Ergodic control of multi-agent systems under communication constraints

We study decentralised ergodic control for multi-agent systems operating over stochastic, lossy communication graphs. Each agent follows controlled stochastic dynamics with uniformly elliptic noise on a compact domain and aims to align its occupation measure with a prescribed target distribution π. We separate two regimes. (i) In the decoupled regime – perfect self-observation, static π, no global coupling – communication is unnecessary: each agent alone can attain μiT⇒π almost surely. (ii) In the team-coupled regime, agents are coupled through a coverage – fairness objective and a shared energy budget, which makes tracking the team-mean occupation measure indispensable. For the coupled regime, we introduce a consensus filter that mixes local coverage estimates over random time-varying graphs while incorporating per-step innovations from fresh samples. Under an explicit Dobrushin-scrambling assumption with exponentially rare mixing failures, square-summable stepsizes and uniformly elliptic dynamics, we prove almost-sure convergence of each agent's empirical measure to π in W1. The analysis replaces informal o(1) arguments with an adapted, summable error sequence and a Robbins–Siegmund-type lemma tailored to measure-valued recursions. We further identify a sharp temporal-connectivity threshold: below it, disagreement persists and team ergodicity fails with non-vanishing probability; above it, deviations obey exponential tails and coverage fluctuations satisfy a large-deviation principle. The framework unifies decentralised decision-making, ergodic coverage and random-graph consensus, and applies to robotic teams, drone swarms and distributed sensing under severe communication constraints.