Departmentof Mathematics

Introduction

Here we present a distributed optimal control problem of the time-dependent Stokes equations. The problem was derived as a test for the paper Güttel and Pearson [2017], which required optimal states and controls that are not polynomial in spatial or time variables. The problem maintains a parameter dependence for the regularization parameter $\beta$ to serve as a test case for the $\beta$ dependence of solvers. This problem and its analytical solution appear in [Güttel and Pearson, 2017, Section 6.2], with computations for final time $T=1$ and control cost parameter $\beta =10^{-2}$.

Variables & Notation

Unknowns

$$\begin{array}{ccccc}\hfill u& \in L^2{\left(\Omega \times I\right)}^2\hfill & \hfill & \text{control variable}\hfill & \hfill \\ \hfill y& =\left(v,p\right)\hfill & \hfill & \text{state variable (velocity, pressure)}\hfill \\ \hfill v& \in L^2\left(I,V\left(\Omega \right)\right)\cap H^1\left(I,V{\left(\Omega \right)}^{\ast }\right)\hfill & \hfill & \text{velocity component of state}\hfill \\ \hfill p& \in L^2\left(I,L_0^2\left(\Omega \right)\right)\hfill & \hfill & \text{pressure component of state}\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill V\left(\Omega \right)& =\left\{\right.v\in H_0^1{\left(\Omega \right)}^2:\mathrm{div}<!--FUNCTION\; APPLICATION-->v=0\left\}\right.,\hfill & \hfill \\ \hfill L_0^2\left(\Omega \right)& =\left\{\right.p\in L^2\left(\Omega \right):{\int \; <!--nolimits-->}_{\Omega }p\mathrm{\text{<mstyle class="text"><mtext class="textup" mathvariant="normal" >d<mi>x</mi><mstyle mathsize="1.61em"><mfenced separators="" open="}" close=""><mrow></mrow></mfenced></mstyle><mo class="MathClass-punc">.</mo> </mtext class="textup" mathvariant="normal" ></mstyle >}}\hfill \end{array}$$

Given Data

$$\begin{array}{ccccc}\hfill T& >0\hfill & \hfill & \text{length of time interval}\hfill & \hfill \\ \hfill \Omega & ={\left(-1,1\right)}^2\hfill & \hfill & \text{spatial domain}\hfill \\ \hfill I& =\left(0,T\right)\hfill & \hfill & \text{time interval}\hfill \\ \hfill Q& ={\left(-1,1\right)}^2\times \left(0,T\right)\hfill & \hfill & \text{space-time domain}\hfill \\ \hfill \Sigma & =\partial \Omega \times \left(0,T\right)\hfill & \hfill & \text{lateral boundary of }Q\hfill \\ \hfill \beta & >0\hfill & \hfill & \text{regularization parameter}\hfill \\ \hfill \zeta & =-\frac{e^T}{4{\pi }^2\beta }\hfill & \hfill & \text{parameter}\hfill \\ \hfill \eta & =\frac{1}{\left(1+4{\pi }^2\right)\beta }\hfill & \hfill & \text{parameter}\hfill \\ \hfill z& ={\left[-4{\pi }^2\left(\zeta +\eta e^t\right)\cos <!--FUNCTION\; APPLICATION-->\left(2\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_2\right),0\right]}^{\top <!--FUNCTION\; APPLICATION-->}\hfill & \hfill & \text{uncontrolled source term}\hfill \\ \hfill v_d& ={\left[v_{d,1},v_{d,2}\right]}^{\top <!--FUNCTION\; APPLICATION-->}\hfill & \hfill & \text{desired state}\hfill \\ \hfill v_{d,1}& =1+\left(-e^t+\left(\zeta +\eta e^t\right)\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_2\right)\hfill \\ \hfill & +2{\pi }^2\left(e^T-e^t\right)\left(1-4{\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_1\right)\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_2\right)\hfill & \hfill & \text{component of desired state}\hfill \\ \hfill v_{d,2}& =1+\left(e^t-\left(\zeta +\eta e^t\right)\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_1\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_2\right)\hfill \\ \hfill & +2{\pi }^2\left(e^T-e^t\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_1\right)\left(4{\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_2\right)-1\right)\hfill & \hfill & \text{component of desired state}\hfill \\ \hfill v_0& ={\left[v_{0,1},v_{0,2}\right]}^{\top <!--FUNCTION\; APPLICATION-->}\hfill & \hfill & \text{initial state}\hfill \\ \hfill v_{0,1}& =\left(\zeta +\eta \right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_2\right)\hfill & \hfill & \text{component of initial state}\hfill \\ \hfill v_{0,2}& =-\left(\zeta +\eta \right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_1\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_2\right)\hfill & \hfill & \text{component of initial state}\hfill \end{array}$$

