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REVIEW 4 major objections 4 minor 52 references

Operator-Theoretic Methods for Differential Games

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper shows that two Koopman-based pipelines — a resolvent-based global feedback search and a data-driven complementarity approach — both reproduce the analytic equilibrium of a zero-sum turret defense game.

desk verdict New and honest Koopman differential-game pipeline, but the resolvent method's aggregate-objective gap undermines the global-feedback equilibrium claim. read the letter →

arxiv 2507.02203 v1 pith:WXBEBHI3 submitted 2025-07-02 math.OC math.DS

Craig Bakker , Adam T. Rupe , Alexander Von Moll , Adam R. Gerlach This is my paper · ORCID
classification math.OCmath.DS MSC 49N7047A1037N35
keywords differentialgamesKoopmanoperatorExtendedDynamicModeDecompositionMixedComplementarityProblemsresolventzero-sumgameturretdefensefeedbackcontrol
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

Two-player zero-sum differential games are normally solved by hand, because nonlinear dynamics produce singular surfaces that resist general numerical treatment. This paper claims that the Koopman operator — a linear operator that evolves functions of the state instead of the state itself — can automate the solution. The first proposed method uses the resolvent of the Koopman generator, $(zI-L)^{-1}$, to propagate terminal and running costs over the entire state domain and solves for a global feedback policy by alternating optimization. The second learns a finite-dimensional Koopman model with control from data and solves each trajectory's equilibrium as a mixed complementarity problem. On a turret defense game with a known analytical solution, both methods reproduce the equilibrium trajectories and the value of the game, with the per-trajectory data-driven method matching the analytic solution more closely and the resolvent method supplying a single feedback law valid over the whole domain.

What carries the argument

The load-bearing object is the Koopman generator $L=f\cdot\nabla_x$ and its resolvent $R_L(z)=(zI-L)^{-1}$, used through the contour-integral identity $K_t=\frac{1}{2\pi i}\int_\gamma e^{zt}R_L(z)\ dz$ that propagates any observable forward in time. The paper folds a running cost into this identity with the factor $\frac{1}{z}(1-e^{-zT})$, so a single resolvent calculation delivers the full cost functional of the game. Both methods then replace the infinite-dimensional operator by a finite dictionary: random Fourier features for the EDMD model, radial basis functions for the resolvent model, with the control policies expanded in the same basis. The EDMD route additionally assumes control-affine lifted dynamics, which lets the players' optimality conditions be assembled into a mixed complementarity problem; the resolvent route avoids that assumption and instead carries the control dependence inside $L$, at the price of nonlinear, alternating optimizations.

What would settle it

Take a zero-sum game with a known pointwise equilibrium solution and a basis rich enough to represent the true value. Minimize the aggregate payoff $J(T)=\int_X J(x,T)\ dx$ with the resolvent method, then test the resulting feedback policy against the pointwise saddle inequalities $J(x,u,v^*)\le V(x)\le J(x,u^*,v)$ at a state where the aggregate optimum compromises one initial condition. If the policy violates either inequality, the method has not found an equilibrium of the original game despite having a perfect basis.

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Extended reading notes

Core claim

The central claim is that Koopman-based reformulations can yield the equilibrium solutions of a zero-sum differential game, at least for the finite approximations of the dynamics that the methods use. In continuous time, the paper expresses the game value as a contour integral of the resolvent, $J(x_0,T)=\frac{1}{2\pi i}\int_\gamma e^{zT}(zI-L)^{-1}\left(g(x_0)+\frac{1}{z}(1-e^{-zT})h(x_0)\right)dz$, and then makes the generator and both feedback policies finite-dimensional through a fixed basis, turning the saddle-point search into a finite-dimensional max-min problem solved by alternating optimizations. In discrete time, it fits an EDMD-with-control model $\Psi(x_{t+1})=K\Psi(x_t)+K_u u_t+K_v v_t$ with quadratic cost surrogates in lifted states, derives the two players' KKT conditions, and solves the resulting mixed complementarity problem trajectory by trajectory. On the turret defense game, the EDMD-MCP solutions are close to the analytic equilibrium for regular, constrained, and universal-line trajectories, while the resolvent-based feedback policy reproduces the structure of the analytic solution with the expected loss of fidelity near the $\alpha=0$ discontinuity.

