Departmentof Mathematics

Introduction

We have a distributed optimal control problem for the Stokes equations with component-wise box constraints for the distributed control. A difficulty in the construction of test examples as the present one is the satisfaction of the divergence-free condition. Moreover, the example is three-dimensional.

This problem and the analytical solution were published in Rösch and Vexler [2006].

Variables & Notation

Unknowns

$$\begin{array}{ccc}\hfill q\in L^2{\left(\Omega \right)}^3& \text{control variable}\hfill & \hfill \\ \hfill u=\left(v,p\right)\in H_0^1{\left(\Omega \right)}^3\times L_0^2\left(\Omega \right)& \text{state variable}\hfill \end{array}$$

Given Data

$$\begin{array}{ccccc}\hfill \Omega & ={\left(0,1\right)}^3\hfill & \hfill & \text{computational domain}\hfill & \hfill \\ \hfill \Gamma & \hfill & \hfill & \text{its boundary}\hfill \\ \hfill a& ={\left(-0.1,-0.1,-0.1\right)}^{\top <!--FUNCTION\; APPLICATION-->}\hfill & \hfill & \text{lower control bound}\hfill \\ \hfill b& ={\left(0.25,0.25,0.01\right)}^{\top <!--FUNCTION\; APPLICATION-->}\hfill & \hfill & \text{upper control bound}\hfill \\ \hfill v_{d,1}& =\left(2-16{\pi }^2\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill & \hfill \\ \hfill & +\left(4{\pi }^2+2\pi \right)\cos <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill & \text{first component of the desired state}\hfill \\ \hfill v_{d,2}& =\left(-1+8{\pi }^2\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill & \hfill \\ \hfill & +\left(-2{\pi }^2+2\pi \right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\cos <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill & \text{second component of the desired state}\hfill \\ \hfill v_{d,3}& =\left(-1+8{\pi }^2\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi z\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\hfill & \hfill & \hfill \\ \hfill & +\left(-2{\pi }^2+2\pi \right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\cos <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill & \text{third component of the desired state}\hfill \\ \hfill f_1& =16{\pi }^2{\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill & \hfill \\ \hfill & +\left(-4{\pi }^2+2\pi \right)\cos <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill & \hfill \\ \hfill & -{\text{proj}}_{\left[a_1,b_1\right]}\left(-2{\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\right)\hfill & \hfill & \text{first component of the given force}\hfill \\ \hfill f_2& =-8{\pi }^2{\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill & \hfill \\ \hfill & +\left(2{\pi }^2+2\pi \right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\cos <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill & \hfill \\ \hfill & -{\text{proj}}_{\left[a_2,b_2\right]}\left({\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\right)\hfill & \hfill & \text{second component of the given force}\hfill \\ \hfill f_3& =-8{\pi }^2{\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi z\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\hfill & \hfill & \hfill \\ \hfill & +\left(2{\pi }^2+2\pi \right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\cos <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill & \hfill \\ \hfill & -{\text{proj}}_{\left[a_3,b_3\right]}\left({\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi z\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\right)\hfill & \hfill & \text{third component of the given force}\hfill \end{array}$$

Problem Description

$$\begin{array}{ccc}\hfill \text{Minimize}& \frac{1}{2}\parallel v-v_d{\parallel }_{L^2{\left(\Omega \right)}^3}^2+\frac{1}{2}\parallel q{\parallel }_{L^2{\left(\Omega \right)}^3}^2\hfill & \hfill \\ \hfill \text{s.t.}& \left\{\begin{array}{ccccc}\hfill -△v+\nabla <!--FUNCTION\; APPLICATION-->p& =q+f\hfill & \hfill & \text{in }\Omega \hfill & \hfill \\ \hfill \nabla <!--FUNCTION\; APPLICATION-->\cdot v& =0\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill v& =0\hfill & \hfill & \text{on }\Gamma \hfill \\ \hfill \end{array}\right.\hfill \\ \hfill \text{and}& a\le q\left(x\right)\le b\text{in }\Omega .\hfill \end{array}$$

Optimality System

The following system for the state (velocity and pressure) $u=\left(v,p\right)\in H_0^1{\left(\Omega \right)}^3\times L_0^2\left(\Omega \right)$, the control $q\in L^2{\left(\Omega \right)}^3$ and the adjoint state $z=\left(w,r\right)\in H_0^1{\left(\Omega \right)}^3\times L_0^2\left(\Omega \right)$, given in strong form, characterizes the unique minimizer. The projection is a component-wise projection onto the interval $\left[a_i,b_i\right]$ for $i=1,2,3$.

$$\begin{array}{ccccc}\hfill -△v+\nabla <!--FUNCTION\; APPLICATION-->p& =q+f\hfill & \hfill & \text{in }\Omega \hfill & \hfill \\ \hfill \nabla <!--FUNCTION\; APPLICATION-->\cdot v& =0\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill v& =0\hfill & \hfill & \text{on }\Gamma \hfill \\ \hfill -△w-\nabla <!--FUNCTION\; APPLICATION-->r& =v-v_d\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill \nabla <!--FUNCTION\; APPLICATION-->\cdot w& =0\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill w& =0\hfill & \hfill & \text{on }\Gamma \hfill \\ \hfill q& ={\text{proj}}_{\left[a,b\right]}\left(-w\right)\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill \end{array}$$

Supplementary Material

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

$$\begin{array}{ccc}\hfill v_1=w_1& =2{\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill & \hfill \\ \hfill v_2=w_2& =-{\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill \\ \hfill v_3=w_3& =-{\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi z\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\hfill \\ \hfill p=r& =\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi y\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi z\right)\hfill \\ \hfill q& ={\text{proj}}_{\left[a,b\right]}\left(-w\right)\hfill \\ \hfill \end{array}$$

References