Conclusion

In continuation of the first part of this paper devoted to boundary control we have developed numerical techniques for solving semilinear distributed control problems with control and state constraints. While three numerical methods, two of interior point type, were compared in [1] for linear problems and homogeneous Dirichlet conditions the emphasis in this work is on treating nonlinearities in both the equations and the boundary equations, see part 1, which are of Dirichlet and Neumann type. The control problem is fully discretized resulting in a large, sparse nonlinear optimization (NLP) problem. Modern NLP software is utilized for its solution and the interior point code LOQO [29] proves to be a robust and efficient tool. While also results for several other NLP programs are given these are only meant to show what a straightforward application, with default options, of these to the problems at hand and through the common interface AMPL [13] yields. The algorithms used are quite different and, for example, LOQO makes use of second derivatives while the quasi-Newton based SQP code SNOPT [14] does not. A total of six problems were solved. The necessary optimality conditions of section 3.1 were checked in all cases. They are given examplarily in one, example 3, with an exponential nonlinearity. In particular, bang-bang controls were computed in two cases and, to the best of our knowledge, for the first time in distributed two-dimensional control. We refer to the conclusion of part 1 concerning the planned generalizations of our work.