Load-bearing premise

The load-bearing premise is that a feedback policy minimizing the payoff averaged over sampled initial states is also the saddle-point policy for each individual initial condition; the paper uses this equivalence without proving it.

Editorial extensions

If this is right

  • If the methods hold, differential games with nonlinear dynamics can be solved numerically without first characterizing their singular surfaces by hand.
  • The resolvent method produces a global feedback policy after one expensive computation, so evaluating it in real time is cheap; this matters for low-latency applications.
  • The EDMD-MCP method computes open-loop trajectories quickly from data and can enforce state and control constraints directly, making it suitable for high-dimensional, data-rich settings.
  • Both methods are approximations of the dynamics, so their equilibrium solutions are equilibria of the approximate model, not necessarily of the original game.
  • The two approaches are complementary: the resolvent method handles control-nonlinear dynamics without control lifting but is limited by smooth feedback bases, while the EDMD-MCP method captures discontinuities and constraints but needs lifted controls or domain knowledge.

Reading between the lines

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

  • A natural reading is that the resolvent method is solving a population-level min-max problem over the sampled domain rather than the pointwise game; whether its feedback law is also a pointwise saddle policy is a testable question, and if not, the method should be presented as an average-case synthesis tool.
  • The poor fidelity near the universal line points to basis resolution rather than method failure; a multi-resolution or discontinuous basis could sharpen the feedback law, though any state-feedback policy cannot represent the time-of-entry-dependent bifurcations that appear in the analytic solution.
  • The success of lifted controls like $\nu=r^2v$ and $\nu_\perp=rv_\perp$ suggests that choosing control coordinates from the structure of the generator $L=\sum_i f_i(x,u)\partial_{x_i}$ could be a general recipe for making EDMD-based game solvers work when naive additive control fails.
  • Combining the MCP read-out with learned dictionaries such as autoencoder-based DMD is a plausible path to higher-dimensional games, while the resolvent approach will likely remain limited by exponential growth in basis size and quadrature nodes.
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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 paper proposes two operator-theoretic approaches to solving two-player zero-sum differential games and demonstrates them on a turret defense game from the literature. The first approach approximates the Koopman generator on a fixed RBF basis and uses a resolvent contour-integral representation of the cost functional to compute a continuous-time global feedback policy by alternating optimization. The second approach builds a discrete-time EDMD-with-control model using random Fourier features, lifts the agent's controls into parallel and perpendicular components, and solves the resulting mixed complementarity problem trajectory by trajectory for open-loop policies. The methods are benchmarked against the analytical solution of Von Moll et al. for the turret defense game. The EDMD-MCP method matches the analytic trajectories closely for most of the approximately 1500 initial conditions tested, while the resolvent method shows qualitative agreement but visible deviations near the Universal Line and around alpha = 3 pi / 4, as the authors acknowledge. The paper also compares computational costs and discusses extensions of both methods.

Significance. The paper is a useful computational demonstration of how Koopman-operator ideas can be combined with complementarity and resolvent methods for differential games, and it provides a comparatively honest and transparent account of where the methods work and where they do not. Its strengths include the use of an external analytical benchmark, validation over a large set of initial conditions, and explicit discussion of the limitations of fixed-basis approximations. However, the central theoretical claim that both methods yield equilibrium solutions is only fully established for the EDMD-MCP method. For the resolvent method, the objective being optimized is an aggregate over initial states, and no proof or verification is supplied that the resulting global feedback policy satisfies the pointwise saddle-point inequalities of the original game. Because of this gap and the acknowledged deviations in the numerical solution, the paper's main claim is stronger than what is demonstrated. The methods are of interest to the computational differential games and Koopman control communities, but the theoretical gap must be closed or the claims substantially qualified.

