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Density control of large-scale multi-agent systems via tuning of agent interactions

Mena Zkhary, Alain Boldini

arXiv:2610.32778v1eess.SYcs.SY

TL;DR

The paper asks whether collective behavior in large-scale multi-agent systems can be regulated by modifying interactions rather than directly controlling agents. It develops a Fourier-based mean-field framework that treats the interaction kernel as a control variable, and reports stabilization of desired densities through kernel feedback. Its conclusions identify controllability conditions and a scope boundary for the approach.

  • Problem

    Existing strategies control individual agents while treating their interaction law as fixed, motivating the question of whether kernel modulation can regulate collective behavior.

    Source: §I · p. 1

  • Method

    For agents moving on a ring, the framework uses a continuified mean-field model and Fourier modes to design time-varying interaction kernels as control inputs.

    Source: §I · p. 2, A. Microscopic and mean-field models · p. 2

  • Results

    LQR kernel feedback shifts the linearized closed-loop poles into the open negative real half-plane, while controlled density errors decrease rapidly for both tested kernels.

    Source: A. Equilibrium stabilization · pp. 7–8

  • Takeaways & Limitations

    The framework is mostly feasible for desired densities with non-sparse spectral content, which the authors identify as the biggest share of densities targeted in practical applications.

    Source: §V · p. 11

    The Fourier approach does not achieve non-periodic densities, and the proposed approach is centralized.

    Source: §V · p. 11

Abstract

from arXiv · show

Collective behavior in biological systems emerges through adaptive interactions among units, enabling large populations to coordinate without centralized control. This mechanism raises the question of whether collective behavior in engineered multi-agent systems can likewise be regulated by modifying the interaction law between agents, rather than by directly controlling individual units. To answer this question, we consider large-scale multi-agent systems evolving on a ring and treat the interaction kernel as a time-varying control variable whose evolution regulates the collective dynamics. Using a mean-field formulation, we develop a framework for the analysis and design of kernel-based controllers. We derive explicit conditions for controllability in application-relevant scenarios, and synthesize a state-feedback controller to regulate the kernel dynamics, validated on the microscopic, discrete dynamics. Numerical simulations demonstrate the approach's effectiveness and robustness to unmodeled perturbations. This work constitutes a first step in the design of bio-inspired control strategies for large-scale multi-agent systems by modulation of interactions between agents.

I. INTRODUCTION

The paper asks whether large-scale collective behavior can be regulated by changing agents’ interaction law rather than applying external controls. It develops a mean-field, Fourier-based framework that designs kernel modulation macroscopically and implements it in the agents’ dynamics.

I. INTRODUCTION

  • Continuification bridges microscopic agent models to scalable continuum descriptions in terms of density and other fields governed by PDEs.The passage compares this bridge to deriving macroscopic fluid equations from microscopic particle interactions –.Source: §I · p. 1
  • Adaptive local interactions in biological and human collectives motivate testing interaction-law modulation as an alternative way to regulate large-scale systems.The paper cites predator avoidance and pedestrian navigation as examples of coordinated behavior emerging without centralized control –.Source: §I · p. 1
  • Existing strategies apply external actions to agents while holding interactions fixed, whereas this framework makes the interaction kernel itself a control variable.Related approaches tune interaction strength without changing the form of interactions.Source: §I · p. 2
  • On a ring, the framework uses Fourier analysis to design kernel modulation at the macroscopic level, then discretizes the resulting interaction law for individual agents.The overview describes this micro–macro control sequence.Source: §I · p. 2, Figure 1 · p. 2

A. Microscopic and mean-field models

  • The microscopic model describes N agents on the one-dimensional unit circle, with positions evolving through coupled ordinary differential equations.The interaction kernel may vary explicitly with time and depends on wrapped angular differences between agent positions.Source: A. Microscopic and mean-field models · p. 2

B. Problem formulation

The problem is to make a prescribed density on the periodic domain an equilibrium through kernel design and, locally, to drive nearby initial densities toward it. Fourier coefficients provide an exact modal description and a complementarity condition for constructing equilibrium kernels.

