Augemented Lagrangian Techniques for Elliptic State Constrained Optimal Control Problems

We propose augmented Lagrangian methods to solve state and control constrained optimal control problems. The approach is based on the Lagrangian formulation of nonsmooth convex optimization in Hilbert spaces developed in [K. Ito and K. Kunisch, Augmented Lagrangian Methods for Nonsmooth Convex Optimization in Hilbert Spaces, preprint, 1994]. We investigate a linear optimal control problem with a boundary control function as in [M. Bergounioux, Numer. Funct. Anal. Optim., 14 (1993), pp. 515--543]. Both the equation and the constraints are augmented. The proposed methods are general and can be adapted to a much wider class of problems.