JournalsowrVol. 5 / No. 1DOI 10.4171/owr/2008/13

Optimal Control of Coupled Systems of PDE

  • Jürgen Sprekels

    Angewandte Analysis und Stochastik, Berlin, Germany
  • Karl Kunisch

    Karl-Franzens-Universität Graz, Austria
  • Günter Leugering

    Universität Erlangen-Nürnberg, Germany
  • Fredi Tröltzsch

    Technische Universität Berlin, Germany
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Abstract

The international conference \emph{Optimal Control of Coupled Systems of PDE}, was held March 2nd--March 8th, 2008, organized by K. Kunisch (Karl-Franzens-Univer\-sity Graz), G. Leugering (University of Erlangen-N\"urnberg), J. Sprekels (Weierstrass Institute of Applied Analysis and Stochastics Berlin) and Fredi Tr\"oltzsch (Technische Universit\"at Berlin). {44} participants attended this meeting and followed {33} talks on optimal control and related topics.

Mathematically, the control of partial differential equations (PDEs) is concerned with the following type of problems: The solution of a PDE (the state of the system) should be influenced in a desired way by the choice of certain control functions or control parameters (the controls), which may occur in different terms of the differential equation. If the controls are to minimize a certain functional related to the state, then an {\em optimal control problem} is posed. If the domain underlying the PDE is subject of the control, then a {\em shape optimization problem} is given. For evolution equations, it can be required to move the solution from a given initial state exactly to a desired final state. This is the question of exact {\em controllability}.

Optimization and control of partial differential equations continues to be a very active field of research. Scientists working in different fields came together to report on their contributions to the numerical analysis of control problems. It is remarkable that optimal control is a challenge for researchers with backgrounds in related fields such as the theory of systems of nonlinear PDEs, numerical methods for solving them, large scale nonlinear optimization, or the numerical simulation and optimization of complex processes in engineering or medical science.

This diversity was reflected by the conference program. Talks were focused on \begin{itemize} \item applications of optimal control to the thermistor problem, crystal growth, quantum mechanics or aviation \item state-constrained optimal control problems \item controllability and observability of the Navier-Stokes equations and of systems for fluid-structure interaction; feedback control \item Hamilton-Bellman-Jacobi equations \item models for the interaction of electromagnetic fields, heat transport and fluid flow \item mesh adaptivity, a-posteriori and a-priori error estimates for the solutions of optimal control problems \item the application of numerical techniques such as semismooth Newton methods, multilevel techniques or domain decomposition \item first- and second-order optimality conditions for the optimal controls of nonlinear systems of PDEs arising from different applications \item modal control \item the optimal shape design of electromagnetic systems or thin shells and on free material optimization. \end{itemize} All these issues are currently subject of active research. In extensive and lively discussions, the participants of the workshop produced new mathematical ideas and tightened connections of joint cooperation.