy the HJB equation to controlled tolerance, the work would enable energy-function computations for a practically relevant class of DAE systems arising in nonlinear control and model reduction. The sparsity-preserving variant is a clear practical strength, and the paper correctly credits the underlying Kronecker-product results without introducing new free parameters or ad-hoc fitting.

major comments (1)

Demonstration section: the two polynomial feedback-control examples are presented as direct support for the extension, yet no quantitative error measures, residual norms on the HJB equation, or comparison against a reference solution are reported; without such verification the claim that the resulting polynomials constitute valid energy-function approximations remains unsubstantiated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comment on our manuscript. We address the major comment below.

read point-by-point responses

Referee: Demonstration section: the two polynomial feedback-control examples are presented as direct support for the extension, yet no quantitative error measures, residual norms on the HJB equation, or comparison against a reference solution are reported; without such verification the claim that the resulting polynomials constitute valid energy-function approximations remains unsubstantiated.

Authors: We agree that quantitative verification strengthens the presentation. In the revised manuscript we will report the residual norms of the HJB equation evaluated at the computed polynomial approximations for both examples. Where the problem dimension permits, we will also include comparisons against reference solutions obtained by direct solution of the HJB PDE or by alternative approximation methods. These additions will directly substantiate that the polynomials satisfy the HJB equation to controlled tolerance. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a methodological extension: the strangeness-index framework is used to partition Stokes-type DAEs into differential equations depending only on differential variables and algebraic equations relating both sets, after which previously published Kronecker-product polynomial approximations to the HJB equation are applied directly. Explicit transformations and a sparsity-preserving variant are supplied for the target class, together with numerical demonstrations on two polynomial feedback-control examples. No equation or claim reduces the new approximation to a quantity defined by the authors themselves, nor does any load-bearing step rest solely on a self-citation whose validity is presupposed within the present manuscript; the prior Kronecker-product results function as independent external inputs whose assumptions do not embed the target DAE result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard mathematical properties of DAEs and polynomial approximation techniques already established in the cited prior work; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract description.

axioms (2)

domain assumption The system admits a strangeness index that permits clean separation into differential and algebraic variables with the stated structural properties.
Invoked when the authors state that the strangeness framework creates separate sets of algebraic and differential variables.
domain assumption The Kronecker-product polynomial approximation technique from prior work extends without modification once the DAE is reduced to the separated form.
Stated as the point at which existing approximations can be applied.

pith-pipeline@v0.9.0 · 5729 in / 1490 out tokens · 39004 ms · 2026-05-19T03:13:12.476559+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

leverages the so-called strangeness framework for DAEs to create separate sets of algebraic and differential variables... existing polynomial approximations using Kronecker products
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Π = I − A12 (A⊤12 E−1 11 A12)−1 A⊤12 E−1 11 ... Θ⊤r E11 Θr ˙xd = ...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

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Moment- matching based model reduction for Navier-Stokes type quadratic-bilinear descrip- tor systems

Mian Ilyas Ahmad, Peter Benner, Pawan Goyal, and Jan Heiland. Moment- matching based model reduction for Navier-Stokes type quadratic-bilinear descrip- tor systems. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f¨ ur Angewandte Mathematik und Mechanik, 315(10):303 – 16, 2017

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[29]

Computation of nonlinear balanced real- ization and model reduction based on Taylor series expansion

Kenji Fujimoto and Daisuke Tsubakino. Computation of nonlinear balanced real- ization and model reduction based on Taylor series expansion. Systems & Control Letters, 57(4):283 – 289, 2008

work page 2008
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Approximation of functions of large matrices with Kronecker structure

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Weichelt

Peter Benner, Matthias Heinkenschloss, Jens Saak, and Heiko K. Weichelt. Ef- ficient solution of large-scale algebraic Riccati equations associated with index-2 DAEs via the inexact low-rank Newton-ADI method. Applied Numerical Mathe- matics,, 2020. 23

work page 2020
[36]

Stoyanov

Miroslav K. Stoyanov. Optimal linear feedback control for incompressible fluid flow. Master’s thesis, Virginia Polytechnic Institute and State University, Blacks- burg, VA USA, 2006

work page 2006
[37]

Brust, Roummel F

Johannes J. Brust, Roummel F. Marcia, and Cosmin G. Petra. Computationally efficient decompositions of oblique projection matrices. SIAM Journal on Matrix Analysis and Applications, 41(2):852–870, 2020

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George Matsaglia and George P. H. Styan. Equalities and inequalities for ranks of matrices. Linear and Multilinear Algebra, 2(3):269–292, 1974. 24

