提出基于得分函数的判据，可严格验证可控扩散过程的集合不变性。
Set Invariance with Probability One for Controlled Diffusion: Score-based Approach
- 通过得分向量场导出集合不变性的充要条件
- 在有限与无限时域下均给出可验证的数学判据
- 适用于需要严格保证概率1不变性的控制系统设计
针对受控扩散过程和一个连通、有界、Lipschitz集，何时能保证集合不变性以概率1成立？本文通过推导两类情形（有限与无限时间范围）下所需的充要条件，利用特定对数似然梯度——即得分向量场——给出了答案。所提出的条件构成一个基于得分的可证明测试，能够准确判定是否存在满足要求的马尔可夫控制器。若问题数据通过测试，则可刻画所有能实现目标集合不变性的控制器；若未通过，则不存在此类控制器。测试计算涉及求解特定的狄利克雷边值问题，有限时域情形还可纳入终时刻命中目标子集的约束。文中通过多个半解析与数值例子验证了结果的有效性。
原文摘要 · Abstract (English)
Given a controlled diffusion and a connected, bounded, Lipschitz set, when is it possible to guarantee controlled set invariance with probability one? In this work, we answer this question by deriving the necessary and sufficient conditions for the same in terms of gradients of certain log-likelihoods -- a.k.a. score vector fields -- for two cases: given finite time horizon and infinite time horizon. The deduced conditions comprise a score-based test that provably certifies or falsifies the existence of Markovian controllers for given controlled set invariance problem data. Our results are constructive in the sense when the problem data passes the proposed test, we characterize all controllers guaranteeing the desired set invariance. When the problem data fails the proposed test, there does not exist a controller that can accomplish the desired set invariance with probability one. The computation in the proposed tests involve solving certain Dirichlet boundary value problems, and in the finite horizon case, can also account for additional constraint of hitting a target subset at the terminal time. We illustrate the results using several semi-analytical and numerical examples.
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