## ccdist2 details:

Keywords: flow control, analytic solution

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Stokes:
• linear elliptic operator of order 2.
• Defined on a 3-dim domain in 3-dim space
• No time dependence.

Design constraints:

• box of order 0

State constraints:

• none

Mixed constraints:

• none

Submitted on 2013-01-04 by Arnd Rösch. Published on 2013-01-04

## ccdist2 description:

### Introduction

We have a distributed optimal control problem for the Stokes equations with component-wise box constraints for the distributed control. A diﬃculty in the construction of test examples as the present one is the satisfaction of the divergence-free condition. Moreover, the example is three-dimensional.

This problem and the analytical solution were published in Rösch and Vexler [2006].

### Optimality System

The following system for the state (velocity and pressure) $u=\left(v,p\right)\in {H}_{0}^{1}{\left(\Omega \right)}^{3}×{L}_{0}^{2}\left(\Omega \right)$, the control $q\in {L}^{2}{\left(\Omega \right)}^{3}$ and the adjoint state $z=\left(w,r\right)\in {H}_{0}^{1}{\left(\Omega \right)}^{3}×{L}_{0}^{2}\left(\Omega \right)$, given in strong form, characterizes the unique minimizer. The projection is a component-wise projection onto the interval $\left[{a}_{i},{b}_{i}\right]$ for $i=1,2,3$.

### Supplementary Material

The optimal state, adjoint state and control are known analytically:

 $\begin{array}{ccc}\hfill {v}_{1}={w}_{1}& =2{sin}^{2}\left(\pi x\right)sin\left(2\pi y\right)sin\left(2\pi z\right)\hfill & \hfill \\ \hfill {v}_{2}={w}_{2}& =-{sin}^{2}\left(\pi y\right)sin\left(2\pi x\right)sin\left(2\pi z\right)\hfill \\ \hfill {v}_{3}={w}_{3}& =-{sin}^{2}\left(\pi z\right)sin\left(2\pi x\right)sin\left(2\pi y\right)\hfill \\ \hfill p=r& =sin\left(2\pi x\right)sin\left(2\pi y\right)sin\left(2\pi z\right)\hfill \\ \hfill q& ={\text{proj}}_{\left[a,b\right]}\left(-w\right)\hfill \\ \hfill \end{array}$

### References

A. Rösch and B. Vexler. Optimal control of the Stokes equations: A priori error analysis for ﬁnite element discretization with postprocessing. SIAM Journal on Numerical Analysis, 44(5):1903–1920, 2006. doi: 10.1137/050637364.