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DFG-Project

Numerical Solution of Optimal Control Problems with Instationary Diffusion-Convection and Diffusion-Reaction Equations



Project director:
Prof. Dr. Peter Benner
Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany
Tel: +49 (0)371 531 8367
email: benner@mathematik.tu-chemnitz.de

Researcher:

Duration: january 2006 - may 2009

Classification:
This is a successor of the sub-project A15 in SFB393 Parallele Numerische Simulation für Physik und Kontinuumsmechanik at Chemnitz University of Technology.

Project description:
In this project we want to develop numerical algorithms of optimal control problems for instationary convection-diffusion and diffusion-reaction equations by using methods of state and output feedback. Linear problems with quadratic cost functional can be interpreted as a linear-quadratic regulator (LQR) problem. For the solutions of LQR problems new efficient methods were developed where Peter Benner was involved. These methods are coupled with solvers for the underlying stationary forward problem by appropriate interfaces.
We obtain nonlinear problems if nonlinear differential operators or nonlinear boundary conditions occur. The solution of nonlinear problems can be found by solving a class of optimal control problems which are called tracking-problems by means of state or output feedback. Since in general the optimal control cannot be computed directly or with untenable effort as done in the lineare case we will use sub-optimal strategies. We will focus on the development of numerical methods for the application of Model Predictive Control (MPC) for 2D and 3D problems. In doing so the whole time horizon will be covered by shorter time frames at which a sub-problem is solved by using an LQR or LQG design. The LQR and LQG designs for the parabolic problems arising after linearization can be solved by the numerical methods mentioned above.

Related Talks:

Sabine Hein
MPC/LQG-basierte Optimalsteuerung für nichtlineare parabolische PDEs 14. Südostdeutsches Kolloquium zur Numerischen Mathematik;Universität Leipzig,
25. April 2008

Jens Saak
Efficient Implementation of Large Scale Lyapunov and Riccati Equation Solvers Computational Methods with applications 2007; Harrachov (Czech Republic);
20.-25. August 2007

Jens Saak
Application of LQR Techniques to the Adaptive Control of Quasilinear Parabolic PDEs ICIAM 2007; ETH Zürich;
16.-20 July 2007

Jens Saak
Numerische Verfahren zur optimalen Steuerung von parabolischen PDEs Workshop Mathematische Systemtheorie Elgersburg (Thüringen); Hotel am Wald Elgersburg;
18.-22. February 2007

Jens Saak
ADI shift parameter computation for large scale algebraic Riccati and Lyapunov equations arising in the LQR problem for parabolic PDEs ALA2006 Düsseldorf; Heinrich Heine University Düsseldorf
Düsseldorf, 23.-27. July 2006

Sabine Hein
MPC for the Burgers Equation Based on an LQG Design; GAMM Annual Meeting 2006; TU Berlin
Berlin, 27.-31. March 2006

Jens Saak
On Adi Parameters For Solving PDE Control-related Matrix Equations; GAMM Annual Meeting 2006; TU Berlin
Berlin, 27.-31. March 2006

Sabine Hein
Nonlinear Optimal Control Problems (Poster); German-Polish Workshop for Young Researchers in Applied and Numerical Linear Algebra; Mathematical Research and Conference Center
Bedlewo (Poland), 02.-04. February 2006

Jens Saak
Efficient numerical solution of large scale LQR problems arising in the optimal control of parabolic PDEs; German-Polish Workshop for Young Researchers in Applied and Numerical Linear Algebra; Mathematical Research and Conference Center
Bedlewo (Poland), 02.-04. February 2006

Related Publications:
Application of LQR Techniques to the Adaptive Control of Quasilinear Parabolic PDEs
Benner, Peter; Saak, Jens;
Proceedings in Applied Mathematics and Mechanics   :  Vol. 7;Issue 1;
December 2007.
On the Parameter Selection Problem in the Newton-ADI Iteration for Large-Scale Riccati Equations;
Benner, Peter; Mena, Hermann; Saak, Jens;
Chemnitz Scientific Computing Preprints 06-03, TU Chemnitz; 2006. ISBN/ISSN: 1864-0087
Numerical Solution of Optimal Control Problems for Parabolic Systems;
Benner, Peter; Görner, Sabine and Saak, Jens;
in Hoffmann, K.H. and Meyer, A.:  Parallel Algorithms and Cluster Computing. Implementations, Algorithms, and Applications   :  vol. 52 of Lecture Notes in Computational Science and Engineering;
Springer Berlin Heidelberg; 2006. ISBN/ISSN: 3-540-33539-0
pp. 151-169.
MPC für nichtlineare partielle Differentialgleichungen parabolischen Typs
Benner, Peter; Görner, Sabine;
Tagungsband GMA-FA 1.40 "Theoretische Verfahren der Regelungstheorie", GAMM-FA "Dynamik und Regelungstheorie", Workshops am Bostalsee, 24.-27.9.2006  :   2006. ISBN/ISSN: 978-3-9810664-2-5
MPC for the Burgers Equation Based on an LGQ Design;
Benner, Peter; Görner, Sabine;
Proceedings in Applied Mathematics and Mechanics   :  Vol. 6, No. 1, pp. 781-782; 2006. ISBN/ISSN: 1617-7061

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Jens Saak, jens.saak@mathematik.tu-chemnitz.de