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

David Arnold, Christian Parkinson, Andrea Bertozzi, Yat Tin Chow, Stanley Osher

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)227-242
Number of pages16
JournalCommunications in Mathematical Sciences
Issue number1
Publication statusPublished - 2019
Externally publishedYes


  • Anisotropic control
  • Hamilton-Jacobi-Bellman equation
  • Level set method
  • Optimal control
  • Optimal path planning


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