major comments (4)
  1. [Section 3.2, Eq. (49)] The resolvent method optimizes the aggregate payoff J(T) = \int_X [g(F_T(x)) + \int_0^T h(F_t(x)) dt] dx (Eq. 23), not the pointwise value function V(x). For a feedback policy to be a saddle-point equilibrium, the inequalities in Eq. (65) must hold at every initial state. Stationarity of the aggregate objective with respect to a control coefficient u_k gives only \sum_i \partial J(x_i)/\partial u_k = 0, which does not imply the pointwise stationarity needed for Eq. (65). No argument via dynamic programming, the Isaacs equation, or the HJI equation is provided. This is not a numerical-fidelity issue: even with an exact finite basis, a policy that is optimal on average over the sampled domain need not be a saddle policy at any particular initial condition, and the small-r region is especially vulnerable because the running and terminal costs are proportional to r. The paper should either prove the equivalence for the considered class of games or verify the pointwise saddle inequalities on a dense grid and restate the result as an approximate global feedback policy.
  2. [Section 6.1] The paper's abstract and Section 1.1 claim that the approaches "yield the equilibrium solutions" and "replicate the behavior of the analytical solution," but Section 6.1 concedes that the resolvent method is "not directly solving for the Nash equilibrium" and instead relies on alternating optimization. This is not merely a wording issue: it means the reported resolvent-based feedback policies have not been shown to satisfy the defining equilibrium property of the game. The authors should either supply a convergence or equivalence argument for the alternating scheme, or explicitly frame the resolvent results as approximate solutions to a related aggregate objective, not as verified equilibrium policies.
  3. [Section 5.2, Figs. 3 and 4] The reported resolvent results show that the smooth fixed basis cannot produce the sharp discontinuity at alpha = 0 (the Universal Line) and that the agent leaves the r = 1 surface near alpha = 3 pi / 4, a region the analytic solution treats as constrained. These are not peripheral artifacts: they are exactly the singular and constrained regimes that define the equilibrium structure of the game. The paper acknowledges these limitations, but the central claim that the method replicates the analytical solution needs to be qualified with a quantitative statement of where in the domain the approximation is valid, or the method needs a basis or representation able to capture the discontinuities.
  4. [Section 4.3.2, Eq. (92)] The lower bound v^2 + v_perp^2 >= 0.9 v_A^2 is introduced because the finite-basis approximation sometimes makes the agent's speed too small, especially at small r. This changes the admissible control set relative to the original game, in which the agent's speed is fixed at v_A. The assertion in Section 4.2 that the relaxed inequality v^2 + v_perp^2 <= v_A^2 is tight at the optimum does not apply to the resolvent solution, since the paper explicitly states that the speed is sometimes well below v_A. The equilibrium status of a policy whose speed is 0.9 v_A in part of the domain is unclear; the authors should either project the final policy onto the feasible set or provide evidence that the lower bound is inactive at the reported solution.
minor comments (4)
  1. [Abstract] There is a typographical double period in "provided in the literature..".
  2. [Section 4.2, Eq. (72)] The inequality in Eq. (72) appears to have the direction reversed: the intended constraint is v^2 + v_perp^2 <= v_A^2, not v_A^2 <= v^2 + v_perp^2.
  3. [Section 4.3.2] The sentence "However, in this particular case, these was a far simpler alternative approach" contains a grammatical error and should be rephrased.
  4. [Section 5.1] The phrase "the constraint ( r = 1)" should read "the constraint r <= 1" for consistency with the problem statement in Section 4.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: both pipelines optimize independently specified costs and dynamics; the only authors' prior work is a post-hoc comparison benchmark.

full rationale

The derivation chain is self-contained. In the EDMD-MCP method, the Koopman dynamics are learned by least-squares regression from gridded data (Section A), and the cost matrices Qg and Qh are fixed a priori from the known running and terminal costs via the explicitly included cos α observable (Eq. 82), not fitted to the equilibrium solution. The MCP then solves the KKT conditions (Eqs. 34-41) for the approximate dynamics, which is exactly an equilibrium computation for those dynamics. In the resolvent method, the objective is computed by propagating the known costs through the resolvent formula (Eqs. 47-49) using a fixed RBF basis, and the control coefficients are the optimization variables; no target value or equilibrium trajectory is used as a training label. The numerical comparison uses the analytical solution of Von Moll et al. [44], which was constructed by backward integration of the equilibrium state and adjoint dynamics and is independent of the Koopman approximation; although that prior work shares authors with the present paper, it is a post-hoc benchmark and not an input to either method. The paper's explicit concession in Section 6.1 that the alternating-optimization resolvent scheme is 'not directly solving for the Nash equilibrium' is a convergence/optimality limitation, not a circular definition. The aggregate objective in Eq. 23/49 is a potential correctness gap if it does not coincide with the pointwise value function, but that is a modeling approximation issue, not a circular reduction. Therefore no circular step was found.

