Departmentof Mathematics

Introduction

This example is taken from Günther and Hinze [2008]. It features a complex active set structure for the inequality constraints on the state.

Variables & Notation

Unknowns

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

Given Data

$$\begin{array}{llllllll}\hfill \Omega & ={\left(0,1\right)}^2& \hfill & \text{computational domain}& \hfill & & \hfill & \\ \hfill \Gamma & & \hfill & \text{its boundary}& \hfill & & \hfill & \\ \hfill u_0& =60& \hfill & \text{desired control}& \hfill & & \hfill & \\ \hfill y_0& =0.5& \hfill & \text{desired state}& \hfill & & \hfill & \\ \hfill a& =0.45& \hfill & \text{lower bound for the state}& \hfill & & \hfill & \\ \hfill b\left(x_1,x_2\right)& =\min <!--FUNCTION\; APPLICATION-->\left\{\right.1,\max <!--FUNCTION\; APPLICATION-->\left\{\right.0.5,50\left(\right.{\left(x_1-0.3\right)}^2+{\left(x_2-0.3\right)}^2\left)\right.\left\}\right.\left\}\right.& \hfill & \text{upper bound for the state}& \hfill & & \hfill & \end{array}$$

Problem Description

$$\begin{array}{llll}\hfill \text{Minimize}& \frac{1}{2}\parallel y-y_0{\parallel }_{L^2\left(\Omega \right)}^2+\frac{1}{2}\parallel u-u_0{\parallel }_{L^2\left(\Omega \right)}^2& \hfill & \\ \hfill \text{s.t.}& \left\{\begin{array}{ccccc}\hfill -△y+y& =u\hfill & \hfill & \text{in }\Omega \hfill & \hfill \\ \hfill \frac{\partial y}{\partial n}& =0\hfill & \hfill & \text{on }\Gamma \hfill \\ \hfill \end{array}\right.& \hfill & \\ \hfill \text{and}& a\le y\left(x\right)\le b\left(x\right)\text{in }\overline{\Omega }.& \hfill & \end{array}$$

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 the Lagrange multipliers ${\mu }^a,{\mu }^b\in \mathcal{ℳ}\left(\Omega \right)=C{\left(\overline{\Omega }\right)}^{\ast }$ for the lower and upper inequality constraint, respectively, given in the strong form, characterizes the unique minimizer.