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Summary: OPTIMAL SOARING
WITH HAMILTON-JACOBI-BELLMAN EQUATIONS
ROBERT ALMGREN AND AGN`ES TOURIN
Abstract. Competition glider flying, like other outdoor sports, is a game of stochastic opti-
mization, 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. This equation comes from a singular stochastic exit time control problem
and involves a gradient constraint, a state constraint and a weak Dirichlet boundary condition. An
accurate numerical solution requires robust monotone numerical methods. The computed results are
of direct applicability for glider flight.
Key words. Hamilton-Jacobi equations, variational inequalities, stochastic control, monotone
approximation, viscosity solutions
AMS subject classifications. 35R35, 35R45, 49L25, 65M06
1. Introduction. Competition glider flying, like other outdoor sports such as
sailboat racing, is a game of probabilities, and of optimization in an uncertain environ-
ment. The pilot must make a continuous series of strategic decisions, with imperfect
information about the conditions to be encountered futher along the course and later
in the day. In a competition, these decisions are made in order to maximize cross-
country speed, and hence final score in the contest; even in noncompetition flying the
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