B. Problem formulation

  • The control objective is to make a prescribed density on the periodic domain an equilibrium by modifying the kernel, with convergence sought from sufficiently close initial densities.The kernel, rather than an external input, is the control parameter.Source: B. Problem formulation · p. 3

C. Fourier representation

  • Periodicity permits Fourier-series representations of both density and kernel, whose coefficients obey Hermitian symmetry because the fields are real-valued.The density and kernel coefficients are defined as complex Fourier coefficients.Source: C. Fourier representation · p. 3
  • Substituting the Fourier expansions into the velocity and continuity equation yields modal dynamics for the density coefficients.The derivation uses the convolution theorem and orthogonality conditions.Source: C. Fourier representation · p. 3
  • The resulting Fourier-mode equations exactly represent the continuity equation and capture nonlinear coupling among modes.This exact representation is infinite-dimensional.Source: C. Fourier representation · p. 3

D. Design of equilibrium kernels

  • An equilibrium kernel is designed so the desired density is an equilibrium, with the Fourier-coefficient condition requiring a zero product for each mode.At equilibrium, the convolution generating velocity must be zero.Source: D. Design of equilibrium kernels · p. 3
  • The equilibrium condition creates complementarity between density and kernel Fourier coefficients; infinitely many kernels are possible, and larger kernel norms correspond to stronger interactions.This modal relationship provides a way to design kernels for desired equilibrium densities.Source: D. Design of equilibrium kernels · p. 3

E. Finite-dimensional, reduced-order dynamic model

The infinite-dimensional modal dynamics is reduced to a finite spatial bandwidth because practical estimation and control concern a limited range of resolvable modes. The truncation preserves dominant interaction dynamics for analysis and controller design.

E. Finite-dimensional, reduced-order dynamic model

  • The model retains a finite spatial bandwidth because smooth densities concentrate variation in low modes, while KDE, sensing and communication resolution, and robot size limit high-mode information.The paper also notes that practical applications typically target low-wavenumber density modes rather than sharp variations.Source: E. Finite-dimensional, reduced-order… · p. 3
  • The retained set M contains non-zero Fourier modes from −N through N, while M0 additionally includes the zero mode.N is the truncation order.Source: E. Finite-dimensional, reduced-order… · p. 4
  • Substituting the truncated expansions into the modal dynamics and setting coefficients outside M0 to zero yields a finite-dimensional approximation.This representation is used for analysis and controller design.Source: E. Finite-dimensional, reduced-order… · pp. 3–4

F. Linearization around equilibria

The system is linearized around a desired density and its equilibrium kernel, with kernel perturbations serving as control inputs. The resulting modal structure separates density and kernel contributions across disjoint Fourier-mode supports.

F. Linearization around equilibria

  • The linearized model treats Fourier coefficients of the kernel perturbation as control inputs for density perturbations near the desired equilibrium.The zero-mode density perturbation remains constant, so the analysis restricts to mass-preserving perturbations with zero initial zero mode.Source: F. Linearization around equilibria · p. 4
  • The linearized dynamics are represented compactly using reduced equilibrium vectors, a diagonal Fourier-mode matrix, and a Toeplitz convolution matrix.The retained nonzero modes are indexed from −N to N, excluding zero.Source: F. Linearization around equilibria · p. 4
  • The equilibrium condition separates the retained Fourier modes into disjoint subsets contributing to the state and input matrices.The matrix factorizations separate contributions of the equilibrium density and equilibrium kernel.Source: F. Linearization around equilibria · pp. 4–5

G. State-feedback controller design

The linearized system is treated as a standard linear time-invariant system, and a state-feedback controller is designed with kernel perturbations as the control input.

G. State-feedback controller design

  • A state-feedback gain Kc can be designed using pole placement or an LQR to obtain desired behavior such as asymptotic stability.The gain is substituted into the linearized system to form the closed-loop dynamics.Source: G. State-feedback controller design · p. 5

III. CONTROLLABILITY ANALYSIS

The analysis establishes controllability of a truncated linearized system under an application-relevant Fourier-mode support pattern and a skew Schur nondegeneracy condition. Its rank argument identifies how the input matrix and its interaction with the system matrix span the state space.