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

The central claim relies on the accuracy of finite-dimensional Koopman approximations and on two unproved equivalences: the aggregate min-max objective representing the pointwise game, and the tightness of the relaxed control constraint. Most background is standard mathematics from the cited literature.

free parameters (9)
  • RFF dictionary variance = 100.0
    Frequencies for random Fourier features sampled from a diagonal Gaussian with variance 100.0, chosen by hand after tests showed low sensitivity (Section 4.2).
  • RFF dictionary size N = not stated
    The number of random Fourier feature observables is never specified, but it determines the fidelity of the EDMDc model.
  • Discrete time step dt = 0.01
    Used for zero-order hold in EDMDc training and MCP discretization (Appendix A.1).
  • RBF kernel grid size = 5x5 on [-0.1,1.1]x[-0.1pi,1.1pi]
    Fixed radial basis function centroids for the resolvent approximation; a denser grid would improve fidelity at higher cost (Section 4.3.1).
  • Evaluation grid size = 25x25 on [0,1]x[0,pi]
    Sampling points for enforcing constraints and approximating the domain integral in the resolvent method.
  • Resolvent quadrature parameters delta, m, h, N = not stated
    Parameters in Eq. 8 for contour integral approximation; the paper cites Horning and Gerlach but does not report the chosen values.
  • Lower bound factor for agent speed constraint = 0.9
    Ad hoc constraint v^2+v_perp^2 >= 0.9 v_A^2 imposed to prevent small speed from basis approximation errors (Section 4.3.2).
  • Alternating optimization iteration cap = 100
    Each player's optimization runs at most 100 iterations before switching; chosen to prevent cycling (Section 4.3.3).
  • Warm-start perturbations = small
    Numerical procedure perturbs warm starts to improve convergence; magnitude not specified.
assumptions (9)
  • standard math Koopman operator theory: for a dynamical system, the KO is a linear operator on L_infinity and its generator is f dot grad_x
    Used throughout Section 2.1 to justify the resolvent representation Eq. 6.
  • domain assumption The finite Galerkin approximation of the Koopman generator via EDMD and RFF converges to the true generator as data and dictionary increase
    Assumed without proof; the paper relies on the EDMDc approximation Eq. 24 being accurate enough for the MCP solutions (Section 4.2).
  • domain assumption The zero-sum game has a value and the saddle-point property holds
    Eq. 65; this is a standard assumption for two-player zero-sum differential games with terminal and running costs.
  • domain assumption The analytic solution of Von Moll et al. is the correct ground truth for v_A=T=1
    The paper uses it as the reference for validating both methods (Section 5).
  • ad hoc to paper The aggregate min-max over sampled initial states is equivalent to solving the differential game's equilibrium feedback policy
    Eq. 49 defines the resolvent optimization as min over v_k, max over u_k of the total cost integrated over all x_i; no proof is given that the resulting feedback policy equals the pointwise saddle-point policy of the original game. This is a load-bearing premise for the resolvent method.
  • ad hoc to paper The relaxed inequality v^2+v_perp^2 <= v_A^2 is tight at the optimum
    Section 4.2 states this without proof; if the inequality is not tight, the lifted control model does not represent the true agent dynamics.
  • ad hoc to paper The basis functions (RFFs for EDMD, RBFs for resolvent) can represent the equilibrium policies and the Koopman generator with sufficient accuracy
    The success of both methods depends on this; the paper's own results show the RBF basis cannot capture discontinuities near the Universal Line and the agent's heading bifurcation (Section 5.2).
  • standard math PATH algorithm solves the nonlinear MCP to a local solution with the provided initial guesses
    The paper relies on PATH for the EDMD-MCP solutions; local solutions are sufficient for their claim, but no global optimality is guaranteed.
  • ad hoc to paper Alternating optimization converges to the saddle point
    The resolvent method uses alternating optimizations with warm starts; the paper itself notes such approaches can fail to converge in general (Section 6, citing reference 51).

