## mpdist3 details:

Keywords: flow control, analytic solution

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Stokes:
• linear parabolic operator of order 2.
• Defined on a 2-dim domain in 2-dim space
• Time dependent.

Design constraints:

• none

State constraints:

• none

Mixed constraints:

• none

Submitted on 2017-06-02 by John Pearson. Published on 2017-12-11

## mpdist3 description:

### Introduction

Here we present a distributed optimal control problem of the time-dependent Stokes equations. The problem was derived as a test for the paper Güttel and Pearson [2017], which required optimal states and controls that are not polynomial in spatial or time variables. The problem maintains a parameter dependence for the regularization parameter $\beta$ to serve as a test case for the $\beta$ dependence of solvers. This problem and its analytical solution appear in [Güttel and Pearson2017, Section 6.2], with computations for ﬁnal time $T=1$ and control cost parameter $\beta =1{0}^{-2}$.

### Variables & Notation

#### Unknowns

where

 $\begin{array}{ccc}\hfill V\left(\Omega \right)& =\left\{\right\v\in {H}_{0}^{1}{\left(\Omega \right)}^{2}:divv=0\left\}\right\,\hfill & \hfill \\ \hfill {L}_{0}^{2}\left(\Omega \right)& =\left\{\right\p\in {L}^{2}\left(\Omega \right):{\int }_{\Omega }p\phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}x\left\}\right\.\hfill \end{array}$

### Optimality System

The following optimality system for the state $y=\left(v,p\right)\in {L}^{2}\left(I,V\left(\Omega \right)\right)\cap {H}^{1}\left(I,V{\left(\Omega \right)}^{\ast }\right)×{L}^{2}\left(I,{L}_{0}^{2}\left(\Omega \right)\right)$, the control $u\in {L}^{2}{\left(\Omega ×I\right)}^{2}$ and the adjoint state $q=\left(\lambda ,\mu \right)\in {L}^{2}\left(I,V\left(\Omega \right)\right)\cap {H}^{1}\left(I,V{\left(\Omega \right)}^{\ast }\right)×{L}^{2}\left(I,{L}_{0}^{2}\left(\Omega \right)\right)$, given in the strong form, characterizes the minimizer.

### Supplementary Material

The optimal state, adjoint state, and control are known analytically, noting that the pressure $p$ and the adjoint pressure $\mu$ are normalized by having mean-value zero:

 $\begin{array}{ccc}\hfill v& ={\left[\left(\zeta +\eta \phantom{\rule{0.3em}{0ex}}{e}^{t}\right)\phantom{\rule{0.3em}{0ex}}{sin}^{2}\left(\pi \phantom{\rule{0.3em}{0ex}}{x}_{1}\right)\phantom{\rule{0.3em}{0ex}}sin\left(2\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{2}\right),\phantom{\rule{3.0235pt}{0ex}}-\left(\zeta +\eta \phantom{\rule{0.3em}{0ex}}{e}^{t}\right)\phantom{\rule{0.3em}{0ex}}sin\left(2\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{1}\right)\phantom{\rule{0.3em}{0ex}}{sin}^{2}\left(\pi \phantom{\rule{0.3em}{0ex}}{x}_{2}\right)\right]}^{\top }\hfill & \hfill \\ \hfill p& =-\pi \phantom{\rule{0.3em}{0ex}}\left(\zeta +\eta \phantom{\rule{0.3em}{0ex}}{e}^{t}\right)\phantom{\rule{0.3em}{0ex}}sin\left(2\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{1}\right)\phantom{\rule{0.3em}{0ex}}sin\left(2\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{2}\right)\hfill \\ \hfill \lambda & =-{\left[-\left({e}^{T}-{e}^{t}\right)\phantom{\rule{0.3em}{0ex}}{sin}^{2}\left(\pi \phantom{\rule{0.3em}{0ex}}{x}_{1}\right)\phantom{\rule{0.3em}{0ex}}sin\left(2\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{2}\right),\phantom{\rule{3.0235pt}{0ex}}\left({e}^{T}-{e}^{t}\right)\phantom{\rule{0.3em}{0ex}}sin\left(2\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{1}\right)\phantom{\rule{0.3em}{0ex}}{sin}^{2}\left(\pi \phantom{\rule{0.3em}{0ex}}{x}_{2}\right)\right]}^{\top }\hfill \\ \hfill \mu & =-\left({x}_{1}+{x}_{2}\right)\hfill \\ \hfill u& =\frac{1}{\beta }{\left[-\left({e}^{T}-{e}^{t}\right)\phantom{\rule{0.3em}{0ex}}{sin}^{2}\left(\pi \phantom{\rule{0.3em}{0ex}}{x}_{1}\right)\phantom{\rule{0.3em}{0ex}}sin\left(2\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{2}\right),\phantom{\rule{3.0235pt}{0ex}}\left({e}^{T}-{e}^{t}\right)\phantom{\rule{0.3em}{0ex}}sin\left(2\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{1}\right)\phantom{\rule{0.3em}{0ex}}{sin}^{2}\left(\pi \phantom{\rule{0.3em}{0ex}}{x}_{2}\right)\right]}^{\top }\hfill \end{array}$

Notice that the sign of $\left(\lambda ,\mu \right)$ is reversed in [Güttel and Pearson2017, Section 6.2]. Consequently, the control law reads $u=\frac{1}{\beta }\phantom{\rule{0.3em}{0ex}}\lambda$ in [Güttel and Pearson2017, Section 6.2].

### References

S. Güttel and J. W. Pearson. A rational deferred correction approach to parabolic optimal control problems. IMA Journal of Numerical Analysis, online-ﬁrst, 2017. doi: 10.1093/imanum/drx046. URL https://academic.oup.com/imajna/advance-article/doi/10.1093/imanum/drx046/4372128.