Safe Multi-Agent Navigation via Constrained HJB-Informed Learning

Multi-agent navigation in unknown and cluttered environments has broad applications, yet remains fundamentally challenging. In particular, dense agent-agent and agent-obstacle reactive interactions can exacerbate the inherent competition between collision-avoidance constraints and goal-reaching objectives. Most existing approaches mitigate this by applying per-step safety filtering on top of a predefined goal-reaching controller or by designing heuristic loss functions that penalizes safety constraints violation gradient. While effective in sparse environments, these methods still suffer from overly-conservative behaviors when interactions become dense. To overcome these limitations, we propose HJB-GNN, a Hamilton-Jacobi-Bellman (HJB)-based learning framework that jointly learns a graph neural network (GNN)-parameterized control barrier function for explicit safety enforcement, a distributed GNN-based navigation policy, and a value function that induces goal-reaching behavior. By exploiting the analytical solution of the constrained HJB equation, the proposed method derives graph-dependent Lagrange multipliers that adaptively balance collision-avoidance and goal-reaching across diverse multi-agent navigation scenarios. Moreover, HJB-GNN supports centralized training with distributed deployment. Extensive simulations and real-world experiments with Crazyflie drone swarms demonstrate its superior safety and goal-reaching performance, as well as strong scalability and generalizability to large-scale teams operating in previously unseen, dense environments.