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Research Group Computational Methods in Systems and Control Theory

# Project

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

The goal of this project is to derive and investigate numerical algorithms for optimal control-based boundary feedback stabilization of multi-field flow problems.

We will follow an approach laid out during the last years in a series of papers by Barbu, Lasiecka, Triggiani, Raymond, and others. They have shown that it is possible to stabilize perturbed flows described by the Navier-Stokes equation by designing a stabilizing controller based on a corresponding linear-quadratic optimal control problem.

Until recently, the numerical solution of these linear-quadratic optimal control problem was a numerical challenge due to the complexity 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.

We suggest three scenarios with increasing difficulty for which we want to demonstrate the applicability of the optimal control-based feedback stabilization approach.

The scenarios are the following:
• #### 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 a homogeneous Dirichlet boundary condition can be used as simplified modell.
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 applications. 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.

### Principal investigators:

Prof. Dr. Peter Benner
Prof. Dr. Eberhard Bänsch
(Applied Mathematics III, FAU Erlangen)

### Researcher:

Anne Katrin Heubner (11/2006-09/2008)
Dr. Jens Saak (10/2009-01/2011)
Heiko Weichelt (since 06/2011)

### Student Assistants:

Martin Köhler (05/09-10/10)
Heiko Weichelt (11/08-12/10)

### Simulations:

#### Navier-Stokes on von Kármán vortex street

 Example: Re=500, t_end=30 Control input over boundary after t_control=10 Openloop control with constant input Download as AVI in higher quality (ca. 12 MB) (Non optimal) feedback Download as AVI in higher quality (ca. 19 MB) Goal: Get laminar flow behind the obstacle.

#### Navier-Stokes coupled with transport of some species

 Example: Re=10, Sc=10, t_end=60 (left picture) Piecewise constant control input (right picture) 3D-Simulation of concentration Download as AVI in higher quality (ca. 60 MB) 2D-Simulation of concentration Download as AVI in higher quality (ca. 80 MB) Goal: Get a fixed rate of reaction on the obstacle.

#### Navier-Stokes on von Kármán vortex street

Example: Re=300, t_end=40

Feedback stabilization over boundary influence after t_control=7.5 for initial and optimal feedback.