Optimization Techniques for Solving Elliptic Control
Problems with Control and State Constraints.
Part 1: Boundary Control

HELMUT MAURER ¹

Westfälische Wilhelms-Universität Münster, Institut für Numerische und
instrumentelle Mathematik, Einsteinstrasse 62, 48149 Münster, Germany

HANS D. MITTELMANN

Arizona State University, Department of Mathematics, Tempe, AZ 85287-1804, USA

Abstract: We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve theses problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.

Key words: Elliptic control problems, boundary control, control and state constraints, discretization techniques, interior point optimization methods