A Least-Squares Method for Optimal Control Problems for a Second-Order Elliptic System in Two Dimensions1

A distributed optimal control problem for an elliptic problem is considered. A penaltyrleast-squares method for optimal control problems is used wherein the constraint equations are enforced via penalization. First, the existence of the optimal control problem is proved. The convergence, as the penalty parameter tends to zero, of the solution to the penalized optimal control problem to that unpenalized one is demonstrated as is the convergence of a gradient method for determining solutions of the penalized optimal control problem. Finally, finite element approximations of the penalized optimal control problem are studied and optimal error estimates are obtained