# Quasi-best approximation in optimization with PDE constraints

## TL;DR

This paper establishes quasi-best approximation bounds for finite element solutions to PDE-constrained quadratic optimization problems, linking error bounds to best approximation errors and analyzing parameter dependencies.

## Contribution

It introduces a quasi-best approximation result for PDE-constrained optimization, including bounds that are independent of regularization parameters under certain conditions.

## Key findings

- Error in state and adjoint state bounded by best approximation error
- Constant depends on inverse square-root of Tikhonov parameter
- Independence of approximation constant when operators are compact

## Abstract

We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square-root of the Tikhonov regularization parameter. Furthermore, if the operators of control-action and observation are compact, this quasi-best-approximation constant becomes independent of the Tikhonov parameter as the meshsize tends to $0$ and we give quantitative relationships between meshsize and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set.