## ccbnd1 details:

Keywords: analytic solution

Geometry: easy, fixed

Design: coupled via boundary values 1st order

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 2013-01-09 by Arnd Rösch. Published on 2013-01-13

## ccbnd1 description:

### Introduction

Here we have a simple boundary optimal control problem of the Poisson equation with pointwise box constraints on the control. The domain is polygonal, and it is the intersection of a square and a circular sector. The regularity of the optimal solution and consequently the approximation properties of numerical solutions depend on the angle $\omega$ of the circular sector.

This problem and the analytical example were published in Mateos and Rösch [2011].

### Variables & Notation

#### Given Data

The given data is chosen in a way which admits an analytic solution. The domain ${\Omega }_{\omega }$ and the solution depend on the angle $0<\omega <2\pi$. The most interesting cases arise when $\omega$ is the largest angle, i.e., in case $\omega \ge \pi ∕2$. The optimal control has the natural low regularity described by the singular exponent $\lambda$, which also depends on $\omega$.

The description of the problem is most convenient when both cartesian coordinates $\left({x}_{1},{x}_{2}\right)$ and polar coordinates $\left(r,\varphi \right)$ are used interchangeably.

The function ${g}_{1}$ can be computed by the formula

 ${g}_{1}=-\frac{\partial }{\partial n}{y}_{d}=-\nabla {y}_{d}\cdot n$

with $n$ denoting the outer normal vector to ${\Gamma }_{\omega }$, and

 $\nabla {y}_{d}=\left(\begin{array}{c}\hfill \frac{\partial {y}_{d}}{\partial {x}_{1}}\hfill \\ & & & & & & & & & \\ \hfill \frac{\partial {y}_{d}}{\partial {x}_{2}}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -\lambda \phantom{\rule{0.3em}{0ex}}{r}^{\lambda -1}cos\left(\lambda \phantom{\rule{0.3em}{0ex}}\varphi \right)\frac{{x}_{1}}{r}-\lambda \phantom{\rule{0.3em}{0ex}}{r}^{\lambda }sin\left(\lambda \phantom{\rule{0.3em}{0ex}}\varphi \right)\frac{{x}_{2}}{{r}^{2}}\hfill \\ & & & & & & & & & \\ \hfill -\lambda \phantom{\rule{0.3em}{0ex}}{r}^{\lambda -1}cos\left(\lambda \phantom{\rule{0.3em}{0ex}}\varphi \right)\frac{{x}_{2}}{r}+\lambda \phantom{\rule{0.3em}{0ex}}{r}^{\lambda }sin\left(\lambda \phantom{\rule{0.3em}{0ex}}\varphi \right)\frac{{x}_{1}}{{r}^{2}}\hfill \end{array}\right).$

Note that ${g}_{1}$ vanishes at the part of ${\Gamma }_{\omega }$ that coincides with the boundary of the circular sector.

### Optimality System

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

### Supplementary Material

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

### References

M. Mateos and A. Rösch. On saturation eﬀects in the Neumann boundary control of elliptic optimal control problems. Computational Optimization and Applications, 49 (2):359–378, 2011. doi: 10.1007/s10589-009-9299-5.