This paper addresses the design of safety certificates for stochastic
systems, with a focus on ensuring long-term safety through fast real-time
control. In stochastic environments, set invariance-based methods that restrict
the probability of risk events in infinitesimal time intervals may exhibit
significant long-term risks due to cumulative uncertainties/risks. On the other
hand, reachability-based approaches that account for the long-term future may
require prohibitive computation in real-time decision making. To overcome this
challenge involving stringent long-term safety vs. computation tradeoffs, we
first introduce a novel technique termed `probabilistic invariance'. This
technique characterizes the invariance conditions of the probability of
interest. When the target probability is defined using long-term trajectories,
this technique can be used to design myopic conditions/controllers with assured
long-term safe probability. Then, we integrate this technique into safe control
and learning. The proposed control methods efficiently assure long-term safety
using neural networks or model predictive controllers with short outlook
horizons. The proposed learning methods can be used to guarantee long-term
safety during and after training. Finally, we demonstrate the performance of
the proposed techniques in numerical simulations.

该论文介绍了一种设计用于随机系统的安全证书的方法，重点是通过快速实时控制来确保长期安全。通过引入一种名为 “概率不变性” 的新颖技术，论文解决了长期安全与计算权衡之间的挑战，并将该技术整合到安全控制和学习中，实现了在随机环境下的长期安全控制和学习。