Regularization and finite element error estimates for elliptic distributed optimal control problems with energy regularization and state or control constraints

TL;DR

This paper analyzes the finite element approximation of elliptic distributed optimal control problems with energy regularization and constraints, establishing error estimates and demonstrating the effectiveness of the approach through numerical results.

Contribution

It extends error analysis to energy regularized control problems with constraints, relating regularization parameters to mesh size for optimal convergence.

Findings

Optimal order of convergence depends on target regularity.

Numerical results confirm the accuracy of the proposed method.

Formulation of variational inequalities for control and state constraints.

Abstract

In this paper we discuss the numerical solution of elliptic distributed optimal control problems with state or control constraints when the control is considered in the energy norm. As in the unconstrained case we can relate the regularization parameter and the finite element mesh size in order to ensure an optimal order of convergence which only depends on the regularity of the given target, also including discontinuous target functions. While in most cases, state or control constraints are discussed for the more common $L^2$ regularization, much less is known in the case of energy regularizations. But in this case, and for both control and state constraints, we can formulate first kind variational inequalities to determine the unknown state, from wich we can compute the control in a post processing step. Related variational inequalities also appear in obstacle problems, and are well established both from a mathematical and a numerical analysis point of view. Numerical results confirm the applicability and accuracy of the proposed approach.