## ccdist1 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:

• box of order 0

State constraints:

• none

Mixed constraints:

• none

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

## ccdist1 description:

### Introduction

Here we have a simple distributed optimal control problem of the Poisson equation with a potential term, and with pointwise bound constraints on the control. 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. Moreover, the non-trivial part of the solution is rotationally symmetric.

This problem and analytical solution appear in [Tröltzsch2010, Section 2.9.2].

### Variables & Notation

#### Given Data

The given data is chosen in a way which admits an analytic solution. This solution is rotationally symmetric on a circle of radius 1/3 strictly contained in $\Omega$.

### 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:

 $\begin{array}{ccc}\hfill y& =1\hfill & \hfill \\ \hfill p& =-12\phantom{\rule{0.3em}{0ex}}\parallel x-\stackrel{̂}{x}{\parallel }^{2}+\frac{1}{3}\hfill \\ \hfill u& =\left\{\begin{array}{cc}1,\phantom{\rule{1em}{0ex}}\hfill & x\in {\Omega }_{3}\hfill \\ 12\phantom{\rule{0.3em}{0ex}}\parallel x-\stackrel{̂}{x}{\parallel }^{2}-1∕3,\phantom{\rule{1em}{0ex}}\hfill & x\in {\Omega }_{2}\hfill \\ 0,\phantom{\rule{1em}{0ex}}\hfill & x\in {\Omega }_{1}\hfill \end{array}\right\\hfill \end{array}$

where the subdomains are deﬁned as follows:

 $\begin{array}{ccc}\hfill {\Omega }_{1}& =\left\{x\in \Omega :\parallel x-\stackrel{̂}{x}\parallel <1∕6\right\}\hfill & \hfill \\ \hfill {\Omega }_{2}& =\left\{x\in \Omega :1∕6\le \parallel x-\stackrel{̂}{x}\parallel <1∕3\right\}\hfill \\ \hfill {\Omega }_{3}& =\left\{x\in \Omega :1∕3\le \parallel x-\stackrel{̂}{x}\parallel \right\}.\hfill \end{array}$

### References

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