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Cite this review

Pith. "Pith review of Operator-Theoretic Methods for Differential Games." pith.science (2026). https://pith.science/paper/WXBEBHI3

@misc{pith2026250702203,
  author       = {Pith},
  title        = {Pith review of: Operator-Theoretic Methods for Differential Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WXBEBHI3}},
  note         = {Machine review of arXiv:2507.02203}
}
read the original abstract

Differential game theory offers an approach for modeling interactions between two or more agents that occur in continuous time. The goal of each agent is to optimize its objective cost functional. In this paper, we present two different methods, based on the Koopman Operator (KO), to solve a zero-sum differential game. The first approach uses the resolvent of the KO to calculate a continuous-time global feedback solution over the entire domain. The second approach uses a discrete-time, data-driven KO representation with control to calculate open-loop control policies one trajectory at a time. We demonstrate these methods on a turret defense game from the literature, and we find that the methods' solutions replicate the behavior of the analytical solution provided in the literature.. Following that demonstration, we highlight the relative advantages and disadvantages of each method and discuss potential future work for this line of research.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

52 extracted references · 43 canonical work pages

  1. [1]

    Wiley, New York (1965)

    Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Optimization, Control and Warfare. Wiley, New York (1965)

    work page 1965

  2. [2]

    Dockner, (2000)

    Dockner, E.J., Jorgensen, S., Long, N.V., Sorger, G.: Differential Games in Economics and Management Science by Engelbert J. Dockner, (2000). /core/books/differential-games-in-economics-and-management- science/F7BC13BC9192CE30AE549D53625B6BBE

    work page 2000

  3. [3]

    In: Dynamic Non- cooperative Game Theory

    Ba¸ sar, T., Olsder, G.J.: Chapter 8: Pursuit-evasion games. In: Dynamic Non- cooperative Game Theory. Mathematics in Science and Engineering, vol. 160, pp. 344–398. Elsevier, London (1982). https://doi.org/10.1016/S0076-5392(08) 62960-4 . http://www.sciencedirect.com/science/article/pii/S0076539208629604

    work page doi:10.1016/s0076-5392(08 1982

  4. [4]

    Gardner, M.: Mathematical carnival - from penny puzzles, card shuffles and tricks of lightning calculators to roller coaster rides into the fourth dimension (1975) 43

    work page 1975

  5. [5]

    PhD thesis, Stanford (1971)

    Merz, A.W.: The homicidal chauffeur - a differential game. PhD thesis, Stanford (1971)

    work page 1971

  6. [6]

    In: 2017 IEEE Conference on Control Technology and Applications (CCTA), pp

    Akilan, Z., Fuchs, Z.: Zero-sum turret defense differential game with singular surfaces. In: 2017 IEEE Conference on Control Technology and Applications (CCTA), pp. 2041–2048 (2017). https://doi.org/10.1109/CCTA.2017.8062754

    work page arXiv 2017

  7. [7]

    (eds.) Multi-Agent Reinforcement Learning: A Selective Overview of Theories and Algorithms, pp

    Zhang, K., Yang, Z., Ba¸ sar, T.: In: Vamvoudakis, K.G., Wan, Y., Lewis, F.L., Cansever, D. (eds.) Multi-Agent Reinforcement Learning: A Selective Overview of Theories and Algorithms, pp. 321–384. Springer, Cham (2021). https://doi.org/ 10.1007/978-3-030-60990-0 12 . https://doi.org/10.1007/978-3-030-60990-0 12

    doi:10.1007/978-3-030-60990-0 2021

  8. [8]

    (eds.) Pursuit-evasion games and zero- sum two-person differential games, pp

    Bernhard, P.: In: Bailleul, J., Samad, T. (eds.) Pursuit-evasion games and zero- sum two-person differential games, pp. 1103–1109. Springer, ??? (2014)

    work page 2014

Show all 52 references
  1. [9]

    Journal of Opti- mization Theory and Applications 3, 283–295 (1969) https://doi.org/10.1007/ BF00931368

    Isaacs, R.: Differential games: Their scope, nature, and future. Journal of Opti- mization Theory and Applications 3, 283–295 (1969) https://doi.org/10.1007/ BF00931368

    1969

  2. [10]

    In: Dynamic Noncooperative Game Theory

    Ba¸ sar, T., Olsder, G.J.: Chapter 6: Nash equilibria of infinite dynamic games. In: Dynamic Noncooperative Game Theory. Mathe- matics in Science and Engineering, vol. 160, pp. 239–304. Elsevier, London (1982). https://doi.org/10.1016/S0076-5392(08)62958-6 . https://www.scienc...

