pith. sign in

arxiv: 2311.01774 · v2 · submitted 2023-11-03 · 🧮 math-ph · cs.NA· math.MP· math.NA· math.OC

Optimal Control of Incompressible Ideal Flows with Obstacle Avoidance

Alexandre Anahory Simoes , Anthony Bloch , Leonardo Colombo This is my paper

Pith reviewed 2026-05-24 06:14 UTC · model grok-4.3

classification 🧮 math-ph cs.NAmath.MPmath.NAmath.OC
keywords optimal controlincompressible ideal flowsEuler equationsobstacle avoidancebarrier potentialvariational formulationpressure shiftfluid dynamics
0
0 comments X

The pith

Introducing a barrier potential into the optimal control of incompressible ideal flows produces modified Euler equations with a shifted effective pressure.

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

The paper shows that adding a barrier-type potential for obstacle avoidance to the optimal control functional for incompressible ideal fluid flows modifies the resulting Euler equations. The barrier acts on the fluid's Lagrangian configuration but manifests in the Eulerian view as an adjustment to the pressure. This extension preserves the variational structure of the original formulation while incorporating avoidance behavior. A reader would care because it offers a variational way to steer inviscid flows around obstacles without introducing viscosity.

Core claim

By incorporating a barrier-type potential into the optimal control functional, the authors derive modified Euler equations for an inviscid incompressible fluid. The barrier term influences the Lagrangian configuration and appears in the Eulerian description as a shift in the effective pressure. Numerical simulations illustrate that this induces a localized deformation of the flow near the obstacle region.

What carries the argument

The barrier-type potential added to the optimal control functional, which penalizes proximity to obstacles and shifts the pressure in the Eulerian equations.

If this is right

  • The flow remains incompressible and inviscid, with only the pressure modified by the barrier.
  • The variational structure is preserved, yielding Euler-type equations with the pressure shift.
  • Numerical reduced Eulerian dynamics show localized deformation near the obstacle consistent with avoidance penalization.
  • The method extends the link between optimal control and fluid dynamics to include obstacle constraints.

Where Pith is reading between the lines

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

  • If the pressure shift works in simulations, it could be tested in physical experiments with controlled flows around barriers.
  • This approach might generalize to other constraints in fluid optimal control problems.
  • Connections to other variational methods in fluid mechanics could be explored for similar modifications.

Load-bearing premise

The barrier-type potential can be added to the optimal control functional while preserving the variational structure that yields Euler-type equations, with only a change in the pressure.

What would settle it

A direct computation or simulation showing whether the effective pressure in the modified equations exactly matches the barrier term's contribution from the Lagrangian side would confirm or refute the shift.

Figures

Figures reproduced from arXiv: 2311.01774 by Alexandre Anahory Simoes, Anthony Bloch, Leonardo Colombo.

Figure 1
Figure 1. Figure 1: Depiction of the initial conditions. The arrows represent [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simulation of modified Euler equations after [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

It was shown in \cite{bloch2000optimal} that an optimal control formulation for incompressible ideal fluid flow yields Euler's equations. In this paper, we consider a variational obstacle-avoidance formulation for incompressible ideal flows by introducing a barrier-type potential in the associated optimal control functional. This leads to \textit{modified Euler equations for an inviscid fluid}, in which the barrier term acts on the Lagrangian configuration and appears in the Eulerian description as a shift in the effective pressure. We also present a numerical illustration of the reduced Eulerian dynamics, showing that the barrier term induces a localized deformation of the flow near the obstacle region, consistent with its role as an obstacle-avoidance penalization.

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 / 2 minor

Summary. The manuscript extends the optimal control formulation of Bloch et al. (2000) for incompressible ideal fluid flow by adding a barrier-type potential to the functional to enforce obstacle avoidance. This yields modified Euler equations for an inviscid fluid in which the barrier acts on the Lagrangian configuration and appears in the Eulerian description as a shift in the effective pressure. A numerical illustration of the reduced Eulerian dynamics is presented, demonstrating localized deformation of the flow near the obstacle.

