Zermelo's Navigation Problem: Optimal Control, Geometry and Computer Algebra

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.