Probabilistic guarantees of safety and performance are important in constrained dynamical systems with stochastic uncertainty. We consider the stochastic reachability problem, which maximizes the probability that the state remains within time-varying state constraints (i.e., a “target tube”), despite bounded control authority. This problem subsumes the stochastic viability and terminal hitting-time stochastic reach-avoid problems. Of special interest is the stochastic reach set, the set of all initial states from which it is possible to stay in the target tube with a probability above a desired threshold. We provide sufficient conditions under which the stochastic reach set is closed, compact, and convex, and provide an underapproximative interpolation technique for stochastic reach sets. Utilizing convex optimization, we propose a scalable and grid-free algorithm that computes a polytopic underapproximation of the stochastic reach set and synthesizes an open-loop controller. This algorithm is anytime, i.e., it produces a valid output even on early termination. We demonstrate the efficacy and scalability of our approach on several numerical examples, and show that our algorithm outperforms existing software tools for verification of linear systems.

安全性和性能的概率保证在具有不确定随机性的受限动力系统中很重要。我们考虑了随机可达性问题，尽管控制权限有限，但该问题使状态保持在时变状态约束（即“目标管”）内的可能性最大化。此问题包含随机生存能力和终端命中时间随机到达避免问题。特别感兴趣的是随机到达范围集，这是所有初始状态的集合，从中可以以高于所需阈值的概率停留在目标管中。我们提供了足够的条件，其中随机到达集是封闭的，紧致的和凸的，并且为随机到达集提供了一种近似逼近的插值技术。利用凸优化，我们提出了一种可扩展且无网格的算法，该算法可计算出随机覆盖集的多主题近似值，并合成开环控制器。该算法是随时可用的，即，即使提前终止，它也会产生有效的输出。我们在几个数值示例上证明了该方法的有效性和可扩展性，并表明我们的算法优于用于验证线性系统的现有软件工具。