REGULARIZED PENALTY METHODS FOR ILL-POSED OPTIMAL-CONTROL PROBLEMS WITH ELLIPTIC-EQUATIONS .2. DISTRIBUTED AND BOUNDARY CONTROL WITH UNBOUNDED CONTROL SET AND STATE CONSTRAINTS

We investigate the application of Prox-Regularization to ill-posed con trol problems for systems governed by elliptic equations. Stable varia nts of Penalty Methods are obtained by means of One- and Multi-Step Re gularization of the penalized problems. Convergence of the resulting m ethods is proved in tile case of distributed control as well as bounda ry control with Dirichlet and Neumann conditions. In the first part of this paper the case of distributed control under Dirichlet boundary c onditions and a bounded set of admissible controls has been considered .