434 RWTH Publication No: 540256        2015        IGPM434.pdf
TITLE Runge-Kutta Schemes for Numerical Discretization of Bilevel Optimal Control Problems
AUTHORS Sonja Steffensen
ABSTRACT In this paper we consider the discretization of bilevel optimal control problems by s-stage Runge-Kutta schemes. Bilevel optimal control problems belong to the class of dynamic or differential games as they are the time-dependent counterpart of finite bilevel optimization problems. The analysis of the Runge-Kutta schemes presented in this paper is based on the continuous optimality system which is derived by replacing the lower-and upper-level control problems, respectively, by their associated necessary optimality conditions. We apply the results of [13, 4, 15] for standard and IMEX Runge-Kutta schemes for general optimal control problems in order to relate the discretization schemes obtained through finite dimensional optimality theory to time-discretizations of the continuous optimality system. Moreover, order conditions up to order three are proven. Finally, we briefly discuss suitable extensions to general leader-follower games.
KEYWORDS Runge-Kutta schemes, optimal control, bilevel program, hierarchical optimal control, game theory