## scdist3 details:

Keywords:

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

## scdist3 description:

### Introduction

This example is taken from Günther and Hinze [2008]. It features a complex active set structure for the inequality constraints on the state.

### 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 the Lagrange multipliers ${\mu }^{a},{\mu }^{b}\in \mathsc{ℳ}\left(\Omega \right)=C{\left(\overline{\Omega }\right)}^{\ast }$ for the lower and upper inequality constraint, respectively, given in the strong form, characterizes the unique minimizer.

### Supplementary Material

A reference value for the functional is provided in Günther and Hinze [2008] as

${J}^{\ast }\approx 1759.04686$

### References

A. Günther and M. Hinze. A-posteriori error control of a state constrained elliptic control problem. Journal of Numerical Mathematics, 16:307–322, 2008. doi: 10.1515/JNUM.2008.014.