On the Solution of Bilevel Optimal Control Problems to Increase the Fairness in Air Races

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.

[1]  Patrice Marcotte,et al.  An overview of bilevel optimization , 2007, Ann. Oper. Res..

[2]  Paul H. Calamai,et al.  Bilevel and multilevel programming: A bibliography review , 1994, J. Glob. Optim..

[3]  Harri Ehtamo,et al.  Visual Aircraft Identification as a Pursuit-Evasion Game , 2000 .

[4]  Ilan Kroo,et al.  Enhanced Collaborative Optimization: Application to an Analytic Test Problem and Aircraft Design , 2008 .

[5]  S. Dempe.  Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints , 2003 .

[6]  Anil V. Rao,et al.  Practical Methods for Optimal Control Using Nonlinear Programming , 1987 .

[7]  Eli Livne,et al.  Coupled Aeroelastic / Trajectory Optimization of Reentry Vehicles , 2007 .

[8]  Jeremy S. Agte,et al.  Bi-Level Integrated System Synthesis , 1998 .

[9]  H. Maurer,et al.  SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control , 2000 .

[10]  John R. Olds,et al.  Evaluation of Multidisciplinary Optimization Techniques Applied to a Reusable Launch Vehicle , 2006 .

[11]  J. Magnus,et al.  Matrix differential calculus with applications to simple, Hadamard, and Kronecker products. , 1985 .

[12]  Kamran Behdinan,et al.  Evaluation of Multidisciplinary Optimization Approaches for Aircraft Conceptual D esign , 2004 .

[13]  J. Betts.  Survey of Numerical Methods for Trajectory Optimization , 1998 .

[14]  T. Raivio,et al.  On Applied Nonlinear and Bilevel Programming or Pursuit-Evasion Games , 2001 .

[15]  Florian Holzapfel,et al.  Airframe and Trajectory Pursuit Modeling for Simulation Assisted Air Race Planning , 2008 .

[16]  R. Callies,et al.  Optimal hypersonic flight trajectories with full safety in case of mission abort , 2000 .

[17]  Ilan Kroo,et al.  Collaborative Approach to Launch Vehicle Design , 1997 .