On the Solution of Bilevel Optimal Control Problems to Increase the Fairness in Air Races
Florian Fisch, Jakob Lenz, Florian Holzapfel, Gottfried Sachs · AIAA Atmospheric Flight Mechanics Conference · 2010
control problems. An efficient way for the solution of such bilevel programming problems is introduced. In every iteration step of the upper-level optimization problem, the two lower-level optimal control problems are solved by applying a multiple shooting method. Furthermore, in each iteration, a sensitivity analysis with respect to selected parameters of the lower-level optimal control problems is carried out. The sensitivity analysis allows for a direct computation of the gradient of the objective of the upper-level parameter optimization problem with respect to the just-mentioned parameters of the lower-level optimal control problems. Thus, a time-consuming evaluation of the gradientoftheupper-leveloptimizationproblemcanbeavoided,allowingforanefficientsolutionoftheentirebilevel optimal control problem. As an illustrative example, the layout of an air racetrack such that two different aircraft have, in fact, exactly the same chance of winning is presented.