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[1] [1]

van der Schaft

A.J. van der Schaft. L2-gain analysis of nonlinear systems and nonlinear state- feedback H∞ control. IEEE Transactions on Automatic Control , 37(6):770–784, 1992

work page 1992

[2] [2]

Scherpen

Jacquelien M. Scherpen. H∞ balancing for nonlinear systems. International Jour- nal of Robust and Nonlinear Control , 6(7):645–668, 1996. 21

work page 1996

[3] [3]

Scherpen and A

Jacquelien M. Scherpen and A. van der Schaft. Normalized coprime factorizations and balancing for unstable nonlinear systems. International Journal of Control , 60(6):1193–1222, 1994

work page 1994

[4] [4]

Scalable computation of energy functions for nonlinear balanced truncation

Boris Kramer, Serkan Gugercin, Jeff Borggaard, and Linus Balecki. Scalable computation of energy functions for nonlinear balanced truncation. Computer Methods in Applied Mechanics and Engineering , 427(117011), 2024

work page 2024

[5] [5]

E. G. Al’Brekht. On the optimal stabilization of nonlinear systems. Journal of Applied Mathematics and Mechanics , 25:1254–1266, 1962

work page 1962

[6] [6]

D. L. Lukes. Optimal regulation of nonlinear dynamical systems. SIAM Journal of Control, 7:75–100, 1969

work page 1969

[7] [7]

C. L. Navasca and A. J. Krener. Solution to Hamiltonian Jacobi Bellman equa- tions. In Conference on Decision and Control , volume 39, pages 570–574. IEEE, 2000

work page 2000

[8] [8]

Sergey Dolgov, Dante Kalise, and Karl K. Kunisch. Tensor decompositions for high-dimensional Hamilton-Jacobi-Bellman equations. SIAM Journal on Scientific Computing, 43(3):A1625–A1650, 2021

work page 2021

[9] [9]

Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semi- linear parabolic PDEs

Dante Kalise and Karl Kunisch. Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semi- linear parabolic PDEs. SIAM Journal on Scientific Computing , 40(2):A629 – A652, 2018

work page 2018

[10] [10]

Taylor expansions of the value function associated with a bilinear optimal control problem

Tobias Breiten, Karl Kunisch, and Laurent Pfeiffer. Taylor expansions of the value function associated with a bilinear optimal control problem. Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 36(5):1361–1399, 2019

work page 2019

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The quadratic-quadratic regulator problem: Approximating feedback controls for quadratic-in-state nonlinear systems

Jeff Borggaard and Lizette Zietsman. The quadratic-quadratic regulator problem: Approximating feedback controls for quadratic-in-state nonlinear systems. In2020 American Control Conference (ACC), pages 818–823. IEEE, 2020

work page 2020

[12] [12]

On approximating polynomial-quadratic regulator problems

Jeff Borggaard and Lizette Zietsman. On approximating polynomial-quadratic regulator problems. IFAC PaersOnLine, 54(9):329–334, 2021

work page 2021

[13] [13]

Recursive blocked algorithms for linear sys- tems with Kronecker product structure

Minhong Chen and Daniel Kressner. Recursive blocked algorithms for linear sys- tems with Kronecker product structure. Numerical Algorithms, 15(9):820 – 20, 2019

work page 2019

[14] [14]

Time-dependent Dirichlet conditions in finite element discretizations

Peter Benner and Jan Heiland. Time-dependent Dirichlet conditions in finite element discretizations. ScienceOpen Research, 2015

work page 2015

[15] [15]

Chan and Tarek P

Tony F. Chan and Tarek P. Mathew. Domain decomposition algorithms. Acta Numerica, 3:61–143, 1994

work page 1994

[16] [16]

Finite Element Methods for Viscous Incompressible Flows

Max Gunzburger. Finite Element Methods for Viscous Incompressible Flows. Aca- demic Press, San Diego, CA, 1989

work page 1989

[17] [17]

Fursikov

Andrei V. Fursikov. Optimal Control of Distributed Systems. Theory and Appli- cations. American Mathematical Society, Providence, RI, 1991

work page 1991

[18] [18]

Linear feedback control of a von k´ arm´ an street by cylinder rotation

Jeff Borggaard, Miroslav Stoyanov, and Lizette Zietsman. Linear feedback control of a von k´ arm´ an street by cylinder rotation. InProceedings of the 2010 American Control Conference, pages 5674–5681, 2010. 22

work page 2010

[19] [19]

D. Cobb. Descriptor variable systems and optimal state regulation. IEEE Trans- actions on Automatic Control , AC-28(5):601 – 611, 1983

work page 1983

[20] [20]

