[ Publications ]

Approximation of optimal control problems governed by PDEs

• Project Leaders:
  • Arnd Rösch (RICAM Linz)
• Project Staff:
  • Klaus Krumbiegel (RICAM Linz)
  • Svetlana Cherednichenko (RICAM Linz)

Project Abstract

Many technical processes are described by systems of partial differential equations. Optimization of such processes or identification of process parameters lead to optimal control problems for partial differential equations. Usually, such problems are characterized by additional constraints, i.e. some quantities of the process have to fulfil certain equations and inequalities. This project is concerned with linear-quadratic optimal control problems: The optimization goal is a quadratic function of the process quantities. Moreover, these quantities occur linear in the equations and inequalities. This project is especially interested in investigating elliptic and parabolic differential equations.
Although this class of problems has a simple structure, it is impossible to solve such problems exactly. Therefore, it is necessary to discretize such problems in a suitable manner. Consequently, approximation properties of the discretized problems with respect to the solution of the continuous problem represent a main focus of the project.
There is a large progress in the theory of control constrained problems in the recent years. In contrast to this, approximation results for state constrained optimal control problems are nearly unknown. This project will lower the large gap between the well investigated control constrained case and the widely unknown field of state constrained problems.
Moreover, the results should be used to construct stopping criteria for iterative methods. Stopping criteria based on error estimates can drastically reduce computational time for optimal control problems in a reliable way. Therefore, it is possible to solve larger and more complicate problems in future.
So-called superconvergence effects appear by solving optimal control problems numerically. A better understanding of superconvergence effects should help to exploit them in numerical algorithms. Such algorithms deliver essential better numerical results for a given discretization.
 
Project Abstract at FWF

Keywords and AMS Classification

• optimal control
• error estimates
• superconvergence
• approximation
• partial differential equations

AMS Subject Classification:

• Rösch, A., Simon, R., Superconvergence properties for optimal control problems discretized by piecewise linear and discontinuous functions,submitted 2006.
• Rösch, A., Wachsmuth, D., Regularity of solutions for an optimal control problem with mixed control-state constraints, accepted 2005.
• S. Cherednichenko, Rösch, A., Error estimates for the regularization of optimal control problems with pointwise control and state constraints, accepted 2006.

---

The Institute is named after the famous Austrian mathematician Johann Radon (1887-1956)

Medieninhaber:
Österreichische Akademie der Wissenschaften
Juristische Person öffentlichen Rechts (BGBl 569/1921 idF BGBl I 130/2003)
Dr. Ignaz Seipel-Platz 2, 1010 Wien
Diese Website dient zur Information über die wissenschaftlichen Aktivitäten der Österreichischen Akademie der Wissenschaften und setzt somit den gesetzlichen Auftrag um, die Wissenschaft in jeder Hinsicht zu fördern.

This RICAM page was made with 100% valid HTML & CSS - Send comments to Webmaster
Today's date and time is 02/09/12 - 23:34 CET and this file ( /projects/PDE_Control_Approximation/index.html ) was last modified on 10/12/06 - 15:45 CEST