    1982 doi

  3. [11]

    Military Operations Research 25(4), 43–61 (2020)

    Quigley, K.J., Gabriel, S.A., Azarm, S.: Multiagent unmanned vehicle trajectories with rolling-horizon games. Military Operations Research 25(4), 43–61 (2020)

    2020

  4. [12]

    Optimization methods and software 5(2), 123–156 (1995)

    Dirkse, S.P., Ferris, M.C.: The path solver: a nommonotone stabilization scheme for mixed complementarity problems. Optimization methods and software 5(2), 123–156 (1995)

    1995

  5. [13]

    EURO Journal on Decision Processes 2, 91–120 (2014)

    Ruiz, C., Conejo, A.J., Fuller, J.D., Gabriel, S.A., Hobbs, B.F.: A tutorial review of complementarity models for decision-making in energy markets. EURO Journal on Decision Processes 2, 91–120 (2014)

    2014

  6. [14]

    In: 2020 IEEE Conference on Control Technology and Applications (CCTA)

    Garcia, E., Casbeer, D.W., Pachter, M.: Defense of a target against intelligent adversaries: A linear quadratic formulation. In: 2020 IEEE Conference on Control Technology and Applications (CCTA). IEEE, ??? (2020). https://doi.org/10.1109/ccta41146.2020.9206368 . http://dx.doi...

    2020

  7. [15]

    Pro- ceedings of the National Academy of Sciences of the United States of America 17(5), 315–318 (1931) 44

    Koopman, B.O.: Hamiltonian Systems and Transformation in Hilbert Space. Pro- ceedings of the National Academy of Sciences of the United States of America 17(5), 315–318 (1931) 44

    1931

  8. [16]

    Chaos: An Interdisci- plinary Journal of Nonlinear Science 22(4), 047510 (2012)

    Budiˇ si´ c, M., Mohr, R., Mezi´ c, I.: Applied koopmanism. Chaos: An Interdisci- plinary Journal of Nonlinear Science 22(4), 047510 (2012)

    2012

  9. [17]

    (eds.): The Koopman Operator in Systems and Control: Concepts, Methodologies, and Applications, 1st ed

    Mauroy, A., Mezi´ c, I., Susuki, Y. (eds.): The Koopman Operator in Systems and Control: Concepts, Methodologies, and Applications, 1st ed. 2020 edition edn. Springer, ??? (2020). https://doi.org/10.1007/978-3-030-35713-9

    2020 doi

  10. [18]

    Machine Learning: Science and Technology 2(3), 035023 (2021) https://doi.org/10.1088/2632-2153/abf0f5 arXiv:1707.01146

    Kaiser, E., Kutz, J.N., Brunton, S.L.: Data-driven discovery of Koopman eigen- functions for control. Machine Learning: Science and Technology 2(3), 035023 (2021) https://doi.org/10.1088/2632-2153/abf0f5 arXiv:1707.01146

    2021 arXiv

  11. [19]

    In: 2022 American Control Conference (ACC), pp

    Krolicki, A., Sutavani, S., Vaidya, U.: Koopman-based policy iteration for robust optimal control. In: 2022 American Control Conference (ACC), pp. 1317–1322 (2022). IEEE

    2022

  12. [20]

    arXiv preprint arXiv:2309.16098 (2023)

    Zhao, Y., Zhu, Q.: Stackelberg game-theoretic trajectory guidance for multi-robot systems with koopman operator. arXiv preprint arXiv:2309.16098 (2023)

    2023 arXiv

  13. [21]

    Reliability Engineering & System Safety 237, 109323 (2023)

    Oster, M.R., King, E., Bakker, C., Bhattacharya, A., Chatterjee, S., Pan, F.: Multi-level optimization with the koopman operator for data-driven, domain- aware, and dynamic system security. Reliability Engineering & System Safety 237, 109323 (2023)

    2023

  14. [22]

    Springer, ??? (1994)

    Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. Springer, ??? (1994). https://doi.org/10.1007/978-1-4612-4286-4

    1994 doi

  15. [23]

    Springer, ??? (2006)

    Engel, K.-J., Nagel, R.: A Short Course on Operator Semigroups. Springer, ??? (2006). https://doi.org/10.1007/0-387-36619-9

    2006 doi

  16. [24]

    arXiv (2024)

    Horning, A., Gerlach, A.R.: A Family of High-Order Accurate Contour Integral Methods for Strongly Continuous Semigroups. arXiv (2024)

