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Banach Control Barrier Functions for Large-Scale Swarm Control

Xuting Gao, Guillem Pascual, Scott Brown, Sonia Martínez

TL;DR

The paper addresses safe control of very large swarms by modeling the swarm as a density $\rho(t,x)$ and enforcing safety via Banach Control Barrier Functions (B-CBFs) in a Banach-space setting. It unifies macroscopic density constraints with microscopic agent actions through a Liouville-type evolution and a projected gradient/OT-based nominal velocity, yielding a tractable infinite-dimensional QP that can be discretized. The work provides scalar and spatially-dependent B-CBF formulations, derives consistency between macroscopic and microscopic laws, and develops distributed algorithms enabling local computation within communication radii while preserving global safety and convergence to targets. Numerical examples in 1D and 2D demonstrate obstacle avoidance, density shaping under constraints, and entropy bounds, illustrating the scalability and robustness of the approach for large swarms with local information exchange.

Abstract

This paper studies the safe control of very large multi-agent systems via a generalized framework that employs so-called Banach Control Barrier Functions (B-CBFs). Modeling a large swarm as probability distribution over a spatial domain, we show how B-CBFs can be used to appropriately capture a variety of macroscopic constraints that can integrate with large-scale swarm objectives. Leveraging this framework, we define stable and filtered gradient flows for large swarms, paying special attention to optimal transport algorithms. Further, we show how to derive agent-level, microscopical algorithms that are consistent with macroscopic counterparts in the large-scale limit. We then identify conditions for which a group of agents can compute a distributed solution that only requires local information from other agents within a communication range. Finally, we showcase the theoretical results over swarm systems in the simulations section.

Banach Control Barrier Functions for Large-Scale Swarm Control

TL;DR

The paper addresses safe control of very large swarms by modeling the swarm as a density and enforcing safety via Banach Control Barrier Functions (B-CBFs) in a Banach-space setting. It unifies macroscopic density constraints with microscopic agent actions through a Liouville-type evolution and a projected gradient/OT-based nominal velocity, yielding a tractable infinite-dimensional QP that can be discretized. The work provides scalar and spatially-dependent B-CBF formulations, derives consistency between macroscopic and microscopic laws, and develops distributed algorithms enabling local computation within communication radii while preserving global safety and convergence to targets. Numerical examples in 1D and 2D demonstrate obstacle avoidance, density shaping under constraints, and entropy bounds, illustrating the scalability and robustness of the approach for large swarms with local information exchange.

Abstract

This paper studies the safe control of very large multi-agent systems via a generalized framework that employs so-called Banach Control Barrier Functions (B-CBFs). Modeling a large swarm as probability distribution over a spatial domain, we show how B-CBFs can be used to appropriately capture a variety of macroscopic constraints that can integrate with large-scale swarm objectives. Leveraging this framework, we define stable and filtered gradient flows for large swarms, paying special attention to optimal transport algorithms. Further, we show how to derive agent-level, microscopical algorithms that are consistent with macroscopic counterparts in the large-scale limit. We then identify conditions for which a group of agents can compute a distributed solution that only requires local information from other agents within a communication range. Finally, we showcase the theoretical results over swarm systems in the simulations section.
Paper Structure (18 sections, 16 theorems, 38 equations, 4 figures)

This paper contains 18 sections, 16 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Measure Spaces and Functionals
  4. Optimal Transport
  5. Problem Formulation
  6. Safe Density Steering Via Quadratic Programming with B-CBF
  7. Scalar-Valued Functionals ($\mathcal{Y} = \mathbb{R}$)
  8. Spatially-Dependent Functions ($\mathcal{Y} = L^1(\mathbb{R}^n)$)
  9. Safe and Stable Density Steering
  10. Nominal Velocity Field
  11. Safety and Stability via Quadratic Programming
  12. Microscopic and Decentralized Algorithms
  13. Numerical Examples
  14. Example: 1D Transport with density constraints
  15. 2D Obstacle Avoidance
  16. ...and 3 more sections

Key Result

Lemma III.1

Let $\mathcal{O} \subseteq \mathbb{R}^n$ be a closed obstacle and let $\mathcal{O}^b = \mathcal{O} \oplus B_{d_{\textup{min}}}$ be a region to avoid, where $d_{\textup{min}}$ is a safety margin or agent-size bound. Consider a functional $H:\mathcal{P}(\mathbb{R}^n) \to \mathbb{R}$ defined as Then, the macroscopic formulation of obstacle avoidance, $H(\rho) \ge 0$, is equivalent to the microscopic

Figures (4)

  • Figure 1: The gray dashed curves represent the initial and terminal density $\rho_0, \rho_*$, and the unsafe area is shaded. The red and blue curves show the evolution of unconstrained OT and safe densities after CBF projection respectively.
  • Figure 2: Heatmap depicting snapshots of the density evolution under obstacle avoidance in 2D. The plots depict the transport from the initial to the target density while respecting the square obstacle constraint. As time progresses, the density flow diverts around the obstacle, successfully avoiding unsafe regions and converging toward the target distribution.
  • Figure 3: Final state of agent distribution for the unconstrained (left) and CBF constrained (right) distributed optimal transport. The target density is shown in grayscale, and the maximum density constraint violation is highlighted in red. The constraint corresponds to a maximum density of $\varepsilon_{\textup{max}}=0.045$, and a minimum density of $\varepsilon_{\textup{min}}=0.01$ .
  • Figure 4: Evolution of the entropy for the unconstrained and B-CBF constrained case, with an entropy threshold of $\varepsilon=3$.

Theorems & Definitions (31)

  • Definition II.1: Fréchet derivative
  • Definition II.2: Transport Map
  • Definition II.3: Monge Problem
  • Lemma III.1: Obstacle Avoidance
  • proof
  • Lemma III.2: Conflict Avoidance
  • proof
  • Corollary III.3: Swarm Cohesion
  • Definition IV.1
  • Theorem IV.2
  • ...and 21 more