III. CONTROLLABILITY ANALYSIS

  • The controllability study focuses on an application-relevant condition, while numerical investigation of more general systems is reserved for Section IV.The authors note that general controllability analysis is challenging.Source: §III · p. 5
  • For a nonnegative normalized desired density, the full Toeplitz matrix is positive definite; removing the boundary columns leaves 2N−1 independent columns.Removing the zero-mode row can reduce the rank by at most one, so the analysis determines when the maximal rank is preserved.Source: §III · p. 5
  • Condition (35) is computationally inexpensive to check: compute the 2N−2 roots and evaluate skew Schur polynomials until one is nonzero.The polynomial values correspond to maximal minors of the banded Toeplitz matrix.Source: §III · pp. 5–6
  • The controllability argument uses AB to supply a column independent of B, completing the state-space span when B has rank 2N−1.The boundary-mode structure makes A act through its two boundary columns.Source: §III · pp. 6–7
  • Under density-kernel support conditions (28) and skew Schur nondegeneracy condition (35), the truncated linearized system is controllable via kernel modifications.The considered application-relevant setup truncates the desired density at mode N−1 and supports the equilibrium kernel only at the boundary modes ±N.Source: §III · pp. 5, 7

IV. RESULTS

The numerical inquiries assess the state-feedback controller's stabilization, robustness, consistency with microscopic dynamics, and relationship to the stated controllability condition.

IV. RESULTS

  • The simulations evaluate whether the state-feedback controller stabilizes the system dynamics.The passage states the evaluation aim but gives no numerical outcome.Source: §IV · p. 7
  • The numerical evaluation tests robustness to large equilibrium perturbations and neglected higher-order Fourier modes.The passage identifies these as perturbations considered in the simulations.Source: §IV · p. 7
  • The simulations compare the reduced mean-field formulation with microscopic multi-agent dynamics and examine controllability beyond Theorem 1's condition.The passage describes these as validation aims, not reported outcomes.Source: §IV · p. 7

A. Equilibrium stabilization

Kernel feedback drives discrete agents toward the desired von Mises density, while a separate sparse-profile example shows control from a uniform initial density. The examples test equilibrium kernels with and without the modal equilibrium condition and report stable linearized closed-loop dynamics.

A. Equilibrium stabilization

  • The examples cover a von Mises target truncated to 15 Fourier modes and a separately prescribed finite-Fourier target, demonstrating applicability to both profile types.For the von Mises case, N = 16 and the truncated representation's maximum pointwise reconstruction error is approximately 10^-14.Source: A. Equilibrium stabilization · p. 7
  • Kernel (61a) meets the modal equilibrium condition and controllability theorem hypotheses; kernel (61b) does not, but can still regulate density with nonzero equilibrium kernel input.For kernel (61b), the linearized model has a nonzero constant forcing term.Source: A. Equilibrium stabilization · p. 7
  • The LQR feedback shifts all closed-loop poles into the open negative-real-part half-plane, confirming asymptotic stability of the linearized closed-loop system.Both kernels have open-loop linearized dynamics that are not asymptotically stable.Source: A. Equilibrium stabilization · p. 7
  • The controlled agents approach the desired von Mises profile and their L2 density errors fall rapidly for both kernels, while the uncontrolled error stays approximately constant.The density is reconstructed from a finite agent population using KDE, which affects achievable accuracy.Source: Figure 2 · p. 8, A. Equilibrium stabilization · p. 8
  • From a uniform initial density, the controller drives the sparse profile toward its target; its error drops by more than one order of magnitude while uncontrolled error stays nearly constant.The error reduction is initially slow and then accelerates.Source: A. Equilibrium stabilization · p. 8, Figure 3 · p. 9

B. Robustness under large initial perturbations and modal truncation

The controller remains effective across large initial perturbations and retains bounded errors under higher-order modal perturbations omitted from synthesis. The latter perturbations raise residual error relative to the nominal case.

