# Optimization Methods for Dirichlet Control Problems

## TL;DR

This paper explores various optimization algorithms for finite element approximations of linear-quadratic Dirichlet optimal control problems governed by elliptic PDEs, analyzing convergence and computational efficiency.

## Contribution

It introduces a preconditioned conjugate method and semismooth Newton methods for constrained and unconstrained problems, with convergence analysis and numerical validation.

## Key findings

- Convergence of optimization procedures without Tikhonov regularization.
- Effective solution methods for control and state constrained problems.
- Numerical examples demonstrating theoretical results.

## Abstract

We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control is discretized explicitely using continuous piecewise linear approximations. Unconstrained, control-constrained, state-constrained and control-and-state constrained problems are analyzed. A preconditioned conjugate method for a reduced problem in the control variable is proposed to solve the unconstrained problem, whereas semismooth Newton methods are discussed for the solution of constrained problems. State constraints are treated via a Moreau-Yosida penalization. Convergence is studied for both the continuous problems and the finite dimensional approximations. In the finite dimensional case, we are able to show convergence of the optimization procedures even in the absence of Tikhonov regularization parameter. Computational aspects are also treated and several numerical examples are included to illustrate the theoretical results.