Optimal Control of Coupled Systems of PDE

Abstract

The international conference Optimal Control of Coupled Systems of PDE, was held March 2nd–March 8th, 2008, organized by K. Kunisch (Karl-Franzens-University Graz), G. Leugering (University of Erlangen-Nürnberg), J. Sprekels (Weierstrass Institute of Applied Analysis and Stochastics Berlin) and Fredi Tröltzsch (Technische Universität 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 optimal control problem is posed. If the domain underlying the PDE is subject of the control, then a 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 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

• applications of optimal control to the thermistor problem, crystal growth, quantum mechanics or aviation
• state-constrained optimal control problems
• controllability and observability of the Navier–Stokes equations and of systems for fluid-structure interaction; feedback control
• Hamilton–Bellman–Jacobi equations
• models for the interaction of electromagnetic fields, heat transport and fluid flow
• mesh adaptivity, a-posteriori and a-priori error estimates for the solutions of optimal control problems the application of numerical techniques such as semismooth Newton methods, multilevel techniques or domain decomposition
• first- and second-order optimality conditions for the optimal controls of nonlinear systems of PDEs arising from different applications
• modal control
• the optimal shape design of electromagnetic systems or thin shells and on free material optimization.

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.