Uwe Prüfert

On two numerical methods for state-constrained elliptic control problems

A linear-quadratic elliptic control problem with pointwise box constraints on the state is considered. The state constraints are treated by a Lavrentiev type regularization. It is shown that the Lagrange multipliers associated with the regularized state constraints are functions in L 2. Moreover, the convergence of the regularized controls is proven for regularization parameter tending to zero. To solve the problem numerically, an interior point method and a primal–dual active set strategy are implemented and treated in function space.

The convergence of an interior point method for an elliptic control problem with mixed control-state constraints

Abstract The paper addresses a primal interior point method for state-constrained PDE optimal control problems in function space. By a Lavrentiev regularization, the state constraint is transformed to a mixed control-state constraint with bounded Lagrange multiplier. Existence and convergence of the central path are established, and linear convergence of a short-step pathfollowing method is shown. The behaviour of the method is demonstrated by numerical examples.

Strategies for time-dependent PDE control with inequality constraints using an integrated modeling and simulation environment

In Neitzel et al. (Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part one: problems without inequality constraints. Technical Report 408, Matheon, Berlin, 2007) we have shown how time-dependent optimal control for partial differential equations can be realized in a modern high-level modeling and simulation package. In this article we extend our approach to (state) constrained problems. “Pure” state constraints in a function space setting lead to non-regular Lagrange multipliers (if they exist), i.e. the Lagrange multipliers are in general Borel measures. This will be overcome by different regularization techniques. To implement inequality constraints, active set methods and barrier methods are widely in use. We show how these techniques can be realized in a modeling and simulation package. We implement a projection method based on active sets as well as a barrier method and a Moreau Yosida regularization, and compare these methods by a program that optimizes the discrete version of the given problem.

A Smooth Regularization of the Projection Formula for Constrained Parabolic Optimal Control Problems

We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, that is, (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here, we show how a standard finite element software environment allows to solve the problem and, thus, to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem.

Modelling the Temperature Evolution During Hot Reversing Strip Rolling of Magnesium Alloys

The paper proposes the approach for the thermal through process modelling of the strip hot rolling chain of magnesium alloys. This strip hot rolling chain comprises the coil reheating after the Twin-Roll Casting (TRC) process, storage or transport operations and reversing hot rolling. The modelling of reversing rolling is implemented in connection with the simultaneous un-/coiling process of a coil. The numerical calculation is based on an object-oriented FEM tool kit written in MATLAB™ and is carried out in three spatial dimensions and time.

Numerical Modeling of Thermal Evolution in Hot Strip Rolling of Magnesium Alloy

This paper introduces a concept for the simulation of temperature distribution in reversing hot rolling of magnesium strip. The technological chain consists of reheating the casted rough strip, carrying the coil form furnace to coiler, setting down on mandrel, and hot reversing rolling. The paper represents the current state of work. The numerical calculation is based on an object-oriented FEM toolkit written in MATLAB™ and is carried out in three spatial dimensions and time.

Temperature Validation of 3D Model for the Reversing Hot Rolling in Connection with a Coil Model

The complex thermal modelling approach of the reversing hot strip rolling in connection with a coil model was developed. The coil model provides the modelling of the (un-) and coiling process. The modelling approach is based on object-orientated principals and implemented in MATLAB using library OOPDE and will be the base for the forecasting of the microstructure and the resulting material properties of strip during the reversing hot rolling. The model was validated using experimental data received after carrying out of the 2-pass reversing hot strip rolling of magnesium alloy AZ31. The results show a sufficient correspondence between experimental and calculated temperatures during the 1st and 2nd rolling pass.