Optimal control in first-order Sobolev spaces with inequality constraints

Abstract

In this paper, an elliptic optimal control problem with controls from $$H^1(\varOmega )$$ which have to satisfy standard box constraints is considered. Thus, Lagrange multipliers associated with the box constraints are, in general, elements of $$H^1(\varOmega )^\star $$ as long as the lower and upper bound belong to $$H^1(\varOmega )$$ as well. If these bounds possess less regularity, the overall existence of a Lagrange multiplier is not even guaranteed. In order to avoid the direct solution of a not necessarily available KKT system, a penalty method is suggested which finds the minimizer of the control-constrained problem. Its convergence properties are analyzed. Furthermore, some numerical strategies for the computation of optimal solutions are suggested and illustrated.