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- ItemOptimal control problems with delays in state and control and mixed control-state constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Göllmann, Laurenz; Kern, Daniela; Maurer, HelmutOptimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control-state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods for the delayed control problem are discussed which amount to solving a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and two numerical examples from chemical engineering and economics illustrate the results.
- ItemThe heat treatment of steel - a mathematical control problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Hömberg, Dietmar; Kern, DanielaThe goal of this paper is to show how the heat treatment of steel can be modelled in terms of a mathematical optimal control problem. The approach is applied to laser surface hardening and the cooling of a steel slab including mechanical effects. Finally, it is shown how the results can be utilized in industrial practice by a coupling with machine-based control.
- ItemOn a thermomechanical model of phase transitions in steel(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Chełminski, Krzysztof; Hömberg, Dietmar; Kern, DanielaWe investigate a thermomechanical model of phase transitions in steel. The strain is assumed to be additively decomposed into an elastic and a thermal part as well as a contribution from transformation induced plasticity. The resulting model can be viewed as an extension of quasistatic linear thermoelasticity. We prove existence of a unique solution and conclude with some numerical simulations.