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Mean-field control barrier functions for stochastic multi-agent systems

Cinzia Tomaselli, Gian Carlo Maffettone, Samy Wu Fung, Levon Nurbekyan, Mario di Bernardo

Abstract

Many applications involving multi-agent systems require fulfilling safety constraints. Control barrier functions offer a systematic framework to enforce forward invariance of safety sets. Recent work extended this paradigm to mean-field scenarios, where the number of agents is large enough to make density-space descriptions a reasonable workaround for the curse of dimensionality. However, an open gap in the recent literature concerns the development of mean-field control barrier functions for Fokker-Planck (advection-diffusion) equations. In this work, we address this gap, enabling safe mean-field control of agents with stochastic microscopic dynamics. We provide bounded stability guarantees under safety corrections and corroborate our results through numerical simulations in two representative scenarios, coverage and shepherding control of multi-agent systems.

Mean-field control barrier functions for stochastic multi-agent systems

Abstract

Many applications involving multi-agent systems require fulfilling safety constraints. Control barrier functions offer a systematic framework to enforce forward invariance of safety sets. Recent work extended this paradigm to mean-field scenarios, where the number of agents is large enough to make density-space descriptions a reasonable workaround for the curse of dimensionality. However, an open gap in the recent literature concerns the development of mean-field control barrier functions for Fokker-Planck (advection-diffusion) equations. In this work, we address this gap, enabling safe mean-field control of agents with stochastic microscopic dynamics. We provide bounded stability guarantees under safety corrections and corroborate our results through numerical simulations in two representative scenarios, coverage and shepherding control of multi-agent systems.
Paper Structure (8 sections, 4 theorems, 40 equations, 2 figures)

This paper contains 8 sections, 4 theorems, 40 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Mean Field Control Barrier Function for advection-diffusion Systems
  3. Bounded convergence with MF-CBFs
  4. Kernel-weighted Overlap as Mean-Field Control Barrier Function
  5. Applications and Numerical Validation
  6. Safe coverage control
  7. Safe shepherding control
  8. Conclusions

Key Result

Theorem 1

Consider a density distribution $\rho(t,\mathbf{x})$ evolving according to the advection--diffusion equation eq:PDE over a domain $\Omega$ subject to boundary conditions ensuring mass conservation, that is $\partial_t\left(\int_\Omega \rho\,\mathrm{d}\mathbf{x}\right)=0$. Let $\mathcal{H}(t, \rho)$ guarantees forward invariance of the safe set $\mathcal{C}$.

Figures (2)

  • Figure 1: Safe coverage control. Effect of MF-CBF enforcement on agents entering dangerous regions. (a) Temporal evolution of the fraction of agents entering the dangerous regions averaged over 50 simulations. (b) Initial and (c) final spatial configurations of the population (magenta disks) when MF-CBF is enforced.
  • Figure 2: Safe shepherding control. Temporal evolution of the average fraction of leaders (a) and followers (b) entering the dangerous regions. (c) Initial and (d) final spatial configurations of the leaders (blue diamonds) and followers (magenta disks) populations when MF-CBF is enforced. At final time, a fraction $0.967$ of followers lies within the goal region

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • proof
  • Corollary 1
  • proof
  • Remark 2