Least-squares finite elements for distributed optimal control problems

TL;DR

This paper introduces a least-squares finite element framework for distributed optimal control problems, providing stable, adaptive, and quasi-optimal solutions for both constrained and unconstrained cases across various PDEs.

Contribution

It develops a unified least-squares finite element approach that is stable, adaptive, and applicable to a broad class of PDE-constrained optimal control problems, including constraints.

Findings

Stable and inf-sup compatible discretization for control problems.

Reliable a posteriori error estimator for adaptive algorithms.

Numerical validation on multiple PDE control problems.

Abstract

We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is $\inf$--$\sup$ stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented.