pith. sign in

arxiv: 2604.27460 · v3 · pith:CWXOXQAFnew · submitted 2026-04-30 · 🧮 math.OC · cs.SY· eess.SY

Solution Sets for Inverse Infinite-Horizon Linear-Quadratic Descriptor Differential Games

Aaditya Kumar , Puduru Viswanadha Reddy This is my paper

Pith reviewed 2026-05-21 01:04 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords inverse differential gamesdescriptor systemslinear-quadratic gamesfeedback Nash equilibriumsolution setsidentifiabilityconvex sets
0
0 comments X

The pith

Descriptor dynamics make the set of costs rationalizing observed Nash strategies rectangular and convex.

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

This paper studies the inverse problem of recovering all quadratic cost functions that explain a given observed feedback strategy profile as a Nash equilibrium in infinite-horizon linear-quadratic differential games whose dynamics include descriptor variables. A reader would care because many physical and engineering systems are naturally described by differential-algebraic equations rather than ordinary differential equations, so the inverse problem must account for algebraic constraints when inferring objectives from play. The authors fully characterize the collection of such costs, prove that the set is both rectangular and convex, and supply an algorithm that returns one valid cost realization whenever the set is nonempty. They further establish that the algebraic constraints alter the geometry of the solution set relative to standard state-space games and can lower the degree to which the underlying costs can be uniquely identified from the observed strategies.

Core claim

Given an observed feedback strategy profile, the set of all quadratic cost functions that render the profile a feedback Nash equilibrium for an infinite-horizon linear-quadratic descriptor differential game is rectangular and convex. An algorithm computes an admissible realization from this set when the set is nonempty. Descriptor dynamics change the geometry of the solution set compared with ordinary state-space dynamics and may reduce identifiability of the costs.

What carries the argument

The rectangular convex solution set of cost matrices whose associated Nash equilibrium conditions are satisfied exactly by the observed feedback strategies under the given descriptor dynamics.

If this is right

  • The solution set of admissible costs is always rectangular and convex when nonempty.
  • An explicit algorithm recovers at least one valid cost pair from any nonempty solution set.
  • The algebraic constraints of descriptor dynamics reshape the geometry of the cost set relative to standard linear-quadratic games.
  • Descriptor structure can strictly reduce the identifiability of the underlying quadratic costs from observed play.

Where Pith is reading between the lines

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

  • The rectangular-convex geometry may allow designers to add secondary selection criteria, such as minimal-norm costs, without losing convexity.
  • The same characterization technique could be tested on descriptor systems arising from mechanical linkages or electrical networks to recover plausible cost functions.
  • Lower identifiability implies that additional measurements or regularization may be required before the recovered costs can be used for prediction.

Load-bearing premise

The observed feedback strategy profile is in fact a feedback Nash equilibrium for the descriptor system under some quadratic costs.

What would settle it

A numerical instance in which the algorithm returns a cost pair that, when the game is solved forward with those costs, yields a different equilibrium strategy profile from the one supplied as input.

read the original abstract

In this letter, we study a model-based inverse problem for infinite-horizon linear-quadratic differential games with descriptor dynamics. Given an observed feedback strategy profile, we seek to identify all cost functions that rationalize it as a feedback Nash equilibrium; this collection is referred to as the solution set. We characterize the solution set, show that it is rectangular and convex, and provide an algorithm for computing an admissible realization whenever it is nonempty. We also show that, compared with the corresponding inverse problem for standard state-space dynamics, descriptor dynamics modify the geometry of the solution set and may reduce identifiability. Finally, we illustrate the results with numerical examples.

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

0 major / 3 minor

Summary. The manuscript studies the inverse problem for infinite-horizon linear-quadratic differential games with descriptor dynamics. Given an observed feedback strategy profile, the authors characterize the solution set of quadratic cost functions that rationalize the profile as a feedback Nash equilibrium, prove that this set is rectangular and convex, and supply an algorithm to compute an admissible realization whenever the set is nonempty. They further show that descriptor dynamics alter the geometry of the solution set relative to the ordinary state-space case and can reduce identifiability, with the analysis resting on regular index-1 pencils and the Weierstrass form; numerical examples are included.

