## mpdist1 details:

Keywords: analytic solution

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

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

Design constraints:

• none

State constraints:

• none

Mixed constraints:

• none

Submitted on 2012-08-21 by Roland Herzog. Published on 2012-08-21

## mpdist1 description:

### Introduction

This is one of the simplest model problems in optimal control of partial diﬀerential equations. Problems of this type are treated extensively in [Tröltzsch2010, Chapter 2], and are sometimes refered to as the mother problem type. The present problem is special in the sense that the control acts in a distributed way on the entire domain $\Omega$, and that the state is observed on the entire domain as well. Furthermore, no constraints beside the elliptic PDE are present.

This problem was adapted from [Tröltzsch2010, Section 2.9.1], where the case $\nu =0$ with additional control constraints was elaborated.

### Variables & Notation

#### Given Data

The given data is chosen in a way which admits an analytic solution.

### Optimality System

The following optimality system for the state $y\in {H}_{0}^{1}\left(\Omega \right)$, the control $u\in {L}^{2}\left(\Omega \right)$ and the adjoint state $p\in {H}_{0}^{1}\left(\Omega \right)$, given in the strong form, characterizes the unique minimizer.

### Supplementary Material

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

 $\begin{array}{ccc}\hfill y& =\mathrm{sin}\left(\pi \phantom{\rule{0.3em}{0ex}}{x}_{1}\right)\phantom{\rule{0.3em}{0ex}}\mathrm{sin}\left(\pi \phantom{\rule{0.3em}{0ex}}{x}_{2}\right)\hfill & \hfill \\ \hfill p& =-\frac{1}{128\phantom{\rule{0.3em}{0ex}}{\pi }^{2}}\mathrm{sin}\left(8\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{1}\right)\phantom{\rule{0.3em}{0ex}}\mathrm{sin}\left(8\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{2}\right)\hfill \\ \hfill u& =-\frac{{\nu }^{-1}}{128\phantom{\rule{0.3em}{0ex}}{\pi }^{2}}\mathrm{sin}\left(8\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{1}\right)\phantom{\rule{0.3em}{0ex}}\mathrm{sin}\left(8\phantom{\rule{0.3em}{0ex}}\pi \phantom{\rule{0.3em}{0ex}}{x}_{2}\right)\hfill \end{array}$

### References

F. Tröltzsch. Optimal Control of Partial Diﬀerential Equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, 2010.