## scdist4 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:

• box of order 0

Mixed constraints:

• none

Submitted on 2014-02-15 by Winnifried Wollner. Published on 2017-01-09

## scdist4 description:

### Introduction

This example is taken from [Cherednichenko et al.2008, Section 5.1]. It features a state constrained problem in which the Lagrange multiplier is given by a Dirac measure.

### Variables & Notation

#### Free Parameters

The solution is parametrized in the Tikhonov parameter $\alpha >0$.

#### Given Data

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

### Optimality System

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

### Supplementary Material

The optimal solution together with adjoint state and Lagrange multiplier for the inequality constraint are known. They are given by

 $\begin{array}{ccc}\hfill y& ={y}_{d},\hfill & \hfill \\ \hfill p& =\frac{1}{2\pi }ln|x|,\hfill \\ \hfill u& =\frac{-1}{2\pi \alpha }ln|x|,\hfill \\ \hfill {\int }_{\Omega }\varphi \phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}\mu & =\varphi \left(0\right)\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{0.3em}{0ex}}\varphi \in C\left(\overline{\Omega }\right).\hfill \end{array}$

### References

S. Cherednichenko, K. Krumbiegel, and A. Rösch. Error estimates for the Lavrentiev regularization of elliptic optimal control problems. Inverse Problems, 24 (5):055003, 21, 2008. doi: 10.1088/0266-5611/24/5/055003.