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Optimal Transport for Time-Varying Multi-Agent Coverage Control

Italo Napolitano, Mario di Bernardo

TL;DR

This work addresses multi-agent coverage under time-varying target densities by formulating an instantaneous, time-dependent semi-discrete optimal transport problem that uses Laguerre partitions to enforce equal-mass regions. It derives coupled dynamics for agent positions and dual variables to track the time-varying Laguerre barycenters, achieving exponential convergence under feedforward compensation. In one dimension, a closed-form control law is obtained, enabling distributed implementations and providing insight into the coupling induced by moving Laguerre boundaries. Across 2D experiments, the TV-OT framework yields superior tracking performance over quasi-static OT and Voronoi-based methods, and distributed approximations demonstrate scalability with acceptable performance losses. Overall, the approach connects optimal transport with probability-space formation control, offering a principled, density-driven paradigm for dynamic formation adaptation and guiding future work on constraints and entropy-regularized methods.

Abstract

Coverage control algorithms have traditionally focused on static target densities, where agents are deployed to optimally cover a fixed spatial distribution. However, many applications involve time-varying densities, including environmental monitoring, surveillance, and adaptive sensor deployment. Although time-varying coverage strategies have been studied within Voronoi-based frameworks, recent works have reformulated static coverage control as a semi-discrete optimal transport problem. Extending this optimal transport perspective to time-varying scenarios has remained an open challenge. This paper presents a rigorous optimal transport formulation for time-varying coverage control, in which agents minimize the instantaneous Wasserstein distance to a continuously evolving target density. The proposed solution relies on a coupled system of differential equations governing agent positions and the dual variables that define Laguerre regions. In one-dimensional domains, the resulting system admits a closed-form analytical solution, offering both computational benefits and theoretical insight into the structure of optimal time-varying coverage. Numerical simulations demonstrate improved tracking performance compared to quasi-static and Voronoi-based methods, validating the proposed framework.

Optimal Transport for Time-Varying Multi-Agent Coverage Control

TL;DR

This work addresses multi-agent coverage under time-varying target densities by formulating an instantaneous, time-dependent semi-discrete optimal transport problem that uses Laguerre partitions to enforce equal-mass regions. It derives coupled dynamics for agent positions and dual variables to track the time-varying Laguerre barycenters, achieving exponential convergence under feedforward compensation. In one dimension, a closed-form control law is obtained, enabling distributed implementations and providing insight into the coupling induced by moving Laguerre boundaries. Across 2D experiments, the TV-OT framework yields superior tracking performance over quasi-static OT and Voronoi-based methods, and distributed approximations demonstrate scalability with acceptable performance losses. Overall, the approach connects optimal transport with probability-space formation control, offering a principled, density-driven paradigm for dynamic formation adaptation and guiding future work on constraints and entropy-regularized methods.

Abstract

Coverage control algorithms have traditionally focused on static target densities, where agents are deployed to optimally cover a fixed spatial distribution. However, many applications involve time-varying densities, including environmental monitoring, surveillance, and adaptive sensor deployment. Although time-varying coverage strategies have been studied within Voronoi-based frameworks, recent works have reformulated static coverage control as a semi-discrete optimal transport problem. Extending this optimal transport perspective to time-varying scenarios has remained an open challenge. This paper presents a rigorous optimal transport formulation for time-varying coverage control, in which agents minimize the instantaneous Wasserstein distance to a continuously evolving target density. The proposed solution relies on a coupled system of differential equations governing agent positions and the dual variables that define Laguerre regions. In one-dimensional domains, the resulting system admits a closed-form analytical solution, offering both computational benefits and theoretical insight into the structure of optimal time-varying coverage. Numerical simulations demonstrate improved tracking performance compared to quasi-static and Voronoi-based methods, validating the proposed framework.
Paper Structure (15 sections, 2 theorems, 41 equations, 9 figures, 3 tables)

This paper contains 15 sections, 2 theorems, 41 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Background
  4. Semi-Discrete Optimal Transport
  5. Coverage Control
  6. Modeling and Problem Statement
  7. Time-varying Optimal Transport Strategy
  8. Proposed solution
  9. Numerical validation
  10. One-dimensional case: Closed-form solution
  11. Numerical validation
  12. Distributed implementation
  13. Comparative analysis
  14. Conclusions
  15. Simulation parameters

Key Result

Theorem 1

Suppose Assumptions 1 and 2 hold. The following continuous-time dynamics for the agent positions $\mathbf{p}_{i}(t)$ and the dual variables $\boldsymbol{\phi}(t)$ achieve local saddle-point tracking for Problem eq:problem: for $i = 1, \dots, N$, where $K_x, K_\phi > 0$ are the primal and dual gains, respectively, and $\mathbf{p}(0)=\mathbf{p}_0$, $\boldsymbol{\phi}(0)=\boldsymbol{\phi}_0$ are arb

Figures (9)

  • Figure 1: Difference between (a) Voronoi and (b) Laguerre regions (colored areas) when $5$ samples (black dots) approximate the same Gaussian distribution (solid black line).
  • Figure 2: Validation in two dimensions with $7$ agents (black dots) covering a time-varying density at (a) $t=0$, (b) $t=0.5$, (c) $t=1.5$, and (d) $t=3$ a.u.
  • Figure 3: Validation with $8$ agents (black dots) covering a time-varying density at (a) $t=1$, (b) $t=2$, (c) $t=4$, and (d) $t=6$ a.u. The agents maintain formation as the density shrinks, expands, and splits into a bimodal distribution.
  • Figure 4: Time evolution of the 2-Wasserstein distance between the empirical and target measures for (a) the first experiment, whose trajectories are reported in Figure \ref{['fig:exp2D']}, and (b) the second experiment, whose trajectories are reported in Figure \ref{['fig:formation_control']}.
  • Figure 5: Validation of the proposed method in one-dimensional domains with $5$ agents. The evolution of densities (background color from white to red) and agent trajectories (solid black lines) is shown in space (x-axis) and time (y-axis). (a) First scenario: agents track a Gaussian distribution whose mean varies linearly over time; (b) Second scenario: both the mean and variance of the Gaussian distribution vary sinusoidally over time; (c) Third scenario: the mean of a multimodal Gaussian distribution varies linearly over time. Panels (d)--(f) show the Wasserstein distances over time between the empirical and target distributions of the three scenarios, respectively, for the proposed strategy (black solid line), for the case without the feedforward term (inoue2020optimal, yellow dotted line), for the Voronoi-based solution (red dotted line), and when the feedforward term is computed numerically (purple dotted line).
  • ...and 4 more figures

Theorems & Definitions (8)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2: Explicit dynamics in 1D
  • Remark 5
  • Remark 6