Significance. If the central derivations hold, the paper makes a useful contribution to inverse optimal control and differential games by extending the framework to descriptor systems that arise in constrained dynamics. The rectangular-convex geometry of the solution set, which follows from the separation of each player's cost matrices in the linear matrix equations once the closed-loop pencil is fixed by the observed strategies, clarifies identifiability questions. The explicit algorithm and the comparison highlighting descriptor-specific effects on geometry and identifiability are practical strengths. The work builds on standard game-theoretic and descriptor-system tools without apparent circularity.

minor comments (3)
  1. The abstract states that the solution set is 'rectangular and convex'; a short parenthetical gloss or forward reference to the precise definition of rectangularity in the parameter space would improve immediate readability for readers outside the immediate subfield.
  2. In the numerical examples, the choice of test pencils and the observed strategies is illustrated, but adding a brief discussion of how the algorithm behaves under small perturbations of the observed feedback gains would strengthen the practical assessment of robustness.
  3. The comparison with the standard state-space inverse problem notes possible loss of identifiability; including one explicit low-dimensional example (with explicit matrix dimensions) where the descriptor case yields a strictly smaller solution set than its state-space counterpart would make the geometric claim more concrete.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive review of our manuscript on the inverse problem for infinite-horizon linear-quadratic descriptor differential games. The summary accurately reflects our contributions, including the characterization of the solution set as rectangular and convex, the algorithm for admissible realizations, and the analysis of how descriptor dynamics affect geometry and identifiability relative to the state-space case. We appreciate the recognition of the work's practical strengths and its foundation in standard tools for games and descriptor systems.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the solution set by reducing the feedback Nash equilibrium conditions for the observed strategies to a system of linear matrix equations in the unknown quadratic cost parameters, with the closed-loop descriptor pencil fixed by the data. The rectangular and convex geometry is a direct algebraic consequence of the players' costs entering these equations separately. Descriptor-specific geometry (via Weierstrass form and index-1 regularity) is handled by standard transformations that do not rely on self-citations or fitted parameters renamed as predictions. The central claims rest on explicit matrix equations and an algorithm for admissible realizations, without any step that reduces by construction to the inputs or to prior self-referential results. The derivation is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the work appears to rely on standard assumptions from linear-quadratic game theory and descriptor systems.

pith-pipeline@v0.9.0 · 5640 in / 1101 out tokens · 45124 ms · 2026-05-21T01:04:21.651155+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Bas ¸ar and G

    T. Bas ¸ar and G. J. Olsder,Dynamic Noncooperative Game Theory. SIAM, 1999

    work page 1999

  2. [2]

    Isaacs,Differential Games

    R. Isaacs,Differential Games. John Wiley and Sons, 1965

    work page 1965

  3. [3]

    Engwerda,LQ Dynamic Optimization and Differential Games

    J. Engwerda,LQ Dynamic Optimization and Differential Games. John Wiley & Sons, 2005

    work page 2005

  4. [4]

    Online Inverse Linear-Quadratic Differential Games Applied to Human Behavior Identification in Shared Control,

    J. Inga, A. Creutz, and S. Hohmann, “Online Inverse Linear-Quadratic Differential Games Applied to Human Behavior Identification in Shared Control,” inEuropean Control Conference (ECC), 2021, pp. 353–360

    work page 2021

  5. [5]

    Learning Human Behavior in Shared Control: Adaptive Inverse Differential Game Approach,

    H.-N. Wu and M. Wang, “Learning Human Behavior in Shared Control: Adaptive Inverse Differential Game Approach,”IEEE Transactions on Cybernetics, vol. 54, no. 6, pp. 3705–3715, 2024

    work page 2024

  6. [6]

    LUCIDGames: Online Unscented Inverse Dynamic Games for Adaptive Trajectory Prediction and Planning,

