Stochastic ergodic control in hybrid systems: A framework for safe autonomy in adversarial environments

Abstract

We study the ergodic control of stochastic hybrid systems by shaping their invariant distribution. Given a regime-switching diffusion with feedback, we pose long-run optimal control with convex constraints on the stationary occupation measure to encode safety, fairness, and coverage requirements. Our analysis establishes (i) verifiable conditions ensuring existence and uniqueness of an invariant measure under controlled switching; (ii) a variational formulation on $P_2$ with a Wasserstein–KKT optimality system that links dual multipliers to sensitivities of stationary constraints; and (iii) sharp bifurcations of the optimal invariant law and dual variables as constraint thresholds, switching rates, or noise intensities vary, revealing phase-transition phenomena predicted by large-deviation heuristics. We further develop a simulation-based primal–dual synthesis method that estimates stationary occupancies from long trajectories and provably converges to a KKT point under standard step–size and convexity assumptions. Case studies — marine navigation in adversarial seas, battery–inverter management under stochastic load, and safety–enveloped medical device scheduling — demonstrate distributional guarantees unattainable by transient or average-cost criteria. The results provide a principled route to safe and interpretable autonomy in hybrid systems, with certificates phrased directly at the level of invariant laws.