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Sparse stabilization of mean-field agent dynamics through a three-operator splitting method

Giacomo Albi, Dante Kalise, Chiara Segala, Franco Zivcovich

TL;DR

This work addresses sparse stabilization of large-scale multi-agent systems by deriving a mean-field optimal control in the form of a $Vlasov$-type PDE and enforcing sparsity with an $\ell_1$ penalty. A three-operator splitting (TOS) algorithm couples gradient steps for the smooth part with proximal shrinkage for the $\ell_1$ term and a projection for budget constraints, using adjoint equations to compute gradients. Computation is scaled via a particle-based Monte Carlo discretization with random batches, preserving mean-field structure. Numerical experiments on the Cucker–Smale model demonstrate consensus with sparse, localized control and confirm the method’s efficiency and robustness.

Abstract

We study the sparse stabilization of nonlinear multi-agent systems within a mean-field optimal control framework. The goal is to drive large populations of interacting agents toward consensus with minimal control effort. In the mean-field limit, the dynamics are described by a Vlasov-type kinetic equation, and sparsity is enforced through an l1-l2 penalization in the cost functional. The resulting nonsmooth optimization problem is solved via a three-operator splitting (TOS) method that separately handles smooth, nonsmooth, and constraint components through gradient, shrinkage, and projection steps. A particle-based Monte Carlo discretization with random batch interactions enables scalable computation while preserving the mean-field structure. Numerical experiments on the Cucker-Smale model demonstrate effective consensus formation with sparse, localized control actions, confirming the efficiency and robustness of the proposed approach.

Sparse stabilization of mean-field agent dynamics through a three-operator splitting method

TL;DR

This work addresses sparse stabilization of large-scale multi-agent systems by deriving a mean-field optimal control in the form of a -type PDE and enforcing sparsity with an penalty. A three-operator splitting (TOS) algorithm couples gradient steps for the smooth part with proximal shrinkage for the term and a projection for budget constraints, using adjoint equations to compute gradients. Computation is scaled via a particle-based Monte Carlo discretization with random batches, preserving mean-field structure. Numerical experiments on the Cucker–Smale model demonstrate consensus with sparse, localized control and confirm the method’s efficiency and robustness.

Abstract

We study the sparse stabilization of nonlinear multi-agent systems within a mean-field optimal control framework. The goal is to drive large populations of interacting agents toward consensus with minimal control effort. In the mean-field limit, the dynamics are described by a Vlasov-type kinetic equation, and sparsity is enforced through an l1-l2 penalization in the cost functional. The resulting nonsmooth optimization problem is solved via a three-operator splitting (TOS) method that separately handles smooth, nonsmooth, and constraint components through gradient, shrinkage, and projection steps. A particle-based Monte Carlo discretization with random batch interactions enables scalable computation while preserving the mean-field structure. Numerical experiments on the Cucker-Smale model demonstrate effective consensus formation with sparse, localized control actions, confirming the efficiency and robustness of the proposed approach.

Paper Structure

This paper contains 10 sections, 1 theorem, 26 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. MEAN-FIELD APPROXIMATION OF OPTIMAL MAS CONTROL
  3. PARTICLE-BASED SCHEME FOR MEAN-FIELD SPARSE STABILIZATION
  4. Discrete optimality conditions
  5. TOS scheme for mean-field sparse control
  6. Numerical experiments
  7. Test 1: Microscopic dynamics
  8. Test 2: Mean-field sparse stabilization
  9. Test 2: Mean-field two dimensional dynamics
  10. Conclusion

Key Result

Proposition IV.1

The proximal operators of $\lambda \mathcal{J}_2$ and $\lambda \mathcal{J}_3$ in eq:tos are given respectively by where the soft-thresholding operator is defined as with $[\,\cdot\,]_+$ the positive part applied componentwise, and the projection map is with $\lambda^*>0$ chosen such that $\|\mathbb{S}_{\lambda^*}(w)\|_1 = \widetilde{B}$.

Figures (6)

  • Figure 1: Test 1.Top row: (left) uncontrolled vs. controlled trajectories for $\alpha=10^{-2}$, $\beta=10^{-1}$; (right) decay of the Lyapunov functional $V(t)={(2N)^{-2}}\sum_{i,j}|v_i(t)-v_j(t)|^2$ for different $\beta$. Bottom row: (left) active control components for $\beta = 0$ and (right) for $\beta=10^{-1}$.
  • Figure 2: Test 2 (uncontrolled). Mean-field dynamics without control: left panel shows the spatial distribution, and right panel the velocity marginal.
  • Figure 3: Test 2 (controlled).Top row: Mean-field dynamics with control parameters $\alpha =10^{-2}, \ \beta = 10^{-1}$: spatial concentration (left) and velocity alignment (right) are achieved. Bottom row: time evolution of control marginals in $x$ (left) and $v$ (right), showing strong activity initially that vanishes as consensus emerges.
  • Figure 4: Test 2.Top row: Initial configuration at $t=0$; Bottoom row: controlled state at $t=3.4$ for $\beta=10^{-1}$.
  • Figure 5: Test 3.Bottom row: Spatial marginals at $t=0.8$ for $\beta=0$ (left) and $\beta=0.1$ (right). Arrows indicate the momentum field, the scale of color the magnitude of the control.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Remark III.1
  • Proposition IV.1
  • Remark IV.2