## scdist5 details:

Keywords: analytic solution

Global classification: linear-quadratic, convex

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Poisson:
• linear elliptic operator of order 2.
• Defined on a 3-dim domain in 3-dim space
• No time dependence.

Design constraints:

• box of order 0

State constraints:

• box of order 0

Mixed constraints:

• none

Submitted on 2014-04-04 by Simeon Steinig. Published on 2014-04-05

## scdist5 description:

### Introduction

Here, we have a distributed optimal control problem of the Poisson equation with pointwise box constraints on the control and a one-sided pointwise state constraint. The present problem is given on the unit ball in $\Omega ={B}_{1}\left(0\right)\subset {ℝ}^{3}$. The control acts in a distributed way on the entire domain $\Omega$ and the state constraint is enforced on the entire domain, too. This problem and the analytical solution appear in [Rösch and Steinig2012, Section 8]. The problem is designed to have a vanishing Lagrange multiplier for the state constraint. It is thus potentially a good test case for a posteriori error estimation.

### Variables & Notation

#### Given Data

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

### Optimality System

The following optimality system, given in the strong form, 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 multiplier $\mu \in \mathsc{ℳ}\left(\Omega \right)$ characterizes the unique minimizer.

### Supplementary Material

The optimal state and control are known analytically:

 $\begin{array}{ccc}\hfill y& =cos\left(\frac{\pi }{2}|x{|}^{2}\right),\hfill & \hfill \\ \hfill p& =sin\left(\pi |x{|}^{2}\right),\hfill \\ \hfill u& =-sin\left(\pi |x{|}^{2}\right),\hfill \\ \hfill \mu & =0.\hfill \end{array}$

Note that the state constraint is active on the ball $|x|\le \frac{1}{2}$. This means that strict complementarity fails on the active set, a fact which makes the problem numerically challenging.

### References

A. Rösch and S Steinig. A priori error estimates for a state-constrained elliptic optimal control problem. ESAIM: Mathematical Modelling and Numerical Analysis, 46(5):1107–1120, 2012. doi: 10.1051/m2an/2011076.