    2024

  17. [25]

    Springer, ??? (2000)

    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, ??? (2000). https://doi.org/10.1007/b97696

    2000 doi

  18. [26]

    2020 Resilience Week (R WS), 34–40 (2020)

    Bakker, C., Bhattacharya, A., Chatterjee, S., Perkins, C.J., Oster, M.R.: The koopman operator: Capabilities and recent advances. 2020 Resilience Week (R WS), 34–40 (2020)

    2020

  19. [27]

    Communications on Pure and Applied Mathematics 77(1), 221–283 (2024) https://doi.org/10.1002/ cpa.22125

    Colbrook, M.J., Townsend, A.: Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. Communications on Pure and Applied Mathematics 77(1), 221–283 (2024) https://doi.org/10.1002/ cpa.22125

    2024

  20. [28]

    Springer, ??? (1995)

    Kato, T.: Perturbation Theory for Linear Operators. Springer, ??? (1995). https: //doi.org/10.1007/978-3-642-66282-9

    1995 doi

  21. [29]

    In: Decision and Control (CDC), 2015 IEEE 54th Annual Conference On, pp

    Mezi´ c, I.: On applications of the spectral theory of the koopman operator in 45 dynamical systems and control theory. In: Decision and Control (CDC), 2015 IEEE 54th Annual Conference On, pp. 7034–7041 (2015). IEEE

    2015

  22. [30]

    IF AC-PapersOnLine 49(18), 716–723 (2016)

    Surana, A., Banaszuk, A.: Linear observer synthesis for nonlinear systems using koopman operator framework. IF AC-PapersOnLine 49(18), 716–723 (2016)

    2016

  23. [31]

    arXiv preprint arXiv:1908.02233 (2019)

    Bakker, C., Rosenthal, S., Nowak, K.E.: Koopman representations of dynamic systems with control. arXiv preprint arXiv:1908.02233 (2019)

    2019 arXiv

  24. [32]

    In: Proceedings of the Twelfth ACM International Conference on Future Energy Systems, pp

    King, E., Bakker, C., Bhattacharya, A., Chatterjee, S., Pan, F., Oster, M.R., Perkins, C.J.: Solving the dynamics-aware economic dispatch problem with the koopman operator. In: Proceedings of the Twelfth ACM International Conference on Future Energy Systems, pp. 137–147 (2021)

    2021

  25. [33]

    Automatica 93, 149–160 (2018)

    Korda, M., Mezi´ c, I.: Linear predictors for nonlinear dynamical systems: Koop- man operator meets model predictive control. Automatica 93, 149–160 (2018)

    2018

  26. [34]

    In: 2022 IEEE 61st Conference on Decision and Control (CDC), pp

    Ramachandran, T., Nandanoori, S.P., Sinha, S., Bakker, C.: Data-driven char- acterization of recovery energy in controlled dynamical systems using koopman operator. In: 2022 IEEE 61st Conference on Decision and Control (CDC), pp. 2230–2235 (2022). IEEE

    2022

  27. [35]

    In: 2022 American Control Conference (ACC), pp

    Sinha, S., Nandanoori, S.P., Ramachandran, T., Bakker, C., Singhal, A.: Data- driven resilience characterization of control dynamical systems. In: 2022 American Control Conference (ACC), pp. 2186–2193 (2022). IEEE

    2022

  28. [36]

    In: 2018 IEEE Conference on Decision and Control (CDC), pp

    Huang, B., Ma, X., Vaidya, U.: Feedback stabilization using koopman operator. In: 2018 IEEE Conference on Decision and Control (CDC), pp. 6434–6439 (2018). IEEE

    2018

  29. [37]

    Journal of Fluid Mechanics 918, 10 (2021)

    Herrmann, B., Baddoo, P.J., Semaan, R., Brunton, S.L., McKeon, B.J.: Data- driven resolvent analysis. Journal of Fluid Mechanics 918, 10 (2021)

    2021

  30. [38]

    In: 2024 European Control Conference (ECC), pp

    Susuki, Y., Mauroy, A., Drmaˇ c, Z.: Koopman resolvents of nonlinear discrete-time systems: Formulation and identification. In: 2024 European Control Conference (ECC), pp. 627–632 (2024). IEEE

    2024

  31. [39]

    Energy 75, 483–500 (2014)

    Huppmann, D., Egging, R.: Market power, fuel substitution and infrastructure– a large-scale equilibrium model of global energy markets. Energy 75, 483–500 (2014)

    2014

  32. [40]

    Agricultural Systems 164, 165–184 (2018)

    Bakker, C., Zaitchik, B.F., Siddiqui, S., Hobbs, B.F., Broaddus, E., Neff, R.A., Haskett, J., Parker, C.L.: Shocks, seasonality, and disaggregation: Modelling food security through the integration of agricultural, transportation, and economic systems. Agricultural Systems 164,...

