Space-time finite element methods for parabolic distributed optimal control problems

TL;DR

This paper introduces a space-time finite element approach for numerically solving distributed optimal control problems governed by parabolic PDEs, ensuring stability and allowing for final-time state specifications.

Contribution

The paper develops a novel space-time variational formulation that satisfies Babuška-Brezzi conditions at both continuous and discrete levels, enabling flexible control problems with reliable error estimation.

Findings

Method is stable and coercive in energy norm.

Allows for final-time desired states.

Numerical experiments confirm theoretical results.

Abstract

We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. We complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrangian multiplier. Our final formulation fulfills the Babu\v{s}ka-Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, we can allow for final-time desired states, and obtain an a-posteriori error estimator which is efficient and reliable. Numerical experiments confirm our theoretical findings.