# A Robust Solver for Distributed Optimal Control for Stokes Flow

## Dipl.-Ing. Markus Kollmann

March 20, 2012, 3:30 p.m. S2 059

In this talk we consider the following optimal control problem:

Minimize $\quad J(u,f) = \frac{1}{2}||u-u_d||_{L^2(\Omega)}^2 + \frac{\alpha}{2}||f||_{L^2(\Omega)}^2 \quad$ (1)

subject to

$-\Delta u + \nabla p = f$ in $\Omega$
div $u = 0$ in $\Omega$
$u = 0$ on $\Gamma$
$+$ inequality constraints.

Here $\Omega$ is an open and bounded domain in $\mathbb{R}^d$ ($d \in \{1,2,3\}$), $\Gamma$ denotes the boundary and $\alpha > 0$ is a cost parameter. We consider two types of inequality constraints:

• inequality constraints on the control $f$,

• inequality constraints on the state $u$.

In both cases, the first order system of necessary and sufficient optimality conditions of (1) is nonlinear. A semi-smooth Newton iteration is applied in order to linearize the system. In every Newton step a linear saddle point system has to be solved (after discretization). For these linear systems solvers are discussed. Numerical examples are given which illustrate the theoretical results.