Departmentof Mathematics

Introduction

Here we have a simple distributed optimal control problem of the Poisson equation with a potential term, and with pointwise bound constraints on the control. 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. Moreover, the non-trivial part of the solution is rotationally symmetric.

This problem and analytical solution appear in [Tröltzsch, 2010, Section 2.9.2].

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

The given data is chosen in a way which admits an analytic solution. This solution is rotationally symmetric on a circle of radius 1/3 strictly contained in $\Omega$.

$$\begin{array}{ccccc}\hfill \Omega & ={\left(0,1\right)}^2\hfill & \hfill & \text{computational domain}\hfill & \hfill \\ \hfill \Gamma & \hfill & \hfill & \text{its boundary}\hfill \\ \hfill \hat{x}& =\left(0.5,0.5\right)\hfill & \hfill & \text{center of }\Omega \hfill \\ \hfill y_{\Omega }& =-\frac{142}{3}+12\parallel x-\hat{x}{\parallel }^2\hfill & \hfill & \text{desired state}\hfill \\ \hfill e_{\Omega }& =1-{\mathrm{proj}<!--FUNCTION\; APPLICATION-->}_{\left[0,1\right]}\left(\right.12\parallel x-\hat{x}{\parallel }^2-\frac{1}{3}\left)\right.\hfill & \hfill & \text{uncontrolled force}\hfill \\ \hfill e_{\Gamma }& =-12\hfill & \hfill & \text{boundary observation coefficient}\hfill \\ \hfill \end{array}$$

Problem Description

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.

$$\begin{array}{ccccc}\hfill -△y+y& =u+f\hfill & \hfill & \text{in }\Omega \hfill & \hfill \\ \hfill \frac{\partial y}{\partial n}& =0\hfill & \hfill & \text{on }\Gamma \hfill \\ \hfill -△p+p& =y-y_{\Omega }\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill \frac{\partial p}{\partial n}& =e_{\Gamma }\hfill & \hfill & \text{on }\Gamma \hfill \\ \hfill u& ={\mathrm{proj}<!--FUNCTION\; APPLICATION-->}_{\left[0,1\right]}\left(-p\right)\hfill & \hfill & \text{in }\Omega \hfill \end{array}$$

Supplementary Material

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

$$\begin{array}{ccc}\hfill y& =1\hfill & \hfill \\ \hfill p& =-12\parallel x-\hat{x}{\parallel }^2+\frac{1}{3}\hfill \\ \hfill u& =\left\{\begin{array}{cc}1,\hfill & x\in {\Omega }_3\hfill \\ 12\parallel x-\hat{x}{\parallel }^2-1∕3,\hfill & x\in {\Omega }_2\hfill \\ 0,\hfill & x\in {\Omega }_1\hfill \end{array}\right.\hfill \end{array}$$

where the subdomains are defined as follows:

$$\begin{array}{ccc}\hfill {\Omega }_1& =\left\{x\in \Omega :\parallel x-\hat{x}\parallel <1∕6\right\}\hfill & \hfill \\ \hfill {\Omega }_2& =\left\{x\in \Omega :1∕6\le \parallel x-\hat{x}\parallel <1∕3\right\}\hfill \\ \hfill {\Omega }_3& =\left\{x\in \Omega :1∕3\le \parallel x-\hat{x}\parallel \right\}.\hfill \end{array}$$

References