Optimal Transport for Time-Varying Multi-Agent Coverage Control

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.

💡 Research Summary

The paper introduces a novel framework for multi‑agent coverage control when the target density evolves over time. Building on recent work that recasts static coverage as a semi‑discrete optimal transport (OT) problem, the authors extend the formulation to the time‑varying case by minimizing, at each instant, the 2‑Wasserstein distance between the agents’ empirical measure (a sum of Dirac deltas) and the continuously changing target measure induced by the density (\bar\rho(x,t)).

Key technical contributions are as follows. First, the authors write the instantaneous OT cost as a max‑over‑dual‑variables problem, where the dual variables (\phi\in\mathbb R^N) define Laguerre (additively weighted Voronoi) cells (V_i(p,\phi)). The mass (a_i(t)=\int_{V_i}\bar\rho(x,t)dx) and barycenter (b_i(t)=\frac{1}{a_i(t)}\int_{V_i}x\bar\rho(x,t)dx) of each cell are computed from the current agent positions (p) and the dual weights.

Second, assuming that the time derivative (\partial_t\bar\rho(x,t)) is known (or can be estimated), the authors propose continuous‑time dynamics for both the primal variables (agent positions) and the dual variables:

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