Scalable Long-Term Safety Certificate for Large-Scale Systems

This letter focuses on safe control problems for high-dimensional systems with large uncertainties. A major challenge is the computation load to account for long outlook horizons in large-scale systems. This challenge is tackled using an integration of probabilistic forward invariance, the comparison theorem, and PDE techniques. Specifically, we propose a probabilistic certificate for long-term safety that only requires myopically ensuring linear control constraints and evaluating two-dimensional PDEs regardless of the system dimension. The certificate is constructed by obtaining a long-term safe probability bound as a solution of the PDE using the comparison theorem and applying a new notion of probabilistic forward invariance on the probability bound. The use of forward invariance directly on probabilistic reachability allows our method to carry the former's computation efficiency and the latter's control over long-term behaviors. Its capability to efficiently ensure long-term safety for high-dimensional systems can be useful in many large-scale distributed autonomous systems operating with limited onboard resources in latency-critical environments.