Departmentof Mathematics

Introduction

Here we have a simple boundary optimal control problem of the Poisson equation with pointwise box constraints on the control. The domain is polygonal, and it is the intersection of a square and a circular sector. The regularity of the optimal solution and consequently the approximation properties of numerical solutions depend on the angle $\omega$ of the circular sector.

This problem and the analytical example were published in Mateos and Rösch [2011].

Variables & Notation

Unknowns

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

Given Data

The given data is chosen in a way which admits an analytic solution. The domain ${\Omega }_{\omega }$ and the solution depend on the angle $0<\omega <2\pi$. The most interesting cases arise when $\omega$ is the largest angle, i.e., in case $\omega \ge \pi ∕2$. The optimal control has the natural low regularity described by the singular exponent $\lambda$, which also depends on $\omega$.

The description of the problem is most convenient when both cartesian coordinates $\left(x_1,x_2\right)$ and polar coordinates $\left(r,\varphi \right)$ are used interchangeably.

$$\begin{array}{ccccc}\hfill S_{\omega }& =\left\{\left(r\cos <!--FUNCTION\; APPLICATION-->\varphi ,r\sin <!--FUNCTION\; APPLICATION-->\varphi \right):r\in \left[0,\sqrt{2}\right),\varphi \in \left(0,\omega \right)\right\}\hfill & \hfill & \text{circular sector}\hfill & \hfill \\ \hfill {\Omega }_{\omega }& ={\left(-1,1\right)}^2\cap S_{\omega }\hfill & \hfill & \text{computational domain}\hfill \\ \hfill {\Gamma }_{\omega }& \hfill & \hfill & \text{its boundary}\hfill \\ \hfill \lambda & =\pi ∕\omega \hfill & \hfill & \text{singular exponent}\hfill \\ \hfill y_d\left(r,\varphi \right)& =-r^{\lambda }\cos <!--FUNCTION\; APPLICATION-->\left(\lambda \varphi \right)\hfill & \hfill & \text{desired state (polar coordinates)}\hfill \\ \hfill g_1\left(x_1,x_2\right)& =-\frac{\partial }{\partial n}{y}_d\left(x_1,x_2\right)\hfill & \hfill & \text{objective term}\hfill \\ \hfill g_2\left(x_1,x_2\right)& =-{\mathrm{proj}<!--FUNCTION\; APPLICATION-->}_{\left[-0.5,0.5\right]}\left(\right.y_d\left(x_1,x_2\right)\left)\right.\hfill & \hfill & \text{boundary term}\hfill \\ \hfill \end{array}$$

The function $g_1$ can be computed by the formula

$$g_1=-\frac{\partial }{\partial n}{y}_d=-\nabla <!--FUNCTION\; APPLICATION-->y_d\cdot n$$

with $n$ denoting the outer normal vector to ${\Gamma }_{\omega }$, and

$$\nabla <!--FUNCTION\; APPLICATION-->y_d=\left(\begin{array}{c}\hfill \frac{\partial y_d}{\partial x_1}\hfill \\ & & & & & & & & & \\ \hfill \frac{\partial y_d}{\partial x_2}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -\lambda r^{\lambda -1}\cos <!--FUNCTION\; APPLICATION-->\left(\lambda \varphi \right)\frac{x_1}{r}-\lambda r^{\lambda }\sin <!--FUNCTION\; APPLICATION-->\left(\lambda \varphi \right)\frac{x_2}{r^2}\hfill \\ & & & & & & & & & \\ \hfill -\lambda r^{\lambda -1}\cos <!--FUNCTION\; APPLICATION-->\left(\lambda \varphi \right)\frac{x_2}{r}+\lambda r^{\lambda }\sin <!--FUNCTION\; APPLICATION-->\left(\lambda \varphi \right)\frac{x_1}{r^2}\hfill \end{array}\right).$$

Note that $g_1$ vanishes at the part of ${\Gamma }_{\omega }$ that coincides with the boundary of the circular sector.

Problem Description

Optimality System

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

$$\begin{array}{ccccc}\hfill -△y+y& =0\hfill & \hfill & \text{in }{\Omega }_{\omega }\hfill & \hfill \\ \hfill \frac{\partial y}{\partial n}& =u+g_2\hfill & \hfill & \text{on }{\Gamma }_{\omega }\hfill \\ \hfill -△p+p& =y-y_d\hfill & \hfill & \text{in }{\Omega }_{\omega }\hfill \\ \hfill \frac{\partial p}{\partial n}& =g_1\hfill & \hfill & \text{on }{\Gamma }_{\omega }\hfill \\ \hfill u& ={\mathrm{proj}<!--FUNCTION\; APPLICATION-->}_{\left[-0.5,0.5\right]}\left(-p{|}_{{\Gamma }_{\omega }}\right)\hfill & \hfill & \text{on }{\Gamma }_{\omega }\hfill \end{array}$$

Supplementary Material

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

$$\begin{array}{ccccc}\hfill y& =0\hfill & \hfill & \text{in }{\Omega }_{\omega }\hfill & \hfill \\ \hfill p& =-y_d\hfill & \hfill & \text{in }{\Omega }_{\omega }\hfill \\ \hfill u& =-g_2\hfill & \hfill & \text{on }{\Gamma }_{\omega }\hfill \end{array}$$

References