We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. Both boundary control and distributed control problems are considered with boundary conditions of Dirichlet or Neumann type. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. Necessary conditions of optimality are discussed both for the continuous and the discretized control problem. It is shown that the recently developed interior point method LOQO of  is capable of solving these problems even for high discretizations. Four 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 controls.
Elliptic control problems, boundary and distributed control, control and state constraints, discretization techniques, interior point optimization methods