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 A {\displaystyle A} to a destination point B {\displaystyle B} . The boat is capable of a certain maximum speed, and the goal is to derive the best possible control to reach B {\displaystyle B} 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 B {\displaystyle B} . Its path then is a line segment from A {\displaystyle A} to B {\displaystyle B} , 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.
History In his 1931 article, Ernst Zermelo formulates the following problem:
In an unbounded plane where the wind distribution is given by a vector field as a function of position and time, a ship moves with constant velocity relative to the surrounding air mass. How must the ship be steered in order to come from a starting point to a given goal in the shortest time?
This is an extension of the classical optimisation problem for geodesics – minimising the length of a curve I [ c ] = ∫ a b 1 + y ′ 2 d x {\displaystyle I[c]=\int _{a}^{b}{\sqrt {1+y'^{2}}}\,dx} connecting points A {\displaystyle A} and B {\displaystyle B} , with the added complexity of considering some wind velocity. Although it is usually impossible to find an exact solution in most cases, the general case was solved by Zermelo himself in the form of a partial differential equation, known as Zermelo's equation, which can be numerically solved. The problem of navigating an airship which is surrounded by air, was presented first in 1929 at a conference by Ernst Zermelo, who was inspired by the circumnavigation of Graf Zeppelin in 1929-08. Other mathematicians have answered the challenge over the following years. The dominant technique for solving the equations is the calculus of variations.
Constant-wind case The case of constant wind is easy to solve exactly. Let d = A B → {\displaystyle \mathbf {d} ={\vec {AB}}} , and suppose that to minimise the travel time the ship travels at a constant maximum speed V {\displaystyle V} . Thus the position of the ship at time t {\displaystyle t} is x = t ( v + w ) {\displaystyle \mathbf {x} =t(\mathbf {v} +\mathbf {w} )} . Let T {\displaystyle T} be the time of arrival at B {\displaystyle B} , so that d = T ( v + w ) {\displaystyle \mathbf {d} =T(\mathbf {v} +\mathbf {w} )} . Taking the dot product of this with w {\displaystyle \mathbf {w} } and d {\displaystyle \mathbf {d} } respectively results in
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