Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints. Part 2: Distributed Control

Part 2 continues the study of optimization techniques for elliptic control problems subject to control and state constraints and is devoted to distributed control. Boundary conditions are of mixed Dirichlet and Neumann type. Necessary conditions of optimality are formally stated in form of a local Pontryagin minimum principle. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. The problems are formulated as AMPL (R. Fourer, D.M. Gay, and B.W. Kernighan, “AMPL: A modeling Language for Mathematical Programming”, Duxbury Press, Brooks-Cole Publishing Company, 1993) scripts and several optimization codes are applied. In particular, 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 controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.