Departmentof Mathematics

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

Unknowns

$$\begin{array}{ccc}\hfill u\in L^2\left(\Omega \right)& \text{control variable}\hfill & \hfill \\ \hfill y\in H^1\left(\Omega \right)& \text{state variable}\hfill \end{array}$$

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.

$$\begin{array}{llllllll}\hfill \Omega & =B_1\left(0\right)=\left\{x\in {ℝ}^2:|x{|}^2<1\right\}& \hfill & \text{computational domain},& \hfill & & \hfill & \\ \hfill \Gamma & & \hfill & \text{its boundary},& \hfill & & \hfill & \\ \hfill y_d\left(x\right)& =\frac{1}{8\pi \alpha }\left(-|x{|}^2+|x{|}^2\ln <!--FUNCTION\; APPLICATION-->\left|x\right|+1\right)& \hfill & \text{desired state},& \hfill & & \hfill & \\ \hfill y_c\left(x\right)& =\frac{1}{8\pi \alpha }\left(-2\left|x\right|+1\right)& \hfill & \text{lower bound}.& \hfill & & \hfill & \end{array}$$

Problem Description

$$\begin{array}{llll}\hfill \text{Minimize}& \frac{1}{2}\parallel y-y_d{\parallel }_{L^2\left(\Omega \right)}^2+\frac{\alpha }{2}\parallel u{\parallel }_{L^2\left(\Omega \right)}^2& \hfill & \\ \hfill \text{s.t.}& \left\{\begin{array}{ccccc}\hfill -△y& =u\hfill & \hfill & \text{in }\Omega ,\hfill & \hfill \\ \hfill y& =0\hfill & \hfill & \text{on }\Gamma ,\hfill \\ \hfill \end{array}\right.& \hfill & \\ \hfill \text{and}& y_c\le y\left(x\right)\text{in }\overline{\Omega }.& \hfill & \end{array}$$

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 $\mu \in \mathcal{ℳ}\left(\Omega \right)=C{\left(\overline{\Omega }\right)}^{\ast }$, given in the strong form, characterizes the unique minimizer.