A priori error estimates for optimal control problems governed by the transient Stokes equations and subject to state constraints pointwise in time

TL;DR

This paper develops a priori error estimates for optimal control problems constrained by transient Stokes equations with pointwise in-time state bounds, using stable discretization schemes and providing numerical validation.

Contribution

It introduces new a priori error estimates for discretized transient Stokes control problems with pointwise state constraints, improving regularity results and validating with numerical experiments.

Findings

Established error estimates for discretized control problems.

Proved improved regularity for the optimal control.

Validated theoretical results with numerical experiments.

Abstract

In this paper, we consider a state constrained optimal control problem governed by the transient Stokes equations. The state constraint is given by an L2 functional in space, which is required to fulfill a pointwise bound in time. The discretization scheme for the Stokes equations consists of inf-sup stable finite elements in space and a discontinuous Galerkin method in time, for which we have recently established best approximation type error estimates. Using these error estimates, for the discrete control problem we establish error estimates and as a by-product we show an improved regularity for the optimal control. We complement our theoretical analysis with numerical results.