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arxiv: 2605.01161 · v1 · submitted 2026-05-01 · 📡 eess.SY · cs.SY· math.DS

Distributed Attraction-Repulsion Potential for Multi-Agent Formation Control

Hemanta Ban , Seddik M. Djouadi , Kevin Tomsovic This is my paper

Pith reviewed 2026-05-09 18:15 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.DS
keywords multi-agent formation controlLennard-Jones potentialcollision avoidancegradient flowLaSalle invariance principleglobal well-posednessdistributed control
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The pith

The gradient flow of the Lennard-Jones potential produces globally well-posed collision-free trajectories that converge to a single equilibrium modulo translations in multi-agent systems.

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

The paper studies a distributed formation control law in which each agent follows the negative gradient of the sum of pairwise Lennard-Jones potentials. Starting from any collision-free initial configuration, the authors establish that solutions exist for all future time and that a positive lower bound on every inter-agent distance is preserved, ruling out finite-time collisions. Treating the total potential energy as a Lyapunov function, LaSalle's invariance principle identifies all omega-limit points as equilibria of the potential; analyticity of the energy along collision-avoiding trajectories then forces the entire system to approach one such equilibrium configuration up to rigid translation.

Core claim

For collision-free initial data, the multi-agent system under the Lennard-Jones gradient flow is globally well-posed with a uniform lower bound on inter-agent distances excluding hard collisions. LaSalle's invariance principle applied to the total energy shows every positive limit point is an equilibrium, and analyticity of the energy along the flow yields convergence to a single equilibrium modulo translations.

What carries the argument

The Lennard-Jones potential, whose gradient supplies the distributed attraction-repulsion control input between every pair of agents.

If this is right

  • All trajectories remain collision-free for all future time whenever they start collision-free.
  • The agents converge to a critical point of the total Lennard-Jones energy.
  • The final configuration is a fixed formation shape that translates rigidly but does not deform or split.
  • Only local neighbor interactions are required to achieve the global convergence property.

Where Pith is reading between the lines

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

  • The same analyticity-plus-invariance argument would apply to any other smooth repulsive-attractive pair potential that blows up at zero distance and decays at infinity.
  • A small additive external potential could be used to bias the system toward a desired target shape while preserving the collision-avoidance guarantee.
  • The result suggests that the potential energy landscape contains no stable clusters that trap subsets of agents away from the global minimum.

Load-bearing premise

That all inter-agent distances remain bounded away from zero uniformly in time, so the total energy stays analytic along every trajectory.

What would settle it

A concrete set of collision-free initial positions for three or more agents whose numerical integration produces a collision (distance zero) in finite time.

Figures

Figures reproduced from arXiv: 2605.01161 by Hemanta Ban, Kevin Tomsovic, Seddik M. Djouadi.

Figure 3
Figure 3. Figure 3: Trajectory plot of 3-agent: (a) Equilateral configura view at source ↗
Figure 2
Figure 2. Figure 2: Stability metrics across 2- and 3-agent cases: (a) view at source ↗
Figure 4
Figure 4. Figure 4: Final configuration and trajectories of N = 8 agents view at source ↗
Figure 5
Figure 5. Figure 5: Stability metrics for N = 8 agent system: (a) Energy over time; (b) Pairwise distances. E. Validation of Exponential Convergence Rate Local linearization of system (12) around any equilibrium (x ⋆ , 0) yields a damped harmonic oscillator structure, and standard eigenvalue analysis gives a predicted exponential decay rate α = min{γ/(2m), λmin(H)/γ}, where H = ∇2 xU(x ⋆ ) [18]. Details of the late-time analy… view at source ↗
read the original abstract

In this paper, a distributed multi-agent formation control driven by the gradient of the Lennard-Jones potential is analyzed. For collision-free initial data, we prove global well-posedness together with a uniform lower bound on all inter-agent distances, thereby excluding hard collisions. Taking the total energy as a Lyapunov function, LaSalle's invariance principle shows that every positive limit point is an equilibrium. Since trajectories remain uniformly away from collisions, the energy is analytic along the flow and an argument yields convergence to a single equilibrium modulo translations. Illustrative numerical examples are presented.

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 paper analyzes a distributed multi-agent formation control law given by the gradient of the Lennard-Jones potential. For collision-free initial data it proves global well-posedness of the closed-loop dynamics together with a uniform positive lower bound on all inter-agent distances. The total energy is used as a Lyapunov function; LaSalle's invariance principle shows that positive limit points are equilibria. Because the distance bound keeps trajectories away from the singularity, the energy remains analytic along the flow, which is then used to strengthen the conclusion to convergence to a single equilibrium configuration modulo translations. Illustrative numerical simulations are included.

Significance. If the proofs are complete, the manuscript supplies a clean, parameter-free theoretical foundation for using singular attraction-repulsion potentials in multi-agent systems. The explicit separation of the global well-posedness / distance-bound step from the subsequent analyticity argument removes the usual circularity risk in LaSalle-type analyses of singular potentials. The result therefore strengthens the standard gradient-flow template and supplies falsifiable predictions (uniform collision avoidance and convergence to equilibria) that can be checked numerically, which is a clear strength for the field of distributed control.

minor comments (3)
  1. The abstract states that 'an argument yields convergence to a single equilibrium modulo translations' but does not name the analytic-function theorem invoked; the main text should cite the precise result (e.g., the identity theorem or a Łojasiewicz-type inequality) and indicate in which section it is applied.
  2. The numerical examples section would benefit from explicit statements of the number of agents, the chosen equilibrium formation, and the precise initial conditions used to generate each figure; without these the reproducibility of the simulations is reduced.
  3. Notation for the Lennard-Jones parameters (ε, σ) and the total energy E should be introduced once in a dedicated 'Preliminaries' subsection and then used consistently; currently the definitions appear piecemeal.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. We are pleased that the separation between the global well-posedness/distance-bound argument and the subsequent analyticity-based convergence result is viewed as removing a common circularity risk in LaSalle analyses of singular potentials.

read point-by-point responses

  1. Referee: No specific major comments are listed in the report, despite the minor_revision recommendation.

    Authors: Since the referee has not identified any particular issues requiring correction or clarification, we have no point-by-point revisions to propose at this stage. We remain ready to address any editorial or minor technical suggestions from the editor or referee once they are provided. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation establishes global well-posedness and a uniform lower bound on inter-agent distances directly from the energy sublevel set {E ≤ E(0)} using the explicit Lennard-Jones potential definition, prior to invoking analyticity of the energy (away from collisions) to apply LaSalle's invariance principle and conclude convergence to a single equilibrium modulo translations. All steps rest on external mathematical theorems and the given potential; no self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes smuggled via prior work appear in the chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard results from dynamical systems theory and the known smoothness properties of the Lennard-Jones potential; no new entities are postulated and no parameters are fitted to data.

axioms (2)
  • standard math LaSalle's invariance principle applies to the closed-loop system
    Invoked to conclude that positive limit points are equilibria.
  • domain assumption The total energy remains analytic along trajectories that stay uniformly away from collisions
    Used to obtain the final convergence argument after the distance bound is established.

pith-pipeline@v0.9.0 · 5393 in / 1325 out tokens · 47018 ms · 2026-05-09T18:15:58.118962+00:00 · methodology

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