Runge-Kutta Schemes for Numerical Discretization of Bilevel Optimal Control Problems

In this paper we consider the discretization of bilevel optimal control problems by s-stage RungeKutta schemes. Bilevel optimal control problem 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.