SIAM Journal on Applied Mathematics, Volume 80, Issue 4, Page 1862-1881, January 2020.
We present a method for optimal path planning of human walking paths in mountainous terrain using a control theoretic formulation and a Hamilton--Jacobi--Bellman equation. Previous models for human navigation were entirely deterministic, assuming perfect knowledge of the ambient elevation data and human walking velocity as a function of the local slope of the terrain. Our model includes a stochastic component which can account for uncertainty in the problem and thus includes a Hamilton--Jacobi--Bellman equation with viscosity. We discuss the model in the presence and absence of stochastic effects and suggest numerical methods for simulating the model. We discuss two different notions of an optimal path when there is uncertainty in the problem. Finally, we compare the optimal paths suggested by the model at different levels of uncertainty and observe that as the size of the uncertainty tends to zero (and thus the viscosity in the equation tends to zero), the optimal path tends toward the deterministic optimal path.

中文翻译：

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具有随机效应的最优人类导航模型

SIAM应用数学杂志，第80卷，第4期，第1862-1881页，2020年1月。
我们提出了一种使用控制理论公式和汉密尔顿-雅各比-贝尔曼方程的山区地形下人的步行路径的最佳路径规划方法。假设完全了解环境海拔数据和人类步行速度随地形局部坡度的变化，以前的人类导航模型是完全确定性的。我们的模型包含一个可以解释问题不确定性的随机成分，因此包含具有粘性的Hamilton-Jacobi-Bellman方程。我们讨论存在和不存在随机效应的模型，并提出模拟模型的数值方法。当问题存在不确定性时，我们讨论最优路径的两种不同概念。最后，