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.