page_title: Zermelo's navigation problem
hierarchical_section_title: Zermelo's navigation problem
caption_attribution_description: English: Figure of Zermelo Navigation example under a constant drift velocity.
context_page_description: In mathematical optimization, Zermelo's navigation problem, proposed in 1931 by Ernst Zermelo, is a classic optimal control problem that deals with a boat navigating on a body of water, originating from a point to a destination point. The boat is capable of a certain maximum speed, and the goal is to derive the best possible control to reach in the least possible time.
  Without considering external forces such as current and wind, the optimal control is for the boat to always head towards. Its path then is a line segment from to, which is trivially optimal. With consideration of current and wind, if the combined force applied to the boat is non-zero the control for no current and wind does not yield the optimal path.
context_section_description: In mathematical optimization, Zermelo's navigation problem, proposed in 1931 by Ernst Zermelo, is a classic optimal control problem that deals with a boat navigating on a body of water, originating from a point  to a destination point . The boat is capable of a certain maximum speed, and the goal is to derive the best possible control to reach  in the least possible time.
  Without considering external forces such as current and wind, the optimal control is for the boat to always head towards . Its path then is a line segment from  to  , which is trivially optimal. With consideration of current and wind, if the combined force applied to the boat is non-zero the control for no current and wind does not yield the optimal path.
caption: Zermelo Navigation with velocity 
    
      
        
          
            v
          
        
      
      {\displaystyle \mathbf {v} }
    
  under constant wind 
    
      
        
          
            w
          
        
      
      {\displaystyle \mathbf {w} }
exif: {}