论文标题
贝尔曼方程将路径计划，避免障碍和不变设置估算的应用概括
A Generalization of Bellman's Equation with Application to Path Planning, Obstacle Avoidance and Invariant Set Estimation
论文作者
论文摘要
标准动态编程（DP）公式可用于用可分离的目标函数解决多阶段优化问题（MSOP）。在本文中，我们考虑了一个更大的MSOP类，具有单调向后可分离的目标函数。可分离的函数是单调向后分离函数的特殊情况。我们提出了一种必要且充分的条件，利用Bellman方程的概括，用于具有单调向后可分离的成本函数的MSOP解决方案，是最佳的。此外，我们表明该提出的条件可用于有效计算两个重要的MSOP的最佳解决方案。 Dubin的汽车具有避免障碍物的最佳路径，以及用于离散时间系统的最大不变设置。
The standard Dynamic Programming (DP) formulation can be used to solve Multi-Stage Optimization Problems (MSOP's) with additively separable objective functions. In this paper we consider a larger class of MSOP's with monotonically backward separable objective functions; additively separable functions being a special case of monotonically backward separable functions. We propose a necessary and sufficient condition, utilizing a generalization of Bellman's equation, for a solution of a MSOP, with a monotonically backward separable cost function, to be optimal. Moreover, we show that this proposed condition can be used to efficiently compute optimal solutions for two important MSOP's; the optimal path for Dubin's car with obstacle avoidance, and the maximal invariant set for discrete time systems.
