Departmentof Mathematics

Introduction

This is one of the simplest model problems in optimal control of partial differential equations. Problems of this type are treated extensively in [Tröltzsch, 2010, Chapter 2], and are sometimes refered to as the mother problem type. The present problem is special in the sense that the control acts in a distributed way on the entire domain $\Omega$, and that the state is observed on the entire domain as well. Furthermore, no constraints beside the elliptic PDE are present.

This problem was adapted from [Tröltzsch, 2010, Section 2.9.1], where the case $\nu =0$ with additional control constraints was elaborated.

Variables & Notation

Unknowns

$$\begin{array}{ccc}\hfill u\in L^2\left(\Omega \right)& \text{control variable}\hfill & \hfill \\ \hfill y\in H_0^1\left(\Omega \right)& \text{state variable}\hfill \end{array}$$

Given Data

The given data is chosen in a way which admits an analytic solution.

$$\begin{array}{ccccc}\hfill \Omega & ={\left(0,1\right)}^2\hfill & \hfill & \text{computational domain}\hfill & \hfill \\ \hfill \nu & =10^{-2}\hfill & \hfill & \text{control cost parameter}\hfill \\ \hfill y_d& =-\sin <!--FUNCTION\; APPLICATION-->\left(8\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(8\pi x_2\right)+\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_2\right)\hfill & \hfill & \text{desired state}\hfill \\ \hfill f& =2{\pi }^2\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_2\right)+\frac{{\nu }^{-1}}{128{\pi }^2}\sin <!--FUNCTION\; APPLICATION-->\left(8\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(8\pi x_2\right)\hfill & \hfill & \text{uncontrolled force}\hfill \end{array}$$

Problem Description

$$\begin{array}{ccc}\hfill \text{Minimize}& \frac{1}{2}\parallel y-y_d{\parallel }_{L^2\left(\Omega \right)}^2+\frac{\nu }{2}\parallel u{\parallel }_{L^2\left(\Omega \right)}^2\hfill & \hfill \\ \hfill \text{s.t.}& \left\{\begin{array}{ccccc}\hfill -△y& =u+f\hfill & \hfill & \text{in }\Omega \hfill & \hfill \\ \hfill y& =0\hfill & \hfill & \text{on }\partial \Omega \hfill \\ \hfill \end{array}\right.\hfill \end{array}$$

Optimality System

The following optimality system for the state $y\in H_0^1\left(\Omega \right)$, the control $u\in L^2\left(\Omega \right)$ and the adjoint state $p\in H_0^1\left(\Omega \right)$, given in the strong form, characterizes the unique minimizer.

$$\begin{array}{ccccc}\hfill -△y& =u+f\hfill & \hfill & \text{in }\Omega \hfill & \hfill \\ \hfill y& =0\hfill & \hfill & \text{on }\partial \Omega \hfill \\ \hfill -△p& =-\left(y-y_d\right)\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill p& =0\hfill & \hfill & \text{on }\partial \Omega \hfill \\ \hfill & \nu u-p=0\hfill & \hfill & \text{in }\Omega \hfill \end{array}$$

Supplementary Material

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

$$\begin{array}{ccc}\hfill y& =\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_2\right)\hfill & \hfill \\ \hfill p& =-\frac{1}{128{\pi }^2}\sin <!--FUNCTION\; APPLICATION-->\left(8\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(8\pi x_2\right)\hfill \\ \hfill u& =-\frac{{\nu }^{-1}}{128{\pi }^2}\sin <!--FUNCTION\; APPLICATION-->\left(8\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(8\pi x_2\right)\hfill \end{array}$$

References