# Optimal Control-Based Feedback Stabilization in Multi-Field Flow Problems

### From DFG-SPP 1253

## Project leaders

Lehrstuhl für Angewandte Mathematik III

Uni Erlangen

Tel: 09131 85-25225

Fax: 09131 85-25228

Email: baensch@am.uni-erlangen.de

Homepage: http://www.am.uni-erlangen.de/am3

Professur Mathematik in Industrie u. Technik
Fakultät für Mathematik

TU Chemnitz

Tel: 0371 531-22540

Tel: 0371 531-22500 (Sekr.)

Fax: 0371 531-22509

Email: benner@mathematik.tu-chemnitz.de

Homepage: http://www.tu-chemnitz.de/~benner/

## DFG funded assistants

(10/2009-01/2011)

Jens Saak (TU Chemnitz)

jens.saak@mathematik.tu-chemnitz.de

(since 06/2011)

Heiko Weichelt (TU Chemnitz)

heiko.weichelt@mathematik.tu-chemnitz.de

## Description

The aim of this project is to develop numerical methods for the stabilization of solutions to flow problems. This is to be achieved by action of boundary control using feedback mechanisms.

In very recent work by Raymond and earlier attempts by Barbu, Triggiani, Lasiecka, and others, it is shown analytically that it is possible to construct a linear-quadratic optimal control problem associated to the Oseen approximation of the Navier-Stokes equation so that the resulting feedback law, applied to the instationary Navier-Stokes equation, is able to exponentially stabilize unstable solution trajectories assuming a certain smallness of initial values.

Until recently, the numerical solution of these linear-quadratic optimal control problem and the associated algebraic Riccati equations (AREs) was an unsolved numerical challenge due to the computational complexity and storage requirements of existing algorithms.

Employing recent advances in reducing these complexities essentially to a cost proportional to the simulation of the forward problem, we plan to apply this methodology to multi-field problems where the flow is coupled with other field equations. In particular, we plan to apply recent ideas to compute the approximate feedback operator directly without ever forming the explicit solution of the ARE.

We consider three different scenarios with increasing complexity for the stabilization of a flow field coupled to further physics. Each of them couples a flow field described by the incompressible Navier-Stokes equations with some other physical phenomenon.

We regard the subsequent examples as prototypes for problems arising from real world applications.

The scenarios are

*** Navier-Stokes coupled with (passive) transport of some (reactive) species**

This example may in a rather crude way model a reactor, where a chemical substance is transported by a flow field and reacts at the surface. The reaction is considered to be fast (compared to diffusion and transport), such that it can be simplified by a homogeneous Dirichlet boundary condition.

The control acts by varying the inflow boundary condition. The idea behind this setting is to stabilize and control the process of reaction, which is strongly influenced by the transport of the substance from the inflow to the reacting surface.

*** Phase transition liquid/solid with convection**

Consider a hot melt that solidifies while flowing through a mould. The objective here is to control the phase boundary between the liquid part of the mould and the solid part. In addition to considering the Navier-Stokes equations in the liquid part, there is a heat equation for the temperature to be solved in the whole domain. The control is given as the temperature distribution on a part of the boundary.

*** Stabilization of a flow with a free capillary surface**

Capillary free surfaces play a decisive role in many technological application. Thus control of the free boundary can be of paramount interest. Here, we consider a model problem, where a fluid is flowing over an obstacle and the upper boundary is a free capillary one. This boundary will be oscillatory due to the Karman vortex shedding in the wake of the obstacle. The goal is to stabilize the free boundary.