Lyapunov-Like Conditions for Tight Exit Probability Bounds through Comparison Theorems for SDEs
Lyapunov-Like Conditions for Tight Exit Probability Bounds through Comparison Theorems for SDEs
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通过 SDE 的比较定理得出紧退出概率界的类李雅普诺夫条件
DOI：
10.23919/acc45564.2020.9147414
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
A. Ames
中科院分区：
文献类型：
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作者：
Petter Nilsson;A. Ames
计算退出概率的上限（系统达到某些“坏”集合的概率）可能有助于控制随机系统的决策。随机微分方程描述的系统的现有分析界限非常宽松，特别是对于低概率事件，这限制了它们在实际情况中的适用性。在本文中，我们分析了为什么现有界限是宽松的，并得出结论，这是基于鞅不等式的基础技术的基本问题。作为替代方案，我们给出了随机微分方程的比较结果，该方程通过类 Lyapunov 函数允许 n 维系统的退出概率以一维 Ornstein-Uhlenbeck 过程的退出概率为上限。尽管后者没有已知的封闭形式表达式，但它取决于三个或四个参数，并且可以为应用程序先验列表。我们将这些想法扩展到受控环境，并给出控制屏障函数的随机模拟。这些界限通过数值示例进行了说明，并且显示出比基于鞅不等式的界限要严格得多。
Computing upper bounds on exit probabilities—the probability that a system reaches certain "bad" sets—may assist decision-making in control of stochastic systems. Existing analytical bounds for systems described by stochastic differential equations are quite loose, especially for low-probability events, which limits their applicability in practical situations. In this paper we analyze why existing bounds are loose, and conclude that it is a fundamental issue with the underlying techniques based on martingale inequalities. As an alternative, we give comparison results for stochastic differential equations that via a Lyapunov-like function allow exit probabilities of an n-dimensional system to be upper-bounded by an exit probability of a one-dimensional Ornstein-Uhlenbeck process. Even though no closed-form expression is known for the latter, it depends on three or four parameters and can be a priori tabulated for applications. We extend these ideas to the controlled setting and state a stochastic analogue of control barrier functions. The bounds are illustrated on numerical examples and are shown to be much tighter than those based on martingale inequalities.