    S. Le Cleac’h, M. Schwager, and Z. Manchester, “LUCIDGames: Online Unscented Inverse Dynamic Games for Adaptive Trajectory Prediction and Planning,”IEEE Robotics and Automation Letters, vol. 6, no. 3, pp. 5485–5492, 2021

    work page 2021

  7. [7]

    Inferring Foresightedness in Dynamic Noncooperative Games,

    C. Armstrong, R. Park, X. Liu, K. Gupta, and D. Fridovich-Keil, “Inferring Foresightedness in Dynamic Noncooperative Games,”IEEE Robotics and Automation Letters, vol. 10, no. 12, pp. 13 193–13 200, 2025

    work page 2025

  8. [8]

    An inverse differential game approach to modelling bird mid-air collision avoidance behaviours,

    T. L. Molloy, G. S. Garden, T. Perez, I. Schiffner, D. Karmaker, and M. V . Srinivasan, “An inverse differential game approach to modelling bird mid-air collision avoidance behaviours,” in18th IFAC Symposium on System Identification (SYSID 2018), vol. 51, no. 15, 2018, pp. 754– 759

    work page 2018

  9. [9]

    Maximum-Entropy Multi-Agent Dynamic Games: Forward and Inverse Solutions,

    N. Mehr, M. Wang, M. Bhatt, and M. Schwager, “Maximum-Entropy Multi-Agent Dynamic Games: Forward and Inverse Solutions,”IEEE Transactions on Robotics, vol. 39, no. 3, pp. 1801–1815, 2023

    work page 2023

  10. [10]

    Inverse Equilibrium Analysis of Oligopolistic Electricity Markets,

    S. Risanger, S.-E. Fleten, and S. A. Gabriel, “Inverse Equilibrium Analysis of Oligopolistic Electricity Markets,”IEEE Transactions on Power Systems, vol. 35, no. 6, pp. 4159–4166, 2020

    work page 2020

  11. [11]

    When Is a Linear Control System Optimal?

    R. E. Kalman, “When Is a Linear Control System Optimal?”Journal of Basic Engineering, vol. 86, no. 1, pp. 51–60, 1964

    work page 1964

  12. [12]

    Vrabie, K

    D. Vrabie, K. Vamvoudakis, and F. Lewis,Optimal Adaptive Control and Differential Games by Reinforcement Learning Principles, ser. Control, Robotics and Sensors. Institution of Engineering and Technology, 2013

    work page 2013

  13. [13]

    From inverse optimal control to inverse reinforcement learning: A historical review,

    N. Ab Azar, A. Shahmansoorian, and M. Davoudi, “From inverse optimal control to inverse reinforcement learning: A historical review,” Annual Reviews in Control, vol. 50, pp. 119–138, 2020

    work page 2020

  14. [14]

    Inverse Noncooperative Dynamic Games,

    T. L. Molloy, J. J. Ford, and T. Perez, “Inverse Noncooperative Dynamic Games,” in20th IFAC World Congress, vol. 50, no. 1, 2017, pp. 11 788– 11 793

    work page 2017

  15. [15]

    Inverse Open-Loop Noncooperative Differential Games and Inverse Optimal Control,

    T. L. Molloy, J. Inga, M. Flad, J. J. Ford, T. Perez, and S. Hohmann, “Inverse Open-Loop Noncooperative Differential Games and Inverse Optimal Control,”IEEE Transactions on Automatic Control, vol. 65, no. 2, pp. 897–904, 2020

    work page 2020

  16. [16]

    Inverse Opti- mal Control for Identification in Non-Cooperative Differential Games,

    S. Rothfuß, J. Inga, F. K ¨opf, M. Flad, and S. Hohmann, “Inverse Opti- mal Control for Identification in Non-Cooperative Differential Games,” IFAC-PapersOnLine, vol. 50, no. 1, pp. 14 909–14 915, 2017

    work page 2017

  17. [17]

    Solution sets for inverse non-cooperative linear-quadratic differential games,

    J. Inga, E. Bischoff, T. L. Molloy, M. Flad, and S. Hohmann, “Solution sets for inverse non-cooperative linear-quadratic differential games,” IEEE Control Systems Letters, vol. 3, no. 4, pp. 871–876, 2019

    work page 2019

  18. [18]

    The Inverse Problem of Linear- Quadratic Differential Games: When is a Control Strategies Profile Nash?