Significance. If the derivation holds, the result supplies a variational mechanism for incorporating configuration-dependent obstacle avoidance into ideal incompressible flow control while preserving the Euler-Poincaré structure; the pressure-shift effect follows directly from the conservative nature of the added potential under the divergence-free constraint. This framework is potentially useful for geometric control problems in fluids. The numerical example provides concrete support for the theoretical claim.

minor comments (2)
  1. [Abstract] The abstract states that the barrier term 'appears in the Eulerian description as a shift in the effective pressure' but does not display the explicit modified equation; adding the precise form (even if derived later) would improve immediate readability.
  2. The numerical illustration is described only qualitatively ('localized deformation'); a brief statement of the discretization scheme, domain, or quantitative measure of the pressure shift would strengthen the supporting evidence without altering the central claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The recognition that the barrier potential preserves the Euler-Poincaré structure while inducing a pressure shift is particularly appreciated.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper cites bloch2000optimal solely to establish the base fact that an optimal-control formulation on volume-preserving diffeomorphisms yields the standard Euler equations. It then introduces a configuration-dependent barrier potential into the control functional and applies standard first-variation arguments; the resulting body force is absorbed into the pressure under the divergence-free constraint, producing a modified Euler equation whose form follows directly from the variational structure without any fitted parameters, self-referential definitions, or load-bearing self-citations. No equation or claim reduces to its own inputs by construction, and the cited prior result is an independent external reference rather than an unverified uniqueness theorem supplied by the present authors.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies insufficient detail to enumerate free parameters, axioms, or invented entities. The barrier potential is introduced but its explicit form and any associated parameters are not given.

pith-pipeline@v0.9.0 · 5657 in / 987 out tokens · 40070 ms · 2026-05-24T06:14:41.926355+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

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

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

25 extracted references · 25 canonical work pages

  1. [1]

    An optimal control formulation for inviscid incompressible ideal fluid flow,

    A. M. Bloch, D. D. Holm, P. E. Crouch, and J. E. Marsden, “An optimal control formulation for inviscid incompressible ideal fluid flow,” in Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No. 00CH37187) , vol. 2. IEEE, 2000, pp. 1273–1278

    work page 2000

  2. [2]

    Sur la géométrie différentielle des groupes de lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits,

    V . Arnold, “Sur la géométrie différentielle des groupes de lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits,” in Annales de l’institut Fourier, vol. 16, no. 1, 1966, pp. 319– 361

    work page 1966

  3. [3]

    Coadjoint orbits, vortices, and clebsch variables for incompressible fluids,

    J. Marsden and A. Weinstein, “Coadjoint orbits, vortices, and clebsch variables for incompressible fluids,” Physica D: Nonlinear Phenomena , vol. 7, no. 1-3, pp. 305–323, 1983

    work page 1983

  4. [4]

    Groups of diffeomorphisms and the solution of the classical euler equations for a perfect fluid,

    D. G. Ebin and J. E. Marsden, “Groups of diffeomorphisms and the solution of the classical euler equations for a perfect fluid,” 1969

    work page 1969

  5. [5]

    Semidirect products and reduction in mechanics,

    J. E. Marsden, T. Ra¸ tiu, and A. Weinstein, “Semidirect products and reduction in mechanics,” Transactions of the american mathematical society, vol. 281, no. 1, pp. 147–177, 1984

    work page 1984

  6. [6]

    Euler’s fluid equations: Optimal control vs optimization,

    D. D. Holm, “Euler’s fluid equations: Optimal control vs optimization,” Physics Letters A , vol. 373, no. 47, pp. 4354–4359, 2009

    work page 2009

  7. [7]

    Optimal control and geodesic flows,

    A. M. Bloch and P. E. Crouch, “Optimal control and geodesic flows,” Systems & control letters , vol. 28, no. 2, pp. 65–72, 1996

    work page 1996

  8. [8]

    Double bracket equations and geodesic flows on symmetric spaces,

    A. M. Bloch, R. W. Brockett, and P. E. Crouch, “Double bracket equations and geodesic flows on symmetric spaces,” Communications in mathematical physics , vol. 187, pp. 357–373, 1997

    work page 1997

  9. [9]

    Discrete rigid body dynamics and optimal control,

    A. M. Bloch, P. E. Crouch, J. E. Marsden, and T. S. Ratiu, “Discrete rigid body dynamics and optimal control,” in Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No. 98CH36171) , vol. 2. IEEE, 1998, pp. 2249–2254

    work page 1998

  10. [10]