D. Cobb. Controllability, observability, and duality in singular systems. IEEE Transactions on Automatic Control, AC-29(12):1076 – 1082, 1984

work page 1984

[21] [21]

F. L. Lewis. Optimal control for singular systems. In Proceedings of the 24th Conference on Decision and Control , pages 266–272, 1985

work page 1985

[22] [22]

Bender and Alan J

Douglas J. Bender and Alan J. Laub. The linear-quadratic optimal regulator for descriptor systems. IEEE Transactions on Automatic Control , AC-32(8), 1987

work page 1987

[23] [23]

Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index

Peter Kunkel and Volker Mehrmann. Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index. Mathematics of Control, Sig- nals, and Systems , 20(3):227–269, August 2008. Publisher: Springer Science and Business Media LLC

work page 2008

[24] [24]

Sj¨ oberg.Optimal Control and Model Reduction of Nonlinear DAE Models

J. Sj¨ oberg.Optimal Control and Model Reduction of Nonlinear DAE Models. PhD thesis, Institutionen f¨ or systemteknik, Link¨ oping, Sweden, 2008

work page 2008

[25] [25]

A differential-algebraic Riccati equation for applications in flow control

Jan Heiland. A differential-algebraic Riccati equation for applications in flow control. SIAM Journal on Control and Optimization , 54(2):718–739, January 2016

work page 2016

[26] [26]

Moment- matching based model reduction for Navier-Stokes type quadratic-bilinear descrip- tor systems

Mian Ilyas Ahmad, Peter Benner, Pawan Goyal, and Jan Heiland. Moment- matching based model reduction for Navier-Stokes type quadratic-bilinear descrip- tor systems. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f¨ ur Angewandte Mathematik und Mechanik, 315(10):303 – 16, 2017

work page 2017

[27] [27]

Kunkel and V

P. Kunkel and V. Mehrmann. Differential-Algebraic Equations—Analysis and Numerical Solution. European Mathematical Society, Z¨ urich, 2006

work page 2006

[28] [28]

Sorensen, and Kai Sun

Matthias Heinkenschloss, Danny C. Sorensen, and Kai Sun. Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations. SIAM Journal on Scientific Computing , 30(2):1038–1063, 2008

work page 2008

[29] [29]

Computation of nonlinear balanced real- ization and model reduction based on Taylor series expansion

Kenji Fujimoto and Daisuke Tsubakino. Computation of nonlinear balanced real- ization and model reduction based on Taylor series expansion. Systems & Control Letters, 57(4):283 – 289, 2008

work page 2008

[30] [30]

Approximation of functions of large matrices with Kronecker structure

Michele Benzi and Valeria Simoncini. Approximation of functions of large matrices with Kronecker structure. Numerische Mathematik, 135(1):1–26, 2017

work page 2017

[31] [31]

Differential/Algebraic Equations are not ODE’s

Linda Petzold. Differential/Algebraic Equations are not ODE’s. SIAM Journal on Scientific and Statistical Computing , 3(3):367–384, 1982

work page 1982

[32] [32]

C. C. Pantelides. The consistent initialization of differential-algebraic systems. SIAM Journal on Scientific Computing , 1998

work page 1998

[33] [33]

Fritzson

P. Fritzson. Principles of Object-Oriented Modeling and Simulation with Modelica 2.1. Wiley-IEEE Press, Hoboken, NJ, 2004

work page 2004

[34] [34]

Navier-Stokes Equations

Roger Temam. Navier-Stokes Equations. North-Holland, New York, 1977

work page 1977

[35] [35]

Weichelt

Peter Benner, Matthias Heinkenschloss, Jens Saak, and Heiko K. Weichelt. Ef- ficient solution of large-scale algebraic Riccati equations associated with index-2 DAEs via the inexact low-rank Newton-ADI method. Applied Numerical Mathe- matics,, 2020. 23

work page 2020

[36] [36]

Stoyanov

Miroslav K. Stoyanov. Optimal linear feedback control for incompressible fluid flow. Master’s thesis, Virginia Polytechnic Institute and State University, Blacks- burg, VA USA, 2006

work page 2006

[37] [37]

Brust, Roummel F

Johannes J. Brust, Roummel F. Marcia, and Cosmin G. Petra. Computationally efficient decompositions of oblique projection matrices. SIAM Journal on Matrix Analysis and Applications, 41(2):852–870, 2020

work page 2020

[38] [38]

George Matsaglia and George P. H. Styan. Equalities and inequalities for ranks of matrices. Linear and Multilinear Algebra, 2(3):269–292, 1974. 24

work page 1974