Solution of bilevel optimal control problems to increase fairness in air races

The focus of this paper lies on the treatment of a new class of bilevel optimal control problems, where the optimal solution of the upper-level parameter optimization problem depends on optimal solutions of two lower-level optimal 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 gradient of the upper-level optimization problem can be avoided, allowing for an efficient solution of the entire bilevel 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.