Project 05: Stokes Distributed Control - NMOPT — Numerical Methods for Optimal Control

Goal¶

Implement a simple linear-quadratic optimal control problem governed by the stationary Stokes equations.

This project introduces vector-valued states, incompressibility, and block preconditioning.

Let $\Omega\subset\mathbb R^d$ , $d=2$ or 3. The state is velocity-pressure $(y,\pi)$ , and the control $u$ is a distributed body force. Minimize

J(y,u) = \frac12\|y-y_d\|_{L^2(\Omega)^d}^2 + \frac\alpha2\|u\|_{L^2(\Omega)^d}^2

subject to

\begin{cases} -\nu\Delta y+\nabla\pi=u+f & \text{in }\Omega,\\ \nabla\cdot y=0 & \text{in }\Omega,\\ y=0 & \text{on }\partial\Omega. \end{cases}

The adjoint variables are $(p,\lambda)$ and satisfy

\begin{cases} -\nu\Delta p+\nabla\lambda=y-y_d & \text{in }\Omega,\\ \nabla\cdot p=0 & \text{in }\Omega,\\ p=0 & \text{on }\partial\Omega. \end{cases}

The control equation is

\alpha u-p=0.

For box constraints on each velocity component,

u_a\le u\le u_b,

replace the last equation by the variational inequality

(\alpha u-p,v-u)_{L^2(\Omega)^d}\ge 0 \qquad \forall v\in U_{ad}.