    Y . Huang, T. Zhang, and Q. Zhu, “The Inverse Problem of Linear- Quadratic Differential Games: When is a Control Strategies Profile Nash?” in2022 58th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2022, pp. 1–7

    work page 2022

  19. [19]

    Inverse reinforcement learning for iden- tification of linear-quadratic zero-sum differential games,

    E. Martirosyan and M. Cao, “Inverse reinforcement learning for iden- tification of linear-quadratic zero-sum differential games,”Systems & Control Letters, vol. 172, 2023

    work page 2023

  20. [20]

    Model-Free Solution for Inverse Linear-Quadratic Nonzero-Sum Differential Games,

    ——, “Model-Free Solution for Inverse Linear-Quadratic Nonzero-Sum Differential Games,”IEEE Control Systems Letters, vol. 8, pp. 2445– 2450, 2024

    work page 2024

  21. [21]

    Feedback Nash equilibria for linear quadratic descriptor differential games,

    J. Engwerda and Salmah, “Feedback Nash equilibria for linear quadratic descriptor differential games,”Automatica, vol. 48, no. 4, pp. 625–631, 2012

    work page 2012

  22. [22]

    Feedback Properties of Descriptor Systems Using Matrix Projectors and Applications to Descriptor Dif- ferential Games,

    P. V . Reddy and J. C. Engwerda, “Feedback Properties of Descriptor Systems Using Matrix Projectors and Applications to Descriptor Dif- ferential Games,”SIAM Journal on Matrix Analysis and Applications, vol. 34, no. 2, pp. 686–708, 2013

    work page 2013

  23. [23]

    Feedback Nash Equilibrium for Randomly Switching Differential–Algebraic Games,

    A. Tanwani and Q. Zhu, “Feedback Nash Equilibrium for Randomly Switching Differential–Algebraic Games,”IEEE Transactions on Auto- matic Control, vol. 65, no. 8, pp. 3286–3301, 2020

    work page 2020

  24. [24]

    Magnus and H

    J. Magnus and H. Neudecker,Matrix Differential Calculus with Appli- cations in Statistics and Econometrics, ser. Wiley Series in Probability and Statistics. Wiley, 2019

    work page 2019

  25. [25]

    A matrix-pencil-based in- terpretation of inconsistent initial conditions and system properties of generalized state-space systems,

    G. Kalogeropoulos and K. G. Arvanitis, “A matrix-pencil-based in- terpretation of inconsistent initial conditions and system properties of generalized state-space systems,”IMA Journal of Mathematical Control and Information, vol. 15, no. 1, pp. 73–91, 1998

    work page 1998

  26. [26]

    Brenan, S

    K. Brenan, S. Campbell, and L. Petzold,Numerical Solution of Initial- Value Problems in Differential-Algebraic Equations, ser. Classics in Applied Mathematics. SIAM, 1996

    work page 1996

  27. [27]

    Kunkel and V

    P. Kunkel and V . Mehrmann,Differential-algebraic equations. Analysis and numerical solution. Z ¨urich: European Mathematical Society Publishing House, 2006

    work page 2006

  28. [28]

    Gantmacher,The Theory of Matrices, Volume 2, ser

    F. Gantmacher,The Theory of Matrices, Volume 2, ser. AMS Chelsea Publishing Series. American Mathematical Society, 2000

    work page 2000

  29. [29]

    Kronecker products and matrix calculus in system theory,

    J. Brewer, “Kronecker products and matrix calculus in system theory,” IEEE Transactions on Circuits and Systems, vol. 25, no. 9, pp. 772–781, 1978

    work page 1978