Zermelo's Navigation Problem: Optimal Control, Geometry and Computer Algebra
Matthias Kawski kawski@asu.edu
Mathematics Arizona State University United States
Abstract
Consider the problem of steering a boat from point P to point Q across a lake
in minimum time. The boat's engine always runs at full throttle, but the
boat's velocity is subject to variable wind or current and the rudder
postion which is our control.. This is basically Zermelo's Navigation
Problem formulated in 1931, and it is a classical example in calculus
of variations and optimal control.
This talk shall start with exploring the problem using interactive graphing
technology. Of particular interest is the emergence of caustics and conjugate
points, i.e., points where geodesics generally stop being the shortest
curves (the fastest trajectories). Everyone is familiar with this effect
in the case of shortest curves on the (curved) sphere (no wind), where
the great circles are length minimizing until they reach the antipodal
point.
After briefly reviewing some classical tools, we survey modern formulations
using geometric optimal control. Of special interest is a notion of curvature
introduced in 2000 by Agrachev, which generalizes Gauss curvature to optimal
control problems such as Zermelo navigation. While geometrically very
natural and intuitive, the algebraic calculations virtually require computer
algebra systems. We demonstrate how only recent improvements (e.g., from
MAPLE release 8 to release 9.5) of the algebraic capabilities of the computer
algebra system allow us to achieve dramatic simplifications of the formulas,
which in turn make it feasible to interactively visualize the curvature
in Zermelo's navigation problem.
