Correspondence Florian Wagener, Universiteit van Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands. Email: f.o.o.wagener@uva.nl Abstract Natural resources are not infinitely resilient and should not be modeled as being such. Finitely resilient resources feature tipping points and history dependence. This paper provides a didactical discussion of mathematical methods that are needed to understand the optimal management of such resources: viscosity solutions of Hamilton–Jacobi–Bellman equations, the costate equation and the associated canonical equations, exact root counting, and geometrical methods to analyze the geometry of the invariant manifolds of the canonical equations.

中文翻译：

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分析临界点动态优化管理的几何方法

通讯 Florian Wagener, Universiteit van Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands。电子邮件：foowagener@uva.nl 摘要 自然资源不是无限弹性的，不应如此建模。有限弹性​​资源具有临界点和历史依赖性。本文提供了理解此类资源优化管理所需的数学方法的教学讨论：Hamilton-Jacobi-Bellman 方程的粘度解、costate 方程和相关的典型方程、精确根计数和几何方法来分析正则方程的不变流形的几何。