This note introduces, and studies, the Minkowski–Bellman inequality and equation. The necessary and sufficient conditions for the characterization and existence of Minkowski functions that satisfy the Minkowski–Bellman inequality, and for the characterization and existence of the unique Minkowski function that satisfies the Minkowski–Bellman equation, are derived. The complete topological characterization of the associated optimal set-valued control map is also derived.

中文翻译：

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Minkowski-Bellman不等式和方程

本说明介绍并研究了Minkowski-Bellman不等式和方程。得出满足Minkowski-Bellman不等式的Minkowski函数的特征和存在，以及满足Minkowski-Bellman方程的唯一Minkowski函数的特征和存在的充要条件。还推导了相关的最佳设定值控制图的完整拓扑特征。