On a SQP-Multigrid Technique for Nonlinear Parabolic Boundary Control Problems

An optimal control problem governed by the heat equation with nonlinear boundary conditions is considered. The objective functional consists of a quadratic terminal part and a quadratic regularization term. It is known, that an SQP method converges quadratically to the optimal solution of the problem. To handle the quadratic optimal control subproblems with high precision, very large scale mathematical programming problems have to be treated. The constrained problem is reduced to an unconstrained one by a method due to Bertsekas. A multigrid approach developed by Hackbusch is applied to solve the unconstrained problems. Some numerical examples illustrate the behaviour of the method. AMS subject classification: 49M40, 49M05 Keywords: optimal control, semilinear parabolic equation, multigrid method, SQP method 1 Introduction The behaviour of Lagrange--Newton--SQP methods for solving nonlinear optimal control problems has been the subject of several recent publications. For instance, th..