    2018

  33. [41]

    Reliability Engineering & System Safety 200, 106958 (2020)

    Bakker, C., Webster, J.B., Nowak, K.E., Chatterjee, S., Perkins, C.J., Brigantic, 46 R.: Multi-game modeling for counter-smuggling. Reliability Engineering & System Safety 200, 106958 (2020)

    2020

  34. [42]

    Mathematical programming 106(1), 25–57 (2006)

    W¨ achter, A., Biegler, L.T.: On the implementation of an interior-point fil- ter line-search algorithm for large-scale nonlinear programming. Mathematical programming 106(1), 25–57 (2006)

    2006

  35. [43]

    Advances in Neural Information Processing Systems 35, 35201– 35212 (2022)

    Wibisono, A., Tao, M., Piliouras, G.: Alternating mirror descent for constrained min-max games. Advances in Neural Information Processing Systems 35, 35201– 35212 (2022)

    2022

  36. [44]

    In: Modeling, Estimation and Control Conference

    Von Moll, A., Gerlach, A.R., Bakker, C., Rupe, A., Pachter, M.: Constrained turret defense with fixed final time. In: Modeling, Estimation and Control Conference. IF AC, ??? (2024). Accepted

    2024

  37. [45]

    Journal of Optimization Theory and Applications (2024) https://doi.org/10.1007/s10957-024-02490-7

    Pachter, M., Weintraub, I.E.: The synthesis of optimal control laws using isaacs’ method for the solution of differential games. Journal of Optimization Theory and Applications (2024) https://doi.org/10.1007/s10957-024-02490-7

    2024 doi

  38. [46]

    Advances in neural information processing systems 20 (2007)

    Rahimi, A., Recht, B.: Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007)

    2007

  39. [47]

    IF AC-PapersOnLine 53(2), 1150–1155 (2020)

    Sinha, S., Nandanoori, S.P., Yeung, E.: Koopman operator methods for global phase space exploration of equivariant dynamical systems. IF AC-PapersOnLine 53(2), 1150–1155 (2020)

    2020

  40. [48]

    Mathematical Programming Computation 15, 581–589 (2023) https: //doi.org/10.1007/s12532-023-00239-3

    Lubin, M., Dowson, O., Dias Garcia, J., Huchette, J., Legat, B., Vielma, J.P.: JuMP 1.0: Recent improvements to a modeling language for mathematical opti- mization. Mathematical Programming Computation 15, 581–589 (2023) https: //doi.org/10.1007/s12532-023-00239-3

    2023 doi

  41. [49]

    In: Proceedings of the 1st Workshop for High Performance Technical Computing in Dynamic Languages – HPTCDL ‘14

    Olver, S., Townsend, A.: A practical framework for infinite-dimensional linear algebra. In: Proceedings of the 1st Workshop for High Performance Technical Computing in Dynamic Languages – HPTCDL ‘14. IEEE, ??? (2014)

    2014

  42. [50]

    Tarek, M.: Nonconvex.jl: A comprehensive julia package for non-convex optimiza- tion (2023) https://doi.org/10.13140/RG.2.2.36120.37121

    2023

  43. [51]

    In: Advances in Neural Information Processing Systems, vol

    Schaefer, F., Anandkumar, A.: Competitive Gradient Descent. In: Advances in Neural Information Processing Systems, vol. 32. Curran Associates, Inc., ??? (2019)

    2019

  44. [52]

    SIAM Journal on Applied Dynamical Systems 15(1), 142–161 (2016) 47

    Proctor, J.L., Brunton, S.L., Kutz, J.N.: Dynamic mode decomposition with control. SIAM Journal on Applied Dynamical Systems 15(1), 142–161 (2016) 47

    2016

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.