Breaking Symmetry-Induced Degeneracy in Multi-Agent Ergodic Coverage via Stochastic Spectral Control

Multi-agent ergodic coverage via Spectral Multiscale Coverage (SMC) provides a principled framework for driving a team of agents so that their collective time-averaged trajectories match a prescribed spatial distribution. While classical SMC has demonstrated empirical success, it can suffer from gradient cancellation, particularly when agents are initialized near symmetry points of the target distribution, leading to undesirable behaviors such as stalling or motion constrained along symmetry axes. In this work, we rigorously characterize the initial conditions and symmetry-induced invariant manifolds that give rise to such directional degeneracy in first-order agent dynamics. To address this, we introduce a stochastic perturbation combined with a contraction term and prove that the resulting dynamics ensure almost-sure escape from zero-gradient manifolds while maintaining mean-square boundedness of agent trajectories. Simulations on symmetric multi-modal reference distributions demonstrate that the proposed stochastic SMC effectively mitigates transient stalling and axis-constrained motion, while ensuring that all agent trajectories remain bounded within the domain.