An Adaptive Multilevel Scheme for Optimal Control Problems Based on a Function Space Interior Point Method

A new approach to the numerical solution of optimal control problems including control and state constraints is presented. Like hybrid methods, the approach aims at combining the advantages of direct and indirect methods. Unlike hybrid methods, however, our method is directly based on interior-point concepts in function space -- realized via an adaptive multilevel scheme applied to the complementarity formulation and to numerical continuation along the central path. Existence of the central path and its continuation towards the solution point is analyzed in some theoretical detail. An adaptive stepsize control with respect to the duality gap parameter is worked out in the framework of affine invariant inexact Newton methods. Finally, the performance of a proto-type of our algorithm is documented by the successful treatment of the well-known intricate windshear problem treated by Bulirsch et al.