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.