Problem Description

Optimality System

The following optimality system for the state $y=\left(v,p\right)\in L^2\left(I,V\left(\Omega \right)\right)\cap H^1\left(I,V{\left(\Omega \right)}^{\ast }\right)\times L^2\left(I,L_0^2\left(\Omega \right)\right)$, the control $u\in L^2{\left(\Omega \times I\right)}^2$ and the adjoint state $q=\left(\lambda ,\mu \right)\in L^2\left(I,V\left(\Omega \right)\right)\cap H^1\left(I,V{\left(\Omega \right)}^{\ast }\right)\times L^2\left(I,L_0^2\left(\Omega \right)\right)$, given in the strong form, characterizes the minimizer.

$$\begin{array}{ccccc}\hfill v_t-△v+\nabla <!--FUNCTION\; APPLICATION-->p& =u+z\hfill & \hfill & \text{in }Q\hfill & \hfill \\ \hfill -\mathrm{div}<!--FUNCTION\; APPLICATION-->v& =0\hfill & \hfill & \text{in }Q\hfill \\ \hfill v& =0\hfill & \hfill & \text{on }\Gamma \hfill \\ \hfill v& =v_0\hfill & \hfill & \text{at }t=0\hfill \\ \hfill -{\lambda }_t-△\lambda +\nabla <!--FUNCTION\; APPLICATION-->\mu & =v-v_d\hfill & \hfill & \text{in }Q\hfill \\ \hfill -\mathrm{div}<!--FUNCTION\; APPLICATION-->\lambda & =0\hfill & \hfill & \text{in }Q\hfill \\ \hfill \lambda & =0\hfill & \hfill & \text{on }\Gamma \hfill \\ \hfill \lambda & =0\hfill & \hfill & \text{at }t=T\hfill \\ \hfill u& =-\frac{1}{\beta }\lambda \hfill & \hfill & \text{in }Q\hfill \end{array}$$

Supplementary Material

The optimal state, adjoint state, and control are known analytically, noting that the pressure $p$ and the adjoint pressure $\mu$ are normalized by having mean-value zero:

$$\begin{array}{ccc}\hfill v& ={\left[\left(\zeta +\eta e^t\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_2\right),-\left(\zeta +\eta e^t\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_1\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_2\right)\right]}^{\top <!--FUNCTION\; APPLICATION-->}\hfill & \hfill \\ \hfill p& =-\pi \left(\zeta +\eta e^t\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_2\right)\hfill \\ \hfill \lambda & =-{\left[-\left(e^T-e^t\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_2\right),\left(e^T-e^t\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_1\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_2\right)\right]}^{\top <!--FUNCTION\; APPLICATION-->}\hfill \\ \hfill \mu & =-\left(x_1+x_2\right)\hfill \\ \hfill u& =\frac{1}{\beta }{\left[-\left(e^T-e^t\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_2\right),\left(e^T-e^t\right)\sin <!--FUNCTION\; APPLICATION-->\left(2\pi x_1\right){\sin <!--FUNCTION\; APPLICATION-->}^2\left(\pi x_2\right)\right]}^{\top <!--FUNCTION\; APPLICATION-->}\hfill \end{array}$$

Notice that the sign of $\left(\lambda ,\mu \right)$ is reversed in [Güttel and Pearson, 2017, Section 6.2]. Consequently, the control law reads $u=\frac{1}{\beta }\lambda$ in [Güttel and Pearson, 2017, Section 6.2].

References