Mean-Field Games for Coordinated Exploration in Dynamic Environments

<div xmlns="http://www.tei-c.org/ns/1.0"> We develop a mean-field game (MFG) framework for coordinated exploration by large swarms of autonomous agents operating in dynamic and uncertain environments. Motivated by applications such as gas leak detection, wildfire containment, and search-and-rescue operations, we model each agent as solving an optimal control problem over a stochastic spatial field, with decentralized feedback based on local observations and a common statistical prior. The collective behavior is characterized by the coupled Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, capturing the equilibrium between exploration cost and coverage density. We introduce a real-time feedback mechanism whereby the agent distribution shapes the evolving environmental field, leading to a closed-loop mean-field system with nonlinear coupling. Our main contributions include: (i) the derivation of existence and uniqueness conditions for feedback mean-field equilibria, (ii) a novel viability-based constraint formulation ensuring safety and spatial coverage guarantees, and (iii) a transport-theoretic reformulation of the exploration problem that enables scalable numerical implementation. Simulations demonstrate robust distributed coverage and adaptive response to time-varying field dynamics. This work bridges optimal transport, stochastic control, and decentralized exploration, offering a foundational perspective on coordinated autonomy under uncertainty. </div>