    Variational obstacle avoidance on riemannian manifolds,

    A. Bloch, M. Camarinha, and L. J. Colombo, “Variational obstacle avoidance on riemannian manifolds,” Proceedings of the 2017 IEEE International Conference on Decision and Control , pp. 146–150, 2017

    work page 2017

  11. [11]

    Dynamic interpolation for obstacle avoidance on riemannian manifolds,

    ——, “Dynamic interpolation for obstacle avoidance on riemannian manifolds,” International Journal of Control , vol. 94, no. 3, pp. 588– 600, 2021

    work page 2021

  12. [12]

    Variational collision avoidance problems on riemannian manifolds,

    M. Assif, R. Banavar, A. Bloch, M. Camarinha, and L. J. Colombo, “Variational collision avoidance problems on riemannian manifolds,” Proceedings of the 2018 IEEE International Conference on Decision and Control, pp. 2791–2796, 2018

    work page 2018

  13. [13]

    Variational collision and obstacle avoidance of multi-agent systems on riemannian manifolds,

    R. Chandrasekaran, L. J. Colombo, M. Camarinha, R. Banavar, and A. Bloch, “Variational collision and obstacle avoidance of multi-agent systems on riemannian manifolds,” Proceedings of the 2020 European Control Conference, 2020

    work page 2020

  14. [14]

    Variational point-obstacle avoidance on riemannian manifolds,

    A. Bloch, M. Camarinha, and L. J. Colombo, “Variational point-obstacle avoidance on riemannian manifolds,” Mathematics of Control, Signals, and Systems, vol. 33, pp. 109–121, 2021

    work page 2021

  15. [15]

    The euler–poincaré equa- tions and semidirect products with applications to continuum theories,

    D. D. Holm, J. E. Marsden, and T. S. Ratiu, “The euler–poincaré equa- tions and semidirect products with applications to continuum theories,” Advances in Mathematics , vol. 137, no. 1, pp. 1–81, 1998

    work page 1998

  16. [16]

    Brand, Vector and tensor analysis

    L. Brand, Vector and tensor analysis. Courier Dover Publications, 2020

    work page 2020

  17. [17]

    J. E. Marsden, A. J. Tromba, A. Weinstein et al. , Basic multivariable calculus. Springer, 1993

    work page 1993

  18. [18]

    Geometric dynamics of optimization,

    F. Gay-Balmaz, D. Holm, and T. Ratiu, “Geometric dynamics of optimization,”Communications in Mathematical Sciences, vol. 11, no. 1, pp. 163–231, 2013

    work page 2013

  19. [19]

    Abraham and J

    R. Abraham and J. E. Marsden, Foundations of mechanics. American Mathematical Soc., 2008, no. 364

    work page 2008

  20. [20]

    Abraham, J

    R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications. Springer Science & Business Media, 2012, vol. 75

    work page 2012

  21. [21]

    Real-time obstacle avoidance for manipulators and mobile robots,

    O. Khatib, “Real-time obstacle avoidance for manipulators and mobile robots,” The international journal of robotics research, vol. 5, no. 1, pp. 90–98, 1986

    work page 1986

  22. [22]

    Robot planning and control via potential functions,

    D. E. Koditschek, “Robot planning and control via potential functions,” The robotics review, p. 349, 1989

    work page 1989

  23. [23]

    A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 2nd ed., ser. Texts in Applied Mathematics. Springer New York, NY , 1990, originally published in the series: Universitext. [Online]. Available: https://doi.org/10.1007/978-1-4684-0364-0

    work page doi:10.1007/978-1-4684-0364-0 1990

  24. [24]

    The helmholtz- hodge decomposition—a survey,

    H. Bhatia, G. Norgard, V . Pascucci, and P.-T. Bremer, “The helmholtz- hodge decomposition—a survey,” IEEE Transactions on Visualization and Computer Graphics , vol. 19, no. 8, pp. 1386–1404, 2013

    work page 2013

  25. [25]

    Structure-preserving discretization of incompressible fluids,

    D. Pavlov, P. Mullen, Y . Tong, E. Kanso, J. Marsden, and M. Desbrun, “Structure-preserving discretization of incompressible fluids,” Physica D: Nonlinear Phenomena , vol. 240, no. 6, pp. 443–458, 2011. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0167278910002873

    work page 2011