Optimal Soaring via Hamilton-Jacobi-Bellman Equations

Abstract

Competition glider flying is a game of stochastic optimization, in which mathematics and quantitative strategies have historically played an important role. We address the problem of uncertain future atmospheric conditions by constructing a nonlinear Hamilton-Jacobi-Bellman equation for the optimal speed to fly, with a free boundary describing the climb/cruise decision. We consider two different forms of knowledge about future atmospheric conditions, the first in which the pilot has complete foreknowledge and the second in which the state of the atmosphere is a Markov process discovered by flying through it. We compute an accurate numerical solution by designing a robust monotone finite difference method. The results obtained are of direct applicability for glider flight.

Keywords: Hamilton-Jacobi equations; glider flying; Variational Inequalities; stochastic control; finite difference; monotone approximation

JEL Classification: C61,C63