Set positive invariance is an important concept in the theory of dynamical systems and one which also has practical applications in areas of computer science, such as formal verification, as well as in control theory. Great progress has been made in understanding positively invariant sets in continuous dynamical systems and powerful computational tools have been developed for reasoning about them; however, many of the insights from recent developments in this area have largely remained folklore and are not elaborated in existing literature. This article contributes an explicit development of modern methods for checking positively invariant sets of ordinary differential equations and describes two possible characterizations of positive invariants: one based on the real induction principle, and a novel alternative based on topological notions. The two characterizations, while in a certain sense equivalent, lead to two different decision procedures for checking whether a given semi-algebraic set is positively invariant under the flow of a system of polynomial ordinary differential equations.

中文翻译：

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表征正不变集：归纳和拓扑方法

集合正不变性是动力系统理论中的一个重要概念，在计算机科学领域也有实际应用，例如形式验证，以及控制理论。在理解连续动力系统中的正不变集方面取得了很大进展，并且已经开发出强大的计算工具来推理它们；然而，该领域最近发展的许多见解在很大程度上仍然是民间传说，在现有文献中没有详细阐述。本文为检验常微分方程的正不变集的现代方法做出了明确的发展，并描述了正不变量的两种可能的表征：一种基于实归纳原理，一种基于拓扑概念的新替代方案。