Analysis and approximations of Dirichlet boundary control of Stokes flows in the energy space

TL;DR

This paper analyzes Dirichlet boundary control of 2D Stokes flows, comparing $L^2$ and energy space regularizations, proving well-posedness, regularity, and error estimates, and highlighting the impact of control space choice on solution regularity.

Contribution

It introduces the first regularity and error analysis for energy space boundary control of Stokes flows, revealing control discontinuities at corners with $L^2$ regularization.

Findings

Control discontinuities occur with $L^2$ regularization at domain corners.

Energy space regularization yields more regular controls.

Finite element error estimates are established for the energy space problem.

Abstract

We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. We consider cost functionals with two different boundary control regularization terms: the $L^2$ norm and an energy space seminorm. We prove well-posedness and regularity results for both problems, develop finite element discretizations for both problems, and prove finite element error estimates for the latter problem. The motivation to study the energy space problem follows from our analysis: we prove that the choice of the control space ${\bm L}^2(\Gamma)$ can lead to an optimal control with discontinuities at the corners, even when the domain is convex. We observe this phenomenon in numerical experiments. This behavior does not occur in Dirichlet boundary control problems for the Poisson equation on convex polygonal domains, and may not be desirable in real applications. For the energy space problem, we derive the first order optimality conditions, and show that the solution of the control problem is more regular than the solution of the problem with the ${\bm L}^2(\Gamma)$ regularization. We also prove a priori error estimates for the control in the energy norm, and present several numerical experiments for both control problems on convex and nonconvex domains.