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.