Communications in Mathematical Sciences

Volume 17 (2019)

Number 1

Optimal human navigation in steep terrain: a Hamilton–Jacobi–Bellman approach

Pages: 227 – 242

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n1.a9

Authors

Christian Parkinson (Department of Mathematics, University of California at Los Angeles)

David Arnold (Department of Mathematics, University of California at Los Angeles)

Andrea L. Bertozzi (Department of Mathematics, University of California at Los Angeles)

Yat Tin Chow (Department of Mathematics, University of California at Riverside)

Stanley Osher (Department of Mathematics, University of California at Los Angeles)

Abstract

We present a method for determining optimal walking paths in steep terrain using the level set method and an optimal control formulation. By viewing the walking direction as a control variable, we can determine the optimal control by solving a Hamilton–Jacobi–Bellman equation. We then calculate the optimal walking path by solving an ordinary differential equation. We demonstrate the effectiveness of our method by computing optimal paths which travel throughout mountainous regions of Yosemite National Park. We include details regarding the numerical implementation of our model and address a specific application of a law enforcement agency patrolling a nationally protected area.

Keywords

optimal path planning, Hamilton–Jacobi–Bellman equation, optimal control, level set method, anisotropic control

2010 Mathematics Subject Classification

00A69, 34H05, 35F21, 49L20

Full Text (PDF format)

Received 9 May 2018

Accepted 5 November 2018

Published 30 May 2019