B. Robustness under large initial perturbations and modal truncation

  • For ε = 10−1, 5 × 10−1, and 100, error trajectories show similar convergence and reach comparable levels.The perturbation direction is randomly selected and held fixed across error scalings.Source: B. Robustness under large initial… · p. 9
  • With the N = 6 controller applied without redesign, mode sets {6, 7, 8} and {7, 8, 9} produce higher residual errors than nominal, but errors decrease and remain bounded.The two perturbed cases show no strong differences; the paper notes KDE spatial smoothing typically filters such perturbations.Source: B. Robustness under large initial… · p. 9
  • The controller remains effective under large initial perturbations and higher-order modal perturbations omitted during synthesis, with bounded errors in the latter cases.The perturbation cases provide numerical evidence of robustness beyond nominal synthesis conditions.Source: B. Robustness under large initial… · p. 9, Figure 4 · p. 9

C. Microscopic validation with finite agent populations

The mean-field-designed feedback controller also reduces density error when applied directly to microscopic populations of 20, 50, 100, and 200 agents. Residual error decreases as population size increases.

C. Microscopic validation with finite agent populations

  • The reduced mean-field controller is evaluated directly on microscopic agent dynamics for populations of 20, 50, 100, and 200.At each evaluation, KDE-reconstructed density Fourier coefficients feed the feedback law and the resulting kernel modification is applied to agent dynamics.Source: C. Microscopic validation with finite… · p. 9
  • Normalized L2 error between reconstructed and desired densities decreases for all tested populations, with residual error falling as agent count increases.The figure compares populations of 20, 50, 100, and 200 agents.Source: Figure 5 · p. 10, C. Microscopic validation with finite… · p. 10
  • The population-dependent residual reflects improved approximation of the continuum density approximation with larger agent populations.Source: C. Microscopic validation with finite… · p. 10

D. Numerical investigation of controllability

The numerical study finds controllability depends strongly on how retained Fourier modes are allocated between equilibrium density and interaction kernel. With more active kernel modes, controllable realizations fall as fewer density modes remain available to the input matrix.

D. Numerical investigation of controllability

  • The study tests all 210 assignments of retained Fourier modes for N = 10, using 100 random realizations per assignment and evaluating controllability-matrix rank.Each conjugate pair is assigned to either the equilibrium density or kernel; assignments with both coefficients zero are excluded.Source: D. Numerical investigation of… · p. 10
  • Controllable equilibria remain close to 100% up to 40% active kernel modes, drop sharply around 50%, and become nearly zero above 60%.The standard deviation rises near the transition, indicating greater variability there.Source: Figure 6 · p. 10, D. Numerical investigation of… · p. 10
  • More active kernel modes leave fewer density modes to populate B, increasing rank deficiency that powers of A cannot recover beyond a threshold.The results extend the numerical investigation beyond Theorem 1's conditions.Source: D. Numerical investigation of… · p. 10

V. CONCLUSION

The paper develops a Fourier-based density-control framework for large-scale multi-agent systems on a ring, using interaction modulation as the control input. Its controllability analysis and simulations support feasibility for many practical target densities, while the approach remains limited to periodic domains and centralized control.

V. CONCLUSION

  • The framework uses a finite-dimensional Fourier approximation to design interaction modulation and state feedback, with simulations showing convergence to desired density despite large initial perturbations and neglected Fourier modes.The approach is grounded in continuified density dynamics on a ring and validated on microscopic and density dynamics.Source: §V · p. 10
  • The approach provides a basis for designing interactions to control and stabilize agent arrangements.The authors describe this as highlighting the feasibility of the approach.Source: §V · pp. 10–11
  • The finite-dimensional approximation is mostly feasible for desired densities with non-sparse spectral content, which represents the largest share of practical target densities.The authors establish controllability requirements under application-relevant conditions and supplement the proof with simulations of random systems beyond the theorem's hypotheses.Source: §V · p. 11
  • The framework does not achieve non-periodic densities and is centralized; future work would address non-periodic domains and decentralization.The authors identify alternative boundary-compatible bases and distributed-estimation or leader-follower approaches as possible future directions.Source: §V · p. 11
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